Integrand size = 35, antiderivative size = 532 \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^3 \sqrt {c+d x}} \, dx=-\frac {A \sqrt {c+d x} \sqrt {a-b x^2}}{2 c x^2}-\frac {(4 B c-3 A d) \sqrt {c+d x} \sqrt {a-b x^2}}{4 c^2 x}+\frac {\sqrt {a} \sqrt {b} \left (8 c^2 C+4 B c d-3 A d^2\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{4 c^2 d \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {\sqrt {a} \sqrt {b} \left (8 c^2 C-4 B c d-A d^2\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{4 c d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\left (4 a c (2 c C-B d)-A \left (4 b c^2-3 a d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{4 c^2 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-1/2*A*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/c/x^2-1/4*(-3*A*d+4*B*c)*(d*x+c)^(1/ 2)*(-b*x^2+a)^(1/2)/c^2/x+1/4*a^(1/2)*b^(1/2)*(-3*A*d^2+4*B*c*d+8*C*c^2)*( d*x+c)^(1/2)*((-b*x^2+a)/a)^(1/2)*EllipticE(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2 )*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/c^2/d/((d*x+c)/ (c+a^(1/2)*d/b^(1/2)))^(1/2)/(-b*x^2+a)^(1/2)-1/4*a^(1/2)*b^(1/2)*(-A*d^2- 4*B*c*d+8*C*c^2)*((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)*((-b*x^2+a)/a)^(1/2 )*EllipticF(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b^ (1/2)*c+a^(1/2)*d))^(1/2))/c/d/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)-1/4*(4*a*c*( -B*d+2*C*c)-A*(-3*a*d^2+4*b*c^2))*((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^(1/2)*(( -b*x^2+a)/a)^(1/2)*EllipticPi(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2,2^ (1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/c^2/(d*x+c)^(1/2)/(-b*x^2+a )^(1/2)
Result contains complex when optimal does not.
Time = 28.76 (sec) , antiderivative size = 1530, normalized size of antiderivative = 2.88 \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^3 \sqrt {c+d x}} \, dx =\text {Too large to display} \] Input:
Integrate[(Sqrt[a - b*x^2]*(A + B*x + C*x^2))/(x^3*Sqrt[c + d*x]),x]
Output:
(Sqrt[a - b*x^2]*(-(((c + d*x)*(2*A*c + 4*B*c*x - 3*A*d*x))/(c^2*x^2)) + ( 8*b*c^5*C*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] + 4*b*B*c^4*d*Sqrt[-c + (Sqrt[a]* d)/Sqrt[b]] - 3*A*b*c^3*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 8*a*c^3*C*d^2 *Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 4*a*B*c^2*d^3*Sqrt[-c + (Sqrt[a]*d)/Sqrt [b]] + 3*a*A*c*d^4*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - 16*b*c^4*C*Sqrt[-c + ( Sqrt[a]*d)/Sqrt[b]]*(c + d*x) - 8*b*B*c^3*d*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] *(c + d*x) + 6*A*b*c^2*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x) + 8*b* c^3*C*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 + 4*b*B*c^2*d*Sqrt[-c + ( Sqrt[a]*d)/Sqrt[b]]*(c + d*x)^2 - 3*A*b*c*d^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b ]]*(c + d*x)^2 + I*Sqrt[b]*c*(Sqrt[b]*c - Sqrt[a]*d)*(-8*c^2*C - 4*B*c*d + 3*A*d^2)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sq rt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-c + (Sq rt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqr t[a]*d)] + I*d*(Sqrt[b]*c - Sqrt[a]*d)*(2*Sqrt[b]*c*(4*B*c - 3*A*d) + Sqrt [a]*(-8*c^2*C + 4*B*c*d - 3*A*d^2))*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d* x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*Ellipti cF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + S qrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)] + (4*I)*A*b*c^2*d^2*Sqrt[(d*(Sqrt[a]/Sq rt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[b] - d*x)/(c + d*x))]*(c + d*x)^(3/2)*EllipticPi[(Sqrt[b]*c)/(Sqrt[b]*c - Sqrt[a]*d), I*ArcSinh[S...
Leaf count is larger than twice the leaf count of optimal. \(1470\) vs. \(2(532)=1064\).
Time = 9.43 (sec) , antiderivative size = 1470, normalized size of antiderivative = 2.76, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {2355, 628, 25, 2352, 25, 2351, 600, 509, 508, 327, 512, 511, 321, 633, 632, 186, 413, 412, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^3 \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 2355 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {\sqrt {a-b x^2}}{x^3 \sqrt {c+d x}}dx+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^3}dx\) |
| \(\Big \downarrow \) 628 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{4} \int -\frac {-\frac {b d x^2}{c}+2 b x+\frac {3 a d}{c}}{x^2 \sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{2 c x^2}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {1}{4} \int \frac {-\frac {b d x^2}{c}+2 b x+\frac {3 a d}{c}}{x^2 \sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{2 c x^2}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^3}dx\) |
| \(\Big \downarrow \) 2352 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{4} \left (\frac {\int -\frac {-\frac {3 a b d^2 x^2}{c}-2 a b d x+a \left (4 b c-\frac {3 a d^2}{c}\right )}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}+\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{c^2 x}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{2 c x^2}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{4} \left (\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{c^2 x}-\frac {\int \frac {-\frac {3 a b d^2 x^2}{c}-2 a b d x+a \left (4 b c-\frac {3 a d^2}{c}\right )}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{2 c x^2}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^3}dx\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{4} \left (\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{c^2 x}-\frac {a \left (4 b c-\frac {3 a d^2}{c}\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+\int \frac {-\frac {3 a b x d^2}{c}-2 a b d}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{2 c x^2}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^3}dx\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{4} \left (\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{c^2 x}-\frac {a \left (4 b c-\frac {3 a d^2}{c}\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+a b d \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {3 a b d \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{c}}{2 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{2 c x^2}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^3}dx\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{4} \left (\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{c^2 x}-\frac {a \left (4 b c-\frac {3 a d^2}{c}\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+a b d \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {3 a b d \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{c \sqrt {a-b x^2}}}{2 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{2 c x^2}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^3}dx\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{4} \left (\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{c^2 x}-\frac {\frac {6 a^{3/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{c \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+a \left (4 b c-\frac {3 a d^2}{c}\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+a b d \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{2 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{2 c x^2}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^3}dx\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{4} \left (\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{c^2 x}-\frac {a \left (4 b c-\frac {3 a d^2}{c}\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+a b d \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {6 a^{3/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{2 c x^2}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^3}dx\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{4} \left (\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{c^2 x}-\frac {a \left (4 b c-\frac {3 a d^2}{c}\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {a b d \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}+\frac {6 a^{3/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{2 c x^2}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^3}dx\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{4} \left (\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{c^2 x}-\frac {-\frac {2 a^{3/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {a-b x^2} \sqrt {c+d x}}+a \left (4 b c-\frac {3 a d^2}{c}\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx+\frac {6 a^{3/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{2 c x^2}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^3}dx\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{4} \left (\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{c^2 x}-\frac {a \left (4 b c-\frac {3 a d^2}{c}\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {2 a^{3/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {6 a^{3/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{2 c x^2}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^3}dx\) |
| \(\Big \downarrow \) 633 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{4} \left (\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{c^2 x}-\frac {\frac {a \sqrt {1-\frac {b x^2}{a}} \left (4 b c-\frac {3 a d^2}{c}\right ) \int \frac {1}{x \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}-\frac {2 a^{3/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {6 a^{3/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{2 c x^2}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^3}dx\) |
| \(\Big \downarrow \) 632 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{4} \left (\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{c^2 x}-\frac {\frac {a \sqrt {1-\frac {b x^2}{a}} \left (4 b c-\frac {3 a d^2}{c}\right ) \int \frac {1}{x \sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}} \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {c+d x}}dx}{\sqrt {a-b x^2}}-\frac {2 a^{3/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {6 a^{3/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{2 c x^2}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^3}dx\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{4} \left (\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{c^2 x}-\frac {-\frac {2 a \sqrt {1-\frac {b x^2}{a}} \left (4 b c-\frac {3 a d^2}{c}\right ) \int \frac {\sqrt {a}}{\sqrt {b} x \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {c+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}}}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {a-b x^2}}-\frac {2 a^{3/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {6 a^{3/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{2 c x^2}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^3}dx\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{4} \left (\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{c^2 x}-\frac {-\frac {2 a \sqrt {1-\frac {b x^2}{a}} \left (4 b c-\frac {3 a d^2}{c}\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \int \frac {\sqrt {a}}{\sqrt {b} x \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} c+\sqrt {a} d}}}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 a^{3/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {6 a^{3/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}}{2 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{2 c x^2}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^3}dx\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^3}dx+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{4} \left (\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{c^2 x}-\frac {-\frac {2 a^{3/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {6 a^{3/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {2 a \sqrt {1-\frac {b x^2}{a}} \left (4 b c-\frac {3 a d^2}{c}\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}}{2 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{2 c x^2}\right )\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\sqrt {c+d x} \sqrt {a-b x^2} C}{d x^2}+\frac {(B d-c C) \sqrt {c+d x} \sqrt {a-b x^2}}{d^2 x^3}\right )dx+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{4} \left (\frac {3 d \sqrt {a-b x^2} \sqrt {c+d x}}{c^2 x}-\frac {-\frac {2 a^{3/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {6 a^{3/2} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {2 a \sqrt {1-\frac {b x^2}{a}} \left (4 b c-\frac {3 a d^2}{c}\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}}{2 a c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{2 c x^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \sqrt {a} \sqrt {b} \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) C}{d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {\sqrt {a} \sqrt {b} c \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) C}{d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {a \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) C}{\sqrt {c+d x} \sqrt {a-b x^2}}-\frac {\sqrt {c+d x} \sqrt {a-b x^2} C}{d x}-\frac {\sqrt {a} \sqrt {b} (c C-B d) \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{4 c d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {5 \sqrt {a} \sqrt {b} (c C-B d) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{4 d \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {(c C-B d) \left (4 b c^2+a d^2\right ) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{4 c d^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{4} \left (\frac {3 d \sqrt {c+d x} \sqrt {a-b x^2}}{c^2 x}-\frac {\frac {6 \sqrt {b} d \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) a^{3/2}}{c \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {2 \sqrt {b} d \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right ) a^{3/2}}{\sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 \left (4 b c-\frac {3 a d^2}{c}\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} c+\sqrt {a} d}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right ) a}{\sqrt {a-b x^2} \sqrt {c+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}}}}{2 a c}\right )-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{2 c x^2}\right )+\frac {(c C-B d) \sqrt {c+d x} \sqrt {a-b x^2}}{4 c d x}+\frac {(c C-B d) \sqrt {c+d x} \sqrt {a-b x^2}}{2 d^2 x^2}\) |
Input:
Int[(Sqrt[a - b*x^2]*(A + B*x + C*x^2))/(x^3*Sqrt[c + d*x]),x]
Output:
((c*C - B*d)*Sqrt[c + d*x]*Sqrt[a - b*x^2])/(2*d^2*x^2) - (C*Sqrt[c + d*x] *Sqrt[a - b*x^2])/(d*x) + ((c*C - B*d)*Sqrt[c + d*x]*Sqrt[a - b*x^2])/(4*c *d*x) + (3*Sqrt[a]*Sqrt[b]*C*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[A rcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2] ) - (Sqrt[a]*Sqrt[b]*(c*C - B*d)*Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*Ellipti cE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[ a] + d)])/(4*c*d*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) - (Sqrt[a]*Sqrt[b]*c*C*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqr t[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a] ]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(d*Sqrt[c + d*x]*Sqrt[a - b* x^2]) - (5*Sqrt[a]*Sqrt[b]*(c*C - B*d)*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/S qrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(4*d*Sqrt[c + d*x]*Sqr t[a - b*x^2]) - (a*C*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqr t[1 - (b*x^2)/a]*EllipticPi[2, ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2 ]], (2*Sqrt[a]*d)/(Sqrt[b]*c + Sqrt[a]*d)])/(Sqrt[c + d*x]*Sqrt[a - b*x^2] ) - ((c*C - B*d)*(4*b*c^2 + a*d^2)*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + S qrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticPi[2, ArcSin[Sqrt[1 - (Sqrt[b]*x)/S qrt[a]]/Sqrt[2]], (2*Sqrt[a]*d)/(Sqrt[b]*c + Sqrt[a]*d)])/(4*c*d^2*Sqrt...
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ )]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2 Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c*f)/d, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*Sqrt[(a_) + (b_.)*(x_)^2], x _Symbol] :> Simp[c^(n - 1/2)*(e*x)^(m + 1)*Sqrt[c + d*x]*(Sqrt[a + b*x^2]/( e*(m + 1))), x] - Simp[1/(2*e*(m + 1)) Int[((e*x)^(m + 1)/(Sqrt[c + d*x]* Sqrt[a + b*x^2]))*ExpandToSum[(2*a*c^(n + 1/2)*(m + 1) + a*c^(n - 1/2)*d*(2 *m + 3)*x + 2*b*c^(n + 1/2)*(m + 2)*x^2 + b*c^(n - 1/2)*d*(2*m + 5)*x^3 - 2 *a*(m + 1)*(c + d*x)^(n + 1/2) - 2*b*(m + 1)*x^2*(c + d*x)^(n + 1/2))/x, x] , x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n + 3/2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > With[{q = Rt[-b/a, 2]}, Simp[1/Sqrt[a] Int[1/(x*Sqrt[c + d*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > Simp[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(x*Sqrt[c + d*x]*Sqrt[1 + b*(x^2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[((Px_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.))/(x_), x_S ymbol] :> Int[PolynomialQuotient[Px, x, x]*(c + d*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, x, x] Int[(c + d*x)^n*((a + b*x^2)^p/x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && PolynomialQ[Px, x]
Int[((Px_)*((e_.)*(x_))^(m_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x _)^2]), x_Symbol] :> With[{Px0 = Coefficient[Px, x, 0]}, Simp[Px0*(e*x)^(m + 1)*Sqrt[c + d*x]*(Sqrt[a + b*x^2]/(a*c*e*(m + 1))), x] + Simp[1/(2*a*c*e* (m + 1)) Int[((e*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[a + b*x^2]))*ExpandToSum[ 2*a*c*(m + 1)*((Px - Px0)/x) - Px0*(a*d*(2*m + 3) + 2*b*c*(m + 2)*x + b*d*( 2*m + 5)*x^2), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[Px, x] && LtQ[m, -1]
Int[(Px_)*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2) ^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, c + d*x, x]*(e*x)^m*(c + d* x)^(n + 1)*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, c + d*x, x] I nt[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p} , x] && PolynomialQ[Px, x] && LtQ[n, 0]
Time = 3.05 (sec) , antiderivative size = 849, normalized size of antiderivative = 1.60
| method | result | size |
| risch | \(-\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}\, \left (-3 A d x +4 B c x +2 A c \right )}{4 c^{2} x^{2}}+\frac {\left (\frac {\left (3 A \,d^{2}-4 B c d -8 C \,c^{2}\right ) \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \left (\left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )-\frac {c \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{d}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {\left (3 A a \,d^{2}-4 b A \,c^{2}-4 B a c d +8 C a \,c^{2}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, 2, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {8 B \,c^{2} \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 A c d \sqrt {a b}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}}{2}, \sqrt {-\frac {2 \sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (d x +c \right )}}{8 c^{2} \sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}\) | \(849\) |
| elliptic | \(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (d x +c \right )}\, \left (-\frac {A \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{2 c \,x^{2}}+\frac {\left (3 A d -4 B c \right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{4 c^{2} x}+\frac {2 \left (-B b +\frac {A b d}{4 c}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (-C b +\frac {\left (3 A d -4 B c \right ) b d}{8 c^{2}}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \left (\left (-\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )+\frac {\sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{b}\right )}{\sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}-\frac {\left (3 A a \,d^{2}-4 b A \,c^{2}-4 B a c d +8 C a \,c^{2}\right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, d \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, -\frac {\left (-\frac {c}{d}+\frac {\sqrt {a b}}{b}\right ) d}{c}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{4 c^{3} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}\) | \(902\) |
| default | \(\text {Expression too large to display}\) | \(3568\) |
Input:
int((-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^3/(d*x+c)^(1/2),x,method=_RETURNVERBO SE)
Output:
-1/4*(-b*x^2+a)^(1/2)*(d*x+c)^(1/2)*(-3*A*d*x+4*B*c*x+2*A*c)/c^2/x^2+1/8/c ^2*((3*A*d^2-4*B*c*d-8*C*c^2)*(a*b)^(1/2)*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/( a*b)^(1/2))^(1/2)*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*(-2*(x-1/b*(a*b)^( 1/2))*b/(a*b)^(1/2))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*((c/d-1/b*(a *b)^(1/2))*EllipticE(1/2*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2) ,(-2/b*(a*b)^(1/2)/(c/d-1/b*(a*b)^(1/2)))^(1/2))-c/d*EllipticF(1/2*2^(1/2) *((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),(-2/b*(a*b)^(1/2)/(c/d-1/b*(a*b )^(1/2)))^(1/2)))-(3*A*a*d^2-4*A*b*c^2-4*B*a*c*d+8*C*a*c^2)*2^(1/2)*((x+1/ b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)* (-2*(x-1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^ (1/2)*EllipticPi(1/2*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2),2,( -2/b*(a*b)^(1/2)/(c/d-1/b*(a*b)^(1/2)))^(1/2))-8*B*c^2*(a*b)^(1/2)*2^(1/2) *((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*((x+c/d)/(c/d-1/b*(a*b)^(1/2))) ^(1/2)*(-2*(x-1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)/(-b*d*x^3-b*c*x^2+a*d* x+a*c)^(1/2)*EllipticF(1/2*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/ 2),(-2/b*(a*b)^(1/2)/(c/d-1/b*(a*b)^(1/2)))^(1/2))+2*A*c*d*(a*b)^(1/2)*2^( 1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)*((x+c/d)/(c/d-1/b*(a*b)^(1/ 2)))^(1/2)*(-2*(x-1/b*(a*b)^(1/2))*b/(a*b)^(1/2))^(1/2)/(-b*d*x^3-b*c*x^2+ a*d*x+a*c)^(1/2)*EllipticF(1/2*2^(1/2)*((x+1/b*(a*b)^(1/2))*b/(a*b)^(1/2)) ^(1/2),(-2/b*(a*b)^(1/2)/(c/d-1/b*(a*b)^(1/2)))^(1/2)))*((-b*x^2+a)*(d*...
Timed out. \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^3 \sqrt {c+d x}} \, dx=\text {Timed out} \] Input:
integrate((-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^3/(d*x+c)^(1/2),x, algorithm="f ricas")
Output:
Timed out
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^3 \sqrt {c+d x}} \, dx=\int \frac {\sqrt {a - b x^{2}} \left (A + B x + C x^{2}\right )}{x^{3} \sqrt {c + d x}}\, dx \] Input:
integrate((-b*x**2+a)**(1/2)*(C*x**2+B*x+A)/x**3/(d*x+c)**(1/2),x)
Output:
Integral(sqrt(a - b*x**2)*(A + B*x + C*x**2)/(x**3*sqrt(c + d*x)), x)
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^3 \sqrt {c+d x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {-b x^{2} + a}}{\sqrt {d x + c} x^{3}} \,d x } \] Input:
integrate((-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^3/(d*x+c)^(1/2),x, algorithm="m axima")
Output:
integrate((C*x^2 + B*x + A)*sqrt(-b*x^2 + a)/(sqrt(d*x + c)*x^3), x)
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^3 \sqrt {c+d x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {-b x^{2} + a}}{\sqrt {d x + c} x^{3}} \,d x } \] Input:
integrate((-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^3/(d*x+c)^(1/2),x, algorithm="g iac")
Output:
integrate((C*x^2 + B*x + A)*sqrt(-b*x^2 + a)/(sqrt(d*x + c)*x^3), x)
Timed out. \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^3 \sqrt {c+d x}} \, dx=\int \frac {\sqrt {a-b\,x^2}\,\left (C\,x^2+B\,x+A\right )}{x^3\,\sqrt {c+d\,x}} \,d x \] Input:
int(((a - b*x^2)^(1/2)*(A + B*x + C*x^2))/(x^3*(c + d*x)^(1/2)),x)
Output:
int(((a - b*x^2)^(1/2)*(A + B*x + C*x^2))/(x^3*(c + d*x)^(1/2)), x)
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^3 \sqrt {c+d x}} \, dx=\int \frac {\sqrt {-b \,x^{2}+a}\, \left (C \,x^{2}+B x +A \right )}{x^{3} \sqrt {d x +c}}d x \] Input:
int((-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^3/(d*x+c)^(1/2),x)
Output:
int((-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^3/(d*x+c)^(1/2),x)