Integrand size = 35, antiderivative size = 505 \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^2 \sqrt {c+d x}} \, dx=\frac {2 C \sqrt {c+d x} \sqrt {a-b x^2}}{3 d}-\frac {A \sqrt {c+d x} \sqrt {a-b x^2}}{c x}-\frac {\sqrt {a} \sqrt {b} \left (4 c^2 C-6 B c d-3 A d^2\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 c d^2 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {\sqrt {a} \left (4 a C d^2-b \left (4 c^2 C-6 B c d+3 A d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{3 \sqrt {b} d^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {a (2 B c-A d) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
2/3*C*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/d-A*(d*x+c)^(1/2)*(-b*x^2+a)^(1/2)/c/ x-1/3*a^(1/2)*b^(1/2)*(-3*A*d^2-6*B*c*d+4*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/d^2/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^( 1/2)/(-b*x^2+a)^(1/2)-1/3*a^(1/2)*(4*a*C*d^2-b*(3*A*d^2-6*B*c*d+4*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))/b^(1/2)/d^2/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)-a*(-A*d+2*B*c)*((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/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 28.08 (sec) , antiderivative size = 994, normalized size of antiderivative = 1.97 \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^2 \sqrt {c+d x}} \, dx =\text {Too large to display} \] Input:
Integrate[(Sqrt[a - b*x^2]*(A + B*x + C*x^2))/(x^2*Sqrt[c + d*x]),x]
Output:
(Sqrt[a - b*x^2]*((c*(-3*A*d + 2*c*C*x)*(c + d*x))/x + (-3*A*b*c^3 - 4*a*c ^3*C + (4*b*c^5*C)/d^2 - (6*b*B*c^4)/d + 6*a*B*c^2*d + 3*a*A*c*d^2 + 6*A*b *c^2*(c + d*x) - (8*b*c^4*C*(c + d*x))/d^2 + (12*b*B*c^3*(c + d*x))/d - 3* A*b*c*(c + d*x)^2 + (4*b*c^3*C*(c + d*x)^2)/d^2 - (6*b*B*c^2*(c + d*x)^2)/ d + (I*Sqrt[b]*c*(Sqrt[b]*c - Sqrt[a]*d)*(-4*c^2*C + 6*B*c*d + 3*A*d^2)*Sq rt[(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)*EllipticE[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqr t[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(d ^2*Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]) - (I*(Sqrt[b]*c - Sqrt[a]*d)*(6*A*Sqrt[ b]*c*d + Sqrt[a]*(4*c^2*C - 6*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)*EllipticF[I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (S qrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(d*Sqrt[-c + (Sqrt[a]*d)/S qrt[b]]) + ((6*I)*a*B*c*d*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)*EllipticPi[(Sqrt[ b]*c)/(Sqrt[b]*c - Sqrt[a]*d), I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sq rt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]] - ((3*I)*a*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)*Ellip ticPi[(Sqrt[b]*c)/(Sqrt[b]*c - Sqrt[a]*d), I*ArcSinh[Sqrt[-c + (Sqrt[a]...
Leaf count is larger than twice the leaf count of optimal. \(1365\) vs. \(2(505)=1010\).
Time = 7.87 (sec) , antiderivative size = 1365, normalized size of antiderivative = 2.70, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.514, Rules used = {2355, 628, 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.
| \(\displaystyle \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^2 \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 2355 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \int \frac {\sqrt {a-b x^2}}{x^2 \sqrt {c+d x}}dx+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^2}dx\) |
| \(\Big \downarrow \) 628 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{2} \int -\frac {\frac {b d x^2}{c}+2 b x+\frac {a d}{c}}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{c x}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (-\frac {1}{2} \int \frac {\frac {b d x^2}{c}+2 b x+\frac {a d}{c}}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{c x}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^2}dx\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{2} \left (-\frac {a d \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{c}-\int \frac {\frac {d x b}{c}+2 b}{\sqrt {c+d x} \sqrt {a-b x^2}}dx\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{c x}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^2}dx\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{2} \left (-b \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {a d \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{c}-\frac {b \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{c}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{c x}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^2}dx\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{2} \left (-b \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {a d \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{c}-\frac {b \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{c \sqrt {a-b x^2}}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{c x}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^2}dx\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{2} \left (-b \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {a d \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{c}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{c \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{c x}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^2}dx\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{2} \left (-b \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx-\frac {a d \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{c}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{c x}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^2}dx\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{2} \left (-\frac {a d \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{c}-\frac {b \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{\sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{c x}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^2}dx\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{2} \left (-\frac {a d \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{c}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{c x}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^2}dx\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{2} \left (-\frac {a d \int \frac {1}{x \sqrt {c+d x} \sqrt {a-b x^2}}dx}{c}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{c x}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^2}dx\) |
| \(\Big \downarrow \) 633 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{2} \left (-\frac {a d \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{x \sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{c \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{c x}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^2}dx\) |
| \(\Big \downarrow \) 632 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{2} \left (-\frac {a d \sqrt {1-\frac {b x^2}{a}} \int \frac {1}{x \sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}} \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {c+d x}}dx}{c \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{c x}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^2}dx\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{2} \left (\frac {2 a d \sqrt {1-\frac {b x^2}{a}} \int \frac {\sqrt {a}}{\sqrt {b} x \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {c+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}}}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{c \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{c x}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^2}dx\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{2} \left (\frac {2 a d \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \int \frac {\sqrt {a}}{\sqrt {b} x \sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} c+\sqrt {a} d}}}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{c \sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{c x}\right )+\int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^2}dx\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \int \frac {\left (\frac {B}{d}+\frac {C x}{d}-\frac {c C}{d^2}\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{x^2}dx+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{2} \left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+\frac {2 a d \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c \sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{c x}\right )\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\sqrt {c+d x} \sqrt {a-b x^2} C}{d x}+\frac {(B d-c C) \sqrt {c+d x} \sqrt {a-b x^2}}{d^2 x^2}\right )dx+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{2} \left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {a-b x^2} \sqrt {c+d x}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{c \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}+\frac {2 a d \sqrt {1-\frac {b x^2}{a}} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{c \sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}\right )-\frac {\sqrt {a-b x^2} \sqrt {c+d x}}{c x}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \sqrt {a} \sqrt {b} c \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}{3 d^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (b c^2+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 {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}{3 \sqrt {b} d^2 \sqrt {c+d x} \sqrt {a-b x^2}}-\frac {2 a c \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}{d \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {2 \sqrt {c+d x} \sqrt {a-b x^2} C}{3 d}-\frac {3 \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 )}{d^2 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {\sqrt {a} \sqrt {b} c (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 )}{d^2 \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {a (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 {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 )}{d \sqrt {c+d x} \sqrt {a-b x^2}}+\left (A+\frac {c (c C-B d)}{d^2}\right ) \left (\frac {1}{2} \left (\frac {2 \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 \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \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 )}{\sqrt {c+d x} \sqrt {a-b x^2}}+\frac {2 a d \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 )}{c \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}}}}\right )-\frac {\sqrt {c+d x} \sqrt {a-b x^2}}{c x}\right )+\frac {(c C-B d) \sqrt {c+d x} \sqrt {a-b x^2}}{d^2 x}\) |
Input:
Int[(Sqrt[a - b*x^2]*(A + B*x + C*x^2))/(x^2*Sqrt[c + d*x]),x]
Output:
(2*C*Sqrt[c + d*x]*Sqrt[a - b*x^2])/(3*d) + ((c*C - B*d)*Sqrt[c + d*x]*Sqr t[a - b*x^2])/(d^2*x) + (2*Sqrt[a]*Sqrt[b]*c*C*Sqrt[c + d*x]*Sqrt[1 - (b*x ^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sq rt[b]*c)/Sqrt[a] + d)])/(3*d^2*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[ a]*d)]*Sqrt[a - b*x^2]) - (3*Sqrt[a]*Sqrt[b]*(c*C - B*d)*Sqrt[c + d*x]*Sqr t[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d)])/(d^2*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[a - b*x^2]) + (Sqrt[a]*Sqrt[b]*c*(c*C - B*d)*Sqrt[(Sqr t[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticF[Arc Sin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqrt[a] + d )])/(d^2*Sqrt[c + d*x]*Sqrt[a - b*x^2]) - (2*Sqrt[a]*C*(b*c^2 + 2*a*d^2)*S qrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*Ellip ticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/((Sqrt[b]*c)/Sqr t[a] + d)])/(3*Sqrt[b]*d^2*Sqrt[c + d*x]*Sqrt[a - b*x^2]) - (2*a*c*C*Sqrt[ (Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticP i[2, ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*Sqrt[a]*d)/(Sqrt[b] *c + Sqrt[a]*d)])/(d*Sqrt[c + d*x]*Sqrt[a - b*x^2]) + (a*(c*C - B*d)*Sqrt[ (Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - (b*x^2)/a]*EllipticP i[2, ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*Sqrt[a]*d)/(Sqrt[b] *c + Sqrt[a]*d)])/(d*Sqrt[c + d*x]*Sqrt[a - b*x^2]) + (A + (c*(c*C - B*...
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_.)*((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]
Leaf count of result is larger than twice the leaf count of optimal. \(866\) vs. \(2(420)=840\).
Time = 3.98 (sec) , antiderivative size = 867, normalized size of antiderivative = 1.72
| method | result | size |
| 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}}{x c}+\frac {2 C \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 d}+\frac {2 \left (-A b +\frac {2 a C}{3}\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 (-B b -\frac {A b d}{2 c}+\frac {2 C b c}{3 d}\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 {a \left (A d -2 B 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}}}\, 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 )}{c^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}\) | \(867\) |
| risch | \(\text {Expression too large to display}\) | \(1492\) |
| default | \(\text {Expression too large to display}\) | \(2696\) |
Input:
int((-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^2/(d*x+c)^(1/2),x,method=_RETURNVERBO SE)
Output:
((-b*x^2+a)*(d*x+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x+c)^(1/2)*(-A/x/c*(-b*d*x^ 3-b*c*x^2+a*d*x+a*c)^(1/2)+2/3*C/d*(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)+2*(- A*b+2/3*a*C)*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*( (x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c /d+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF((( x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b )^(1/2)))^(1/2))+2*(-B*b-1/2*A*b*d/c+2/3*C/d*b*c)*(c/d-1/b*(a*b)^(1/2))*(( x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^( 1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/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(((x+c/d)/(c/d-1 /b*(a*b)^(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/ 2))+1/b*(a*b)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),((-c/d +1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)))+a*(A*d-2*B*c)/c^2*(c/d-1 /b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2)) /(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)) )^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*d*EllipticPi(((x+c/d)/(c/d-1/b* (a*b)^(1/2)))^(1/2),-(-c/d+1/b*(a*b)^(1/2))/c*d,((-c/d+1/b*(a*b)^(1/2))/(- c/d-1/b*(a*b)^(1/2)))^(1/2)))
Timed out. \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^2 \sqrt {c+d x}} \, dx=\text {Timed out} \] Input:
integrate((-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^2/(d*x+c)^(1/2),x, algorithm="f ricas")
Output:
Timed out
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^2 \sqrt {c+d x}} \, dx=\int \frac {\sqrt {a - b x^{2}} \left (A + B x + C x^{2}\right )}{x^{2} \sqrt {c + d x}}\, dx \] Input:
integrate((-b*x**2+a)**(1/2)*(C*x**2+B*x+A)/x**2/(d*x+c)**(1/2),x)
Output:
Integral(sqrt(a - b*x**2)*(A + B*x + C*x**2)/(x**2*sqrt(c + d*x)), x)
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^2 \sqrt {c+d x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {-b x^{2} + a}}{\sqrt {d x + c} x^{2}} \,d x } \] Input:
integrate((-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^2/(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^2), x)
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^2 \sqrt {c+d x}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {-b x^{2} + a}}{\sqrt {d x + c} x^{2}} \,d x } \] Input:
integrate((-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^2/(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^2), x)
Timed out. \[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^2 \sqrt {c+d x}} \, dx=\int \frac {\sqrt {a-b\,x^2}\,\left (C\,x^2+B\,x+A\right )}{x^2\,\sqrt {c+d\,x}} \,d x \] Input:
int(((a - b*x^2)^(1/2)*(A + B*x + C*x^2))/(x^2*(c + d*x)^(1/2)),x)
Output:
int(((a - b*x^2)^(1/2)*(A + B*x + C*x^2))/(x^2*(c + d*x)^(1/2)), x)
\[ \int \frac {\sqrt {a-b x^2} \left (A+B x+C x^2\right )}{x^2 \sqrt {c+d x}} \, dx=\int \frac {\sqrt {-b \,x^{2}+a}\, \left (C \,x^{2}+B x +A \right )}{x^{2} \sqrt {d x +c}}d x \] Input:
int((-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^2/(d*x+c)^(1/2),x)
Output:
int((-b*x^2+a)^(1/2)*(C*x^2+B*x+A)/x^2/(d*x+c)^(1/2),x)