Integrand size = 35, antiderivative size = 493 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=-\frac {2 \left (25 a C d^2+b \left (57 c^2 C-49 B c d+35 A d^2\right )\right ) \sqrt {c+d x} \sqrt {a-b x^2}}{105 b^2 d^3}+\frac {2 (16 c C-7 B d) (c+d x)^{3/2} \sqrt {a-b x^2}}{35 b d^3}-\frac {2 C (c+d x)^{5/2} \sqrt {a-b x^2}}{7 b d^3}+\frac {2 \sqrt {a} \left (a d^2 (44 c C-63 B d)+2 b c \left (24 c^2 C-28 B c d+35 A d^2\right )\right ) \sqrt {c+d x} \sqrt {\frac {a-b x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{105 b^{3/2} d^4 \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {a-b x^2}}-\frac {2 \sqrt {a} \left (25 a^2 C d^4+a b d^2 \left (32 c^2 C-49 B c d+35 A d^2\right )+2 b^2 c^2 \left (24 c^2 C-28 B c d+35 A d^2\right )\right ) \sqrt {\frac {c+d x}{c+\frac {\sqrt {a} d}{\sqrt {b}}}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{105 b^{5/2} d^4 \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:
-2/105*(25*a*C*d^2+b*(35*A*d^2-49*B*c*d+57*C*c^2))*(d*x+c)^(1/2)*(-b*x^2+a )^(1/2)/b^2/d^3+2/35*(-7*B*d+16*C*c)*(d*x+c)^(3/2)*(-b*x^2+a)^(1/2)/b/d^3- 2/7*C*(d*x+c)^(5/2)*(-b*x^2+a)^(1/2)/b/d^3+2/105*a^(1/2)*(a*d^2*(-63*B*d+4 4*C*c)+2*b*c*(35*A*d^2-28*B*c*d+24*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))/b^(3/2)/d^4/((d*x+c)/(c+a^(1/2)*d/b^(1/2)))^( 1/2)/(-b*x^2+a)^(1/2)-2/105*a^(1/2)*(25*a^2*C*d^4+a*b*d^2*(35*A*d^2-49*B*c *d+32*C*c^2)+2*b^2*c^2*(35*A*d^2-28*B*c*d+24*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^(5/2)/d^4 /(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 26.95 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.25 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\frac {2 \sqrt {a-b x^2} \left (a d^2 (44 c C-63 B d)+2 b c \left (24 c^2 C-28 B c d+35 A d^2\right )-(c+d x) \left (25 a C d^2+b \left (24 c^2 C-2 c d (14 B+9 C x)+d^2 (35 A+3 x (7 B+5 C x))\right )\right )-\frac {i \sqrt {b} \left (\sqrt {b} c-\sqrt {a} d\right ) \left (a d^2 (44 c C-63 B d)+2 b c \left (24 c^2 C-28 B c d+35 A d^2\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right )|\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d^2 \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}-\frac {i \sqrt {a} \left (25 a^{3/2} C d^3+a \sqrt {b} d^2 (44 c C-63 B d)+2 b^{3/2} c \left (24 c^2 C-28 B c d+35 A d^2\right )+\sqrt {a} b d \left (-12 c^2 C+14 B c d+35 A d^2\right )\right ) \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} \left (-a+b x^2\right )}\right )}{105 b^2 d^3 \sqrt {c+d x}} \] Input:
Integrate[(x^2*(A + B*x + C*x^2))/(Sqrt[c + d*x]*Sqrt[a - b*x^2]),x]
Output:
(2*Sqrt[a - b*x^2]*(a*d^2*(44*c*C - 63*B*d) + 2*b*c*(24*c^2*C - 28*B*c*d + 35*A*d^2) - (c + d*x)*(25*a*C*d^2 + b*(24*c^2*C - 2*c*d*(14*B + 9*C*x) + d^2*(35*A + 3*x*(7*B + 5*C*x)))) - (I*Sqrt[b]*(Sqrt[b]*c - Sqrt[a]*d)*(a*d ^2*(44*c*C - 63*B*d) + 2*b*c*(24*c^2*C - 28*B*c*d + 35*A*d^2))*Sqrt[(d*(Sq rt[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)/Sqrt[b]]/Sqr t[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(d^2*Sqrt[- c + (Sqrt[a]*d)/Sqrt[b]]*(-a + b*x^2)) - (I*Sqrt[a]*(25*a^(3/2)*C*d^3 + a* Sqrt[b]*d^2*(44*c*C - 63*B*d) + 2*b^(3/2)*c*(24*c^2*C - 28*B*c*d + 35*A*d^ 2) + Sqrt[a]*b*d*(-12*c^2*C + 14*B*c*d + 35*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]] , (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)])/(d*Sqrt[-c + (Sqrt[a]* d)/Sqrt[b]]*(-a + b*x^2))))/(105*b^2*d^3*Sqrt[c + d*x])
Time = 2.46 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {2185, 27, 2185, 27, 2185, 27, 600, 509, 508, 327, 512, 511, 321}
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.
| \(\displaystyle \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {a-b x^2} \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle -\frac {2 \int -\frac {-b d^3 (16 c C-7 B d) x^3-d^2 \left (11 b C c^2-7 A b d^2-5 a C d^2\right ) x^2-2 c C d \left (b c^2-5 a d^2\right ) x+5 a c^2 C d^2}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{7 b d^4}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {-b d^3 (16 c C-7 B d) x^3-d^2 \left (11 b C c^2-7 A b d^2-5 a C d^2\right ) x^2-2 c C d \left (b c^2-5 a d^2\right ) x+5 a c^2 C d^2}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{7 b d^4}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (16 c C-7 B d)-\frac {2 \int \frac {-b \left (25 a C d^2+b \left (57 C c^2-49 B d c+35 A d^2\right )\right ) x^2 d^5+a b c (23 c C-21 B d) d^5-b \left (2 b (11 c C-7 B d) c^2+a d^2 (2 c C+21 B d)\right ) x d^4}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{5 b d^3}}{7 b d^4}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (16 c C-7 B d)-\frac {\int \frac {-b \left (25 a C d^2+b \left (57 C c^2-49 B d c+35 A d^2\right )\right ) x^2 d^5+a b c (23 c C-21 B d) d^5-b \left (2 b (11 c C-7 B d) c^2+a d^2 (2 c C+21 B d)\right ) x d^4}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{5 b d^3}}{7 b d^4}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\) |
| \(\Big \downarrow \) 2185 |
| \(\displaystyle \frac {\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (16 c C-7 B d)-\frac {\frac {2}{3} d^4 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a C d^2+b \left (35 A d^2-49 B c d+57 c^2 C\right )\right )-\frac {2 \int \frac {b d^6 \left (a d \left (25 a C d^2-b \left (12 C c^2-14 B d c-35 A d^2\right )\right )-b \left (a (44 c C-63 B d) d^2+2 b c \left (24 C c^2-28 B d c+35 A d^2\right )\right ) x\right )}{2 \sqrt {c+d x} \sqrt {a-b x^2}}dx}{3 b d^2}}{5 b d^3}}{7 b d^4}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (16 c C-7 B d)-\frac {\frac {2}{3} d^4 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a C d^2+b \left (35 A d^2-49 B c d+57 c^2 C\right )\right )-\frac {1}{3} d^4 \int \frac {a d \left (25 a C d^2-b \left (12 C c^2-14 B d c-35 A d^2\right )\right )-b \left (a (44 c C-63 B d) d^2+2 b c \left (24 C c^2-28 B d c+35 A d^2\right )\right ) x}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{5 b d^3}}{7 b d^4}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (16 c C-7 B d)-\frac {\frac {2}{3} d^4 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a C d^2+b \left (35 A d^2-49 B c d+57 c^2 C\right )\right )-\frac {1}{3} d^4 \left (\frac {\left (25 a^2 C d^4+a b d^2 \left (35 A d^2-49 B c d+32 c^2 C\right )+2 b^2 c^2 \left (35 A d^2-28 B c d+24 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {b \left (a d^2 (44 c C-63 B d)+2 b c \left (35 A d^2-28 B c d+24 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a-b x^2}}dx}{d}\right )}{5 b d^3}}{7 b d^4}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (16 c C-7 B d)-\frac {\frac {2}{3} d^4 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a C d^2+b \left (35 A d^2-49 B c d+57 c^2 C\right )\right )-\frac {1}{3} d^4 \left (\frac {\left (25 a^2 C d^4+a b d^2 \left (35 A d^2-49 B c d+32 c^2 C\right )+2 b^2 c^2 \left (35 A d^2-28 B c d+24 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}-\frac {b \sqrt {1-\frac {b x^2}{a}} \left (a d^2 (44 c C-63 B d)+2 b c \left (35 A d^2-28 B c d+24 c^2 C\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}\right )}{5 b d^3}}{7 b d^4}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (16 c C-7 B d)-\frac {\frac {2}{3} d^4 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a C d^2+b \left (35 A d^2-49 B c d+57 c^2 C\right )\right )-\frac {1}{3} d^4 \left (\frac {\left (25 a^2 C d^4+a b d^2 \left (35 A d^2-49 B c d+32 c^2 C\right )+2 b^2 c^2 \left (35 A d^2-28 B c d+24 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (a d^2 (44 c C-63 B d)+2 b c \left (35 A d^2-28 B c d+24 c^2 C\right )\right ) \int \frac {\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{5 b d^3}}{7 b d^4}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (16 c C-7 B d)-\frac {\frac {2}{3} d^4 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a C d^2+b \left (35 A d^2-49 B c d+57 c^2 C\right )\right )-\frac {1}{3} d^4 \left (\frac {\left (25 a^2 C d^4+a b d^2 \left (35 A d^2-49 B c d+32 c^2 C\right )+2 b^2 c^2 \left (35 A d^2-28 B c d+24 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {a-b x^2}}dx}{d}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (a d^2 (44 c C-63 B d)+2 b c \left (35 A d^2-28 B c d+24 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{5 b d^3}}{7 b d^4}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (16 c C-7 B d)-\frac {\frac {2}{3} d^4 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a C d^2+b \left (35 A d^2-49 B c d+57 c^2 C\right )\right )-\frac {1}{3} d^4 \left (\frac {\sqrt {1-\frac {b x^2}{a}} \left (25 a^2 C d^4+a b d^2 \left (35 A d^2-49 B c d+32 c^2 C\right )+2 b^2 c^2 \left (35 A d^2-28 B c d+24 c^2 C\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}+\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (a d^2 (44 c C-63 B d)+2 b c \left (35 A d^2-28 B c d+24 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}\right )}{5 b d^3}}{7 b d^4}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (16 c C-7 B d)-\frac {\frac {2}{3} d^4 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a C d^2+b \left (35 A d^2-49 B c d+57 c^2 C\right )\right )-\frac {1}{3} d^4 \left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (a d^2 (44 c C-63 B d)+2 b c \left (35 A d^2-28 B c d+24 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (25 a^2 C d^4+a b d^2 \left (35 A d^2-49 B c d+32 c^2 C\right )+2 b^2 c^2 \left (35 A d^2-28 B c d+24 c^2 C\right )\right ) \int \frac {1}{\sqrt {1-\frac {d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\frac {\sqrt {b} c}{\sqrt {a}}+d}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {b} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}\right )}{5 b d^3}}{7 b d^4}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {2}{5} d \sqrt {a-b x^2} (c+d x)^{3/2} (16 c C-7 B d)-\frac {\frac {2}{3} d^4 \sqrt {a-b x^2} \sqrt {c+d x} \left (25 a C d^2+b \left (35 A d^2-49 B c d+57 c^2 C\right )\right )-\frac {1}{3} d^4 \left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x} \left (a d^2 (44 c C-63 B d)+2 b c \left (35 A d^2-28 B c d+24 c^2 C\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{d \sqrt {a-b x^2} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}}}-\frac {2 \sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \left (25 a^2 C d^4+a b d^2 \left (35 A d^2-49 B c d+32 c^2 C\right )+2 b^2 c^2 \left (35 A d^2-28 B c d+24 c^2 C\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x}}\right )}{5 b d^3}}{7 b d^4}-\frac {2 C \sqrt {a-b x^2} (c+d x)^{5/2}}{7 b d^3}\) |
Input:
Int[(x^2*(A + B*x + C*x^2))/(Sqrt[c + d*x]*Sqrt[a - b*x^2]),x]
Output:
(-2*C*(c + d*x)^(5/2)*Sqrt[a - b*x^2])/(7*b*d^3) + ((2*d*(16*c*C - 7*B*d)* (c + d*x)^(3/2)*Sqrt[a - b*x^2])/5 - ((2*d^4*(25*a*C*d^2 + b*(57*c^2*C - 4 9*B*c*d + 35*A*d^2))*Sqrt[c + d*x]*Sqrt[a - b*x^2])/3 - (d^4*((2*Sqrt[a]*S qrt[b]*(a*d^2*(44*c*C - 63*B*d) + 2*b*c*(24*c^2*C - 28*B*c*d + 35*A*d^2))* Sqrt[c + d*x]*Sqrt[1 - (b*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sq rt[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]) - (2*Sqrt[a]*(25*a^2*C*d^4 + a*b*d^2*(32*c^2*C - 49*B*c*d + 35*A*d^2) + 2*b^2*c^2*(24*c^2*C - 28*B*c* d + 35*A*d^2))*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)/Sqrt[a]]/Sqrt[2]], (2*d)/ ((Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*d*Sqrt[c + d*x]*Sqrt[a - b*x^2])))/3) /(5*b*d^3))/(7*b*d^4)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[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[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) ^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x )^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p )*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d , e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && !(EqQ[d, 0] && True) && !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Time = 6.03 (sec) , antiderivative size = 778, normalized size of antiderivative = 1.58
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (d x +c \right )}\, \left (-\frac {2 C \,x^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{7 b d}-\frac {2 \left (B -\frac {6 c C}{7 d}\right ) x \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{5 b d}-\frac {2 \left (A +\frac {5 a C}{7 b}-\frac {4 \left (B -\frac {6 c C}{7 d}\right ) c}{5 d}\right ) \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}{3 b d}+\frac {2 \left (\frac {2 \left (B -\frac {6 c C}{7 d}\right ) a c}{5 b d}+\frac {\left (A +\frac {5 a C}{7 b}-\frac {4 \left (B -\frac {6 c C}{7 d}\right ) c}{5 d}\right ) a}{3 b}\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 (\frac {4 C a c}{7 b d}+\frac {3 \left (B -\frac {6 c C}{7 d}\right ) a}{5 b}-\frac {2 \left (A +\frac {5 a C}{7 b}-\frac {4 \left (B -\frac {6 c C}{7 d}\right ) c}{5 d}\right ) 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}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}\) | \(778\) |
| risch | \(\text {Expression too large to display}\) | \(1068\) |
| default | \(\text {Expression too large to display}\) | \(3200\) |
Input:
int(x^2*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(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)*(-2/7*C/b/d*x^2* (-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)-2/5*(B-6/7*c*C/d)/b/d*x*(-b*d*x^3-b*c*x ^2+a*d*x+a*c)^(1/2)-2/3*(A+5/7*a*C/b-4/5*(B-6/7*c*C/d)/d*c)/b/d*(-b*d*x^3- b*c*x^2+a*d*x+a*c)^(1/2)+2*(2/5*(B-6/7*c*C/d)/b/d*a*c+1/3*(A+5/7*a*C/b-4/5 *(B-6/7*c*C/d)/d*c)/b*a)*(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*(4/7*C/b/d*a*c+3/5*(B-6/7*c*C/d)/b*a-2/3*(A +5/7*a*C/b-4/5*(B-6/7*c*C/d)/d*c)/d*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))))
Time = 0.09 (sec) , antiderivative size = 381, normalized size of antiderivative = 0.77 \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=-\frac {2 \, {\left ({\left (48 \, C b^{2} c^{4} - 56 \, B b^{2} c^{3} d - 21 \, B a b c d^{3} + 2 \, {\left (4 \, C a b + 35 \, A b^{2}\right )} c^{2} d^{2} + 15 \, {\left (5 \, C a^{2} + 7 \, A a b\right )} d^{4}\right )} \sqrt {-b d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right ) + 3 \, {\left (48 \, C b^{2} c^{3} d - 56 \, B b^{2} c^{2} d^{2} - 63 \, B a b d^{4} + 2 \, {\left (22 \, C a b + 35 \, A b^{2}\right )} c d^{3}\right )} \sqrt {-b d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b c^{2} + 3 \, a d^{2}\right )}}{3 \, b d^{2}}, -\frac {8 \, {\left (b c^{3} - 9 \, a c d^{2}\right )}}{27 \, b d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) + 3 \, {\left (15 \, C b^{2} d^{4} x^{2} + 24 \, C b^{2} c^{2} d^{2} - 28 \, B b^{2} c d^{3} + 5 \, {\left (5 \, C a b + 7 \, A b^{2}\right )} d^{4} - 3 \, {\left (6 \, C b^{2} c d^{3} - 7 \, B b^{2} d^{4}\right )} x\right )} \sqrt {-b x^{2} + a} \sqrt {d x + c}\right )}}{315 \, b^{3} d^{5}} \] Input:
integrate(x^2*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="f ricas")
Output:
-2/315*((48*C*b^2*c^4 - 56*B*b^2*c^3*d - 21*B*a*b*c*d^3 + 2*(4*C*a*b + 35* A*b^2)*c^2*d^2 + 15*(5*C*a^2 + 7*A*a*b)*d^4)*sqrt(-b*d)*weierstrassPInvers e(4/3*(b*c^2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3 *d*x + c)/d) + 3*(48*C*b^2*c^3*d - 56*B*b^2*c^2*d^2 - 63*B*a*b*d^4 + 2*(22 *C*a*b + 35*A*b^2)*c*d^3)*sqrt(-b*d)*weierstrassZeta(4/3*(b*c^2 + 3*a*d^2) /(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), weierstrassPInverse(4/3*(b*c^ 2 + 3*a*d^2)/(b*d^2), -8/27*(b*c^3 - 9*a*c*d^2)/(b*d^3), 1/3*(3*d*x + c)/d )) + 3*(15*C*b^2*d^4*x^2 + 24*C*b^2*c^2*d^2 - 28*B*b^2*c*d^3 + 5*(5*C*a*b + 7*A*b^2)*d^4 - 3*(6*C*b^2*c*d^3 - 7*B*b^2*d^4)*x)*sqrt(-b*x^2 + a)*sqrt( d*x + c))/(b^3*d^5)
\[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int \frac {x^{2} \left (A + B x + C x^{2}\right )}{\sqrt {a - b x^{2}} \sqrt {c + d x}}\, dx \] Input:
integrate(x**2*(C*x**2+B*x+A)/(d*x+c)**(1/2)/(-b*x**2+a)**(1/2),x)
Output:
Integral(x**2*(A + B*x + C*x**2)/(sqrt(a - b*x**2)*sqrt(c + d*x)), x)
\[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} x^{2}}{\sqrt {-b x^{2} + a} \sqrt {d x + c}} \,d x } \] Input:
integrate(x^2*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="m axima")
Output:
integrate((C*x^2 + B*x + A)*x^2/(sqrt(-b*x^2 + a)*sqrt(d*x + c)), x)
\[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} x^{2}}{\sqrt {-b x^{2} + a} \sqrt {d x + c}} \,d x } \] Input:
integrate(x^2*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x, algorithm="g iac")
Output:
integrate((C*x^2 + B*x + A)*x^2/(sqrt(-b*x^2 + a)*sqrt(d*x + c)), x)
Timed out. \[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int \frac {x^2\,\left (C\,x^2+B\,x+A\right )}{\sqrt {a-b\,x^2}\,\sqrt {c+d\,x}} \,d x \] Input:
int((x^2*(A + B*x + C*x^2))/((a - b*x^2)^(1/2)*(c + d*x)^(1/2)),x)
Output:
int((x^2*(A + B*x + C*x^2))/((a - b*x^2)^(1/2)*(c + d*x)^(1/2)), x)
\[ \int \frac {x^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {a-b x^2}} \, dx=\int \frac {x^{2} \left (C \,x^{2}+B x +A \right )}{\sqrt {d x +c}\, \sqrt {-b \,x^{2}+a}}d x \] Input:
int(x^2*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x)
Output:
int(x^2*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2),x)