Integrand size = 21, antiderivative size = 497 \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^{3/2} \, dx=\frac {4 \sqrt {d+e x} \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4-3 c d e \left (c d^2-31 a e^2\right ) x\right ) \sqrt {a+c x^2}}{1155 c e^3}+\frac {2 \sqrt {d+e x} \left (c d^2-3 a e^2+28 c d e x\right ) \left (a+c x^2\right )^{3/2}}{231 c e}+\frac {2 e \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 c}+\frac {32 \sqrt {-a} d \left (c d^2-3 a e^2\right ) \left (c d^2+9 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{1155 \sqrt {c} e^4 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {8 \sqrt {-a} \left (c d^2+a e^2\right ) \left (4 c^2 d^4+21 a c d^2 e^2-15 a^2 e^4\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{1155 c^{3/2} e^4 \sqrt {d+e x} \sqrt {a+c x^2}} \]
2/231*(28*c*d*e*x-3*a*e^2+c*d^2)*(c*x^2+a)^(3/2)*(e*x+d)^(1/2)/c/e+2/11*e* (c*x^2+a)^(5/2)*(e*x+d)^(1/2)/c+4/1155*(4*c^2*d^4+21*a*c*d^2*e^2-15*a^2*e^ 4-3*c*d*e*(-31*a*e^2+c*d^2)*x)*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)/c/e^3+32/1155 *d*(-3*a*e^2+c*d^2)*(9*a*e^2+c*d^2)*EllipticE(1/2*(1-x*c^(1/2)/(-a)^(1/2)) ^(1/2)*2^(1/2),(-2*a*e/(-a*e+d*(-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*(e*x +d)^(1/2)*(1+c*x^2/a)^(1/2)/e^4/c^(1/2)/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/( e*(-a)^(1/2)+d*c^(1/2)))^(1/2)-8/1155*(a*e^2+c*d^2)*(-15*a^2*e^4+21*a*c*d^ 2*e^2+4*c^2*d^4)*EllipticF(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2* a*e/(-a*e+d*(-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*(1+c*x^2/a)^(1/2)*((e*x +d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)/c^(3/2)/e^4/(e*x+d)^(1/2)/(c*x ^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 25.71 (sec) , antiderivative size = 698, normalized size of antiderivative = 1.40 \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^{3/2} \, dx=\frac {\sqrt {d+e x} \left (\frac {2 \left (a+c x^2\right ) \left (60 a^2 e^4+a c e^2 \left (47 d^2+326 d e x+195 e^2 x^2\right )+c^2 \left (8 d^4-6 d^3 e x+5 d^2 e^2 x^2+140 d e^3 x^3+105 e^4 x^4\right )\right )}{c e^3}-\frac {8 \left (4 d e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (c^2 d^4+6 a c d^2 e^2-27 a^2 e^4\right ) \left (a+c x^2\right )+4 \sqrt {c} d \left (-i c^{5/2} d^5+\sqrt {a} c^2 d^4 e-6 i a c^{3/2} d^3 e^2+6 a^{3/2} c d^2 e^3+27 i a^2 \sqrt {c} d e^4-27 a^{5/2} e^5\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {a} e \left (-4 c^{5/2} d^5-i \sqrt {a} c^2 d^4 e-24 a c^{3/2} d^3 e^2-114 i a^{3/2} c d^2 e^3+108 a^2 \sqrt {c} d e^4+15 i a^{5/2} e^5\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{c e^5 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} (d+e x)}\right )}{1155 \sqrt {a+c x^2}} \]
(Sqrt[d + e*x]*((2*(a + c*x^2)*(60*a^2*e^4 + a*c*e^2*(47*d^2 + 326*d*e*x + 195*e^2*x^2) + c^2*(8*d^4 - 6*d^3*e*x + 5*d^2*e^2*x^2 + 140*d*e^3*x^3 + 1 05*e^4*x^4)))/(c*e^3) - (8*(4*d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(c^2* d^4 + 6*a*c*d^2*e^2 - 27*a^2*e^4)*(a + c*x^2) + 4*Sqrt[c]*d*((-I)*c^(5/2)* d^5 + Sqrt[a]*c^2*d^4*e - (6*I)*a*c^(3/2)*d^3*e^2 + 6*a^(3/2)*c*d^2*e^3 + (27*I)*a^2*Sqrt[c]*d*e^4 - 27*a^(5/2)*e^5)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^ (3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + Sqrt[a]*e*(-4*c^(5 /2)*d^5 - I*Sqrt[a]*c^2*d^4*e - 24*a*c^(3/2)*d^3*e^2 - (114*I)*a^(3/2)*c*d ^2*e^3 + 108*a^2*Sqrt[c]*d*e^4 + (15*I)*a^(5/2)*e^5)*Sqrt[(e*((I*Sqrt[a])/ Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]* (d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[ d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(c*e^5*S qrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(1155*Sqrt[a + c*x^2])
Time = 1.05 (sec) , antiderivative size = 849, normalized size of antiderivative = 1.71, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {497, 27, 682, 27, 682, 27, 599, 25, 1511, 1416, 1509}
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.
| \(\displaystyle \int \left (a+c x^2\right )^{3/2} (d+e x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 497 |
| \(\displaystyle \frac {2 \int \frac {\left (11 c d^2+12 c e x d-a e^2\right ) \left (c x^2+a\right )^{3/2}}{2 \sqrt {d+e x}}dx}{11 c}+\frac {2 e \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (11 c d^2+12 c e x d-a e^2\right ) \left (c x^2+a\right )^{3/2}}{\sqrt {d+e x}}dx}{11 c}+\frac {2 e \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {4 \int \frac {3 c e \left (a e \left (29 c d^2-3 a e^2\right )-c d \left (c d^2-31 a e^2\right ) x\right ) \sqrt {c x^2+a}}{2 \sqrt {d+e x}}dx}{21 c e^2}+\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (-3 a e^2+c d^2+28 c d e x\right )}{21 e}}{11 c}+\frac {2 e \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \int \frac {\left (a e \left (29 c d^2-3 a e^2\right )-c d \left (c d^2-31 a e^2\right ) x\right ) \sqrt {c x^2+a}}{\sqrt {d+e x}}dx}{7 e}+\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (-3 a e^2+c d^2+28 c d e x\right )}{21 e}}{11 c}+\frac {2 e \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {2 \left (\frac {4 \int \frac {c \left (a e \left (c^2 d^4+114 a c e^2 d^2-15 a^2 e^4\right )-4 c d \left (c d^2-3 a e^2\right ) \left (c d^2+9 a e^2\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+a}}dx}{15 c e^2}+\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (-15 a^2 e^4-3 c d e x \left (c d^2-31 a e^2\right )+21 a c d^2 e^2+4 c^2 d^4\right )}{15 e^2}\right )}{7 e}+\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (-3 a e^2+c d^2+28 c d e x\right )}{21 e}}{11 c}+\frac {2 e \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \left (\frac {2 \int \frac {a e \left (c^2 d^4+114 a c e^2 d^2-15 a^2 e^4\right )-4 c d \left (c d^2-3 a e^2\right ) \left (c d^2+9 a e^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{15 e^2}+\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (-15 a^2 e^4-3 c d e x \left (c d^2-31 a e^2\right )+21 a c d^2 e^2+4 c^2 d^4\right )}{15 e^2}\right )}{7 e}+\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (-3 a e^2+c d^2+28 c d e x\right )}{21 e}}{11 c}+\frac {2 e \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {\frac {2 \left (\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (-15 a^2 e^4-3 c d e x \left (c d^2-31 a e^2\right )+21 a c d^2 e^2+4 c^2 d^4\right )}{15 e^2}-\frac {4 \int -\frac {\left (c d^2+a e^2\right ) \left (4 c^2 d^4+21 a c e^2 d^2-15 a^2 e^4\right )-4 c d \left (c d^2-3 a e^2\right ) \left (c d^2+9 a e^2\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{15 e^4}\right )}{7 e}+\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (-3 a e^2+c d^2+28 c d e x\right )}{21 e}}{11 c}+\frac {2 e \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 \left (\frac {4 \int \frac {\left (c d^2+a e^2\right ) \left (4 c^2 d^4+21 a c e^2 d^2-15 a^2 e^4\right )-4 c d \left (c d^2-3 a e^2\right ) \left (c d^2+9 a e^2\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{15 e^4}+\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (-15 a^2 e^4-3 c d e x \left (c d^2-31 a e^2\right )+21 a c d^2 e^2+4 c^2 d^4\right )}{15 e^2}\right )}{7 e}+\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (-3 a e^2+c d^2+28 c d e x\right )}{21 e}}{11 c}+\frac {2 e \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {\frac {2 \left (\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (-15 a^2 e^4-3 c d e x \left (c d^2-31 a e^2\right )+21 a c d^2 e^2+4 c^2 d^4\right )}{15 e^2}-\frac {4 \left (\sqrt {a e^2+c d^2} \left (4 \sqrt {c} d \left (c d^2-3 a e^2\right ) \left (9 a e^2+c d^2\right )-\sqrt {a e^2+c d^2} \left (-15 a^2 e^4+21 a c d^2 e^2+4 c^2 d^4\right )\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}-4 \sqrt {c} d \left (c d^2-3 a e^2\right ) \sqrt {a e^2+c d^2} \left (9 a e^2+c d^2\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}\right )}{15 e^4}\right )}{7 e}+\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (-3 a e^2+c d^2+28 c d e x\right )}{21 e}}{11 c}+\frac {2 e \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\frac {2 \left (\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (-15 a^2 e^4-3 c d e x \left (c d^2-31 a e^2\right )+21 a c d^2 e^2+4 c^2 d^4\right )}{15 e^2}-\frac {4 \left (\frac {\left (a e^2+c d^2\right )^{3/4} \left (4 \sqrt {c} d \left (c d^2-3 a e^2\right ) \left (9 a e^2+c d^2\right )-\sqrt {a e^2+c d^2} \left (-15 a^2 e^4+21 a c d^2 e^2+4 c^2 d^4\right )\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-4 \sqrt {c} d \left (c d^2-3 a e^2\right ) \sqrt {a e^2+c d^2} \left (9 a e^2+c d^2\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}\right )}{15 e^4}\right )}{7 e}+\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (-3 a e^2+c d^2+28 c d e x\right )}{21 e}}{11 c}+\frac {2 e \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{11 c}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2 e \sqrt {d+e x} \left (c x^2+a\right )^{5/2}}{11 c}+\frac {\frac {2 \sqrt {d+e x} \left (c d^2+28 c e x d-3 a e^2\right ) \left (c x^2+a\right )^{3/2}}{21 e}+\frac {2 \left (\frac {2 \sqrt {d+e x} \left (4 c^2 d^4+21 a c e^2 d^2-3 c e \left (c d^2-31 a e^2\right ) x d-15 a^2 e^4\right ) \sqrt {c x^2+a}}{15 e^2}-\frac {4 \left (\frac {\left (c d^2+a e^2\right )^{3/4} \left (4 \sqrt {c} d \left (c d^2-3 a e^2\right ) \left (c d^2+9 a e^2\right )-\sqrt {c d^2+a e^2} \left (4 c^2 d^4+21 a c e^2 d^2-15 a^2 e^4\right )\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right ) \sqrt {\frac {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}{\left (\frac {c d^2}{e^2}+a\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}-4 \sqrt {c} d \left (c d^2-3 a e^2\right ) \sqrt {c d^2+a e^2} \left (c d^2+9 a e^2\right ) \left (\frac {\sqrt [4]{c d^2+a e^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right ) \sqrt {\frac {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}{\left (\frac {c d^2}{e^2}+a\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}-\frac {\sqrt {d+e x} \sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}{\left (\frac {c d^2}{e^2}+a\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right )}\right )\right )}{15 e^4}\right )}{7 e}}{11 c}\) |
(2*e*Sqrt[d + e*x]*(a + c*x^2)^(5/2))/(11*c) + ((2*Sqrt[d + e*x]*(c*d^2 - 3*a*e^2 + 28*c*d*e*x)*(a + c*x^2)^(3/2))/(21*e) + (2*((2*Sqrt[d + e*x]*(4* c^2*d^4 + 21*a*c*d^2*e^2 - 15*a^2*e^4 - 3*c*d*e*(c*d^2 - 31*a*e^2)*x)*Sqrt [a + c*x^2])/(15*e^2) - (4*(-4*Sqrt[c]*d*(c*d^2 - 3*a*e^2)*Sqrt[c*d^2 + a* e^2]*(c*d^2 + 9*a*e^2)*(-((Sqrt[d + e*x]*Sqrt[a + (c*d^2)/e^2 - (2*c*d*(d + e*x))/e^2 + (c*(d + e*x)^2)/e^2])/((a + (c*d^2)/e^2)*(1 + (Sqrt[c]*(d + e*x))/Sqrt[c*d^2 + a*e^2]))) + ((c*d^2 + a*e^2)^(1/4)*(1 + (Sqrt[c]*(d + e *x))/Sqrt[c*d^2 + a*e^2])*Sqrt[(a + (c*d^2)/e^2 - (2*c*d*(d + e*x))/e^2 + (c*(d + e*x)^2)/e^2)/((a + (c*d^2)/e^2)*(1 + (Sqrt[c]*(d + e*x))/Sqrt[c*d^ 2 + a*e^2])^2)]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[d + e*x])/(c*d^2 + a*e^2) ^(1/4)], (1 + (Sqrt[c]*d)/Sqrt[c*d^2 + a*e^2])/2])/(c^(1/4)*Sqrt[a + (c*d^ 2)/e^2 - (2*c*d*(d + e*x))/e^2 + (c*(d + e*x)^2)/e^2])) + ((c*d^2 + a*e^2) ^(3/4)*(4*Sqrt[c]*d*(c*d^2 - 3*a*e^2)*(c*d^2 + 9*a*e^2) - Sqrt[c*d^2 + a*e ^2]*(4*c^2*d^4 + 21*a*c*d^2*e^2 - 15*a^2*e^4))*(1 + (Sqrt[c]*(d + e*x))/Sq rt[c*d^2 + a*e^2])*Sqrt[(a + (c*d^2)/e^2 - (2*c*d*(d + e*x))/e^2 + (c*(d + e*x)^2)/e^2)/((a + (c*d^2)/e^2)*(1 + (Sqrt[c]*(d + e*x))/Sqrt[c*d^2 + a*e ^2])^2)]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[d + e*x])/(c*d^2 + a*e^2)^(1/4)] , (1 + (Sqrt[c]*d)/Sqrt[c*d^2 + a*e^2])/2])/(2*c^(1/4)*Sqrt[a + (c*d^2)/e^ 2 - (2*c*d*(d + e*x))/e^2 + (c*(d + e*x)^2)/e^2])))/(15*e^4)))/(7*e))/(11* c)
3.7.63.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[1/(b *(n + 2*p + 1)) Int[(c + d*x)^(n - 2)*(a + b*x^2)^p*Simp[b*c^2*(n + 2*p + 1) - a*d^2*(n - 1) + 2*b*c*d*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, n , p}, x] && If[RationalQ[n], GtQ[n, 1], SumSimplerQ[n, -2]] && NeQ[n + 2*p + 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[-2/d^2 Subst[Int[(B*c - A*d - B*x^2)/Sqrt[(b*c^2 + a *d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2)], x], x, Sqrt[c + d*x]], x] /; Fr eeQ[{a, b, c, d, A, B}, x] && PosQ[b/a]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Timed out.
hanged
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.74 \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^{3/2} \, dx=\frac {2 \, {\left (4 \, {\left (4 \, c^{3} d^{6} + 27 \, a c^{2} d^{4} e^{2} + 234 \, a^{2} c d^{2} e^{4} - 45 \, a^{3} e^{6}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 48 \, {\left (c^{3} d^{5} e + 6 \, a c^{2} d^{3} e^{3} - 27 \, a^{2} c d e^{5}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (105 \, c^{3} e^{6} x^{4} + 140 \, c^{3} d e^{5} x^{3} + 8 \, c^{3} d^{4} e^{2} + 47 \, a c^{2} d^{2} e^{4} + 60 \, a^{2} c e^{6} + 5 \, {\left (c^{3} d^{2} e^{4} + 39 \, a c^{2} e^{6}\right )} x^{2} - 2 \, {\left (3 \, c^{3} d^{3} e^{3} - 163 \, a c^{2} d e^{5}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{3465 \, c^{2} e^{5}} \]
2/3465*(4*(4*c^3*d^6 + 27*a*c^2*d^4*e^2 + 234*a^2*c*d^2*e^4 - 45*a^3*e^6)* sqrt(c*e)*weierstrassPInverse(4/3*(c*d^2 - 3*a*e^2)/(c*e^2), -8/27*(c*d^3 + 9*a*d*e^2)/(c*e^3), 1/3*(3*e*x + d)/e) + 48*(c^3*d^5*e + 6*a*c^2*d^3*e^3 - 27*a^2*c*d*e^5)*sqrt(c*e)*weierstrassZeta(4/3*(c*d^2 - 3*a*e^2)/(c*e^2) , -8/27*(c*d^3 + 9*a*d*e^2)/(c*e^3), weierstrassPInverse(4/3*(c*d^2 - 3*a* e^2)/(c*e^2), -8/27*(c*d^3 + 9*a*d*e^2)/(c*e^3), 1/3*(3*e*x + d)/e)) + 3*( 105*c^3*e^6*x^4 + 140*c^3*d*e^5*x^3 + 8*c^3*d^4*e^2 + 47*a*c^2*d^2*e^4 + 6 0*a^2*c*e^6 + 5*(c^3*d^2*e^4 + 39*a*c^2*e^6)*x^2 - 2*(3*c^3*d^3*e^3 - 163* a*c^2*d*e^5)*x)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(c^2*e^5)
\[ \int (d+e x)^{3/2} \left (a+c x^2\right )^{3/2} \, dx=\int \left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {3}{2}}\, dx \]
\[ \int (d+e x)^{3/2} \left (a+c x^2\right )^{3/2} \, dx=\int { {\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}} \,d x } \]
\[ \int (d+e x)^{3/2} \left (a+c x^2\right )^{3/2} \, dx=\int { {\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int (d+e x)^{3/2} \left (a+c x^2\right )^{3/2} \, dx=\int {\left (c\,x^2+a\right )}^{3/2}\,{\left (d+e\,x\right )}^{3/2} \,d x \]