Integrand size = 21, antiderivative size = 494 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {8 \sqrt {d+e x} \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4-24 c d e \left (c d^2+2 a e^2\right ) x\right ) \sqrt {a+c x^2}}{693 e^5}+\frac {20 \sqrt {d+e x} \left (8 c d^2+9 a e^2-7 c d e x\right ) \left (a+c x^2\right )^{3/2}}{693 e^3}+\frac {2 \sqrt {d+e x} \left (a+c x^2\right )^{5/2}}{11 e}+\frac {16 \sqrt {-a} \sqrt {c} d \left (32 c^2 d^4+93 a c d^2 e^2+93 a^2 e^4\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{693 e^6 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {16 \sqrt {-a} \left (c d^2+a e^2\right ) \left (32 c^2 d^4+69 a c d^2 e^2+45 a^2 e^4\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{693 \sqrt {c} e^6 \sqrt {d+e x} \sqrt {a+c x^2}} \]
20/693*(-7*c*d*e*x+9*a*e^2+8*c*d^2)*(c*x^2+a)^(3/2)*(e*x+d)^(1/2)/e^3+2/11 *(c*x^2+a)^(5/2)*(e*x+d)^(1/2)/e+8/693*(32*c^2*d^4+69*a*c*d^2*e^2+45*a^2*e ^4-24*c*d*e*(2*a*e^2+c*d^2)*x)*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)/e^5+16/693*d* (93*a^2*e^4+93*a*c*d^2*e^2+32*c^2*d^4)*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)*c ^(1/2)*(e*x+d)^(1/2)*(1+c*x^2/a)^(1/2)/e^6/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2 )/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)-16/693*(a*e^2+c*d^2)*(45*a^2*e^4+69*a*c* d^2*e^2+32*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)/e^6/c^(1/2)/(e*x+d)^(1/2)/( c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.96 (sec) , antiderivative size = 634, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (-\frac {8 d e^2 \left (32 c^2 d^4+93 a c d^2 e^2+93 a^2 e^4\right ) \left (a+c x^2\right )}{d+e x}+e^2 \left (a+c x^2\right ) \left (333 a^2 e^4+2 a c e^2 \left (178 d^2-131 d e x+108 e^2 x^2\right )+c^2 \left (128 d^4-96 d^3 e x+80 d^2 e^2 x^2-70 d e^3 x^3+63 e^4 x^4\right )\right )-8 i c d \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (32 c^2 d^4+93 a c d^2 e^2+93 a^2 e^4\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} \sqrt {d+e x} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\frac {8 \sqrt {a} e \left (32 c^{5/2} d^5+8 i \sqrt {a} c^2 d^4 e+93 a c^{3/2} d^3 e^2+21 i a^{3/2} c d^2 e^3+93 a^2 \sqrt {c} d e^4+45 i a^{5/2} e^5\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} \sqrt {d+e x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{693 e^7 \sqrt {a+c x^2}} \]
(2*Sqrt[d + e*x]*((-8*d*e^2*(32*c^2*d^4 + 93*a*c*d^2*e^2 + 93*a^2*e^4)*(a + c*x^2))/(d + e*x) + e^2*(a + c*x^2)*(333*a^2*e^4 + 2*a*c*e^2*(178*d^2 - 131*d*e*x + 108*e^2*x^2) + c^2*(128*d^4 - 96*d^3*e*x + 80*d^2*e^2*x^2 - 70 *d*e^3*x^3 + 63*e^4*x^4)) - (8*I)*c*d*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(32 *c^2*d^4 + 93*a*c*d^2*e^2 + 93*a^2*e^4)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x)) /(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*Sqrt[d + e*x] *EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqr t[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + (8*Sqrt[a]*e*(32*c^(5/2 )*d^5 + (8*I)*Sqrt[a]*c^2*d^4*e + 93*a*c^(3/2)*d^3*e^2 + (21*I)*a^(3/2)*c* d^2*e^3 + 93*a^2*Sqrt[c]*d*e^4 + (45*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))]* Sqrt[d + e*x]*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)])/Sqrt[-d - ( I*Sqrt[a]*e)/Sqrt[c]]))/(693*e^7*Sqrt[a + c*x^2])
Time = 0.99 (sec) , antiderivative size = 854, normalized size of antiderivative = 1.73, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {493, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 493 |
| \(\displaystyle \frac {10 \int \frac {(a e-c d x) \left (c x^2+a\right )^{3/2}}{\sqrt {d+e x}}dx}{11 e}+\frac {2 \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{11 e}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {10 \left (\frac {4 \int \frac {c \left (a e \left (c d^2+9 a e^2\right )-8 c d \left (c d^2+2 a e^2\right ) x\right ) \sqrt {c x^2+a}}{2 \sqrt {d+e x}}dx}{21 c e^2}+\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (9 a e^2+8 c d^2-7 c d e x\right )}{63 e^2}\right )}{11 e}+\frac {2 \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{11 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10 \left (\frac {2 \int \frac {\left (a e \left (c d^2+9 a e^2\right )-8 c d \left (c d^2+2 a e^2\right ) x\right ) \sqrt {c x^2+a}}{\sqrt {d+e x}}dx}{21 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (9 a e^2+8 c d^2-7 c d e x\right )}{63 e^2}\right )}{11 e}+\frac {2 \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{11 e}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {10 \left (\frac {2 \left (\frac {4 \int \frac {c \left (a e \left (8 c^2 d^4+21 a c e^2 d^2+45 a^2 e^4\right )-c d \left (32 c^2 d^4+93 a c e^2 d^2+93 a^2 e^4\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+a}}dx}{15 c e^2}+\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (45 a^2 e^4-24 c d e x \left (2 a e^2+c d^2\right )+69 a c d^2 e^2+32 c^2 d^4\right )}{15 e^2}\right )}{21 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (9 a e^2+8 c d^2-7 c d e x\right )}{63 e^2}\right )}{11 e}+\frac {2 \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{11 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10 \left (\frac {2 \left (\frac {2 \int \frac {a e \left (8 c^2 d^4+21 a c e^2 d^2+45 a^2 e^4\right )-c d \left (32 c^2 d^4+93 a c e^2 d^2+93 a^2 e^4\right ) x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{15 e^2}+\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (45 a^2 e^4-24 c d e x \left (2 a e^2+c d^2\right )+69 a c d^2 e^2+32 c^2 d^4\right )}{15 e^2}\right )}{21 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (9 a e^2+8 c d^2-7 c d e x\right )}{63 e^2}\right )}{11 e}+\frac {2 \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{11 e}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {10 \left (\frac {2 \left (\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (45 a^2 e^4-24 c d e x \left (2 a e^2+c d^2\right )+69 a c d^2 e^2+32 c^2 d^4\right )}{15 e^2}-\frac {4 \int -\frac {\left (c d^2+a e^2\right ) \left (32 c^2 d^4+69 a c e^2 d^2+45 a^2 e^4\right )-c d \left (32 c^2 d^4+93 a c e^2 d^2+93 a^2 e^4\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{15 e^4}\right )}{21 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (9 a e^2+8 c d^2-7 c d e x\right )}{63 e^2}\right )}{11 e}+\frac {2 \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{11 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {10 \left (\frac {2 \left (\frac {4 \int \frac {\left (c d^2+a e^2\right ) \left (32 c^2 d^4+69 a c e^2 d^2+45 a^2 e^4\right )-c d \left (32 c^2 d^4+93 a c e^2 d^2+93 a^2 e^4\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{15 e^4}+\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (45 a^2 e^4-24 c d e x \left (2 a e^2+c d^2\right )+69 a c d^2 e^2+32 c^2 d^4\right )}{15 e^2}\right )}{21 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (9 a e^2+8 c d^2-7 c d e x\right )}{63 e^2}\right )}{11 e}+\frac {2 \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{11 e}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {10 \left (\frac {2 \left (\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (45 a^2 e^4-24 c d e x \left (2 a e^2+c d^2\right )+69 a c d^2 e^2+32 c^2 d^4\right )}{15 e^2}-\frac {4 \left (-\sqrt {a e^2+c d^2} \left (\sqrt {a e^2+c d^2} \left (45 a^2 e^4+69 a c d^2 e^2+32 c^2 d^4\right )-\sqrt {c} d \left (93 a^2 e^4+93 a c d^2 e^2+32 c^2 d^4\right )\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}-\sqrt {c} d \sqrt {a e^2+c d^2} \left (93 a^2 e^4+93 a c d^2 e^2+32 c^2 d^4\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}\right )}{15 e^4}\right )}{21 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (9 a e^2+8 c d^2-7 c d e x\right )}{63 e^2}\right )}{11 e}+\frac {2 \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{11 e}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {10 \left (\frac {2 \left (\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (45 a^2 e^4-24 c d e x \left (2 a e^2+c d^2\right )+69 a c d^2 e^2+32 c^2 d^4\right )}{15 e^2}-\frac {4 \left (-\sqrt {c} d \sqrt {a e^2+c d^2} \left (93 a^2 e^4+93 a c d^2 e^2+32 c^2 d^4\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}-\frac {\left (a e^2+c d^2\right )^{3/4} \left (\sqrt {a e^2+c d^2} \left (45 a^2 e^4+69 a c d^2 e^2+32 c^2 d^4\right )-\sqrt {c} d \left (93 a^2 e^4+93 a c d^2 e^2+32 c^2 d^4\right )\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}\right )}{15 e^4}\right )}{21 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} \sqrt {d+e x} \left (9 a e^2+8 c d^2-7 c d e x\right )}{63 e^2}\right )}{11 e}+\frac {2 \left (a+c x^2\right )^{5/2} \sqrt {d+e x}}{11 e}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2 \sqrt {d+e x} \left (c x^2+a\right )^{5/2}}{11 e}+\frac {10 \left (\frac {2 \sqrt {d+e x} \left (8 c d^2-7 c e x d+9 a e^2\right ) \left (c x^2+a\right )^{3/2}}{63 e^2}+\frac {2 \left (\frac {2 \sqrt {d+e x} \left (32 c^2 d^4+69 a c e^2 d^2-24 c e \left (c d^2+2 a e^2\right ) x d+45 a^2 e^4\right ) \sqrt {c x^2+a}}{15 e^2}-\frac {4 \left (-\sqrt {c} d \sqrt {c d^2+a e^2} \left (32 c^2 d^4+93 a c e^2 d^2+93 a^2 e^4\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 )-\frac {\left (c d^2+a e^2\right )^{3/4} \left (\sqrt {c d^2+a e^2} \left (32 c^2 d^4+69 a c e^2 d^2+45 a^2 e^4\right )-\sqrt {c} d \left (32 c^2 d^4+93 a c e^2 d^2+93 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}}\right )}{15 e^4}\right )}{21 e^2}\right )}{11 e}\) |
(2*Sqrt[d + e*x]*(a + c*x^2)^(5/2))/(11*e) + (10*((2*Sqrt[d + e*x]*(8*c*d^ 2 + 9*a*e^2 - 7*c*d*e*x)*(a + c*x^2)^(3/2))/(63*e^2) + (2*((2*Sqrt[d + e*x ]*(32*c^2*d^4 + 69*a*c*d^2*e^2 + 45*a^2*e^4 - 24*c*d*e*(c*d^2 + 2*a*e^2)*x )*Sqrt[a + c*x^2])/(15*e^2) - (4*(-(Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(32*c^2* d^4 + 93*a*c*d^2*e^2 + 93*a^2*e^4)*(-((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 + (S qrt[c]*(d + e*x))/Sqrt[c*d^2 + a*e^2]))) + ((c*d^2 + a*e^2)^(1/4)*(1 + (Sq rt[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)*Sq rt[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)*(Sqrt[c*d^2 + a*e^2]*(32*c^2*d^4 + 69*a*c*d^2*e^2 + 45 *a^2*e^4) - Sqrt[c]*d*(32*c^2*d^4 + 93*a*c*d^2*e^2 + 93*a^2*e^4))*(1 + (Sq rt[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)]*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)))/(21*e^2)))/(11*e)
3.7.71.3.1 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[ (c + d*x)^(n + 1)*((a + b*x^2)^p/(d*(n + 2*p + 1))), x] + Simp[2*(p/(d*(n + 2*p + 1))) Int[(c + d*x)^n*(a + b*x^2)^(p - 1)*(a*d - b*c*x), x], x] /; FreeQ[{a, b, c, d, n}, x] && GtQ[p, 0] && NeQ[n + 2*p + 1, 0] && ( !Rationa lQ[n] || LtQ[n, 1]) && !ILtQ[n + 2*p, 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]
Leaf count of result is larger than twice the leaf count of optimal. \(1147\) vs. \(2(416)=832\).
Time = 3.22 (sec) , antiderivative size = 1148, normalized size of antiderivative = 2.32
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1148\) |
| elliptic | \(\text {Expression too large to display}\) | \(1151\) |
| default | \(\text {Expression too large to display}\) | \(1970\) |
2/693*(63*c^2*e^4*x^4-70*c^2*d*e^3*x^3+216*a*c*e^4*x^2+80*c^2*d^2*e^2*x^2- 262*a*c*d*e^3*x-96*c^2*d^3*e*x+333*a^2*e^4+356*a*c*d^2*e^2+128*c^2*d^4)*(e *x+d)^(1/2)*(c*x^2+a)^(1/2)/e^5+8/693/e^5*(90*a^3*e^5*(d/e-(-a*c)^(1/2)/c) *((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/ 2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(-d/e+(-a*c)^(1/2)/c))^(1/2)/(c*e*x^3+c*d *x^2+a*e*x+a*d)^(1/2)*EllipticF(((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2),((-d/ e+(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2))+16*a*c^2*d^4*e*(d/e-(-a*c) ^(1/2)/c)*((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-d/e-( -a*c)^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(-d/e+(-a*c)^(1/2)/c))^(1/2)/(c* e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)*EllipticF(((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1 /2),((-d/e+(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2))+42*d^2*e^3*a^2*c* (d/e-(-a*c)^(1/2)/c)*((x+d/e)/(d/e-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2) /c)/(-d/e-(-a*c)^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(-d/e+(-a*c)^(1/2)/c) )^(1/2)/(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)*EllipticF(((x+d/e)/(d/e-(-a*c)^( 1/2)/c))^(1/2),((-d/e+(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2))-2*(93* a^2*c*d*e^4+93*a*c^2*d^3*e^2+32*c^3*d^5)*(d/e-(-a*c)^(1/2)/c)*((x+d/e)/(d/ e-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2)* ((x+(-a*c)^(1/2)/c)/(-d/e+(-a*c)^(1/2)/c))^(1/2)/(c*e*x^3+c*d*x^2+a*e*x+a* d)^(1/2)*((-d/e-(-a*c)^(1/2)/c)*EllipticE(((x+d/e)/(d/e-(-a*c)^(1/2)/c))^( 1/2),((-d/e+(-a*c)^(1/2)/c)/(-d/e-(-a*c)^(1/2)/c))^(1/2))+(-a*c)^(1/2)/...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 369, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (8 \, {\left (32 \, c^{3} d^{6} + 117 \, a c^{2} d^{4} e^{2} + 156 \, a^{2} c d^{2} e^{4} + 135 \, a^{3} e^{6}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 24 \, {\left (32 \, c^{3} d^{5} e + 93 \, a c^{2} d^{3} e^{3} + 93 \, a^{2} c d e^{5}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (63 \, c^{3} e^{6} x^{4} - 70 \, c^{3} d e^{5} x^{3} + 128 \, c^{3} d^{4} e^{2} + 356 \, a c^{2} d^{2} e^{4} + 333 \, a^{2} c e^{6} + 8 \, {\left (10 \, c^{3} d^{2} e^{4} + 27 \, a c^{2} e^{6}\right )} x^{2} - 2 \, {\left (48 \, c^{3} d^{3} e^{3} + 131 \, a c^{2} d e^{5}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{2079 \, c e^{7}} \]
2/2079*(8*(32*c^3*d^6 + 117*a*c^2*d^4*e^2 + 156*a^2*c*d^2*e^4 + 135*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) + 24*(32*c^3*d^5*e + 93*a*c^2* d^3*e^3 + 93*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*(63*c^3*e^6*x^4 - 70*c^3*d*e^5*x^3 + 128*c^3*d^4*e^2 + 356*a*c^2*d^2 *e^4 + 333*a^2*c*e^6 + 8*(10*c^3*d^2*e^4 + 27*a*c^2*e^6)*x^2 - 2*(48*c^3*d ^3*e^3 + 131*a*c^2*d*e^5)*x)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(c*e^7)
\[ \int \frac {\left (a+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {5}{2}}}{\sqrt {d + e x}}\, dx \]
\[ \int \frac {\left (a+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{\sqrt {e x + d}} \,d x } \]
\[ \int \frac {\left (a+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{\sqrt {e x + d}} \,d x } \]
Timed out. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{\sqrt {d+e x}} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{5/2}}{\sqrt {d+e\,x}} \,d x \]