Integrand size = 21, antiderivative size = 457 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=-\frac {8 c \sqrt {d+e x} \left (d \left (32 c d^2+33 a e^2\right )-3 e \left (8 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{63 e^5}-\frac {20 c (8 d-7 e x) \sqrt {d+e x} \left (a+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}-\frac {16 \sqrt {-a} \sqrt {c} \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {16 \sqrt {-a} \sqrt {c} d \left (c d^2+a e^2\right ) \left (32 c d^2+33 a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{63 e^6 \sqrt {d+e x} \sqrt {a+c x^2}} \]
-2*(c*x^2+a)^(5/2)/e/(e*x+d)^(1/2)-20/63*c*(-7*e*x+8*d)*(c*x^2+a)^(3/2)*(e *x+d)^(1/2)/e^3-8/63*c*(d*(33*a*e^2+32*c*d^2)-3*e*(7*a*e^2+8*c*d^2)*x)*(e* x+d)^(1/2)*(c*x^2+a)^(1/2)/e^5-16/63*(21*a^2*e^4+57*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/ 63*d*(a*e^2+c*d^2)*(33*a*e^2+32*c*d^2)*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)*c ^(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/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 25.27 (sec) , antiderivative size = 684, normalized size of antiderivative = 1.50 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {\sqrt {d+e x} \left (-\frac {2 \left (a+c x^2\right ) \left (63 a^2 e^4+2 a c e^2 \left (106 d^2+29 d e x-14 e^2 x^2\right )+c^2 \left (128 d^4+32 d^3 e x-16 d^2 e^2 x^2+10 d e^3 x^3-7 e^4 x^4\right )\right )}{e^5 (d+e x)}+\frac {16 \left (e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (32 c^2 d^4+57 a c d^2 e^2+21 a^2 e^4\right ) \left (a+c x^2\right )+\sqrt {c} \left (-32 i c^{5/2} d^5+32 \sqrt {a} c^2 d^4 e-57 i a c^{3/2} d^3 e^2+57 a^{3/2} c d^2 e^3-21 i a^2 \sqrt {c} d e^4+21 a^{5/2} e^5\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )-\sqrt {a} \sqrt {c} e \left (32 c^2 d^4+8 i \sqrt {a} c^{3/2} d^3 e+57 a c d^2 e^2+12 i a^{3/2} \sqrt {c} d e^3+21 a^2 e^4\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{e^7 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} (d+e x)}\right )}{63 \sqrt {a+c x^2}} \]
(Sqrt[d + e*x]*((-2*(a + c*x^2)*(63*a^2*e^4 + 2*a*c*e^2*(106*d^2 + 29*d*e* x - 14*e^2*x^2) + c^2*(128*d^4 + 32*d^3*e*x - 16*d^2*e^2*x^2 + 10*d*e^3*x^ 3 - 7*e^4*x^4)))/(e^5*(d + e*x)) + (16*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c ]]*(32*c^2*d^4 + 57*a*c*d^2*e^2 + 21*a^2*e^4)*(a + c*x^2) + Sqrt[c]*((-32* I)*c^(5/2)*d^5 + 32*Sqrt[a]*c^2*d^4*e - (57*I)*a*c^(3/2)*d^3*e^2 + 57*a^(3 /2)*c*d^2*e^3 - (21*I)*a^2*Sqrt[c]*d*e^4 + 21*a^(5/2)*e^5)*Sqrt[(e*((I*Sqr t[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)] - Sq rt[a]*Sqrt[c]*e*(32*c^2*d^4 + (8*I)*Sqrt[a]*c^(3/2)*d^3*e + 57*a*c*d^2*e^2 + (12*I)*a^(3/2)*Sqrt[c]*d*e^3 + 21*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))]*(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)]))/(e^7*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(63*Sqrt[a + c*x^2])
Time = 1.02 (sec) , antiderivative size = 811, normalized size of antiderivative = 1.77, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {492, 591, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 492 |
| \(\displaystyle \frac {10 c \int \frac {x \left (c x^2+a\right )^{3/2}}{\sqrt {d+e x}}dx}{e}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 591 |
| \(\displaystyle \frac {10 c \left (\frac {4 \int -\frac {\left (a d e-\left (8 c d^2+7 a e^2\right ) x\right ) \sqrt {c x^2+a}}{2 \sqrt {d+e x}}dx}{21 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (8 d-7 e x) \sqrt {d+e x}}{63 e^2}\right )}{e}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10 c \left (-\frac {2 \int \frac {\left (a d e-\left (8 c d^2+7 a e^2\right ) x\right ) \sqrt {c x^2+a}}{\sqrt {d+e x}}dx}{21 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (8 d-7 e x) \sqrt {d+e x}}{63 e^2}\right )}{e}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {10 c \left (-\frac {2 \left (\frac {4 \int \frac {c \left (4 a d e \left (2 c d^2+3 a e^2\right )-\left (32 c^2 d^4+57 a c e^2 d^2+21 a^2 e^4\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+a}}dx}{15 c e^2}+\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (d \left (33 a e^2+32 c d^2\right )-3 e x \left (7 a e^2+8 c d^2\right )\right )}{15 e^2}\right )}{21 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (8 d-7 e x) \sqrt {d+e x}}{63 e^2}\right )}{e}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {10 c \left (-\frac {2 \left (\frac {2 \int \frac {4 a d e \left (2 c d^2+3 a e^2\right )-\left (32 c^2 d^4+57 a c e^2 d^2+21 a^2 e^4\right ) x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{15 e^2}+\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (d \left (33 a e^2+32 c d^2\right )-3 e x \left (7 a e^2+8 c d^2\right )\right )}{15 e^2}\right )}{21 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (8 d-7 e x) \sqrt {d+e x}}{63 e^2}\right )}{e}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {10 c \left (-\frac {2 \left (\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (d \left (33 a e^2+32 c d^2\right )-3 e x \left (7 a e^2+8 c d^2\right )\right )}{15 e^2}-\frac {4 \int -\frac {d \left (c d^2+a e^2\right ) \left (32 c d^2+33 a e^2\right )-\left (32 c^2 d^4+57 a c e^2 d^2+21 a^2 e^4\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{15 e^4}\right )}{21 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (8 d-7 e x) \sqrt {d+e x}}{63 e^2}\right )}{e}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {10 c \left (-\frac {2 \left (\frac {4 \int \frac {d \left (c d^2+a e^2\right ) \left (32 c d^2+33 a e^2\right )-\left (32 c^2 d^4+57 a c e^2 d^2+21 a^2 e^4\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{15 e^4}+\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (d \left (33 a e^2+32 c d^2\right )-3 e x \left (7 a e^2+8 c d^2\right )\right )}{15 e^2}\right )}{21 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (8 d-7 e x) \sqrt {d+e x}}{63 e^2}\right )}{e}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {10 c \left (-\frac {2 \left (\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (d \left (33 a e^2+32 c d^2\right )-3 e x \left (7 a e^2+8 c d^2\right )\right )}{15 e^2}-\frac {4 \left (\frac {\sqrt {a e^2+c d^2} \left (21 a^2 e^4+57 a c d^2 e^2-\sqrt {c} d \sqrt {a e^2+c d^2} \left (33 a e^2+32 c d^2\right )+32 c^2 d^4\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}-\frac {\sqrt {a e^2+c d^2} \left (21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}\right )}{15 e^4}\right )}{21 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (8 d-7 e x) \sqrt {d+e x}}{63 e^2}\right )}{e}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {10 c \left (-\frac {2 \left (\frac {2 \sqrt {a+c x^2} \sqrt {d+e x} \left (d \left (33 a e^2+32 c d^2\right )-3 e x \left (7 a e^2+8 c d^2\right )\right )}{15 e^2}-\frac {4 \left (\frac {\left (a e^2+c d^2\right )^{3/4} \left (21 a^2 e^4+57 a c d^2 e^2-\sqrt {c} d \sqrt {a e^2+c d^2} \left (33 a e^2+32 c d^2\right )+32 c^2 d^4\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {a e^2+c d^2} \left (21 a^2 e^4+57 a c d^2 e^2+32 c^2 d^4\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}\right )}{15 e^4}\right )}{21 e^2}-\frac {2 \left (a+c x^2\right )^{3/2} (8 d-7 e x) \sqrt {d+e x}}{63 e^2}\right )}{e}-\frac {2 \left (a+c x^2\right )^{5/2}}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {10 c \left (-\frac {2 (8 d-7 e x) \sqrt {d+e x} \left (c x^2+a\right )^{3/2}}{63 e^2}-\frac {2 \left (\frac {2 \sqrt {d+e x} \left (d \left (32 c d^2+33 a e^2\right )-3 e \left (8 c d^2+7 a e^2\right ) x\right ) \sqrt {c x^2+a}}{15 e^2}-\frac {4 \left (\frac {\left (c d^2+a e^2\right )^{3/4} \left (32 c^2 d^4+57 a c e^2 d^2-\sqrt {c} \sqrt {c d^2+a e^2} \left (32 c d^2+33 a e^2\right ) d+21 a^2 e^4\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right ) \sqrt {\frac {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}{\left (\frac {c d^2}{e^2}+a\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}-\frac {\sqrt {c d^2+a e^2} \left (32 c^2 d^4+57 a c e^2 d^2+21 a^2 e^4\right ) \left (\frac {\sqrt [4]{c d^2+a e^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right ) \sqrt {\frac {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}{\left (\frac {c d^2}{e^2}+a\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}-\frac {\sqrt {d+e x} \sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}{\left (\frac {c d^2}{e^2}+a\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right )}\right )}{\sqrt {c}}\right )}{15 e^4}\right )}{21 e^2}\right )}{e}-\frac {2 \left (c x^2+a\right )^{5/2}}{e \sqrt {d+e x}}\) |
(-2*(a + c*x^2)^(5/2))/(e*Sqrt[d + e*x]) + (10*c*((-2*(8*d - 7*e*x)*Sqrt[d + e*x]*(a + c*x^2)^(3/2))/(63*e^2) - (2*((2*Sqrt[d + e*x]*(d*(32*c*d^2 + 33*a*e^2) - 3*e*(8*c*d^2 + 7*a*e^2)*x)*Sqrt[a + c*x^2])/(15*e^2) - (4*(-(( Sqrt[c*d^2 + a*e^2]*(32*c^2*d^4 + 57*a*c*d^2*e^2 + 21*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 + (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])))/Sqrt[c]) + ((c*d^2 + a*e^2)^(3/4)*(32*c^2*d^4 + 57* a*c*d^2*e^2 + 21*a^2*e^4 - Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(32*c*d^2 + 33*a* e^2))*(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 + (Sq rt[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^(3/4)*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)))/e
3.7.72.3.1 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 + 1))), x] - Simp[2*b*(p/(d*(n + 1)) ) Int[x*(c + d*x)^(n + 1)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && GtQ[p, 0] && (IntegerQ[p] || LtQ[n, -1]) && NeQ[n, -1] && !IL tQ[n + 2*p + 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
Int[(x_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c + d*x)^(n + 1))*(a + b*x^2)^p*((c*(2*p + 1) - d*(n + 2*p + 1)*x)/ (d^2*(n + 2*p + 1)*(n + 2*p + 2))), x] + Simp[2*(p/(d^2*(n + 2*p + 1)*(n + 2*p + 2))) Int[(c + d*x)^n*(a + b*x^2)^(p - 1)*Simp[a*c*d*n + (b*c^2*(2*p + 1) + a*d^2*(n + 2*p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && GtQ[p, 0] && LeQ[-1, n, 0] && !ILtQ[n + 2*p, 0]
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. \(1262\) vs. \(2(381)=762\).
Time = 4.90 (sec) , antiderivative size = 1263, normalized size of antiderivative = 2.76
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1263\) |
| default | \(\text {Expression too large to display}\) | \(1736\) |
| risch | \(\text {Expression too large to display}\) | \(1745\) |
((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*(-2*(c*e*x^2+a*e)* (a^2*e^4+2*a*c*d^2*e^2+c^2*d^4)/e^6/((x+d/e)*(c*e*x^2+a*e))^(1/2)+2/9*c^2/ e^2*x^3*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)-34/63*c^2/e^3*d*x^2*(c*e*x^3+c*d *x^2+a*e*x+a*d)^(1/2)+2/5*(c^2/e^3*(3*a*e^2+c*d^2)-7/9*c^2*a/e+34/21*c^3/e ^3*d^2)/c/e*x*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)+2/3*(-c^2*d/e^4*(3*a*e^2+c *d^2)+43/63*d/e^2*c^2*a-4/5*(c^2/e^3*(3*a*e^2+c*d^2)-7/9*c^2*a/e+34/21*c^3 /e^3*d^2)/e*d)/c/e*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)+2*(-c*d*(3*a^2*e^4+3* a*c*d^2*e^2+c^2*d^4)/e^6+(a^2*e^4+2*a*c*d^2*e^2+c^2*d^4)*c/e^6*d-2/5*(c^2/ e^3*(3*a*e^2+c*d^2)-7/9*c^2*a/e+34/21*c^3/e^3*d^2)/c/e*a*d-1/3*(-c^2*d/e^4 *(3*a*e^2+c*d^2)+43/63*d/e^2*c^2*a-4/5*(c^2/e^3*(3*a*e^2+c*d^2)-7/9*c^2*a/ e+34/21*c^3/e^3*d^2)/e*d)/c*a)*(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)*E llipticF(((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*(c/e^5*(3*a^2*e^4+3*a*c*d^2*e^2+c^2*d^4)+(a^2*e ^4+2*a*c*d^2*e^2+c^2*d^4)*c/e^5+68/63*d^2/e^3*c^2*a-3/5*(c^2/e^3*(3*a*e^2+ c*d^2)-7/9*c^2*a/e+34/21*c^3/e^3*d^2)/c*a-2/3*(-c^2*d/e^4*(3*a*e^2+c*d^2)+ 43/63*d/e^2*c^2*a-4/5*(c^2/e^3*(3*a*e^2+c*d^2)-7/9*c^2*a/e+34/21*c^3/e^3*d ^2)/e*d)/e*d)*(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/...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 418, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left (8 \, {\left (32 \, c^{2} d^{6} + 81 \, a c d^{4} e^{2} + 57 \, a^{2} d^{2} e^{4} + {\left (32 \, c^{2} d^{5} e + 81 \, a c d^{3} e^{3} + 57 \, a^{2} d e^{5}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 24 \, {\left (32 \, c^{2} d^{5} e + 57 \, a c d^{3} e^{3} + 21 \, a^{2} d e^{5} + {\left (32 \, c^{2} d^{4} e^{2} + 57 \, a c d^{2} e^{4} + 21 \, a^{2} e^{6}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (7 \, c^{2} e^{6} x^{4} - 10 \, c^{2} d e^{5} x^{3} - 128 \, c^{2} d^{4} e^{2} - 212 \, a c d^{2} e^{4} - 63 \, a^{2} e^{6} + 4 \, {\left (4 \, c^{2} d^{2} e^{4} + 7 \, a c e^{6}\right )} x^{2} - 2 \, {\left (16 \, c^{2} d^{3} e^{3} + 29 \, a c d e^{5}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{189 \, {\left (e^{8} x + d e^{7}\right )}} \]
-2/189*(8*(32*c^2*d^6 + 81*a*c*d^4*e^2 + 57*a^2*d^2*e^4 + (32*c^2*d^5*e + 81*a*c*d^3*e^3 + 57*a^2*d*e^5)*x)*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^2*d^5*e + 57*a*c*d^3*e^3 + 21*a^2*d*e^5 + (32*c^2*d^4*e^2 + 57 *a*c*d^2*e^4 + 21*a^2*e^6)*x)*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*(7*c^2*e^6*x^4 - 10*c^2*d*e^5*x^3 - 128*c^2*d^4*e^2 - 212*a*c*d^ 2*e^4 - 63*a^2*e^6 + 4*(4*c^2*d^2*e^4 + 7*a*c*e^6)*x^2 - 2*(16*c^2*d^3*e^3 + 29*a*c*d*e^5)*x)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(e^8*x + d*e^7)
\[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{5/2}}{{\left (d+e\,x\right )}^{3/2}} \,d x \]