Integrand size = 21, antiderivative size = 420 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=-\frac {8 c \left (32 c d^2+9 a e^2+8 c d e x\right ) \sqrt {a+c x^2}}{15 e^5 \sqrt {d+e x}}+\frac {4 c (8 d+3 e x) \left (a+c x^2\right )^{3/2}}{15 e^3 (d+e x)^{3/2}}-\frac {2 \left (a+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}-\frac {16 \sqrt {-a} c^{3/2} \left (32 c d^2+9 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^6 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {16 \sqrt {-a} c^{3/2} d \left (32 c d^2+17 a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^6 \sqrt {d+e x} \sqrt {a+c x^2}} \]
4/15*c*(3*e*x+8*d)*(c*x^2+a)^(3/2)/e^3/(e*x+d)^(3/2)-2/5*(c*x^2+a)^(5/2)/e /(e*x+d)^(5/2)-8/15*c*(8*c*d*e*x+9*a*e^2+32*c*d^2)*(c*x^2+a)^(1/2)/e^5/(e* x+d)^(1/2)-16/15*c^(3/2)*(9*a*e^2+32*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^6/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/( e*(-a)^(1/2)+d*c^(1/2)))^(1/2)+16/15*c^(3/2)*d*(17*a*e^2+32*c*d^2)*Ellipti cF(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/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.66 (sec) , antiderivative size = 603, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\frac {\sqrt {d+e x} \left (-\frac {2 \left (a+c x^2\right ) \left (3 a^2 e^4+2 a c e^2 \left (10 d^2+25 d e x+18 e^2 x^2\right )+c^2 \left (128 d^4+288 d^3 e x+176 d^2 e^2 x^2+10 d e^3 x^3-3 e^4 x^4\right )\right )}{e^5 (d+e x)^3}+\frac {16 c \left (e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (32 c d^2+9 a e^2\right ) \left (a+c x^2\right )+\sqrt {c} \left (-32 i c^{3/2} d^3+32 \sqrt {a} c d^2 e-9 i a \sqrt {c} d e^2+9 a^{3/2} e^3\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )-\sqrt {a} \sqrt {c} e \left (32 c d^2+8 i \sqrt {a} \sqrt {c} d e+9 a e^2\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{e^7 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} (d+e x)}\right )}{15 \sqrt {a+c x^2}} \]
(Sqrt[d + e*x]*((-2*(a + c*x^2)*(3*a^2*e^4 + 2*a*c*e^2*(10*d^2 + 25*d*e*x + 18*e^2*x^2) + c^2*(128*d^4 + 288*d^3*e*x + 176*d^2*e^2*x^2 + 10*d*e^3*x^ 3 - 3*e^4*x^4)))/(e^5*(d + e*x)^3) + (16*c*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sq rt[c]]*(32*c*d^2 + 9*a*e^2)*(a + c*x^2) + Sqrt[c]*((-32*I)*c^(3/2)*d^3 + 3 2*Sqrt[a]*c*d^2*e - (9*I)*a*Sqrt[c]*d*e^2 + 9*a^(3/2)*e^3)*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*d^2 + (8*I)*Sqrt[a]*Sqrt[c]*d*e + 9*a*e^2)*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)/Sq rt[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))))/(15*Sqrt[a + c*x^2] )
Time = 0.90 (sec) , antiderivative size = 772, normalized size of antiderivative = 1.84, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {492, 590, 27, 681, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 492 |
| \(\displaystyle \frac {2 c \int \frac {x \left (c x^2+a\right )^{3/2}}{(d+e x)^{5/2}}dx}{e}-\frac {2 \left (a+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 590 |
| \(\displaystyle \frac {2 c \left (\frac {2 \left (a+c x^2\right )^{3/2} (8 d+3 e x)}{15 e^2 (d+e x)^{3/2}}-\frac {4 \int -\frac {(3 a e-8 c d x) \sqrt {c x^2+a}}{2 (d+e x)^{3/2}}dx}{5 e^2}\right )}{e}-\frac {2 \left (a+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c \left (\frac {2 \int \frac {(3 a e-8 c d x) \sqrt {c x^2+a}}{(d+e x)^{3/2}}dx}{5 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} (8 d+3 e x)}{15 e^2 (d+e x)^{3/2}}\right )}{e}-\frac {2 \left (a+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 681 |
| \(\displaystyle \frac {2 c \left (\frac {2 \left (-\frac {2 \int \frac {c \left (8 a d e-\left (32 c d^2+9 a e^2\right ) x\right )}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{3 e^2}-\frac {2 \sqrt {a+c x^2} \left (9 a e^2+32 c d^2+8 c d e x\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} (8 d+3 e x)}{15 e^2 (d+e x)^{3/2}}\right )}{e}-\frac {2 \left (a+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c \left (\frac {2 \left (-\frac {2 c \int \frac {8 a d e-\left (32 c d^2+9 a e^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{3 e^2}-\frac {2 \sqrt {a+c x^2} \left (9 a e^2+32 c d^2+8 c d e x\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} (8 d+3 e x)}{15 e^2 (d+e x)^{3/2}}\right )}{e}-\frac {2 \left (a+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {2 c \left (\frac {2 \left (\frac {4 c \int -\frac {d \left (32 c d^2+17 a e^2\right )-\left (32 c d^2+9 a e^2\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{3 e^4}-\frac {2 \sqrt {a+c x^2} \left (9 a e^2+32 c d^2+8 c d e x\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} (8 d+3 e x)}{15 e^2 (d+e x)^{3/2}}\right )}{e}-\frac {2 \left (a+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (\frac {2 \left (-\frac {4 c \int \frac {d \left (32 c d^2+17 a e^2\right )-\left (32 c d^2+9 a e^2\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{3 e^4}-\frac {2 \sqrt {a+c x^2} \left (9 a e^2+32 c d^2+8 c d e x\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} (8 d+3 e x)}{15 e^2 (d+e x)^{3/2}}\right )}{e}-\frac {2 \left (a+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {2 c \left (\frac {2 \left (\frac {4 c \left (\frac {\sqrt {a e^2+c d^2} \left (-\frac {\sqrt {c} d \left (17 a e^2+32 c d^2\right )}{\sqrt {a e^2+c d^2}}+9 a e^2+32 c d^2\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}-\frac {\sqrt {a e^2+c d^2} \left (9 a e^2+32 c d^2\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}\right )}{3 e^4}-\frac {2 \sqrt {a+c x^2} \left (9 a e^2+32 c d^2+8 c d e x\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} (8 d+3 e x)}{15 e^2 (d+e x)^{3/2}}\right )}{e}-\frac {2 \left (a+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {2 c \left (\frac {2 \left (\frac {4 c \left (\frac {\left (a e^2+c d^2\right )^{3/4} \left (-\frac {\sqrt {c} d \left (17 a e^2+32 c d^2\right )}{\sqrt {a e^2+c d^2}}+9 a e^2+32 c d^2\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {a e^2+c d^2} \left (9 a e^2+32 c d^2\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}\right )}{3 e^4}-\frac {2 \sqrt {a+c x^2} \left (9 a e^2+32 c d^2+8 c d e x\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} (8 d+3 e x)}{15 e^2 (d+e x)^{3/2}}\right )}{e}-\frac {2 \left (a+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2 c \left (\frac {2 \left (\frac {4 c \left (\frac {\left (a e^2+c d^2\right )^{3/4} \left (-\frac {\sqrt {c} d \left (17 a e^2+32 c d^2\right )}{\sqrt {a e^2+c d^2}}+9 a e^2+32 c d^2\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {a e^2+c d^2} \left (9 a e^2+32 c d^2\right ) \left (\frac {\sqrt [4]{a e^2+c d^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {d+e x} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )}\right )}{\sqrt {c}}\right )}{3 e^4}-\frac {2 \sqrt {a+c x^2} \left (9 a e^2+32 c d^2+8 c d e x\right )}{3 e^2 \sqrt {d+e x}}\right )}{5 e^2}+\frac {2 \left (a+c x^2\right )^{3/2} (8 d+3 e x)}{15 e^2 (d+e x)^{3/2}}\right )}{e}-\frac {2 \left (a+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}\) |
(-2*(a + c*x^2)^(5/2))/(5*e*(d + e*x)^(5/2)) + (2*c*((2*(8*d + 3*e*x)*(a + c*x^2)^(3/2))/(15*e^2*(d + e*x)^(3/2)) + (2*((-2*(32*c*d^2 + 9*a*e^2 + 8* c*d*e*x)*Sqrt[a + c*x^2])/(3*e^2*Sqrt[d + e*x]) + (4*c*(-((Sqrt[c*d^2 + a* e^2]*(32*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])))/Sqrt[c]) + ((c *d^2 + a*e^2)^(3/4)*(32*c*d^2 + 9*a*e^2 - (Sqrt[c]*d*(32*c*d^2 + 17*a*e^2) )/Sqrt[c*d^2 + 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 + (Sqrt[c]*(d + e*x))/Sqrt[c*d^2 + a*e^2])^2)]*EllipticF[2*Arc Tan[(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])))/(3*e^4)))/(5*e^2)))/e
3.7.74.3.1 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 + 1)*x)/(d^2*( n + 1)*(n + 2*p + 2))), x] + Simp[2*(p/(d^2*(n + 1)*(n + 2*p + 2))) Int[( c + d*x)^(n + 1)*(a + b*x^2)^(p - 1)*(a*d*(n + 1) + b*c*(2*p + 1)*x), x], x ] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && LtQ[n, -1] && !ILtQ[n + 2*p + 1, 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)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/ (e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Sim p[g*(2*a*e + 2*a*e*m) + (g*(2*c*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x] , x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ[m + 2 *p + 1, 0] && (IntegerQ[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. \(911\) vs. \(2(342)=684\).
Time = 6.35 (sec) , antiderivative size = 912, normalized size of antiderivative = 2.17
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{5 e^{8} \left (x +\frac {d}{e}\right )^{3}}+\frac {44 \left (e^{2} a +c \,d^{2}\right ) c d \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{15 e^{7} \left (x +\frac {d}{e}\right )^{2}}-\frac {8 \left (c e \,x^{2}+a e \right ) \left (9 e^{2} a +32 c \,d^{2}\right ) c}{15 e^{6} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+a e \right )}}+\frac {2 c^{2} x \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{5 e^{4}}-\frac {38 c^{2} d \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{15 e^{5}}+\frac {2 \left (-\frac {c^{2} d \left (9 e^{2} a +10 c \,d^{2}\right )}{e^{6}}+\frac {22 d \left (e^{2} a +c \,d^{2}\right ) c^{2}}{15 e^{6}}+\frac {4 \left (9 e^{2} a +32 c \,d^{2}\right ) c^{2} d}{15 e^{6}}+\frac {13 d \,c^{2} a}{15 e^{4}}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}+\frac {2 \left (\frac {3 c^{2} \left (e^{2} a +2 c \,d^{2}\right )}{e^{5}}+\frac {4 c^{2} \left (9 e^{2} a +32 c \,d^{2}\right )}{15 e^{5}}-\frac {3 c^{2} a}{5 e^{3}}+\frac {38 c^{3} d^{2}}{15 e^{5}}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\) | \(912\) |
| risch | \(\text {Expression too large to display}\) | \(2973\) |
| default | \(\text {Expression too large to display}\) | \(3421\) |
((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*(-2/5*(a^2*e^4+2*a *c*d^2*e^2+c^2*d^4)/e^8*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)/(x+d/e)^3+44/15/ e^7*(a*e^2+c*d^2)*c*d*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)/(x+d/e)^2-8/15*(c* e*x^2+a*e)*(9*a*e^2+32*c*d^2)*c/e^6/((x+d/e)*(c*e*x^2+a*e))^(1/2)+2/5*c^2/ e^4*x*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)-38/15*c^2/e^5*d*(c*e*x^3+c*d*x^2+a *e*x+a*d)^(1/2)+2*(-c^2*d*(9*a*e^2+10*c*d^2)/e^6+22/15*d/e^6*(a*e^2+c*d^2) *c^2+4/15*(9*a*e^2+32*c*d^2)*c^2*d/e^6+13/15*d/e^4*c^2*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)*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*(3*c^2/e^5*(a*e^2+2* c*d^2)+4/15*c^2/e^5*(9*a*e^2+32*c*d^2)-3/5/e^3*c^2*a+38/15*c^3/e^5*d^2)*(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)/c*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))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.20 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=-\frac {2 \, {\left (8 \, {\left (32 \, c^{2} d^{6} + 33 \, a c d^{4} e^{2} + {\left (32 \, c^{2} d^{3} e^{3} + 33 \, a c d e^{5}\right )} x^{3} + 3 \, {\left (32 \, c^{2} d^{4} e^{2} + 33 \, a c d^{2} e^{4}\right )} x^{2} + 3 \, {\left (32 \, c^{2} d^{5} e + 33 \, a c d^{3} e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 24 \, {\left (32 \, c^{2} d^{5} e + 9 \, a c d^{3} e^{3} + {\left (32 \, c^{2} d^{2} e^{4} + 9 \, a c e^{6}\right )} x^{3} + 3 \, {\left (32 \, c^{2} d^{3} e^{3} + 9 \, a c d e^{5}\right )} x^{2} + 3 \, {\left (32 \, c^{2} d^{4} e^{2} + 9 \, a c d^{2} e^{4}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (3 \, c^{2} e^{6} x^{4} - 10 \, c^{2} d e^{5} x^{3} - 128 \, c^{2} d^{4} e^{2} - 20 \, a c d^{2} e^{4} - 3 \, a^{2} e^{6} - 4 \, {\left (44 \, c^{2} d^{2} e^{4} + 9 \, a c e^{6}\right )} x^{2} - 2 \, {\left (144 \, c^{2} d^{3} e^{3} + 25 \, a c d e^{5}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{45 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \]
-2/45*(8*(32*c^2*d^6 + 33*a*c*d^4*e^2 + (32*c^2*d^3*e^3 + 33*a*c*d*e^5)*x^ 3 + 3*(32*c^2*d^4*e^2 + 33*a*c*d^2*e^4)*x^2 + 3*(32*c^2*d^5*e + 33*a*c*d^3 *e^3)*x)*sqrt(c*e)*weierstrassPInverse(4/3*(c*d^2 - 3*a*e^2)/(c*e^2), -8/2 7*(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 + 9*a *c*d^3*e^3 + (32*c^2*d^2*e^4 + 9*a*c*e^6)*x^3 + 3*(32*c^2*d^3*e^3 + 9*a*c* d*e^5)*x^2 + 3*(32*c^2*d^4*e^2 + 9*a*c*d^2*e^4)*x)*sqrt(c*e)*weierstrassZe ta(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), weier strassPInverse(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*(3*c^2*e^6*x^4 - 10*c^2*d*e^5*x^3 - 128*c^2 *d^4*e^2 - 20*a*c*d^2*e^4 - 3*a^2*e^6 - 4*(44*c^2*d^2*e^4 + 9*a*c*e^6)*x^2 - 2*(144*c^2*d^3*e^3 + 25*a*c*d*e^5)*x)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(e ^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)
\[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \]
\[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (a+c x^2\right )^{5/2}}{(d+e x)^{7/2}} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{5/2}}{{\left (d+e\,x\right )}^{7/2}} \,d x \]