Integrand size = 21, antiderivative size = 450 \[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^{5/2}} \, dx=\frac {(a e+c d x) \sqrt {d+e x}}{3 a \left (c d^2+a e^2\right ) \left (a+c x^2\right )^{3/2}}+\frac {\sqrt {d+e x} \left (a e \left (c d^2+5 a e^2\right )+4 c d \left (c d^2+2 a e^2\right ) x\right )}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^2}}+\frac {2 \sqrt {c} d \left (c d^2+2 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 (-a)^{3/2} \left (c d^2+a e^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {\left (4 c d^2+5 a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{6 (-a)^{3/2} \sqrt {c} \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}} \]
1/3*(c*d*x+a*e)*(e*x+d)^(1/2)/a/(a*e^2+c*d^2)/(c*x^2+a)^(3/2)+1/6*(a*e*(5* a*e^2+c*d^2)+4*c*d*(2*a*e^2+c*d^2)*x)*(e*x+d)^(1/2)/a^2/(a*e^2+c*d^2)^2/(c *x^2+a)^(1/2)+2/3*d*(2*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))*c^(1/2)*(e*x+d) ^(1/2)*(1+c*x^2/a)^(1/2)/(-a)^(3/2)/(a*e^2+c*d^2)^2/(c*x^2+a)^(1/2)/((e*x+ d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)-1/6*(5*a*e^2+4*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))*(1+c*x^2/a)^(1/2)*((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2))) ^(1/2)/(-a)^(3/2)/(a*e^2+c*d^2)/c^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 12.27 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.23 \[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^{5/2}} \, dx=\frac {\sqrt {d+e x} \left (a c d^2 e+5 a^2 e^3+4 c^2 d^3 x+8 a c d e^2 x+\frac {2 a \left (c d^2+a e^2\right ) (a e+c d x)}{a+c x^2}-\frac {4 d e \left (c d^2+2 a e^2\right ) \left (a+c x^2\right )}{d+e x}-\frac {4 i c d \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (c d^2+2 a e^2\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} \sqrt {d+e x} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{e}+\frac {\sqrt {a} \left (4 c^{3/2} d^3+i \sqrt {a} c d^2 e+8 a \sqrt {c} d e^2+5 i a^{3/2} e^3\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} \sqrt {d+e x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{6 a^2 \left (c d^2+a e^2\right )^2 \sqrt {a+c x^2}} \]
(Sqrt[d + e*x]*(a*c*d^2*e + 5*a^2*e^3 + 4*c^2*d^3*x + 8*a*c*d*e^2*x + (2*a *(c*d^2 + a*e^2)*(a*e + c*d*x))/(a + c*x^2) - (4*d*e*(c*d^2 + 2*a*e^2)*(a + c*x^2))/(d + e*x) - ((4*I)*c*d*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(c*d^2 + 2*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))]*Sqrt[d + e*x]*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)])/e + (Sqrt[a]*(4*c^(3/2)*d^3 + I*Sqrt[a]*c*d^2*e + 8*a* Sqrt[c]*d*e^2 + (5*I)*a^(3/2)*e^3)*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]*Elli pticF[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]]))/(6*a^2*(c*d^2 + a*e^2)^2*Sqrt[a + c*x^2])
Time = 0.86 (sec) , antiderivative size = 803, normalized size of antiderivative = 1.78, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {496, 27, 686, 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.
| \(\displaystyle \int \frac {1}{\left (a+c x^2\right )^{5/2} \sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 496 |
| \(\displaystyle \frac {\sqrt {d+e x} (a e+c d x)}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}-\frac {\int -\frac {4 c d^2+3 c e x d+5 a e^2}{2 \sqrt {d+e x} \left (c x^2+a\right )^{3/2}}dx}{3 a \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {4 c d^2+3 c e x d+5 a e^2}{\sqrt {d+e x} \left (c x^2+a\right )^{3/2}}dx}{6 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} (a e+c d x)}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {\frac {\sqrt {d+e x} \left (4 c d x \left (2 a e^2+c d^2\right )+a e \left (5 a e^2+c d^2\right )\right )}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}-\frac {\int -\frac {c e \left (a e \left (c d^2+5 a e^2\right )-4 c d \left (c d^2+2 a e^2\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+a}}dx}{a c \left (a e^2+c d^2\right )}}{6 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} (a e+c d x)}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {e \int \frac {a e \left (c d^2+5 a e^2\right )-4 c d \left (c d^2+2 a e^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{2 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} \left (4 c d x \left (2 a e^2+c d^2\right )+a e \left (5 a e^2+c d^2\right )\right )}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}}{6 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} (a e+c d x)}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {\frac {\sqrt {d+e x} \left (4 c d x \left (2 a e^2+c d^2\right )+a e \left (5 a e^2+c d^2\right )\right )}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}-\frac {\int -\frac {\left (c d^2+a e^2\right ) \left (4 c d^2+5 a e^2\right )-4 c d \left (c d^2+2 a e^2\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{a e \left (a e^2+c d^2\right )}}{6 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} (a e+c d x)}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\left (c d^2+a e^2\right ) \left (4 c d^2+5 a e^2\right )-4 c d \left (c d^2+2 a e^2\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{a e \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} \left (4 c d x \left (2 a e^2+c d^2\right )+a e \left (5 a e^2+c d^2\right )\right )}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}}{6 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} (a e+c d x)}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {\frac {\sqrt {d+e x} \left (4 c d x \left (2 a e^2+c d^2\right )+a e \left (5 a e^2+c d^2\right )\right )}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}-\frac {\sqrt {a e^2+c d^2} \left (4 \sqrt {c} d \left (2 a e^2+c d^2\right )-\sqrt {a e^2+c d^2} \left (5 a e^2+4 c d^2\right )\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}-4 \sqrt {c} d \sqrt {a e^2+c d^2} \left (2 a e^2+c d^2\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{a e \left (a e^2+c d^2\right )}}{6 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} (a e+c d x)}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\frac {\sqrt {d+e x} \left (4 c d x \left (2 a e^2+c d^2\right )+a e \left (5 a e^2+c d^2\right )\right )}{a \sqrt {a+c x^2} \left (a e^2+c d^2\right )}-\frac {\frac {\left (a e^2+c d^2\right )^{3/4} \left (4 \sqrt {c} d \left (2 a e^2+c d^2\right )-\sqrt {a e^2+c d^2} \left (5 a e^2+4 c d^2\right )\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-4 \sqrt {c} d \sqrt {a e^2+c d^2} \left (2 a e^2+c d^2\right ) \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{a e \left (a e^2+c d^2\right )}}{6 a \left (a e^2+c d^2\right )}+\frac {\sqrt {d+e x} (a e+c d x)}{3 a \left (a+c x^2\right )^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {\sqrt {d+e x} (a e+c d x)}{3 a \left (c d^2+a e^2\right ) \left (c x^2+a\right )^{3/2}}+\frac {\frac {\sqrt {d+e x} \left (a e \left (c d^2+5 a e^2\right )+4 c d \left (c d^2+2 a e^2\right ) x\right )}{a \left (c d^2+a e^2\right ) \sqrt {c x^2+a}}-\frac {\frac {\left (c d^2+a e^2\right )^{3/4} \left (4 \sqrt {c} d \left (c d^2+2 a e^2\right )-\sqrt {c d^2+a e^2} \left (4 c d^2+5 a e^2\right )\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right ) \sqrt {\frac {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}{\left (\frac {c d^2}{e^2}+a\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}-4 \sqrt {c} d \sqrt {c d^2+a e^2} \left (c d^2+2 a e^2\right ) \left (\frac {\sqrt [4]{c d^2+a e^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right ) \sqrt {\frac {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}{\left (\frac {c d^2}{e^2}+a\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}-\frac {\sqrt {d+e x} \sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}{\left (\frac {c d^2}{e^2}+a\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}+1\right )}\right )}{a e \left (c d^2+a e^2\right )}}{6 a \left (c d^2+a e^2\right )}\) |
((a*e + c*d*x)*Sqrt[d + e*x])/(3*a*(c*d^2 + a*e^2)*(a + c*x^2)^(3/2)) + (( Sqrt[d + e*x]*(a*e*(c*d^2 + 5*a*e^2) + 4*c*d*(c*d^2 + 2*a*e^2)*x))/(a*(c*d ^2 + a*e^2)*Sqrt[a + c*x^2]) - (-4*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2]*(c*d^2 + 2*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 + 2*a*e^2) - Sqrt[c*d^2 + a*e^2]*(4*c*d^2 + 5*a*e^2))*(1 + (S qrt[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]))/(a* e*(c*d^2 + a*e^2)))/(6*a*(c*d^2 + a*e^2))
3.7.97.3.1 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[ (-(a*d + b*c*x))*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1)*(b*c^2 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 *p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad raticQ[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))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ [p, -1] && (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. \(857\) vs. \(2(378)=756\).
Time = 3.55 (sec) , antiderivative size = 858, normalized size of antiderivative = 1.91
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {\left (\frac {d x}{3 c a \left (e^{2} a +c \,d^{2}\right )}+\frac {e}{3 \left (e^{2} a +c \,d^{2}\right ) c^{2}}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{\left (x^{2}+\frac {a}{c}\right )^{2}}-\frac {2 \left (c e x +c d \right ) \left (-\frac {d \left (2 e^{2} a +c \,d^{2}\right ) x}{3 a^{2} \left (e^{2} a +c \,d^{2}\right )^{2}}-\frac {e \left (5 e^{2} a +c \,d^{2}\right )}{12 \left (e^{2} a +c \,d^{2}\right )^{2} a c}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) \left (c e x +c d \right )}}+\frac {2 \left (\frac {5 e^{2} a +4 c \,d^{2}}{6 \left (e^{2} a +c \,d^{2}\right ) a^{2}}-\frac {e^{2} \left (5 e^{2} a +c \,d^{2}\right )}{12 \left (e^{2} a +c \,d^{2}\right )^{2} a}-\frac {2 c \,d^{2} \left (2 e^{2} a +c \,d^{2}\right )}{3 a^{2} \left (e^{2} a +c \,d^{2}\right )^{2}}\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 c d e \left (2 e^{2} a +c \,d^{2}\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 )}{3 \left (e^{2} a +c \,d^{2}\right )^{2} a^{2} \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\) | \(858\) |
| default | \(\text {Expression too large to display}\) | \(2673\) |
((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*((1/3/c*d/a/(a*e^2 +c*d^2)*x+1/3*e/(a*e^2+c*d^2)/c^2)*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)/(x^2+ 1/c*a)^2-2*(c*e*x+c*d)*(-1/3*d*(2*a*e^2+c*d^2)/a^2/(a*e^2+c*d^2)^2*x-1/12* e*(5*a*e^2+c*d^2)/(a*e^2+c*d^2)^2/a/c)/((x^2+1/c*a)*(c*e*x+c*d))^(1/2)+2*( 1/6/(a*e^2+c*d^2)*(5*a*e^2+4*c*d^2)/a^2-1/12*e^2*(5*a*e^2+c*d^2)/(a*e^2+c* d^2)^2/a-2/3*c*d^2*(2*a*e^2+c*d^2)/a^2/(a*e^2+c*d^2)^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)*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*d*e*(2*a*e^2+c*d^2 )/(a*e^2+c*d^2)^2/a^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 = 0.11 (sec) , antiderivative size = 578, normalized size of antiderivative = 1.28 \[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^{5/2}} \, dx=\frac {{\left (4 \, a^{2} c^{2} d^{4} + 11 \, a^{3} c d^{2} e^{2} + 15 \, a^{4} e^{4} + {\left (4 \, c^{4} d^{4} + 11 \, a c^{3} d^{2} e^{2} + 15 \, a^{2} c^{2} e^{4}\right )} x^{4} + 2 \, {\left (4 \, a c^{3} d^{4} + 11 \, a^{2} c^{2} d^{2} e^{2} + 15 \, a^{3} c e^{4}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 12 \, {\left (a^{2} c^{2} d^{3} e + 2 \, a^{3} c d e^{3} + {\left (c^{4} d^{3} e + 2 \, a c^{3} d e^{3}\right )} x^{4} + 2 \, {\left (a c^{3} d^{3} e + 2 \, a^{2} c^{2} d e^{3}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (3 \, a^{2} c^{2} d^{2} e^{2} + 7 \, a^{3} c e^{4} + 4 \, {\left (c^{4} d^{3} e + 2 \, a c^{3} d e^{3}\right )} x^{3} + {\left (a c^{3} d^{2} e^{2} + 5 \, a^{2} c^{2} e^{4}\right )} x^{2} + 2 \, {\left (3 \, a c^{3} d^{3} e + 5 \, a^{2} c^{2} d e^{3}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}}{18 \, {\left (a^{4} c^{3} d^{4} e + 2 \, a^{5} c^{2} d^{2} e^{3} + a^{6} c e^{5} + {\left (a^{2} c^{5} d^{4} e + 2 \, a^{3} c^{4} d^{2} e^{3} + a^{4} c^{3} e^{5}\right )} x^{4} + 2 \, {\left (a^{3} c^{4} d^{4} e + 2 \, a^{4} c^{3} d^{2} e^{3} + a^{5} c^{2} e^{5}\right )} x^{2}\right )}} \]
1/18*((4*a^2*c^2*d^4 + 11*a^3*c*d^2*e^2 + 15*a^4*e^4 + (4*c^4*d^4 + 11*a*c ^3*d^2*e^2 + 15*a^2*c^2*e^4)*x^4 + 2*(4*a*c^3*d^4 + 11*a^2*c^2*d^2*e^2 + 1 5*a^3*c*e^4)*x^2)*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) + 12*(a^2*c^2*d ^3*e + 2*a^3*c*d*e^3 + (c^4*d^3*e + 2*a*c^3*d*e^3)*x^4 + 2*(a*c^3*d^3*e + 2*a^2*c^2*d*e^3)*x^2)*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*(3*a^2*c^2*d^2*e^2 + 7*a^3*c*e^4 + 4*(c^4*d^3*e + 2*a*c^3*d*e^3)*x^3 + ( a*c^3*d^2*e^2 + 5*a^2*c^2*e^4)*x^2 + 2*(3*a*c^3*d^3*e + 5*a^2*c^2*d*e^3)*x )*sqrt(c*x^2 + a)*sqrt(e*x + d))/(a^4*c^3*d^4*e + 2*a^5*c^2*d^2*e^3 + a^6* c*e^5 + (a^2*c^5*d^4*e + 2*a^3*c^4*d^2*e^3 + a^4*c^3*e^5)*x^4 + 2*(a^3*c^4 *d^4*e + 2*a^4*c^3*d^2*e^3 + a^5*c^2*e^5)*x^2)
\[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (a + c x^{2}\right )^{\frac {5}{2}} \sqrt {d + e x}}\, dx \]
\[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )}^{\frac {5}{2}} \sqrt {e x + d}} \,d x } \]
\[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + a\right )}^{\frac {5}{2}} \sqrt {e x + d}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {d+e x} \left (a+c x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (c\,x^2+a\right )}^{5/2}\,\sqrt {d+e\,x}} \,d x \]