Integrand size = 21, antiderivative size = 358 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {4 c (4 d+e x) \sqrt {a+c x^2}}{3 e^3 \sqrt {d+e x}}-\frac {2 \left (a+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}+\frac {32 \sqrt {-a} c^{3/2} d \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 e^4 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {8 \sqrt {-a} \sqrt {c} \left (4 c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 e^4 \sqrt {d+e x} \sqrt {a+c x^2}} \]
-2/3*(c*x^2+a)^(3/2)/e/(e*x+d)^(3/2)+4/3*c*(e*x+4*d)*(c*x^2+a)^(1/2)/e^3/( e*x+d)^(1/2)+32/3*c^(3/2)*d*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*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c ^(1/2)))^(1/2)-8/3*(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))*(-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^4/ (e*x+d)^(1/2)/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 23.60 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.38 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \sqrt {a+c x^2} \left (-a e^2+c \left (8 d^2+10 d e x+e^2 x^2\right )\right )}{3 e^3 (d+e x)^{3/2}}-\frac {8 c \left (4 d e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (a+c x^2\right )+4 \sqrt {c} d \left (-i \sqrt {c} d+\sqrt {a} e\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )-\sqrt {a} e \left (4 \sqrt {c} d+i \sqrt {a} e\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{3 e^5 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \sqrt {d+e x} \sqrt {a+c x^2}} \]
(2*Sqrt[a + c*x^2]*(-(a*e^2) + c*(8*d^2 + 10*d*e*x + e^2*x^2)))/(3*e^3*(d + e*x)^(3/2)) - (8*c*(4*d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(a + c*x^2) + 4*Sqrt[c]*d*((-I)*Sqrt[c]*d + Sqrt[a]*e)*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*Sqrt [c]*d + I*Sqrt[a]*e)*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*ArcS inh[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)]))/(3*e^5*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]* Sqrt[d + e*x]*Sqrt[a + c*x^2])
Time = 0.71 (sec) , antiderivative size = 690, normalized size of antiderivative = 1.93, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {492, 590, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 492 |
| \(\displaystyle \frac {2 c \int \frac {x \sqrt {c x^2+a}}{(d+e x)^{3/2}}dx}{e}-\frac {2 \left (a+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 590 |
| \(\displaystyle \frac {2 c \left (\frac {2 \sqrt {a+c x^2} (4 d+e x)}{3 e^2 \sqrt {d+e x}}-\frac {4 \int -\frac {a e-4 c d x}{2 \sqrt {d+e x} \sqrt {c x^2+a}}dx}{3 e^2}\right )}{e}-\frac {2 \left (a+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c \left (\frac {2 \int \frac {a e-4 c d x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{3 e^2}+\frac {2 \sqrt {a+c x^2} (4 d+e x)}{3 e^2 \sqrt {d+e x}}\right )}{e}-\frac {2 \left (a+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {2 c \left (\frac {2 \sqrt {a+c x^2} (4 d+e x)}{3 e^2 \sqrt {d+e x}}-\frac {4 \int -\frac {4 c d^2-4 c (d+e x) d+a e^2}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{3 e^4}\right )}{e}-\frac {2 \left (a+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 c \left (\frac {4 \int \frac {4 c d^2-4 c (d+e x) d+a e^2}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{3 e^4}+\frac {2 \sqrt {a+c x^2} (4 d+e x)}{3 e^2 \sqrt {d+e x}}\right )}{e}-\frac {2 \left (a+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {2 c \left (\frac {2 \sqrt {a+c x^2} (4 d+e x)}{3 e^2 \sqrt {d+e x}}-\frac {4 \left (-\left (\left (-4 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+4 c d^2\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}\right )-4 \sqrt {c} d \sqrt {a e^2+c d^2} \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}\right )}{3 e^4}\right )}{e}-\frac {2 \left (a+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {2 c \left (\frac {2 \sqrt {a+c x^2} (4 d+e x)}{3 e^2 \sqrt {d+e x}}-\frac {4 \left (-4 \sqrt {c} d \sqrt {a e^2+c d^2} \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}-\frac {\sqrt [4]{a e^2+c d^2} \left (-4 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+4 c d^2\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}\right )}{3 e^4}\right )}{e}-\frac {2 \left (a+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2 c \left (\frac {2 \sqrt {a+c x^2} (4 d+e x)}{3 e^2 \sqrt {d+e x}}-\frac {4 \left (-\frac {\sqrt [4]{a e^2+c d^2} \left (-4 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+4 c d^2\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-4 \sqrt {c} d \sqrt {a e^2+c d^2} \left (\frac {\sqrt [4]{a e^2+c d^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {d+e x} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )}\right )\right )}{3 e^4}\right )}{e}-\frac {2 \left (a+c x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}\) |
(-2*(a + c*x^2)^(3/2))/(3*e*(d + e*x)^(3/2)) + (2*c*((2*(4*d + e*x)*Sqrt[a + c*x^2])/(3*e^2*Sqrt[d + e*x]) - (4*(-4*Sqrt[c]*d*Sqrt[c*d^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)^(1/4)*(4*c*d^2 + a*e^2 - 4*Sqrt[c]*d*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)]*Elli pticF[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])))/(3*e^4)))/e
3.7.67.3.1 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[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. \(726\) vs. \(2(286)=572\).
Time = 4.37 (sec) , antiderivative size = 727, normalized size of antiderivative = 2.03
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (e^{2} a +c \,d^{2}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 e^{5} \left (x +\frac {d}{e}\right )^{2}}+\frac {16 \left (c e \,x^{2}+a e \right ) c d}{3 e^{4} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+a e \right )}}+\frac {2 c \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 e^{3}}+\frac {2 \left (\frac {c \left (2 e^{2} a +3 c \,d^{2}\right )}{e^{4}}-\frac {\left (e^{2} a +c \,d^{2}\right ) c}{3 e^{4}}-\frac {8 c^{2} d^{2}}{3 e^{4}}-\frac {a c}{3 e^{2}}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}-\frac {32 c^{2} d \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{3 e^{3} \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\) | \(727\) |
| default | \(\text {Expression too large to display}\) | \(1597\) |
| risch | \(\text {Expression too large to display}\) | \(2126\) |
((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*(-2/3*(a*e^2+c*d^2 )/e^5*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)/(x+d/e)^2+16/3*(c*e*x^2+a*e)/e^4*c *d/((x+d/e)*(c*e*x^2+a*e))^(1/2)+2/3*c/e^3*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/ 2)+2*(c*(2*a*e^2+3*c*d^2)/e^4-1/3*(a*e^2+c*d^2)*c/e^4-8/3*c^2/e^4*d^2-1/3/ e^2*a*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 ))-32/3*c^2/e^3*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 /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 = 0.09 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (4 \, {\left (4 \, c d^{4} + 3 \, a d^{2} e^{2} + {\left (4 \, c d^{2} e^{2} + 3 \, a e^{4}\right )} x^{2} + 2 \, {\left (4 \, c d^{3} e + 3 \, a d e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 48 \, {\left (c d e^{3} x^{2} + 2 \, c d^{2} e^{2} x + c d^{3} e\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (c e^{4} x^{2} + 10 \, c d e^{3} x + 8 \, c d^{2} e^{2} - a e^{4}\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{9 \, {\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} \]
2/9*(4*(4*c*d^4 + 3*a*d^2*e^2 + (4*c*d^2*e^2 + 3*a*e^4)*x^2 + 2*(4*c*d^3*e + 3*a*d*e^3)*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) + 48*(c*d*e^3*x^ 2 + 2*c*d^2*e^2*x + c*d^3*e)*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*(c*e^4*x^2 + 10*c*d*e^3*x + 8*c*d^2*e^2 - a*e^4)*sqrt(c*x^2 + a)* sqrt(e*x + d))/(e^7*x^2 + 2*d*e^6*x + d^2*e^5)
\[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{3/2}}{{\left (d+e\,x\right )}^{5/2}} \,d x \]