Integrand size = 21, antiderivative size = 382 \[ \int \frac {1}{(d+e x)^{5/2} \sqrt {a+c x^2}} \, dx=-\frac {2 e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {8 c d e \sqrt {a+c x^2}}{3 \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}-\frac {8 \sqrt {-a} c^{3/2} d \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \left (c d^2+a e^2\right )^2 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {2 \sqrt {-a} \sqrt {c} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 \left (c d^2+a e^2\right ) \sqrt {d+e x} \sqrt {a+c x^2}} \]
-2/3*e*(c*x^2+a)^(1/2)/(a*e^2+c*d^2)/(e*x+d)^(3/2)-8/3*c*d*e*(c*x^2+a)^(1/ 2)/(a*e^2+c*d^2)^2/(e*x+d)^(1/2)-8/3*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)/(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)+2/3*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)/(a*e^2+c*d^2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.71 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.29 \[ \int \frac {1}{(d+e x)^{5/2} \sqrt {a+c x^2}} \, dx=\frac {2 \left (-e^2 \left (a+c x^2\right ) \left (a e^2+c d (5 d+4 e x)\right )+\frac {c (d+e x) \left (4 d e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (a+c x^2\right )+4 \sqrt {c} d \left (-i \sqrt {c} d+\sqrt {a} e\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+i \left (3 c d^2+4 i \sqrt {a} \sqrt {c} d e-a e^2\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{3 e \left (c d^2+a e^2\right )^2 (d+e x)^{3/2} \sqrt {a+c x^2}} \]
(2*(-(e^2*(a + c*x^2)*(a*e^2 + c*d*(5*d + 4*e*x))) + (c*(d + e*x)*(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*Sq rt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sq rt[-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)] + I*(3*c*d^2 + (4*I)*Sqrt[a]*Sqrt[c]*d*e - 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*Sq rt[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]]))/(3*e*(c*d^2 + a*e^2)^2*( d + e*x)^(3/2)*Sqrt[a + c*x^2])
Time = 0.75 (sec) , antiderivative size = 718, normalized size of antiderivative = 1.88, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {498, 27, 688, 27, 599, 27, 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.
| \(\displaystyle \int \frac {1}{\sqrt {a+c x^2} (d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 498 |
| \(\displaystyle -\frac {2 c \int -\frac {3 d-e x}{2 (d+e x)^{3/2} \sqrt {c x^2+a}}dx}{3 \left (a e^2+c d^2\right )}-\frac {2 e \sqrt {a+c x^2}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \int \frac {3 d-e x}{(d+e x)^{3/2} \sqrt {c x^2+a}}dx}{3 \left (a e^2+c d^2\right )}-\frac {2 e \sqrt {a+c x^2}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {c \left (-\frac {2 \int -\frac {3 c d^2+4 c e x d-a e^2}{2 \sqrt {d+e x} \sqrt {c x^2+a}}dx}{a e^2+c d^2}-\frac {8 d e \sqrt {a+c x^2}}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\right )}{3 \left (a e^2+c d^2\right )}-\frac {2 e \sqrt {a+c x^2}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \left (\frac {\int \frac {3 c d^2+4 c e x d-a e^2}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{a e^2+c d^2}-\frac {8 d e \sqrt {a+c x^2}}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\right )}{3 \left (a e^2+c d^2\right )}-\frac {2 e \sqrt {a+c x^2}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {c \left (-\frac {2 \int \frac {e \left (c d^2-4 c (d+e x) d+a e^2\right )}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{e^2 \left (a e^2+c d^2\right )}-\frac {8 d e \sqrt {a+c x^2}}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\right )}{3 \left (a e^2+c d^2\right )}-\frac {2 e \sqrt {a+c x^2}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \left (-\frac {2 \int \frac {c d^2-4 c (d+e x) d+a e^2}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{e \left (a e^2+c d^2\right )}-\frac {8 d e \sqrt {a+c x^2}}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\right )}{3 \left (a e^2+c d^2\right )}-\frac {2 e \sqrt {a+c x^2}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {c \left (-\frac {2 \left (4 \sqrt {c} d \sqrt {a e^2+c d^2} \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}-\sqrt {a e^2+c d^2} \left (4 \sqrt {c} d-\sqrt {a e^2+c d^2}\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}\right )}{e \left (a e^2+c d^2\right )}-\frac {8 d e \sqrt {a+c x^2}}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\right )}{3 \left (a e^2+c d^2\right )}-\frac {2 e \sqrt {a+c x^2}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {c \left (-\frac {2 \left (4 \sqrt {c} d \sqrt {a e^2+c d^2} \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}-\frac {\left (a e^2+c d^2\right )^{3/4} \left (4 \sqrt {c} d-\sqrt {a e^2+c d^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}\right )}{e \left (a e^2+c d^2\right )}-\frac {8 d e \sqrt {a+c x^2}}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\right )}{3 \left (a e^2+c d^2\right )}-\frac {2 e \sqrt {a+c x^2}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {c \left (-\frac {2 \left (4 \sqrt {c} d \sqrt {a e^2+c d^2} \left (\frac {\sqrt [4]{a e^2+c d^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {d+e x} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )}\right )-\frac {\left (a e^2+c d^2\right )^{3/4} \left (4 \sqrt {c} d-\sqrt {a e^2+c d^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}\right )}{e \left (a e^2+c d^2\right )}-\frac {8 d e \sqrt {a+c x^2}}{\sqrt {d+e x} \left (a e^2+c d^2\right )}\right )}{3 \left (a e^2+c d^2\right )}-\frac {2 e \sqrt {a+c x^2}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}\) |
(-2*e*Sqrt[a + c*x^2])/(3*(c*d^2 + a*e^2)*(d + e*x)^(3/2)) + (c*((-8*d*e*S qrt[a + c*x^2])/((c*d^2 + a*e^2)*Sqrt[d + e*x]) - (2*(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)^(3/4)*(4*S qrt[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)]*Ellipt icF[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])))/(e*(c*d^2 + a*e^2))))/(3*(c*d^2 + a*e ^2))
3.7.83.3.1 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[ d*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + S imp[b/((n + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p*(c*(n + 1) - d*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[n , -1] && ((LtQ[n, -1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]) || (SumSimp lerQ[n, 1] && IntegerQ[p]) || ILtQ[Simplify[n + 2*p + 3], 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[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( (m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -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. \(705\) vs. \(2(310)=620\).
Time = 2.87 (sec) , antiderivative size = 706, normalized size of antiderivative = 1.85
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{3 e \left (e^{2} a +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {8 \left (c e \,x^{2}+a e \right ) c d}{3 \left (e^{2} a +c \,d^{2}\right )^{2} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+a e \right )}}+\frac {2 \left (-\frac {c}{3 \left (e^{2} a +c \,d^{2}\right )}+\frac {4 c^{2} d^{2}}{3 \left (e^{2} a +c \,d^{2}\right )^{2}}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}+\frac {8 d e \,c^{2} \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{3 \left (e^{2} a +c \,d^{2}\right )^{2} \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\) | \(706\) |
| default | \(\text {Expression too large to display}\) | \(1904\) |
((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*(-2/3/e/(a*e^2+c*d ^2)*(c*e*x^3+c*d*x^2+a*e*x+a*d)^(1/2)/(x+d/e)^2-8/3*(c*e*x^2+a*e)/(a*e^2+c *d^2)^2*c*d/((x+d/e)*(c*e*x^2+a*e))^(1/2)+2*(-1/3*c/(a*e^2+c*d^2)+4/3*c^2* d^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))+8/3*d*e*c^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)*((-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 = 380, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(d+e x)^{5/2} \sqrt {a+c x^2}} \, dx=\frac {2 \, {\left ({\left (5 \, c d^{4} - 3 \, a d^{2} e^{2} + {\left (5 \, c d^{2} e^{2} - 3 \, a e^{4}\right )} x^{2} + 2 \, {\left (5 \, c d^{3} e - 3 \, a d e^{3}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) - 12 \, {\left (c d e^{3} x^{2} + 2 \, c d^{2} e^{2} x + c d^{3} e\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (4 \, c d e^{3} x + 5 \, c d^{2} e^{2} + a e^{4}\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}\right )}}{9 \, {\left (c^{2} d^{6} e + 2 \, a c d^{4} e^{3} + a^{2} d^{2} e^{5} + {\left (c^{2} d^{4} e^{3} + 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e^{2} + 2 \, a c d^{3} e^{4} + a^{2} d e^{6}\right )} x\right )}} \]
2/9*((5*c*d^4 - 3*a*d^2*e^2 + (5*c*d^2*e^2 - 3*a*e^4)*x^2 + 2*(5*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) - 12*(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*(4*c*d*e^3*x + 5*c*d^2*e^2 + a*e^4)*sqrt(c*x^2 + a)*sqrt(e*x + d))/ (c^2*d^6*e + 2*a*c*d^4*e^3 + a^2*d^2*e^5 + (c^2*d^4*e^3 + 2*a*c*d^2*e^5 + a^2*e^7)*x^2 + 2*(c^2*d^5*e^2 + 2*a*c*d^3*e^4 + a^2*d*e^6)*x)
\[ \int \frac {1}{(d+e x)^{5/2} \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {a + c x^{2}} \left (d + e x\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {1}{(d+e x)^{5/2} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{(d+e x)^{5/2} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(d+e x)^{5/2} \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^{5/2}} \,d x \]