Integrand size = 21, antiderivative size = 368 \[ \int \frac {(d+e x)^{3/2}}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {(a e-c d x) \sqrt {d+e x}}{3 a c \left (a+c x^2\right )^{3/2}}+\frac {(a e+4 c d x) \sqrt {d+e x}}{6 a^2 c \sqrt {a+c x^2}}+\frac {2 d \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{3 (-a)^{3/2} \sqrt {c} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}-\frac {\left (4 c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{6 (-a)^{3/2} c^{3/2} \sqrt {d+e x} \sqrt {a+c x^2}} \]
-1/3*(-c*d*x+a*e)*(e*x+d)^(1/2)/a/c/(c*x^2+a)^(3/2)+1/6*(4*c*d*x+a*e)*(e*x +d)^(1/2)/a^2/c/(c*x^2+a)^(1/2)+2/3*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))*(e*x+d)^(1/2) *(1+c*x^2/a)^(1/2)/(-a)^(3/2)/c^(1/2)/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/(e* (-a)^(1/2)+d*c^(1/2)))^(1/2)-1/6*(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 )/c^(3/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 24.87 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.37 \[ \int \frac {(d+e x)^{3/2}}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {\sqrt {d+e x} \left (\frac {-2 a^2 e+8 c^2 d x^3+2 a c x (6 d+e x)}{a^2 c \left (a+c x^2\right )}+\frac {(d+e x) \left (-\frac {8 d e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (a+c x^2\right )}{(d+e x)^2}+\frac {8 i \sqrt {c} d \left (\sqrt {c} d+i \sqrt {a} e\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {d+e x}}+\frac {2 \sqrt {a} e \left (4 \sqrt {c} d+i \sqrt {a} e\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {d+e x}}\right )}{a^2 c e \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{12 \sqrt {a+c x^2}} \]
(Sqrt[d + e*x]*((-2*a^2*e + 8*c^2*d*x^3 + 2*a*c*x*(6*d + e*x))/(a^2*c*(a + c*x^2)) + ((d + e*x)*((-8*d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(a + c*x ^2))/(d + e*x)^2 + ((8*I)*Sqrt[c]*d*(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))]*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[d + e*x] + (2*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))]*EllipticF[I* ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sq rt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[d + e*x]))/(a^2*c*e*Sqrt[-d - (I *Sqrt[a]*e)/Sqrt[c]])))/(12*Sqrt[a + c*x^2])
Time = 0.78 (sec) , antiderivative size = 706, normalized size of antiderivative = 1.92, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {495, 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.
| \(\displaystyle \int \frac {(d+e x)^{3/2}}{\left (a+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 495 |
| \(\displaystyle \frac {\int \frac {4 c d^2+3 c e x d+a e^2}{2 \sqrt {d+e x} \left (c x^2+a\right )^{3/2}}dx}{3 a c}-\frac {\sqrt {d+e x} (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {4 c d^2+3 c e x d+a e^2}{\sqrt {d+e x} \left (c x^2+a\right )^{3/2}}dx}{6 a c}-\frac {\sqrt {d+e x} (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {\frac {\sqrt {d+e x} (a e+4 c d x)}{a \sqrt {a+c x^2}}-\frac {\int -\frac {c e \left (c d^2+a e^2\right ) (a e-4 c d x)}{2 \sqrt {d+e x} \sqrt {c x^2+a}}dx}{a c \left (a e^2+c d^2\right )}}{6 a c}-\frac {\sqrt {d+e x} (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {e \int \frac {a e-4 c d x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx}{2 a}+\frac {\sqrt {d+e x} (a e+4 c d x)}{a \sqrt {a+c x^2}}}{6 a c}-\frac {\sqrt {d+e x} (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {\frac {\sqrt {d+e x} (a e+4 c d x)}{a \sqrt {a+c x^2}}-\frac {\int -\frac {4 c d^2-4 c (d+e x) d+a e^2}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{a e}}{6 a c}-\frac {\sqrt {d+e x} (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {4 c d^2-4 c (d+e x) d+a e^2}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{a e}+\frac {\sqrt {d+e x} (a e+4 c d x)}{a \sqrt {a+c x^2}}}{6 a c}-\frac {\sqrt {d+e x} (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {\frac {\sqrt {d+e x} (a e+4 c d x)}{a \sqrt {a+c x^2}}-\frac {-\left (\left (-4 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+4 c d^2\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}\right )-4 \sqrt {c} d \sqrt {a e^2+c d^2} \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{a e}}{6 a c}-\frac {\sqrt {d+e x} (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {\frac {\sqrt {d+e x} (a e+4 c d x)}{a \sqrt {a+c x^2}}-\frac {-4 \sqrt {c} d \sqrt {a e^2+c d^2} \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}-\frac {\sqrt [4]{a e^2+c d^2} \left (-4 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+4 c d^2\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}}{a e}}{6 a c}-\frac {\sqrt {d+e x} (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {\frac {\sqrt {d+e x} (a e+4 c d x)}{a \sqrt {a+c x^2}}-\frac {-\frac {\sqrt [4]{a e^2+c d^2} \left (-4 \sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+4 c d^2\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 \sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-4 \sqrt {c} d \sqrt {a e^2+c d^2} \left (\frac {\sqrt [4]{a e^2+c d^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {d+e x} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )}\right )}{a e}}{6 a c}-\frac {\sqrt {d+e x} (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
-1/3*((a*e - c*d*x)*Sqrt[d + e*x])/(a*c*(a + c*x^2)^(3/2)) + (((a*e + 4*c* d*x)*Sqrt[d + e*x])/(a*Sqrt[a + c*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)]*Ellip ticE[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)]* EllipticF[2*ArcTan[(c^(1/4)*Sqrt[d + e*x])/(c*d^2 + a*e^2)^(1/4)], (1 + (S qrt[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))/(6*a*c)
3.7.95.3.1 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*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(c + d*x)^(n - 2)*(a + b*x^2)^(p + 1)*Simp[a* d^2*(n - 1) - b*c^2*(2*p + 3) - b*c*d*(n + 2*p + 2)*x, x], x], x] /; FreeQ[ {a, b, c, d}, x] && LtQ[p, -1] && GtQ[n, 1] && IntQuadraticQ[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. \(699\) vs. \(2(296)=592\).
Time = 2.06 (sec) , antiderivative size = 700, normalized size of antiderivative = 1.90
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {\left (\frac {d x}{3 a \,c^{2}}-\frac {e}{3 c^{3}}\right ) \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}{\left (x^{2}+\frac {a}{c}\right )^{2}}-\frac {2 \left (c e x +c d \right ) \left (-\frac {d x}{3 c \,a^{2}}-\frac {e}{12 c^{2} a}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) \left (c e x +c d \right )}}+\frac {2 \left (\frac {e^{2} a +4 c \,d^{2}}{6 c \,a^{2}}-\frac {e^{2}}{12 a c}-\frac {2 d^{2}}{3 a^{2}}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}-\frac {2 d e \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{3 a^{2} \sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\) | \(700\) |
| default | \(\text {Expression too large to display}\) | \(1633\) |
((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*((1/3/a/c^2*d*x-1/ 3*e/c^3)*(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/c/a^2*x-1/12*e/c^2/a)/((x^2+1/c*a)*(c*e*x+c*d))^(1/2)+2*(1/6/c*(a*e^2 +4*c*d^2)/a^2-1/12/a/c*e^2-2/3*d^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)*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*d*e/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 = 0.19 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.95 \[ \int \frac {(d+e x)^{3/2}}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {{\left (4 \, a^{2} c d^{2} + 3 \, a^{3} e^{2} + {\left (4 \, c^{3} d^{2} + 3 \, a c^{2} e^{2}\right )} x^{4} + 2 \, {\left (4 \, a c^{2} d^{2} + 3 \, a^{2} c e^{2}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 12 \, {\left (c^{3} d e x^{4} + 2 \, a c^{2} d e x^{2} + a^{2} c d e\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + 3 \, {\left (4 \, c^{3} d e x^{3} + a c^{2} e^{2} x^{2} + 6 \, a c^{2} d e x - a^{2} c e^{2}\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}}{18 \, {\left (a^{2} c^{4} e x^{4} + 2 \, a^{3} c^{3} e x^{2} + a^{4} c^{2} e\right )}} \]
1/18*((4*a^2*c*d^2 + 3*a^3*e^2 + (4*c^3*d^2 + 3*a*c^2*e^2)*x^4 + 2*(4*a*c^ 2*d^2 + 3*a^2*c*e^2)*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*(c ^3*d*e*x^4 + 2*a*c^2*d*e*x^2 + a^2*c*d*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), weierstrassPIn verse(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^3*d*e*x^3 + a*c^2*e^2*x^2 + 6*a*c^2*d*e*x - a^2 *c*e^2)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(a^2*c^4*e*x^4 + 2*a^3*c^3*e*x^2 + a^4*c^2*e)
\[ \int \frac {(d+e x)^{3/2}}{\left (a+c x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}}}{\left (a + c x^{2}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {(d+e x)^{3/2}}{\left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(d+e x)^{3/2}}{\left (a+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(d+e x)^{3/2}}{\left (a+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{3/2}}{{\left (c\,x^2+a\right )}^{5/2}} \,d x \]