Integrand size = 21, antiderivative size = 363 \[ \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^{3/2}} \, dx=-\frac {(a e-c d x) (d+e x)^{3/2}}{a c \sqrt {a+c x^2}}-\frac {d e \sqrt {d+e x} \sqrt {a+c x^2}}{a c}-\frac {\left (c d^2-3 a e^2\right ) \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {-a} c^{3/2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {a+c x^2}}+\frac {d \left (c d^2+a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{\sqrt {-a} c^{3/2} \sqrt {d+e x} \sqrt {a+c x^2}} \]
-(-c*d*x+a*e)*(e*x+d)^(3/2)/a/c/(c*x^2+a)^(1/2)-d*e*(e*x+d)^(1/2)*(c*x^2+a )^(1/2)/a/c-(-3*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))*(e*x+d)^(1/2)*(1+c*x^2 /a)^(1/2)/c^(3/2)/(-a)^(1/2)/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/(e*(-a)^(1/2 )+d*c^(1/2)))^(1/2)+d*(a*e^2+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)/c^(3/2)/(-a)^(1/2)/(e* x+d)^(1/2)/(c*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 13.13 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.32 \[ \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {\sqrt {d+e x} \left (\frac {e \left (-c d^2+3 a e^2\right ) \left (a+c x^2\right )}{d+e x}+c \left (c d^2 x-a e (2 d+e x)\right )-\frac {i c \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (c d^2-3 a e^2\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} \sqrt {d+e x} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{e}+\frac {\sqrt {a} \sqrt {c} \left (c d^2+4 i \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \sqrt {\frac {e \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {i \sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} \sqrt {d+e x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{a c^2 \sqrt {a+c x^2}} \]
(Sqrt[d + e*x]*((e*(-(c*d^2) + 3*a*e^2)*(a + c*x^2))/(d + e*x) + c*(c*d^2* x - a*e*(2*d + e*x)) - (I*c*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(c*d^2 - 3*a* e^2)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/S qrt[c] - e*x)/(d + e*x))]*Sqrt[d + e*x]*EllipticE[I*ArcSinh[Sqrt[-d - (I*S qrt[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]*Sqrt[c]*(c*d^2 + (4*I)*Sqrt[a]*Sqrt[c]*d*e - 3 *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]*EllipticF[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]]))/(a*c^2*Sqrt[a + c*x^2 ])
Time = 0.78 (sec) , antiderivative size = 707, normalized size of antiderivative = 1.95, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {495, 27, 687, 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.
| \(\displaystyle \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 495 |
| \(\displaystyle \frac {\int \frac {3 e (a e-c d x) \sqrt {d+e x}}{2 \sqrt {c x^2+a}}dx}{a c}-\frac {(d+e x)^{3/2} (a e-c d x)}{a c \sqrt {a+c x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 e \int \frac {(a e-c d x) \sqrt {d+e x}}{\sqrt {c x^2+a}}dx}{2 a c}-\frac {(d+e x)^{3/2} (a e-c d x)}{a c \sqrt {a+c x^2}}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {3 e \left (\frac {2 \int \frac {c \left (4 a d e-\left (c d^2-3 a e^2\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+a}}dx}{3 c}-\frac {2}{3} d \sqrt {a+c x^2} \sqrt {d+e x}\right )}{2 a c}-\frac {(d+e x)^{3/2} (a e-c d x)}{a c \sqrt {a+c x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 e \left (\frac {1}{3} \int \frac {4 a d e-\left (c d^2-3 a e^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+a}}dx-\frac {2}{3} d \sqrt {a+c x^2} \sqrt {d+e x}\right )}{2 a c}-\frac {(d+e x)^{3/2} (a e-c d x)}{a c \sqrt {a+c x^2}}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle \frac {3 e \left (-\frac {2 \int -\frac {d \left (c d^2+a e^2\right )-\left (c d^2-3 a e^2\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{3 e^2}-\frac {2}{3} d \sqrt {a+c x^2} \sqrt {d+e x}\right )}{2 a c}-\frac {(d+e x)^{3/2} (a e-c d x)}{a c \sqrt {a+c x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 e \left (\frac {2 \int \frac {d \left (c d^2+a e^2\right )-\left (c d^2-3 a e^2\right ) (d+e x)}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{3 e^2}-\frac {2}{3} d \sqrt {a+c x^2} \sqrt {d+e x}\right )}{2 a c}-\frac {(d+e x)^{3/2} (a e-c d x)}{a c \sqrt {a+c x^2}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {3 e \left (-\frac {2 \left (\frac {\sqrt {a e^2+c d^2} \left (-\sqrt {c} d \sqrt {a e^2+c d^2}-3 a e^2+c d^2\right ) \int \frac {1}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}-\frac {\left (c d^2-3 a e^2\right ) \sqrt {a e^2+c d^2} \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}\right )}{3 e^2}-\frac {2}{3} d \sqrt {a+c x^2} \sqrt {d+e x}\right )}{2 a c}-\frac {(d+e x)^{3/2} (a e-c d x)}{a c \sqrt {a+c x^2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {3 e \left (-\frac {2 \left (\frac {\left (a e^2+c d^2\right )^{3/4} \left (-\sqrt {c} d \sqrt {a e^2+c d^2}-3 a e^2+c d^2\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\left (c d^2-3 a e^2\right ) \sqrt {a e^2+c d^2} \int \frac {1-\frac {\sqrt {c} (d+e x)}{\sqrt {c d^2+a e^2}}}{\sqrt {\frac {c d^2}{e^2}-\frac {2 c (d+e x) d}{e^2}+\frac {c (d+e x)^2}{e^2}+a}}d\sqrt {d+e x}}{\sqrt {c}}\right )}{3 e^2}-\frac {2}{3} d \sqrt {a+c x^2} \sqrt {d+e x}\right )}{2 a c}-\frac {(d+e x)^{3/2} (a e-c d x)}{a c \sqrt {a+c x^2}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {3 e \left (-\frac {2 \left (\frac {\left (a e^2+c d^2\right )^{3/4} \left (-\sqrt {c} d \sqrt {a e^2+c d^2}-3 a e^2+c d^2\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{2 c^{3/4} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\left (c d^2-3 a e^2\right ) \sqrt {a e^2+c d^2} \left (\frac {\sqrt [4]{a e^2+c d^2} \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right ) \sqrt {\frac {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt [4]{c d^2+a e^2}}\right )|\frac {1}{2} \left (\frac {\sqrt {c} d}{\sqrt {c d^2+a e^2}}+1\right )\right )}{\sqrt [4]{c} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}-\frac {\sqrt {d+e x} \sqrt {a+\frac {c d^2}{e^2}-\frac {2 c d (d+e x)}{e^2}+\frac {c (d+e x)^2}{e^2}}}{\left (a+\frac {c d^2}{e^2}\right ) \left (\frac {\sqrt {c} (d+e x)}{\sqrt {a e^2+c d^2}}+1\right )}\right )}{\sqrt {c}}\right )}{3 e^2}-\frac {2}{3} d \sqrt {a+c x^2} \sqrt {d+e x}\right )}{2 a c}-\frac {(d+e x)^{3/2} (a e-c d x)}{a c \sqrt {a+c x^2}}\) |
-(((a*e - c*d*x)*(d + e*x)^(3/2))/(a*c*Sqrt[a + c*x^2])) + (3*e*((-2*d*Sqr t[d + e*x]*Sqrt[a + c*x^2])/3 - (2*(-(((c*d^2 - 3*a*e^2)*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)]*Elli pticE[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)*(c* d^2 - 3*a*e^2 - Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*(1 + (Sqrt[c]*(d + e*x))/Sq rt[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^(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^2)))/(2*a*c)
3.7.86.3.1 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[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) ), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp [c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x ] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 0])
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. \(692\) vs. \(2(299)=598\).
Time = 2.77 (sec) , antiderivative size = 693, normalized size of antiderivative = 1.91
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+a \right )}\, \left (-\frac {2 \left (c e x +c d \right ) \left (\frac {\left (e^{2} a -c \,d^{2}\right ) x}{2 c^{2} a}+\frac {d e}{c^{2}}\right )}{\sqrt {\left (x^{2}+\frac {a}{c}\right ) \left (c e x +c d \right )}}+\frac {2 \left (\frac {4 d \,e^{2}}{c}-\frac {d \left (3 e^{2} a -c \,d^{2}\right )}{a c}+\frac {d \left (e^{2} a -c \,d^{2}\right )}{c a}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}+\frac {2 \left (\frac {e^{3}}{c}+\frac {\left (e^{2} a -c \,d^{2}\right ) e}{2 a c}\right ) \left (\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {-a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c e \,x^{3}+c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\) | \(693\) |
| default | \(\text {Expression too large to display}\) | \(1150\) |
((e*x+d)*(c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+a)^(1/2)*(-2*(c*e*x+c*d)*(1 /2*(a*e^2-c*d^2)/c^2/a*x+d*e/c^2)/((x^2+1/c*a)*(c*e*x+c*d))^(1/2)+2*(4*d*e ^2/c-d*(3*a*e^2-c*d^2)/a/c+1/c*d*(a*e^2-c*d^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*(e^3/c+1/2*(a*e^2-c*d^2)*e/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)*((-d/e-(-a*c)^(1/2)/c)*Ellip ticE(((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 = 0.22 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.83 \[ \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {{\left (a c d^{3} + 9 \, a^{2} d e^{2} + {\left (c^{2} d^{3} + 9 \, a c d e^{2}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 3 \, {\left (a c d^{2} e - 3 \, a^{2} e^{3} + {\left (c^{2} d^{2} e - 3 \, a c e^{3}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} - 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} + 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (2 \, a c d e^{2} - {\left (c^{2} d^{2} e - a c e^{3}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}}{3 \, {\left (a c^{3} e x^{2} + a^{2} c^{2} e\right )}} \]
1/3*((a*c*d^3 + 9*a^2*d*e^2 + (c^2*d^3 + 9*a*c*d*e^2)*x^2)*sqrt(c*e)*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*(a*c*d^2*e - 3*a^2*e^3 + (c^2*d^2*e - 3*a*c* 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*(2*a*c*d*e ^2 - (c^2*d^2*e - a*c*e^3)*x)*sqrt(c*x^2 + a)*sqrt(e*x + d))/(a*c^3*e*x^2 + a^2*c^2*e)
\[ \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {5}{2}}}{\left (a + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(d+e x)^{5/2}}{\left (a+c x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{5/2}}{{\left (c\,x^2+a\right )}^{3/2}} \,d x \]