Integrand size = 19, antiderivative size = 191 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {(a e-c d x) (d+e x)^4}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {2 (d+e x)^2 \left (2 a^2 e^3-c d \left (c d^2+3 a e^2\right ) x\right )}{3 a^2 c^2 \sqrt {a+c x^2}}-\frac {e \left (4 \left (c^2 d^4+4 a c d^2 e^2-2 a^2 e^4\right )+c d e \left (2 c d^2+7 a e^2\right ) x\right ) \sqrt {a+c x^2}}{3 a^2 c^3}+\frac {5 d e^4 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{c^{5/2}} \]
-1/3*(-c*d*x+a*e)*(e*x+d)^4/a/c/(c*x^2+a)^(3/2)+5*d*e^4*arctanh(x*c^(1/2)/ (c*x^2+a)^(1/2))/c^(5/2)-2/3*(e*x+d)^2*(2*a^2*e^3-c*d*(3*a*e^2+c*d^2)*x)/a ^2/c^2/(c*x^2+a)^(1/2)-1/3*e*(-8*a^2*e^4+16*a*c*d^2*e^2+4*c^2*d^4+c*d*e*(7 *a*e^2+2*c*d^2)*x)*(c*x^2+a)^(1/2)/a^2/c^3
Time = 1.00 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.94 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^{5/2}} \, dx=\frac {8 a^4 e^5+2 c^4 d^5 x^3+a c^3 d^3 x \left (3 d^2+10 e^2 x^2\right )+a^3 c e^3 \left (-20 d^2-15 d e x+12 e^2 x^2\right )+a^2 c^2 e \left (-5 d^4-30 d^2 e^2 x^2-20 d e^3 x^3+3 e^4 x^4\right )-15 a^2 \sqrt {c} d e^4 \left (a+c x^2\right )^{3/2} \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{3 a^2 c^3 \left (a+c x^2\right )^{3/2}} \]
(8*a^4*e^5 + 2*c^4*d^5*x^3 + a*c^3*d^3*x*(3*d^2 + 10*e^2*x^2) + a^3*c*e^3* (-20*d^2 - 15*d*e*x + 12*e^2*x^2) + a^2*c^2*e*(-5*d^4 - 30*d^2*e^2*x^2 - 2 0*d*e^3*x^3 + 3*e^4*x^4) - 15*a^2*Sqrt[c]*d*e^4*(a + c*x^2)^(3/2)*Log[-(Sq rt[c]*x) + Sqrt[a + c*x^2]])/(3*a^2*c^3*(a + c*x^2)^(3/2))
Time = 0.38 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.15, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {495, 27, 684, 25, 27, 676, 224, 219}
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.
| \(\displaystyle \int \frac {(d+e x)^5}{\left (a+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 495 |
| \(\displaystyle \frac {\int \frac {2 (d+e x)^3 \left (c d^2-c e x d+2 a e^2\right )}{\left (c x^2+a\right )^{3/2}}dx}{3 a c}-\frac {(d+e x)^4 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \int \frac {(d+e x)^3 \left (c d^2-c e x d+2 a e^2\right )}{\left (c x^2+a\right )^{3/2}}dx}{3 a c}-\frac {(d+e x)^4 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 684 |
| \(\displaystyle \frac {2 \left (\frac {\int -\frac {e (d+e x) \left (a e \left (c d^2-4 a e^2\right )+c d \left (2 c d^2+7 a e^2\right ) x\right )}{\sqrt {c x^2+a}}dx}{a c}-\frac {(d+e x)^2 \left (2 a^2 e^3-c d x \left (3 a e^2+c d^2\right )\right )}{a c \sqrt {a+c x^2}}\right )}{3 a c}-\frac {(d+e x)^4 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (-\frac {\int \frac {e (d+e x) \left (a e \left (c d^2-4 a e^2\right )+c d \left (2 c d^2+7 a e^2\right ) x\right )}{\sqrt {c x^2+a}}dx}{a c}-\frac {(d+e x)^2 \left (2 a^2 e^3-c d x \left (3 a e^2+c d^2\right )\right )}{a c \sqrt {a+c x^2}}\right )}{3 a c}-\frac {(d+e x)^4 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (-\frac {e \int \frac {(d+e x) \left (a e \left (c d^2-4 a e^2\right )+c d \left (2 c d^2+7 a e^2\right ) x\right )}{\sqrt {c x^2+a}}dx}{a c}-\frac {(d+e x)^2 \left (2 a^2 e^3-c d x \left (3 a e^2+c d^2\right )\right )}{a c \sqrt {a+c x^2}}\right )}{3 a c}-\frac {(d+e x)^4 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 676 |
| \(\displaystyle \frac {2 \left (-\frac {e \left (-\frac {15}{2} a^2 d e^3 \int \frac {1}{\sqrt {c x^2+a}}dx+\frac {2 \sqrt {a+c x^2} \left (-2 a^2 e^4+4 a c d^2 e^2+c^2 d^4\right )}{c}+\frac {1}{2} d e x \sqrt {a+c x^2} \left (7 a e^2+2 c d^2\right )\right )}{a c}-\frac {(d+e x)^2 \left (2 a^2 e^3-c d x \left (3 a e^2+c d^2\right )\right )}{a c \sqrt {a+c x^2}}\right )}{3 a c}-\frac {(d+e x)^4 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {2 \left (-\frac {e \left (-\frac {15}{2} a^2 d e^3 \int \frac {1}{1-\frac {c x^2}{c x^2+a}}d\frac {x}{\sqrt {c x^2+a}}+\frac {2 \sqrt {a+c x^2} \left (-2 a^2 e^4+4 a c d^2 e^2+c^2 d^4\right )}{c}+\frac {1}{2} d e x \sqrt {a+c x^2} \left (7 a e^2+2 c d^2\right )\right )}{a c}-\frac {(d+e x)^2 \left (2 a^2 e^3-c d x \left (3 a e^2+c d^2\right )\right )}{a c \sqrt {a+c x^2}}\right )}{3 a c}-\frac {(d+e x)^4 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 \left (-\frac {e \left (-\frac {15 a^2 d e^3 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 \sqrt {c}}+\frac {2 \sqrt {a+c x^2} \left (-2 a^2 e^4+4 a c d^2 e^2+c^2 d^4\right )}{c}+\frac {1}{2} d e x \sqrt {a+c x^2} \left (7 a e^2+2 c d^2\right )\right )}{a c}-\frac {(d+e x)^2 \left (2 a^2 e^3-c d x \left (3 a e^2+c d^2\right )\right )}{a c \sqrt {a+c x^2}}\right )}{3 a c}-\frac {(d+e x)^4 (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}}\) |
-1/3*((a*e - c*d*x)*(d + e*x)^4)/(a*c*(a + c*x^2)^(3/2)) + (2*(-(((d + e*x )^2*(2*a^2*e^3 - c*d*(c*d^2 + 3*a*e^2)*x))/(a*c*Sqrt[a + c*x^2])) - (e*((2 *(c^2*d^4 + 4*a*c*d^2*e^2 - 2*a^2*e^4)*Sqrt[a + c*x^2])/c + (d*e*(2*c*d^2 + 7*a*e^2)*x*Sqrt[a + c*x^2])/2 - (15*a^2*d*e^3*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(2*Sqrt[c])))/(a*c)))/(3*a*c)
3.6.77.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
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[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p + 3))/(c*(2*p + 3)) Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g , p}, x] && !LeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*((a*(e*f + d*g ) - (c*d*f - a*e*g)*x)/(2*a*c*(p + 1))), x] - Simp[1/(2*a*c*(p + 1)) Int[ (d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^ 2*f*(2*p + 3) + e*(a*e*g*m - c*d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a , c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) || !ILtQ[m + 2*p + 3, 0])
Time = 2.36 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.47
| method | result | size |
| default | \(d^{5} \left (\frac {x}{3 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {c \,x^{2}+a}}\right )+e^{5} \left (\frac {x^{4}}{c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 a \left (-\frac {x^{2}}{c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a}{3 c^{2} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\right )}{c}\right )+5 d \,e^{4} \left (-\frac {x^{3}}{3 c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{c \sqrt {c \,x^{2}+a}}+\frac {\ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}}}}{c}\right )-\frac {5 d^{4} e}{3 c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+10 d^{2} e^{3} \left (-\frac {x^{2}}{c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a}{3 c^{2} \left (c \,x^{2}+a \right )^{\frac {3}{2}}}\right )+10 d^{3} e^{2} \left (-\frac {x}{2 c \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {a \left (\frac {x}{3 a \left (c \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {2 x}{3 a^{2} \sqrt {c \,x^{2}+a}}\right )}{2 c}\right )\) | \(281\) |
| risch | \(\frac {e^{5} \sqrt {c \,x^{2}+a}}{c^{3}}+\frac {\frac {5 d \,e^{4} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+a}\right )}{\sqrt {c}}+\frac {\left (-15 d \,e^{4} a^{2} c +10 d^{3} e^{2} c^{2} a -4 \sqrt {-a c}\, a^{2} e^{5}+20 \sqrt {-a c}\, a c \,d^{2} e^{3}+d^{5} c^{3}\right ) \sqrt {c \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}+2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}}{4 a^{2} c^{2} \left (x -\frac {\sqrt {-a c}}{c}\right )}-\frac {\left (-4 \sqrt {-a c}\, a^{2} e^{5}+20 \sqrt {-a c}\, a c \,d^{2} e^{3}+15 d \,e^{4} a^{2} c -10 d^{3} e^{2} c^{2} a -d^{5} c^{3}\right ) \sqrt {c \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}-2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}}{4 a^{2} c^{2} \left (x +\frac {\sqrt {-a c}}{c}\right )}+\frac {\left (-\sqrt {-a c}\, a^{2} e^{5}+10 \sqrt {-a c}\, a c \,d^{2} e^{3}-5 \sqrt {-a c}\, c^{2} d^{4} e -5 d \,e^{4} a^{2} c +10 d^{3} e^{2} c^{2} a -d^{5} c^{3}\right ) \left (-\frac {\sqrt {c \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}+2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}}{3 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}}-\frac {\sqrt {c \left (x -\frac {\sqrt {-a c}}{c}\right )^{2}+2 \sqrt {-a c}\, \left (x -\frac {\sqrt {-a c}}{c}\right )}}{3 a \left (x -\frac {\sqrt {-a c}}{c}\right )}\right )}{4 a \,c^{2}}+\frac {\left (\sqrt {-a c}\, a^{2} e^{5}-10 \sqrt {-a c}\, a c \,d^{2} e^{3}+5 \sqrt {-a c}\, c^{2} d^{4} e -5 d \,e^{4} a^{2} c +10 d^{3} e^{2} c^{2} a -d^{5} c^{3}\right ) \left (\frac {\sqrt {c \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}-2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}}{3 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}}-\frac {\sqrt {c \left (x +\frac {\sqrt {-a c}}{c}\right )^{2}-2 \sqrt {-a c}\, \left (x +\frac {\sqrt {-a c}}{c}\right )}}{3 a \left (x +\frac {\sqrt {-a c}}{c}\right )}\right )}{4 a \,c^{2}}}{c^{2}}\) | \(707\) |
d^5*(1/3*x/a/(c*x^2+a)^(3/2)+2/3*x/a^2/(c*x^2+a)^(1/2))+e^5*(x^4/c/(c*x^2+ a)^(3/2)-4*a/c*(-x^2/c/(c*x^2+a)^(3/2)-2/3*a/c^2/(c*x^2+a)^(3/2)))+5*d*e^4 *(-1/3*x^3/c/(c*x^2+a)^(3/2)+1/c*(-x/c/(c*x^2+a)^(1/2)+1/c^(3/2)*ln(c^(1/2 )*x+(c*x^2+a)^(1/2))))-5/3*d^4*e/c/(c*x^2+a)^(3/2)+10*d^2*e^3*(-x^2/c/(c*x ^2+a)^(3/2)-2/3*a/c^2/(c*x^2+a)^(3/2))+10*d^3*e^2*(-1/2*x/c/(c*x^2+a)^(3/2 )+1/2*a/c*(1/3*x/a/(c*x^2+a)^(3/2)+2/3*x/a^2/(c*x^2+a)^(1/2)))
Time = 0.29 (sec) , antiderivative size = 488, normalized size of antiderivative = 2.55 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^{5/2}} \, dx=\left [\frac {15 \, {\left (a^{2} c^{2} d e^{4} x^{4} + 2 \, a^{3} c d e^{4} x^{2} + a^{4} d e^{4}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (3 \, a^{2} c^{2} e^{5} x^{4} - 5 \, a^{2} c^{2} d^{4} e - 20 \, a^{3} c d^{2} e^{3} + 8 \, a^{4} e^{5} + 2 \, {\left (c^{4} d^{5} + 5 \, a c^{3} d^{3} e^{2} - 10 \, a^{2} c^{2} d e^{4}\right )} x^{3} - 6 \, {\left (5 \, a^{2} c^{2} d^{2} e^{3} - 2 \, a^{3} c e^{5}\right )} x^{2} + 3 \, {\left (a c^{3} d^{5} - 5 \, a^{3} c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{6 \, {\left (a^{2} c^{5} x^{4} + 2 \, a^{3} c^{4} x^{2} + a^{4} c^{3}\right )}}, -\frac {15 \, {\left (a^{2} c^{2} d e^{4} x^{4} + 2 \, a^{3} c d e^{4} x^{2} + a^{4} d e^{4}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (3 \, a^{2} c^{2} e^{5} x^{4} - 5 \, a^{2} c^{2} d^{4} e - 20 \, a^{3} c d^{2} e^{3} + 8 \, a^{4} e^{5} + 2 \, {\left (c^{4} d^{5} + 5 \, a c^{3} d^{3} e^{2} - 10 \, a^{2} c^{2} d e^{4}\right )} x^{3} - 6 \, {\left (5 \, a^{2} c^{2} d^{2} e^{3} - 2 \, a^{3} c e^{5}\right )} x^{2} + 3 \, {\left (a c^{3} d^{5} - 5 \, a^{3} c d e^{4}\right )} x\right )} \sqrt {c x^{2} + a}}{3 \, {\left (a^{2} c^{5} x^{4} + 2 \, a^{3} c^{4} x^{2} + a^{4} c^{3}\right )}}\right ] \]
[1/6*(15*(a^2*c^2*d*e^4*x^4 + 2*a^3*c*d*e^4*x^2 + a^4*d*e^4)*sqrt(c)*log(- 2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 2*(3*a^2*c^2*e^5*x^4 - 5*a^2* c^2*d^4*e - 20*a^3*c*d^2*e^3 + 8*a^4*e^5 + 2*(c^4*d^5 + 5*a*c^3*d^3*e^2 - 10*a^2*c^2*d*e^4)*x^3 - 6*(5*a^2*c^2*d^2*e^3 - 2*a^3*c*e^5)*x^2 + 3*(a*c^3 *d^5 - 5*a^3*c*d*e^4)*x)*sqrt(c*x^2 + a))/(a^2*c^5*x^4 + 2*a^3*c^4*x^2 + a ^4*c^3), -1/3*(15*(a^2*c^2*d*e^4*x^4 + 2*a^3*c*d*e^4*x^2 + a^4*d*e^4)*sqrt (-c)*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - (3*a^2*c^2*e^5*x^4 - 5*a^2*c^2*d ^4*e - 20*a^3*c*d^2*e^3 + 8*a^4*e^5 + 2*(c^4*d^5 + 5*a*c^3*d^3*e^2 - 10*a^ 2*c^2*d*e^4)*x^3 - 6*(5*a^2*c^2*d^2*e^3 - 2*a^3*c*e^5)*x^2 + 3*(a*c^3*d^5 - 5*a^3*c*d*e^4)*x)*sqrt(c*x^2 + a))/(a^2*c^5*x^4 + 2*a^3*c^4*x^2 + a^4*c^ 3)]
\[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{5}}{\left (a + c x^{2}\right )^{\frac {5}{2}}}\, dx \]
Time = 0.21 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.47 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {5}{3} \, d e^{4} x {\left (\frac {3 \, x^{2}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {2 \, a}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2}}\right )} + \frac {e^{5} x^{4}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} - \frac {10 \, d^{2} e^{3} x^{2}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {4 \, a e^{5} x^{2}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2}} + \frac {2 \, d^{5} x}{3 \, \sqrt {c x^{2} + a} a^{2}} + \frac {d^{5} x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {10 \, d^{3} e^{2} x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {10 \, d^{3} e^{2} x}{3 \, \sqrt {c x^{2} + a} a c} - \frac {5 \, d e^{4} x}{3 \, \sqrt {c x^{2} + a} c^{2}} + \frac {5 \, d e^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{c^{\frac {5}{2}}} - \frac {5 \, d^{4} e}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c} - \frac {20 \, a d^{2} e^{3}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2}} + \frac {8 \, a^{2} e^{5}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{3}} \]
-5/3*d*e^4*x*(3*x^2/((c*x^2 + a)^(3/2)*c) + 2*a/((c*x^2 + a)^(3/2)*c^2)) + e^5*x^4/((c*x^2 + a)^(3/2)*c) - 10*d^2*e^3*x^2/((c*x^2 + a)^(3/2)*c) + 4* a*e^5*x^2/((c*x^2 + a)^(3/2)*c^2) + 2/3*d^5*x/(sqrt(c*x^2 + a)*a^2) + 1/3* d^5*x/((c*x^2 + a)^(3/2)*a) - 10/3*d^3*e^2*x/((c*x^2 + a)^(3/2)*c) + 10/3* d^3*e^2*x/(sqrt(c*x^2 + a)*a*c) - 5/3*d*e^4*x/(sqrt(c*x^2 + a)*c^2) + 5*d* e^4*arcsinh(c*x/sqrt(a*c))/c^(5/2) - 5/3*d^4*e/((c*x^2 + a)^(3/2)*c) - 20/ 3*a*d^2*e^3/((c*x^2 + a)^(3/2)*c^2) + 8/3*a^2*e^5/((c*x^2 + a)^(3/2)*c^3)
Time = 0.29 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.08 \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^{5/2}} \, dx=-\frac {5 \, d e^{4} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{c^{\frac {5}{2}}} + \frac {{\left ({\left ({\left (\frac {3 \, e^{5} x}{c} + \frac {2 \, {\left (c^{6} d^{5} + 5 \, a c^{5} d^{3} e^{2} - 10 \, a^{2} c^{4} d e^{4}\right )}}{a^{2} c^{5}}\right )} x - \frac {6 \, {\left (5 \, a^{2} c^{4} d^{2} e^{3} - 2 \, a^{3} c^{3} e^{5}\right )}}{a^{2} c^{5}}\right )} x + \frac {3 \, {\left (a c^{5} d^{5} - 5 \, a^{3} c^{3} d e^{4}\right )}}{a^{2} c^{5}}\right )} x - \frac {5 \, a^{2} c^{4} d^{4} e + 20 \, a^{3} c^{3} d^{2} e^{3} - 8 \, a^{4} c^{2} e^{5}}{a^{2} c^{5}}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} \]
-5*d*e^4*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(5/2) + 1/3*((((3*e^5*x/ c + 2*(c^6*d^5 + 5*a*c^5*d^3*e^2 - 10*a^2*c^4*d*e^4)/(a^2*c^5))*x - 6*(5*a ^2*c^4*d^2*e^3 - 2*a^3*c^3*e^5)/(a^2*c^5))*x + 3*(a*c^5*d^5 - 5*a^3*c^3*d* e^4)/(a^2*c^5))*x - (5*a^2*c^4*d^4*e + 20*a^3*c^3*d^2*e^3 - 8*a^4*c^2*e^5) /(a^2*c^5))/(c*x^2 + a)^(3/2)
Timed out. \[ \int \frac {(d+e x)^5}{\left (a+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^5}{{\left (c\,x^2+a\right )}^{5/2}} \,d x \]