Integrand size = 26, antiderivative size = 354 \[ \int \frac {x^5}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=-\frac {\left (b^2 c^2+5 a d (2 b c-7 a d)\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{16 b^4 d e^2}-\frac {\left (b^2 c^2+5 a d (2 b c-7 a d)\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^2}{24 b^3 d (b c-a d) e^2}-\frac {a^2 \left (c+d x^2\right )^3}{b (b c-a d)^2 e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}+\frac {\left (b^2 c^2-2 a b c d+7 a^2 d^2\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )^3}{6 b^2 d (b c-a d)^2 e^2}-\frac {(b c-a d) \left (b^2 c^2+5 a d (2 b c-7 a d)\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{16 b^{9/2} d^{3/2} e^{3/2}} \]
-1/16*(-a*d+b*c)*(b^2*c^2+5*a*d*(-7*a*d+2*b*c))*arctanh(d^(1/2)*(e*(b*x^2+ a)/(d*x^2+c))^(1/2)/b^(1/2)/e^(1/2))/b^(9/2)/d^(3/2)/e^(3/2)-a^2*(d*x^2+c) ^3/b/(-a*d+b*c)^2/e/(e*(b*x^2+a)/(d*x^2+c))^(1/2)-1/16*(b^2*c^2+5*a*d*(-7* a*d+2*b*c))*(d*x^2+c)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/b^4/d/e^2-1/24*(b^2*c^ 2+5*a*d*(-7*a*d+2*b*c))*(d*x^2+c)^2*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/b^3/d/(- a*d+b*c)/e^2+1/6*(7*a^2*d^2-2*a*b*c*d+b^2*c^2)*(d*x^2+c)^3*(e*(b*x^2+a)/(d *x^2+c))^(1/2)/b^2/d/(-a*d+b*c)^2/e^2
Time = 4.43 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.70 \[ \int \frac {x^5}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\frac {\sqrt {d} \sqrt {\frac {b \left (c+d x^2\right )}{b c-a d}} \left (105 a^3 d^2+5 a^2 b d \left (-20 c+7 d x^2\right )+a b^2 \left (3 c^2-38 c d x^2-14 d^2 x^4\right )+b^3 x^2 \left (3 c^2+14 c d x^2+8 d^2 x^4\right )\right )-3 \sqrt {b c-a d} \left (b^2 c^2+10 a b c d-35 a^2 d^2\right ) \sqrt {a+b x^2} \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b c-a d}}\right )}{48 b^4 d^{3/2} e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \sqrt {\frac {b \left (c+d x^2\right )}{b c-a d}}} \]
(Sqrt[d]*Sqrt[(b*(c + d*x^2))/(b*c - a*d)]*(105*a^3*d^2 + 5*a^2*b*d*(-20*c + 7*d*x^2) + a*b^2*(3*c^2 - 38*c*d*x^2 - 14*d^2*x^4) + b^3*x^2*(3*c^2 + 1 4*c*d*x^2 + 8*d^2*x^4)) - 3*Sqrt[b*c - a*d]*(b^2*c^2 + 10*a*b*c*d - 35*a^2 *d^2)*Sqrt[a + b*x^2]*ArcSinh[(Sqrt[d]*Sqrt[a + b*x^2])/Sqrt[b*c - a*d]])/ (48*b^4*d^(3/2)*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[(b*(c + d*x^2))/( b*c - a*d)])
Time = 0.43 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.82, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {2053, 2052, 365, 25, 27, 298, 215, 215, 221}
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.
| \(\displaystyle \int \frac {x^5}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2053 |
| \(\displaystyle \frac {1}{2} \int \frac {x^4}{\left (\frac {e \left (b x^2+a\right )}{d x^2+c}\right )^{3/2}}dx^2\) |
| \(\Big \downarrow \) 2052 |
| \(\displaystyle e (b c-a d) \int \frac {\left (a e-c x^4\right )^2}{x^4 \left (b e-d x^4\right )^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}\) |
| \(\Big \downarrow \) 365 |
| \(\displaystyle e (b c-a d) \left (\frac {\int -\frac {e \left (a (2 b c-7 a d) e-b c^2 x^4\right )}{\left (b e-d x^4\right )^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{b e}-\frac {a^2 e}{b x^2 \left (b e-d x^4\right )^3}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle e (b c-a d) \left (-\frac {\int \frac {e \left (a (2 b c-7 a d) e-b c^2 x^4\right )}{\left (b e-d x^4\right )^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{b e}-\frac {a^2 e}{b x^2 \left (b e-d x^4\right )^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle e (b c-a d) \left (-\frac {\int \frac {a (2 b c-7 a d) e-b c^2 x^4}{\left (b e-d x^4\right )^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{b}-\frac {a^2 e}{b x^2 \left (b e-d x^4\right )^3}\right )\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle e (b c-a d) \left (-\frac {\frac {1}{6} \left (\frac {5 a (2 b c-7 a d)}{b}+\frac {b c^2}{d}\right ) \int \frac {1}{\left (b e-d x^4\right )^3}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}-\frac {\left (\frac {b c^2}{d}-\frac {a (2 b c-7 a d)}{b}\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{6 \left (b e-d x^4\right )^3}}{b}-\frac {a^2 e}{b x^2 \left (b e-d x^4\right )^3}\right )\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle e (b c-a d) \left (-\frac {\frac {1}{6} \left (\frac {5 a (2 b c-7 a d)}{b}+\frac {b c^2}{d}\right ) \left (\frac {3 \int \frac {1}{\left (b e-d x^4\right )^2}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{4 b e}+\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 b e \left (b e-d x^4\right )^2}\right )-\frac {\left (\frac {b c^2}{d}-\frac {a (2 b c-7 a d)}{b}\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{6 \left (b e-d x^4\right )^3}}{b}-\frac {a^2 e}{b x^2 \left (b e-d x^4\right )^3}\right )\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle e (b c-a d) \left (-\frac {\frac {1}{6} \left (\frac {5 a (2 b c-7 a d)}{b}+\frac {b c^2}{d}\right ) \left (\frac {3 \left (\frac {\int \frac {1}{b e-d x^4}d\sqrt {\frac {e \left (b x^2+a\right )}{d x^2+c}}}{2 b e}+\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 b e \left (b e-d x^4\right )}\right )}{4 b e}+\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 b e \left (b e-d x^4\right )^2}\right )-\frac {\left (\frac {b c^2}{d}-\frac {a (2 b c-7 a d)}{b}\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{6 \left (b e-d x^4\right )^3}}{b}-\frac {a^2 e}{b x^2 \left (b e-d x^4\right )^3}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle e (b c-a d) \left (-\frac {a^2 e}{b x^2 \left (b e-d x^4\right )^3}-\frac {\frac {1}{6} \left (\frac {5 a (2 b c-7 a d)}{b}+\frac {b c^2}{d}\right ) \left (\frac {3 \left (\frac {\text {arctanh}\left (\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{\sqrt {b} \sqrt {e}}\right )}{2 b^{3/2} \sqrt {d} e^{3/2}}+\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{2 b e \left (b e-d x^4\right )}\right )}{4 b e}+\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{4 b e \left (b e-d x^4\right )^2}\right )-\frac {\left (\frac {b c^2}{d}-\frac {a (2 b c-7 a d)}{b}\right ) \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{6 \left (b e-d x^4\right )^3}}{b}\right )\) |
(b*c - a*d)*e*(-((a^2*e)/(b*x^2*(b*e - d*x^4)^3)) - (-1/6*(((b*c^2)/d - (a *(2*b*c - 7*a*d))/b)*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(b*e - d*x^4)^3 + (((b*c^2)/d + (5*a*(2*b*c - 7*a*d))/b)*(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]/ (4*b*e*(b*e - d*x^4)^2) + (3*(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]/(2*b*e*(b* e - d*x^4)) + ArcTanh[(Sqrt[d]*Sqrt[(e*(a + b*x^2))/(c + d*x^2)])/(Sqrt[b] *Sqrt[e])]/(2*b^(3/2)*Sqrt[d]*e^(3/2))))/(4*b*e)))/6)/b)
3.4.7.3.1 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)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x _Symbol] :> Simp[c^2*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^p*Simp[2*b*c^2*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*d^2*(m + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1]
Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)))/((c_) + (d_.)*(x_)))^(p_), x_S ymbol] :> With[{q = Denominator[p]}, Simp[q*e*(b*c - a*d) Subst[Int[x^(q* (p + 1) - 1)*(((-a)*e + c*x^q)^m/(b*e - d*x^q)^(m + 2)), x], x, (e*((a + b* x)/(c + d*x)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m}, x] && FractionQ[p] && IntegerQ[m]
Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.) ))^(p_), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(e*( (a + b*x)/(c + d*x)))^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.24 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.90
| method | result | size |
| risch | \(\frac {\left (8 b^{2} d^{2} x^{4}-22 a b \,d^{2} x^{2}+14 b^{2} c d \,x^{2}+57 a^{2} d^{2}-52 a b c d +3 b^{2} c^{2}\right ) \left (b \,x^{2}+a \right )}{48 d \,b^{4} e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}-\frac {\left (\frac {\left (35 a^{2} d^{2}-10 a b c d -b^{2} c^{2}\right ) \left (a d -b c \right ) \ln \left (\frac {\frac {1}{2} e d a +\frac {1}{2} e b c +b d e \,x^{2}}{\sqrt {b d e}}+\sqrt {b d e \,x^{4}+\left (e d a +e b c \right ) x^{2}+a c e}\right )}{2 \sqrt {b d e}}-\frac {16 a^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d \left (d \,x^{2}+c \right )}{\left (a d -b c \right ) \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}\right ) \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{16 d \,b^{4} e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right )}\) | \(318\) |
| default | \(\text {Expression too large to display}\) | \(1027\) |
1/48/d*(8*b^2*d^2*x^4-22*a*b*d^2*x^2+14*b^2*c*d*x^2+57*a^2*d^2-52*a*b*c*d+ 3*b^2*c^2)*(b*x^2+a)/b^4/e/(e*(b*x^2+a)/(d*x^2+c))^(1/2)-1/16/d/b^4*(1/2*( 35*a^2*d^2-10*a*b*c*d-b^2*c^2)*(a*d-b*c)*ln((1/2*e*d*a+1/2*e*b*c+b*d*e*x^2 )/(b*d*e)^(1/2)+(b*d*e*x^4+(a*d*e+b*c*e)*x^2+a*c*e)^(1/2))/(b*d*e)^(1/2)-1 6*a^2*(a^2*d^2-2*a*b*c*d+b^2*c^2)*d*(d*x^2+c)/(a*d-b*c)/(b*d*e*x^4+a*d*e*x ^2+b*c*e*x^2+a*c*e)^(1/2))/e/(e*(b*x^2+a)/(d*x^2+c))^(1/2)*((d*x^2+c)*e*(b *x^2+a))^(1/2)/(d*x^2+c)
Time = 1.25 (sec) , antiderivative size = 781, normalized size of antiderivative = 2.21 \[ \int \frac {x^5}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (a b^{3} c^{3} + 9 \, a^{2} b^{2} c^{2} d - 45 \, a^{3} b c d^{2} + 35 \, a^{4} d^{3} + {\left (b^{4} c^{3} + 9 \, a b^{3} c^{2} d - 45 \, a^{2} b^{2} c d^{2} + 35 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {b d e} \log \left (8 \, b^{2} d^{2} e x^{4} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} e x^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} e - 4 \, {\left (2 \, b d^{2} x^{4} + b c^{2} + a c d + {\left (3 \, b c d + a d^{2}\right )} x^{2}\right )} \sqrt {b d e} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}\right ) + 4 \, {\left (8 \, b^{4} d^{4} x^{8} + 3 \, a b^{3} c^{3} d - 100 \, a^{2} b^{2} c^{2} d^{2} + 105 \, a^{3} b c d^{3} + 2 \, {\left (11 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{6} + {\left (17 \, b^{4} c^{2} d^{2} - 52 \, a b^{3} c d^{3} + 35 \, a^{2} b^{2} d^{4}\right )} x^{4} + {\left (3 \, b^{4} c^{3} d - 35 \, a b^{3} c^{2} d^{2} - 65 \, a^{2} b^{2} c d^{3} + 105 \, a^{3} b d^{4}\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{192 \, {\left (b^{6} d^{2} e^{2} x^{2} + a b^{5} d^{2} e^{2}\right )}}, \frac {3 \, {\left (a b^{3} c^{3} + 9 \, a^{2} b^{2} c^{2} d - 45 \, a^{3} b c d^{2} + 35 \, a^{4} d^{3} + {\left (b^{4} c^{3} + 9 \, a b^{3} c^{2} d - 45 \, a^{2} b^{2} c d^{2} + 35 \, a^{3} b d^{3}\right )} x^{2}\right )} \sqrt {-b d e} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {-b d e} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{2 \, {\left (b^{2} d e x^{2} + a b d e\right )}}\right ) + 2 \, {\left (8 \, b^{4} d^{4} x^{8} + 3 \, a b^{3} c^{3} d - 100 \, a^{2} b^{2} c^{2} d^{2} + 105 \, a^{3} b c d^{3} + 2 \, {\left (11 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{6} + {\left (17 \, b^{4} c^{2} d^{2} - 52 \, a b^{3} c d^{3} + 35 \, a^{2} b^{2} d^{4}\right )} x^{4} + {\left (3 \, b^{4} c^{3} d - 35 \, a b^{3} c^{2} d^{2} - 65 \, a^{2} b^{2} c d^{3} + 105 \, a^{3} b d^{4}\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{96 \, {\left (b^{6} d^{2} e^{2} x^{2} + a b^{5} d^{2} e^{2}\right )}}\right ] \]
[1/192*(3*(a*b^3*c^3 + 9*a^2*b^2*c^2*d - 45*a^3*b*c*d^2 + 35*a^4*d^3 + (b^ 4*c^3 + 9*a*b^3*c^2*d - 45*a^2*b^2*c*d^2 + 35*a^3*b*d^3)*x^2)*sqrt(b*d*e)* log(8*b^2*d^2*e*x^4 + 8*(b^2*c*d + a*b*d^2)*e*x^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*e - 4*(2*b*d^2*x^4 + b*c^2 + a*c*d + (3*b*c*d + a*d^2)*x^2)*sqrt (b*d*e)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c))) + 4*(8*b^4*d^4*x^8 + 3*a*b^3*c^ 3*d - 100*a^2*b^2*c^2*d^2 + 105*a^3*b*c*d^3 + 2*(11*b^4*c*d^3 - 7*a*b^3*d^ 4)*x^6 + (17*b^4*c^2*d^2 - 52*a*b^3*c*d^3 + 35*a^2*b^2*d^4)*x^4 + (3*b^4*c ^3*d - 35*a*b^3*c^2*d^2 - 65*a^2*b^2*c*d^3 + 105*a^3*b*d^4)*x^2)*sqrt((b*e *x^2 + a*e)/(d*x^2 + c)))/(b^6*d^2*e^2*x^2 + a*b^5*d^2*e^2), 1/96*(3*(a*b^ 3*c^3 + 9*a^2*b^2*c^2*d - 45*a^3*b*c*d^2 + 35*a^4*d^3 + (b^4*c^3 + 9*a*b^3 *c^2*d - 45*a^2*b^2*c*d^2 + 35*a^3*b*d^3)*x^2)*sqrt(-b*d*e)*arctan(1/2*(2* b*d*x^2 + b*c + a*d)*sqrt(-b*d*e)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c))/(b^2*d *e*x^2 + a*b*d*e)) + 2*(8*b^4*d^4*x^8 + 3*a*b^3*c^3*d - 100*a^2*b^2*c^2*d^ 2 + 105*a^3*b*c*d^3 + 2*(11*b^4*c*d^3 - 7*a*b^3*d^4)*x^6 + (17*b^4*c^2*d^2 - 52*a*b^3*c*d^3 + 35*a^2*b^2*d^4)*x^4 + (3*b^4*c^3*d - 35*a*b^3*c^2*d^2 - 65*a^2*b^2*c*d^3 + 105*a^3*b*d^4)*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)) )/(b^6*d^2*e^2*x^2 + a*b^5*d^2*e^2)]
Timed out. \[ \int \frac {x^5}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x^5}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Exception generated. \[ \int \frac {x^5}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%%{2,[1,0,0]%%%},[2,1,0]%%%}+%%%{%%{[-4,0]:[1,0,%%%{-1,[1 ,1,1]%%%}
Timed out. \[ \int \frac {x^5}{\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}} \, dx=\int \frac {x^5}{{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2}} \,d x \]