Integrand size = 24, antiderivative size = 174 \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {(2 b c+3 a d) x}{6 b (b c-a d)^2 \left (c+d x^2\right )^{3/2}}+\frac {a x}{2 b (b c-a d) \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}}+\frac {(4 b c+11 a d) x}{6 (b c-a d)^3 \sqrt {c+d x^2}}-\frac {\sqrt {a} (3 b c+2 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{2 (b c-a d)^{7/2}} \]
1/6*(3*a*d+2*b*c)*x/b/(-a*d+b*c)^2/(d*x^2+c)^(3/2)+1/2*a*x/b/(-a*d+b*c)/(b *x^2+a)/(d*x^2+c)^(3/2)-1/2*(2*a*d+3*b*c)*arctan(x*(-a*d+b*c)^(1/2)/a^(1/2 )/(d*x^2+c)^(1/2))*a^(1/2)/(-a*d+b*c)^(7/2)+1/6*(11*a*d+4*b*c)*x/(-a*d+b*c )^3/(d*x^2+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 11.01 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.76 \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {x^5 \left (9 c \left (7 c+2 d x^2\right ) \operatorname {Hypergeometric2F1}\left (1,2,\frac {9}{2},\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )+\frac {8 (b c-a d) x^2 \left (c+d x^2\right ) \operatorname {Hypergeometric2F1}\left (2,3,\frac {11}{2},\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )}{a+b x^2}\right )}{315 c^3 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2}} \]
(x^5*(9*c*(7*c + 2*d*x^2)*Hypergeometric2F1[1, 2, 9/2, ((b*c - a*d)*x^2)/( c*(a + b*x^2))] + (8*(b*c - a*d)*x^2*(c + d*x^2)*Hypergeometric2F1[2, 3, 1 1/2, ((b*c - a*d)*x^2)/(c*(a + b*x^2))])/(a + b*x^2)))/(315*c^3*(a + b*x^2 )^2*(c + d*x^2)^(3/2))
Time = 0.34 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.16, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {372, 402, 27, 402, 27, 291, 218}
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.
| \(\displaystyle \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\int \frac {a c-2 (b c+a d) x^2}{\left (b x^2+a\right ) \left (d x^2+c\right )^{5/2}}dx}{2 b (b c-a d)}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\frac {\int \frac {b c \left (5 a c-2 (2 b c+3 a d) x^2\right )}{\left (b x^2+a\right ) \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}-\frac {x (3 a d+2 b c)}{3 \left (c+d x^2\right )^{3/2} (b c-a d)}}{2 b (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\frac {b \int \frac {5 a c-2 (2 b c+3 a d) x^2}{\left (b x^2+a\right ) \left (d x^2+c\right )^{3/2}}dx}{3 (b c-a d)}-\frac {x (3 a d+2 b c)}{3 \left (c+d x^2\right )^{3/2} (b c-a d)}}{2 b (b c-a d)}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\frac {b \left (\frac {\int \frac {3 a c (3 b c+2 a d)}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{c (b c-a d)}-\frac {x (11 a d+4 b c)}{\sqrt {c+d x^2} (b c-a d)}\right )}{3 (b c-a d)}-\frac {x (3 a d+2 b c)}{3 \left (c+d x^2\right )^{3/2} (b c-a d)}}{2 b (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\frac {b \left (\frac {3 a (2 a d+3 b c) \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b c-a d}-\frac {x (11 a d+4 b c)}{\sqrt {c+d x^2} (b c-a d)}\right )}{3 (b c-a d)}-\frac {x (3 a d+2 b c)}{3 \left (c+d x^2\right )^{3/2} (b c-a d)}}{2 b (b c-a d)}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\frac {b \left (\frac {3 a (2 a d+3 b c) \int \frac {1}{a-\frac {(a d-b c) x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{b c-a d}-\frac {x (11 a d+4 b c)}{\sqrt {c+d x^2} (b c-a d)}\right )}{3 (b c-a d)}-\frac {x (3 a d+2 b c)}{3 \left (c+d x^2\right )^{3/2} (b c-a d)}}{2 b (b c-a d)}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\frac {b \left (\frac {3 \sqrt {a} (2 a d+3 b c) \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{(b c-a d)^{3/2}}-\frac {x (11 a d+4 b c)}{\sqrt {c+d x^2} (b c-a d)}\right )}{3 (b c-a d)}-\frac {x (3 a d+2 b c)}{3 \left (c+d x^2\right )^{3/2} (b c-a d)}}{2 b (b c-a d)}\) |
(a*x)/(2*b*(b*c - a*d)*(a + b*x^2)*(c + d*x^2)^(3/2)) - (-1/3*((2*b*c + 3* a*d)*x)/((b*c - a*d)*(c + d*x^2)^(3/2)) + (b*(-(((4*b*c + 11*a*d)*x)/((b*c - a*d)*Sqrt[c + d*x^2])) + (3*Sqrt[a]*(3*b*c + 2*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(b*c - a*d)^(3/2)))/(3*(b*c - a*d)))/( 2*b*(b*c - a*d))
3.8.76.3.1 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[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a , b, c, d, e, m, 2, p, q, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Time = 3.13 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.84
| method | result | size |
| pseudoelliptic | \(\frac {\left (\frac {\left (b \,x^{2}+a \right ) \left (a d +\frac {3 b c}{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}-\frac {\sqrt {d \,x^{2}+c}\, b x}{2}\right ) a}{\left (b \,x^{2}+a \right ) \left (a d -b c \right )^{3}}-\frac {\left (a d +b c \right ) x}{\left (a d -b c \right )^{3} \sqrt {d \,x^{2}+c}}-\frac {d \,x^{3}}{3 \left (a d -b c \right )^{2} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}\) | \(146\) |
| default | \(\text {Expression too large to display}\) | \(3522\) |
((b*x^2+a)/((a*d-b*c)*a)^(1/2)*(a*d+3/2*b*c)*arctanh((d*x^2+c)^(1/2)/x*a/( (a*d-b*c)*a)^(1/2))-1/2*(d*x^2+c)^(1/2)*b*x)*a/(b*x^2+a)/(a*d-b*c)^3-(a*d+ b*c)*x/(a*d-b*c)^3/(d*x^2+c)^(1/2)-1/3*d/(a*d-b*c)^2/(d*x^2+c)^(3/2)*x^3
Leaf count of result is larger than twice the leaf count of optimal. 458 vs. \(2 (150) = 300\).
Time = 0.70 (sec) , antiderivative size = 1008, normalized size of antiderivative = 5.79 \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left ({\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{6} + 3 \, a b c^{3} + 2 \, a^{2} c^{2} d + {\left (6 \, b^{2} c^{2} d + 7 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} x^{4} + {\left (3 \, b^{2} c^{3} + 8 \, a b c^{2} d + 4 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {-\frac {a}{b c - a d}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (b^{2} c^{2} - 3 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{3} - {\left (a b c^{2} - a^{2} c d\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{b c - a d}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left ({\left (4 \, b^{2} c d + 11 \, a b d^{2}\right )} x^{5} + 2 \, {\left (3 \, b^{2} c^{2} + 8 \, a b c d + 4 \, a^{2} d^{2}\right )} x^{3} + 3 \, {\left (3 \, a b c^{2} + 2 \, a^{2} c d\right )} x\right )} \sqrt {d x^{2} + c}}{24 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\right )}}, \frac {3 \, {\left ({\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{6} + 3 \, a b c^{3} + 2 \, a^{2} c^{2} d + {\left (6 \, b^{2} c^{2} d + 7 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} x^{4} + {\left (3 \, b^{2} c^{3} + 8 \, a b c^{2} d + 4 \, a^{2} c d^{2}\right )} x^{2}\right )} \sqrt {\frac {a}{b c - a d}} \arctan \left (-\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {a}{b c - a d}}}{2 \, {\left (a d x^{3} + a c x\right )}}\right ) + 2 \, {\left ({\left (4 \, b^{2} c d + 11 \, a b d^{2}\right )} x^{5} + 2 \, {\left (3 \, b^{2} c^{2} + 8 \, a b c d + 4 \, a^{2} d^{2}\right )} x^{3} + 3 \, {\left (3 \, a b c^{2} + 2 \, a^{2} c d\right )} x\right )} \sqrt {d x^{2} + c}}{12 \, {\left (a b^{3} c^{5} - 3 \, a^{2} b^{2} c^{4} d + 3 \, a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3} + {\left (b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}\right )} x^{6} + {\left (2 \, b^{4} c^{4} d - 5 \, a b^{3} c^{3} d^{2} + 3 \, a^{2} b^{2} c^{2} d^{3} + a^{3} b c d^{4} - a^{4} d^{5}\right )} x^{4} + {\left (b^{4} c^{5} - a b^{3} c^{4} d - 3 \, a^{2} b^{2} c^{3} d^{2} + 5 \, a^{3} b c^{2} d^{3} - 2 \, a^{4} c d^{4}\right )} x^{2}\right )}}\right ] \]
[-1/24*(3*((3*b^2*c*d^2 + 2*a*b*d^3)*x^6 + 3*a*b*c^3 + 2*a^2*c^2*d + (6*b^ 2*c^2*d + 7*a*b*c*d^2 + 2*a^2*d^3)*x^4 + (3*b^2*c^3 + 8*a*b*c^2*d + 4*a^2* c*d^2)*x^2)*sqrt(-a/(b*c - a*d))*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^ 4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 + 4*((b^2*c^2 - 3*a*b*c*d + 2* a^2*d^2)*x^3 - (a*b*c^2 - a^2*c*d)*x)*sqrt(d*x^2 + c)*sqrt(-a/(b*c - a*d)) )/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*((4*b^2*c*d + 11*a*b*d^2)*x^5 + 2*(3*b^ 2*c^2 + 8*a*b*c*d + 4*a^2*d^2)*x^3 + 3*(3*a*b*c^2 + 2*a^2*c*d)*x)*sqrt(d*x ^2 + c))/(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3 + (b ^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*x^6 + (2*b^4*c ^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*x^4 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4 )*x^2), 1/12*(3*((3*b^2*c*d^2 + 2*a*b*d^3)*x^6 + 3*a*b*c^3 + 2*a^2*c^2*d + (6*b^2*c^2*d + 7*a*b*c*d^2 + 2*a^2*d^3)*x^4 + (3*b^2*c^3 + 8*a*b*c^2*d + 4*a^2*c*d^2)*x^2)*sqrt(a/(b*c - a*d))*arctan(-1/2*((b*c - 2*a*d)*x^2 - a*c )*sqrt(d*x^2 + c)*sqrt(a/(b*c - a*d))/(a*d*x^3 + a*c*x)) + 2*((4*b^2*c*d + 11*a*b*d^2)*x^5 + 2*(3*b^2*c^2 + 8*a*b*c*d + 4*a^2*d^2)*x^3 + 3*(3*a*b*c^ 2 + 2*a^2*c*d)*x)*sqrt(d*x^2 + c))/(a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b* c^3*d^2 - a^4*c^2*d^3 + (b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*x^6 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3 *b*c*d^4 - a^4*d^5)*x^4 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + ...
\[ \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {x^{4}}{\left (a + b x^{2}\right )^{2} \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {x^{4}}{{\left (b x^{2} + a\right )}^{2} {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 594 vs. \(2 (150) = 300\).
Time = 0.88 (sec) , antiderivative size = 594, normalized size of antiderivative = 3.41 \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\frac {{\left (\frac {2 \, {\left (b^{4} c^{5} d^{2} - a b^{3} c^{4} d^{3} - 3 \, a^{2} b^{2} c^{3} d^{4} + 5 \, a^{3} b c^{2} d^{5} - 2 \, a^{4} c d^{6}\right )} x^{2}}{b^{6} c^{7} d - 6 \, a b^{5} c^{6} d^{2} + 15 \, a^{2} b^{4} c^{5} d^{3} - 20 \, a^{3} b^{3} c^{4} d^{4} + 15 \, a^{4} b^{2} c^{3} d^{5} - 6 \, a^{5} b c^{2} d^{6} + a^{6} c d^{7}} + \frac {3 \, {\left (b^{4} c^{6} d - 2 \, a b^{3} c^{5} d^{2} + 2 \, a^{3} b c^{3} d^{4} - a^{4} c^{2} d^{5}\right )}}{b^{6} c^{7} d - 6 \, a b^{5} c^{6} d^{2} + 15 \, a^{2} b^{4} c^{5} d^{3} - 20 \, a^{3} b^{3} c^{4} d^{4} + 15 \, a^{4} b^{2} c^{3} d^{5} - 6 \, a^{5} b c^{2} d^{6} + a^{6} c d^{7}}\right )} x}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} + \frac {{\left (3 \, a b c \sqrt {d} + 2 \, a^{2} d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a b c d - a^{2} d^{2}}} - \frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c \sqrt {d} - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} d^{\frac {3}{2}} - a b c^{2} \sqrt {d}}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b - 2 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c + 4 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a d + b c^{2}\right )}} \]
1/3*(2*(b^4*c^5*d^2 - a*b^3*c^4*d^3 - 3*a^2*b^2*c^3*d^4 + 5*a^3*b*c^2*d^5 - 2*a^4*c*d^6)*x^2/(b^6*c^7*d - 6*a*b^5*c^6*d^2 + 15*a^2*b^4*c^5*d^3 - 20* a^3*b^3*c^4*d^4 + 15*a^4*b^2*c^3*d^5 - 6*a^5*b*c^2*d^6 + a^6*c*d^7) + 3*(b ^4*c^6*d - 2*a*b^3*c^5*d^2 + 2*a^3*b*c^3*d^4 - a^4*c^2*d^5)/(b^6*c^7*d - 6 *a*b^5*c^6*d^2 + 15*a^2*b^4*c^5*d^3 - 20*a^3*b^3*c^4*d^4 + 15*a^4*b^2*c^3* d^5 - 6*a^5*b*c^2*d^6 + a^6*c*d^7))*x/(d*x^2 + c)^(3/2) + 1/2*(3*a*b*c*sqr t(d) + 2*a^2*d^(3/2))*arctan(1/2*((sqrt(d)*x - sqrt(d*x^2 + c))^2*b - b*c + 2*a*d)/sqrt(a*b*c*d - a^2*d^2))/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^ 2 - a^3*d^3)*sqrt(a*b*c*d - a^2*d^2)) - ((sqrt(d)*x - sqrt(d*x^2 + c))^2*a *b*c*sqrt(d) - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^2*d^(3/2) - a*b*c^2*sqr t(d))/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*((sqrt(d)*x - s qrt(d*x^2 + c))^4*b - 2*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c + 4*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*d + b*c^2))
Timed out. \[ \int \frac {x^4}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {x^4}{{\left (b\,x^2+a\right )}^2\,{\left (d\,x^2+c\right )}^{5/2}} \,d x \]