Integrand size = 24, antiderivative size = 109 \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=-\frac {c x}{d (b c-a d) \sqrt {c+d x^2}}+\frac {a^{3/2} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b (b c-a d)^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b d^{3/2}} \]
a^(3/2)*arctan(x*(-a*d+b*c)^(1/2)/a^(1/2)/(d*x^2+c)^(1/2))/b/(-a*d+b*c)^(3 /2)+arctanh(x*d^(1/2)/(d*x^2+c)^(1/2))/b/d^(3/2)-c*x/d/(-a*d+b*c)/(d*x^2+c )^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(666\) vs. \(2(109)=218\).
Time = 2.43 (sec) , antiderivative size = 666, normalized size of antiderivative = 6.11 \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\frac {c x \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}{d (-b c+a d) \left (c+d x^2-\sqrt {c} \sqrt {c+d x^2}\right )}+\frac {a^{3/2} \sqrt {c} \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (\sqrt {c}-\sqrt {c+d x^2}\right )}\right )}{\sqrt {b} (b c-a d)^{3/2} \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {a^{3/2} \arctan \left (\frac {\sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{\left (b^2 c-a b d\right ) \sqrt {2 b c-a d-2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {a^{3/2} \sqrt {c} \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{\sqrt {b} (b c-a d)^{3/2} \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {a^{3/2} \arctan \left (\frac {\sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}} x}{\sqrt {a} \left (-\sqrt {c}+\sqrt {c+d x^2}\right )}\right )}{\left (b^2 c-a b d\right ) \sqrt {2 b c-a d+2 \sqrt {b} \sqrt {c} \sqrt {b c-a d}}}+\frac {2 a \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c}-\sqrt {c+d x^2}}\right )}{\sqrt {d} \left (b^2 c-a b d\right )}+\frac {2 c \text {arctanh}\left (\frac {\sqrt {d} x}{-\sqrt {c}+\sqrt {c+d x^2}}\right )}{d^{3/2} (b c-a d)} \]
(c*x*(-Sqrt[c] + Sqrt[c + d*x^2]))/(d*(-(b*c) + a*d)*(c + d*x^2 - Sqrt[c]* Sqrt[c + d*x^2])) + (a^(3/2)*Sqrt[c]*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqrt[b]* Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(Sqrt[c] - Sqrt[c + d*x^2]))])/(Sqrt[ b]*(b*c - a*d)^(3/2)*Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]] ) + (a^(3/2)*ArcTan[(Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]] *x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c + d*x^2]))])/((b^2*c - a*b*d)*Sqrt[2*b*c - a*d - 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (a^(3/2)*Sqrt[c]*ArcTan[(Sqrt [2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c + d*x^2]))])/(Sqrt[b]*(b*c - a*d)^(3/2)*Sqrt[2*b*c - a*d + 2*Sqrt[b ]*Sqrt[c]*Sqrt[b*c - a*d]]) + (a^(3/2)*ArcTan[(Sqrt[2*b*c - a*d + 2*Sqrt[b ]*Sqrt[c]*Sqrt[b*c - a*d]]*x)/(Sqrt[a]*(-Sqrt[c] + Sqrt[c + d*x^2]))])/((b ^2*c - a*b*d)*Sqrt[2*b*c - a*d + 2*Sqrt[b]*Sqrt[c]*Sqrt[b*c - a*d]]) + (2* a*ArcTanh[(Sqrt[d]*x)/(Sqrt[c] - Sqrt[c + d*x^2])])/(Sqrt[d]*(b^2*c - a*b* d)) + (2*c*ArcTanh[(Sqrt[d]*x)/(-Sqrt[c] + Sqrt[c + d*x^2])])/(d^(3/2)*(b* c - a*d))
Time = 0.27 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.22, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {372, 398, 224, 219, 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.
| \(\displaystyle \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle \frac {\int \frac {(b c-a d) x^2+a c}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{d (b c-a d)}-\frac {c x}{d \sqrt {c+d x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {\frac {a^2 d \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}+\frac {(b c-a d) \int \frac {1}{\sqrt {d x^2+c}}dx}{b}}{d (b c-a d)}-\frac {c x}{d \sqrt {c+d x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {a^2 d \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}+\frac {(b c-a d) \int \frac {1}{1-\frac {d x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{b}}{d (b c-a d)}-\frac {c x}{d \sqrt {c+d x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {a^2 d \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{b}+\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b \sqrt {d}}}{d (b c-a d)}-\frac {c x}{d \sqrt {c+d x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {\frac {a^2 d \int \frac {1}{a-\frac {(a d-b c) x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{b}+\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b \sqrt {d}}}{d (b c-a d)}-\frac {c x}{d \sqrt {c+d x^2} (b c-a d)}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {a^{3/2} d \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{b \sqrt {b c-a d}}+\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} x}{\sqrt {c+d x^2}}\right )}{b \sqrt {d}}}{d (b c-a d)}-\frac {c x}{d \sqrt {c+d x^2} (b c-a d)}\) |
-((c*x)/(d*(b*c - a*d)*Sqrt[c + d*x^2])) + ((a^(3/2)*d*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(b*Sqrt[b*c - a*d]) + ((b*c - a*d)*Arc Tanh[(Sqrt[d]*x)/Sqrt[c + d*x^2]])/(b*Sqrt[d]))/(d*(b*c - a*d))
3.8.13.3.1 Defintions of rubi rules used
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[((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[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[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
Time = 3.02 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.07
| method | result | size |
| pseudoelliptic | \(-c \left (-\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}}{x \sqrt {d}}\right )}{d^{\frac {3}{2}} c b}-\frac {x}{\left (a d -b c \right ) d \sqrt {d \,x^{2}+c}}+\frac {a^{2} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{2}+c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\left (a d -b c \right ) b c \sqrt {\left (a d -b c \right ) a}}\right )\) | \(117\) |
| default | \(\frac {-\frac {x}{d \sqrt {d \,x^{2}+c}}+\frac {\ln \left (x \sqrt {d}+\sqrt {d \,x^{2}+c}\right )}{d^{\frac {3}{2}}}}{b}-\frac {a x}{b^{2} c \sqrt {d \,x^{2}+c}}+\frac {a^{2} \left (-\frac {b}{\left (a d -b c \right ) \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}+\frac {2 d \sqrt {-a b}\, \left (2 d \left (x -\frac {\sqrt {-a b}}{b}\right )+\frac {2 d \sqrt {-a b}}{b}\right )}{\left (a d -b c \right ) \left (-\frac {4 d \left (a d -b c \right )}{b}+\frac {4 d^{2} a}{b}\right ) \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}+\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{2 b^{2} \sqrt {-a b}}-\frac {a^{2} \left (-\frac {b}{\left (a d -b c \right ) \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}-\frac {2 d \sqrt {-a b}\, \left (2 d \left (x +\frac {\sqrt {-a b}}{b}\right )-\frac {2 d \sqrt {-a b}}{b}\right )}{\left (a d -b c \right ) \left (-\frac {4 d \left (a d -b c \right )}{b}+\frac {4 d^{2} a}{b}\right ) \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}+\frac {b \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{2 b^{2} \sqrt {-a b}}\) | \(804\) |
-c*(-1/d^(3/2)/c/b*arctanh((d*x^2+c)^(1/2)/x/d^(1/2))-1/(a*d-b*c)/d/(d*x^2 +c)^(1/2)*x+1/(a*d-b*c)*a^2/b/c/((a*d-b*c)*a)^(1/2)*arctanh((d*x^2+c)^(1/2 )/x*a/((a*d-b*c)*a)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (91) = 182\).
Time = 0.40 (sec) , antiderivative size = 977, normalized size of antiderivative = 8.96 \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\left [-\frac {4 \, \sqrt {d x^{2} + c} b c d x - 2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right ) + {\left (a d^{3} x^{2} + a c d^{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 (b^{2} c^{2} d^{2} - a b c d^{3} + {\left (b^{2} c d^{3} - a b d^{4}\right )} x^{2}\right )}}, -\frac {4 \, \sqrt {d x^{2} + c} b c d x + 4 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (a d^{3} x^{2} + a c d^{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 (b^{2} c^{2} d^{2} - a b c d^{3} + {\left (b^{2} c d^{3} - a b d^{4}\right )} x^{2}\right )}}, -\frac {2 \, \sqrt {d x^{2} + c} b c d x + {\left (a d^{3} x^{2} + a c d^{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 ) - {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )} \sqrt {d} \log \left (-2 \, d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {d} x - c\right )}{2 \, {\left (b^{2} c^{2} d^{2} - a b c d^{3} + {\left (b^{2} c d^{3} - a b d^{4}\right )} x^{2}\right )}}, -\frac {2 \, \sqrt {d x^{2} + c} b c d x + 2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x}{\sqrt {d x^{2} + c}}\right ) + {\left (a d^{3} x^{2} + a c d^{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 (b^{2} c^{2} d^{2} - a b c d^{3} + {\left (b^{2} c d^{3} - a b d^{4}\right )} x^{2}\right )}}\right ] \]
[-1/4*(4*sqrt(d*x^2 + c)*b*c*d*x - 2*(b*c^2 - a*c*d + (b*c*d - a*d^2)*x^2) *sqrt(d)*log(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) + (a*d^3*x^2 + a* c*d^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)))/(b^2*c^2*d^2 - a*b*c*d^3 + (b^2*c*d^3 - a*b*d^4 )*x^2), -1/4*(4*sqrt(d*x^2 + c)*b*c*d*x + 4*(b*c^2 - a*c*d + (b*c*d - a*d^ 2)*x^2)*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) + (a*d^3*x^2 + a*c*d^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)))/(b^2*c^2*d^2 - a*b*c*d^3 + (b^2*c*d^3 - a*b*d^4)*x^2 ), -1/2*(2*sqrt(d*x^2 + c)*b*c*d*x + (a*d^3*x^2 + a*c*d^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)) - (b*c^2 - a*c*d + (b*c*d - a*d^2)*x^2)*sqrt(d)*lo g(-2*d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c))/(b^2*c^2*d^2 - a*b*c*d^3 + (b^2*c*d^3 - a*b*d^4)*x^2), -1/2*(2*sqrt(d*x^2 + c)*b*c*d*x + 2*(b*c^2 - a *c*d + (b*c*d - a*d^2)*x^2)*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) + (a*d^3*x^2 + a*c*d^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)))/(b^2*c^2*...
\[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {x^{4}}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {x^{4}}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]
Exception generated. \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {x^4}{\left (a+b x^2\right ) \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {x^4}{\left (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \]