Optimal. Leaf size=108 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} (b c-a d)^2}+\frac{\sqrt{c} (b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 d^{3/2} (b c-a d)^2}-\frac{c x}{2 d \left (c+d x^2\right ) (b c-a d)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.243666, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} (b c-a d)^2}+\frac{\sqrt{c} (b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 d^{3/2} (b c-a d)^2}-\frac{c x}{2 d \left (c+d x^2\right ) (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[x^4/((a + b*x^2)*(c + d*x^2)^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 38.5092, size = 94, normalized size = 0.87 \[ \frac{a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{\sqrt{b} \left (a d - b c\right )^{2}} - \frac{\sqrt{c} \left (3 a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{2 d^{\frac{3}{2}} \left (a d - b c\right )^{2}} + \frac{c x}{2 d \left (c + d x^{2}\right ) \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(b*x**2+a)/(d*x**2+c)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.218645, size = 108, normalized size = 1. \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} (a d-b c)^2}+\frac{\sqrt{c} (b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 d^{3/2} (b c-a d)^2}+\frac{c x}{2 d \left (c+d x^2\right ) (a d-b c)} \]
Antiderivative was successfully verified.
[In] Integrate[x^4/((a + b*x^2)*(c + d*x^2)^2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.014, size = 144, normalized size = 1.3 \[{\frac{acx}{2\, \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{{c}^{2}xb}{2\, \left ( ad-bc \right ) ^{2}d \left ( d{x}^{2}+c \right ) }}-{\frac{3\,ac}{2\, \left ( ad-bc \right ) ^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{c}^{2}b}{2\, \left ( ad-bc \right ) ^{2}d}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{{a}^{2}}{ \left ( ad-bc \right ) ^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(b*x^2+a)/(d*x^2+c)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/((b*x^2 + a)*(d*x^2 + c)^2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.306612, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (a d^{2} x^{2} + a c d\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) -{\left (b c^{2} - 3 \, a c d +{\left (b c d - 3 \, a d^{2}\right )} x^{2}\right )} \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{2} - 2 \, d x \sqrt{-\frac{c}{d}} - c}{d x^{2} + c}\right ) - 2 \,{\left (b c^{2} - a c d\right )} x}{4 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2}\right )}}, \frac{4 \,{\left (a d^{2} x^{2} + a c d\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) -{\left (b c^{2} - 3 \, a c d +{\left (b c d - 3 \, a d^{2}\right )} x^{2}\right )} \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{2} - 2 \, d x \sqrt{-\frac{c}{d}} - c}{d x^{2} + c}\right ) - 2 \,{\left (b c^{2} - a c d\right )} x}{4 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2}\right )}}, \frac{{\left (b c^{2} - 3 \, a c d +{\left (b c d - 3 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c}{d}} \arctan \left (\frac{x}{\sqrt{\frac{c}{d}}}\right ) +{\left (a d^{2} x^{2} + a c d\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) -{\left (b c^{2} - a c d\right )} x}{2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2}\right )}}, \frac{2 \,{\left (a d^{2} x^{2} + a c d\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) +{\left (b c^{2} - 3 \, a c d +{\left (b c d - 3 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c}{d}} \arctan \left (\frac{x}{\sqrt{\frac{c}{d}}}\right ) -{\left (b c^{2} - a c d\right )} x}{2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/((b*x^2 + a)*(d*x^2 + c)^2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 42.6211, size = 1850, normalized size = 17.13 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(b*x**2+a)/(d*x**2+c)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.288677, size = 163, normalized size = 1.51 \[ \frac{a^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{a b}} + \frac{{\left (b c^{2} - 3 \, a c d\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{2 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \sqrt{c d}} - \frac{c x}{2 \,{\left (b c d - a d^{2}\right )}{\left (d x^{2} + c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/((b*x^2 + a)*(d*x^2 + c)^2),x, algorithm="giac")
[Out]