Optimal. Leaf size=207 \[ \frac{3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 \sqrt{c} \sqrt{d} (b c-a d)^4}+\frac{3 x (3 a d+b c)}{8 \left (c+d x^2\right ) (b c-a d)^3}+\frac{x (2 a d+b c)}{4 b \left (c+d x^2\right )^2 (b c-a d)^2}+\frac{a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac{3 \sqrt{a} \sqrt{b} (a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 (b c-a d)^4} \]
[Out]
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Rubi [A] time = 0.629957, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{8 \sqrt{c} \sqrt{d} (b c-a d)^4}+\frac{3 x (3 a d+b c)}{8 \left (c+d x^2\right ) (b c-a d)^3}+\frac{x (2 a d+b c)}{4 b \left (c+d x^2\right )^2 (b c-a d)^2}+\frac{a x}{2 b \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)}-\frac{3 \sqrt{a} \sqrt{b} (a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 (b c-a d)^4} \]
Antiderivative was successfully verified.
[In] Int[x^4/((a + b*x^2)^2*(c + d*x^2)^3),x]
[Out]
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Rubi in Sympy [A] time = 119.91, size = 187, normalized size = 0.9 \[ - \frac{3 \sqrt{a} \sqrt{b} \left (a d + b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \left (a d - b c\right )^{4}} - \frac{a x}{2 b \left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{2} \left (a d - b c\right )} - \frac{3 x \left (3 a d + b c\right )}{8 \left (c + d x^{2}\right ) \left (a d - b c\right )^{3}} + \frac{3 \left (a^{2} d^{2} + 6 a b c d + b^{2} c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}}{8 \sqrt{c} \sqrt{d} \left (a d - b c\right )^{4}} + \frac{x \left (2 a d + b c\right )}{4 b \left (c + d x^{2}\right )^{2} \left (a d - b c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(b*x**2+a)**2/(d*x**2+c)**3,x)
[Out]
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Mathematica [A] time = 0.611585, size = 166, normalized size = 0.8 \[ \frac{\frac{3 \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} \sqrt{d}}+\frac{2 c x (b c-a d)^2}{\left (c+d x^2\right )^2}+\frac{4 a b x (b c-a d)}{a+b x^2}+\frac{x (5 a d+3 b c) (b c-a d)}{c+d x^2}-12 \sqrt{a} \sqrt{b} (a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 (b c-a d)^4} \]
Antiderivative was successfully verified.
[In] Integrate[x^4/((a + b*x^2)^2*(c + d*x^2)^3),x]
[Out]
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Maple [B] time = 0.023, size = 388, normalized size = 1.9 \[ -{\frac{5\,{x}^{3}{a}^{2}{d}^{3}}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{x}^{3}abc{d}^{2}}{4\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,{x}^{3}{b}^{2}{c}^{2}d}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{3\,x{a}^{2}c{d}^{2}}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{xab{c}^{2}d}{4\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{5\,x{b}^{2}{c}^{3}}{8\, \left ( ad-bc \right ) ^{4} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{3\,{a}^{2}{d}^{2}}{8\, \left ( ad-bc \right ) ^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{9\,cabd}{4\, \left ( ad-bc \right ) ^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{3\,{b}^{2}{c}^{2}}{8\, \left ( ad-bc \right ) ^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{a}^{2}bxd}{2\, \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{a{b}^{2}xc}{2\, \left ( ad-bc \right ) ^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,{a}^{2}bd}{2\, \left ( ad-bc \right ) ^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,a{b}^{2}c}{2\, \left ( ad-bc \right ) ^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(b*x^2+a)^2/(d*x^2+c)^3,x)
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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)^2*(d*x^2 + c)^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 1.10676, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/((b*x^2 + a)^2*(d*x^2 + c)^3),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(b*x**2+a)**2/(d*x**2+c)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.247166, size = 406, normalized size = 1.96 \[ \frac{a b x}{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\left (b x^{2} + a\right )}} - \frac{3 \,{\left (a b^{2} c + a^{2} b d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{a b}} + \frac{3 \,{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{8 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt{c d}} + \frac{3 \, b c d x^{3} + 5 \, a d^{2} x^{3} + 5 \, b c^{2} x + 3 \, a c d x}{8 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}{\left (d x^{2} + c\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/((b*x^2 + a)^2*(d*x^2 + c)^3),x, algorithm="giac")
[Out]