Optimal. Leaf size=74 \[ \frac{a}{2 b \left (a+b x^2\right ) (b c-a d)}+\frac{c \log \left (a+b x^2\right )}{2 (b c-a d)^2}-\frac{c \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]
[Out]
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Rubi [A] time = 0.159262, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{a}{2 b \left (a+b x^2\right ) (b c-a d)}+\frac{c \log \left (a+b x^2\right )}{2 (b c-a d)^2}-\frac{c \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[x^3/((a + b*x^2)^2*(c + d*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 24.5667, size = 58, normalized size = 0.78 \[ - \frac{a}{2 b \left (a + b x^{2}\right ) \left (a d - b c\right )} + \frac{c \log{\left (a + b x^{2} \right )}}{2 \left (a d - b c\right )^{2}} - \frac{c \log{\left (c + d x^{2} \right )}}{2 \left (a d - b c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(b*x**2+a)**2/(d*x**2+c),x)
[Out]
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Mathematica [A] time = 0.055822, size = 74, normalized size = 1. \[ \frac{a}{2 b \left (a+b x^2\right ) (b c-a d)}+\frac{c \log \left (a+b x^2\right )}{2 (b c-a d)^2}-\frac{c \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/((a + b*x^2)^2*(c + d*x^2)),x]
[Out]
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Maple [A] time = 0.018, size = 95, normalized size = 1.3 \[ -{\frac{c\ln \left ( d{x}^{2}+c \right ) }{2\, \left ( ad-bc \right ) ^{2}}}+{\frac{c\ln \left ( b{x}^{2}+a \right ) }{2\, \left ( ad-bc \right ) ^{2}}}-{\frac{{a}^{2}d}{2\, \left ( ad-bc \right ) ^{2}b \left ( b{x}^{2}+a \right ) }}+{\frac{ac}{2\, \left ( ad-bc \right ) ^{2} \left ( b{x}^{2}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(b*x^2+a)^2/(d*x^2+c),x)
[Out]
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Maxima [A] time = 1.35115, size = 142, normalized size = 1.92 \[ \frac{c \log \left (b x^{2} + a\right )}{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} - \frac{c \log \left (d x^{2} + c\right )}{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} + \frac{a}{2 \,{\left (a b^{2} c - a^{2} b d +{\left (b^{3} c - a b^{2} d\right )} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x^2 + a)^2*(d*x^2 + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.233411, size = 158, normalized size = 2.14 \[ \frac{a b c - a^{2} d +{\left (b^{2} c x^{2} + a b c\right )} \log \left (b x^{2} + a\right ) -{\left (b^{2} c x^{2} + a b c\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x^2 + a)^2*(d*x^2 + c)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.54609, size = 253, normalized size = 3.42 \[ - \frac{a}{2 a^{2} b d - 2 a b^{2} c + x^{2} \left (2 a b^{2} d - 2 b^{3} c\right )} - \frac{c \log{\left (x^{2} + \frac{- \frac{a^{3} c d^{3}}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b c^{2} d^{2}}{\left (a d - b c\right )^{2}} - \frac{3 a b^{2} c^{3} d}{\left (a d - b c\right )^{2}} + a c d + \frac{b^{3} c^{4}}{\left (a d - b c\right )^{2}} + b c^{2}}{2 b c d} \right )}}{2 \left (a d - b c\right )^{2}} + \frac{c \log{\left (x^{2} + \frac{\frac{a^{3} c d^{3}}{\left (a d - b c\right )^{2}} - \frac{3 a^{2} b c^{2} d^{2}}{\left (a d - b c\right )^{2}} + \frac{3 a b^{2} c^{3} d}{\left (a d - b c\right )^{2}} + a c d - \frac{b^{3} c^{4}}{\left (a d - b c\right )^{2}} + b c^{2}}{2 b c d} \right )}}{2 \left (a d - b c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(b*x**2+a)**2/(d*x**2+c),x)
[Out]
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GIAC/XCAS [A] time = 0.242386, size = 124, normalized size = 1.68 \[ -\frac{\frac{b^{2} c{\rm ln}\left ({\left | \frac{b c}{b x^{2} + a} - \frac{a d}{b x^{2} + a} + d \right |}\right )}{b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}} - \frac{a b}{{\left (b^{2} c - a b d\right )}{\left (b x^{2} + a\right )}}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x^2 + a)^2*(d*x^2 + c)),x, algorithm="giac")
[Out]