Optimal. Leaf size=74 \[ -\frac{c}{2 d \left (c+d x^2\right ) (b c-a d)}-\frac{a \log \left (a+b x^2\right )}{2 (b c-a d)^2}+\frac{a \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]
[Out]
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Rubi [A] time = 0.160293, 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 \[ -\frac{c}{2 d \left (c+d x^2\right ) (b c-a d)}-\frac{a \log \left (a+b x^2\right )}{2 (b c-a d)^2}+\frac{a \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[x^3/((a + b*x^2)*(c + d*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 25.5599, size = 58, normalized size = 0.78 \[ - \frac{a \log{\left (a + b x^{2} \right )}}{2 \left (a d - b c\right )^{2}} + \frac{a \log{\left (c + d x^{2} \right )}}{2 \left (a d - b c\right )^{2}} + \frac{c}{2 d \left (c + d x^{2}\right ) \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(b*x**2+a)/(d*x**2+c)**2,x)
[Out]
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Mathematica [A] time = 0.0556604, size = 74, normalized size = 1. \[ \frac{c}{2 d \left (c+d x^2\right ) (a d-b c)}-\frac{a \log \left (a+b x^2\right )}{2 (b c-a d)^2}+\frac{a \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/((a + b*x^2)*(c + d*x^2)^2),x]
[Out]
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Maple [A] time = 0.017, size = 95, normalized size = 1.3 \[{\frac{ac}{2\, \left ( ad-bc \right ) ^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{b{c}^{2}}{2\, \left ( ad-bc \right ) ^{2}d \left ( d{x}^{2}+c \right ) }}+{\frac{a\ln \left ( d{x}^{2}+c \right ) }{2\, \left ( ad-bc \right ) ^{2}}}-{\frac{a\ln \left ( b{x}^{2}+a \right ) }{2\, \left ( ad-bc \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(b*x^2+a)/(d*x^2+c)^2,x)
[Out]
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Maxima [A] time = 1.35796, size = 142, normalized size = 1.92 \[ -\frac{a \log \left (b x^{2} + a\right )}{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} + \frac{a \log \left (d x^{2} + c\right )}{2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} - \frac{c}{2 \,{\left (b c^{2} d - a c d^{2} +{\left (b c d^{2} - a d^{3}\right )} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x^2 + a)*(d*x^2 + c)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.224049, size = 158, normalized size = 2.14 \[ -\frac{b c^{2} - a c d +{\left (a d^{2} x^{2} + a c d\right )} \log \left (b x^{2} + a\right ) -{\left (a d^{2} x^{2} + a c d\right )} \log \left (d x^{2} + c\right )}{2 \,{\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3} +{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x^2 + a)*(d*x^2 + c)^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.44839, size = 253, normalized size = 3.42 \[ \frac{a \log{\left (x^{2} + \frac{- \frac{a^{4} d^{3}}{\left (a d - b c\right )^{2}} + \frac{3 a^{3} b c d^{2}}{\left (a d - b c\right )^{2}} - \frac{3 a^{2} b^{2} c^{2} d}{\left (a d - b c\right )^{2}} + a^{2} d + \frac{a b^{3} c^{3}}{\left (a d - b c\right )^{2}} + a b c}{2 a b d} \right )}}{2 \left (a d - b c\right )^{2}} - \frac{a \log{\left (x^{2} + \frac{\frac{a^{4} d^{3}}{\left (a d - b c\right )^{2}} - \frac{3 a^{3} b c d^{2}}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b^{2} c^{2} d}{\left (a d - b c\right )^{2}} + a^{2} d - \frac{a b^{3} c^{3}}{\left (a d - b c\right )^{2}} + a b c}{2 a b d} \right )}}{2 \left (a d - b c\right )^{2}} + \frac{c}{2 a c d^{2} - 2 b c^{2} d + x^{2} \left (2 a d^{3} - 2 b c d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(b*x**2+a)/(d*x**2+c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.295666, size = 123, normalized size = 1.66 \[ -\frac{\frac{a d^{2}{\rm ln}\left ({\left | b - \frac{b c}{d x^{2} + c} + \frac{a d}{d x^{2} + c} \right |}\right )}{b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}} + \frac{c d}{{\left (b c d - a d^{2}\right )}{\left (d x^{2} + c\right )}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x^2 + a)*(d*x^2 + c)^2),x, algorithm="giac")
[Out]