Optimal. Leaf size=100 \[ -\frac{a}{2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{c}{4 d \left (c+d x^2\right )^2 (b c-a d)}-\frac{a b \log \left (a+b x^2\right )}{2 (b c-a d)^3}+\frac{a b \log \left (c+d x^2\right )}{2 (b c-a d)^3} \]
[Out]
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Rubi [A] time = 0.217837, antiderivative size = 100, 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 \[ -\frac{a}{2 \left (c+d x^2\right ) (b c-a d)^2}-\frac{c}{4 d \left (c+d x^2\right )^2 (b c-a d)}-\frac{a b \log \left (a+b x^2\right )}{2 (b c-a d)^3}+\frac{a b \log \left (c+d x^2\right )}{2 (b c-a d)^3} \]
Antiderivative was successfully verified.
[In] Int[x^3/((a + b*x^2)*(c + d*x^2)^3),x]
[Out]
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Rubi in Sympy [A] time = 35.0775, size = 82, normalized size = 0.82 \[ \frac{a b \log{\left (a + b x^{2} \right )}}{2 \left (a d - b c\right )^{3}} - \frac{a b \log{\left (c + d x^{2} \right )}}{2 \left (a d - b c\right )^{3}} - \frac{a}{2 \left (c + d x^{2}\right ) \left (a d - b c\right )^{2}} + \frac{c}{4 d \left (c + d x^{2}\right )^{2} \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(b*x**2+a)/(d*x**2+c)**3,x)
[Out]
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Mathematica [A] time = 0.217999, size = 77, normalized size = 0.77 \[ \frac{\frac{(a d-b c) \left (a d \left (c+2 d x^2\right )+b c^2\right )}{d \left (c+d x^2\right )^2}+2 a b \log \left (c+d x^2\right )-2 a b \log \left (a+b x^2\right )}{4 (b c-a d)^3} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/((a + b*x^2)*(c + d*x^2)^3),x]
[Out]
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Maple [A] time = 0.019, size = 177, normalized size = 1.8 \[ -{\frac{{a}^{2}d}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{abc}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{{a}^{2}cd}{4\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{ab{c}^{2}}{2\, \left ( ad-bc \right ) ^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{b}^{2}{c}^{3}}{4\, \left ( ad-bc \right ) ^{3}d \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{ab\ln \left ( d{x}^{2}+c \right ) }{2\, \left ( ad-bc \right ) ^{3}}}+{\frac{ab\ln \left ( b{x}^{2}+a \right ) }{2\, \left ( ad-bc \right ) ^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(b*x^2+a)/(d*x^2+c)^3,x)
[Out]
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Maxima [A] time = 1.35306, size = 293, normalized size = 2.93 \[ -\frac{a b \log \left (b x^{2} + a\right )}{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} + \frac{a b \log \left (d x^{2} + c\right )}{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} - \frac{2 \, a d^{2} x^{2} + b c^{2} + a c d}{4 \,{\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3} +{\left (b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )} x^{4} + 2 \,{\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x^2 + a)*(d*x^2 + c)^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.244985, size = 346, normalized size = 3.46 \[ -\frac{b^{2} c^{3} - a^{2} c d^{2} + 2 \,{\left (a b c d^{2} - a^{2} d^{3}\right )} x^{2} + 2 \,{\left (a b d^{3} x^{4} + 2 \, a b c d^{2} x^{2} + a b c^{2} d\right )} \log \left (b x^{2} + a\right ) - 2 \,{\left (a b d^{3} x^{4} + 2 \, a b c d^{2} x^{2} + a b c^{2} d\right )} \log \left (d x^{2} + c\right )}{4 \,{\left (b^{3} c^{5} d - 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{3} d^{3} - a^{3} c^{2} d^{4} +{\left (b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}\right )} x^{4} + 2 \,{\left (b^{3} c^{4} d^{2} - 3 \, a b^{2} c^{3} d^{3} + 3 \, a^{2} b c^{2} d^{4} - a^{3} c d^{5}\right )} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x^2 + a)*(d*x^2 + c)^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 16.3175, size = 410, normalized size = 4.1 \[ - \frac{a b \log{\left (x^{2} + \frac{- \frac{a^{5} b d^{4}}{\left (a d - b c\right )^{3}} + \frac{4 a^{4} b^{2} c d^{3}}{\left (a d - b c\right )^{3}} - \frac{6 a^{3} b^{3} c^{2} d^{2}}{\left (a d - b c\right )^{3}} + \frac{4 a^{2} b^{4} c^{3} d}{\left (a d - b c\right )^{3}} + a^{2} b d - \frac{a b^{5} c^{4}}{\left (a d - b c\right )^{3}} + a b^{2} c}{2 a b^{2} d} \right )}}{2 \left (a d - b c\right )^{3}} + \frac{a b \log{\left (x^{2} + \frac{\frac{a^{5} b d^{4}}{\left (a d - b c\right )^{3}} - \frac{4 a^{4} b^{2} c d^{3}}{\left (a d - b c\right )^{3}} + \frac{6 a^{3} b^{3} c^{2} d^{2}}{\left (a d - b c\right )^{3}} - \frac{4 a^{2} b^{4} c^{3} d}{\left (a d - b c\right )^{3}} + a^{2} b d + \frac{a b^{5} c^{4}}{\left (a d - b c\right )^{3}} + a b^{2} c}{2 a b^{2} d} \right )}}{2 \left (a d - b c\right )^{3}} - \frac{a c d + 2 a d^{2} x^{2} + b c^{2}}{4 a^{2} c^{2} d^{3} - 8 a b c^{3} d^{2} + 4 b^{2} c^{4} d + x^{4} \left (4 a^{2} d^{5} - 8 a b c d^{4} + 4 b^{2} c^{2} d^{3}\right ) + x^{2} \left (8 a^{2} c d^{4} - 16 a b c^{2} d^{3} + 8 b^{2} c^{3} d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(b*x**2+a)/(d*x**2+c)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.275597, size = 235, normalized size = 2.35 \[ -\frac{a b^{2}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )}} + \frac{a b d{\rm ln}\left ({\left | d x^{2} + c \right |}\right )}{2 \,{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )}} - \frac{b^{2} c^{3} - a^{2} c d^{2} + 2 \,{\left (a b c d^{2} - a^{2} d^{3}\right )} x^{2}}{4 \,{\left (d x^{2} + c\right )}^{2}{\left (b c - a d\right )}^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x^2 + a)*(d*x^2 + c)^3),x, algorithm="giac")
[Out]