Optimal. Leaf size=112 \[ \frac{a^3}{b^2 (a+b x) (b c-a d)^2}+\frac{a^2 (3 b c-a d) \log (a+b x)}{b^2 (b c-a d)^3}+\frac{c^3}{d^2 (c+d x) (b c-a d)^2}+\frac{c^2 (b c-3 a d) \log (c+d x)}{d^2 (b c-a d)^3} \]
[Out]
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Rubi [A] time = 0.235, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{a^3}{b^2 (a+b x) (b c-a d)^2}+\frac{a^2 (3 b c-a d) \log (a+b x)}{b^2 (b c-a d)^3}+\frac{c^3}{d^2 (c+d x) (b c-a d)^2}+\frac{c^2 (b c-3 a d) \log (c+d x)}{d^2 (b c-a d)^3} \]
Antiderivative was successfully verified.
[In] Int[x^3/((a + b*x)^2*(c + d*x)^2),x]
[Out]
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Rubi in Sympy [A] time = 37.9358, size = 100, normalized size = 0.89 \[ \frac{a^{3}}{b^{2} \left (a + b x\right ) \left (a d - b c\right )^{2}} + \frac{a^{2} \left (a d - 3 b c\right ) \log{\left (a + b x \right )}}{b^{2} \left (a d - b c\right )^{3}} + \frac{c^{3}}{d^{2} \left (c + d x\right ) \left (a d - b c\right )^{2}} + \frac{c^{2} \left (3 a d - b c\right ) \log{\left (c + d x \right )}}{d^{2} \left (a d - b c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(b*x+a)**2/(d*x+c)**2,x)
[Out]
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Mathematica [A] time = 0.286956, size = 105, normalized size = 0.94 \[ \frac{\frac{a^3}{b^2 (a+b x)}+\frac{c^3}{d^2 (c+d x)}}{(b c-a d)^2}+\frac{a^2 (3 b c-a d) \log (a+b x)}{b^2 (b c-a d)^3}+\frac{c^2 (3 a d-b c) \log (c+d x)}{d^2 (a d-b c)^3} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/((a + b*x)^2*(c + d*x)^2),x]
[Out]
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Maple [A] time = 0.019, size = 149, normalized size = 1.3 \[ 3\,{\frac{{c}^{2}\ln \left ( dx+c \right ) a}{ \left ( ad-bc \right ) ^{3}d}}-{\frac{{c}^{3}\ln \left ( dx+c \right ) b}{ \left ( ad-bc \right ) ^{3}{d}^{2}}}+{\frac{{c}^{3}}{{d}^{2} \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}+{\frac{{a}^{3}\ln \left ( bx+a \right ) d}{ \left ( ad-bc \right ) ^{3}{b}^{2}}}-3\,{\frac{{a}^{2}\ln \left ( bx+a \right ) c}{ \left ( ad-bc \right ) ^{3}b}}+{\frac{{a}^{3}}{{b}^{2} \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(b*x+a)^2/(d*x+c)^2,x)
[Out]
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Maxima [A] time = 1.3542, size = 383, normalized size = 3.42 \[ \frac{{\left (3 \, a^{2} b c - a^{3} d\right )} \log \left (b x + a\right )}{b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}} + \frac{{\left (b c^{3} - 3 \, a c^{2} d\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}} + \frac{a b^{2} c^{3} + a^{3} c d^{2} +{\left (b^{3} c^{3} + a^{3} d^{3}\right )} x}{a b^{4} c^{3} d^{2} - 2 \, a^{2} b^{3} c^{2} d^{3} + a^{3} b^{2} c d^{4} +{\left (b^{5} c^{2} d^{3} - 2 \, a b^{4} c d^{4} + a^{2} b^{3} d^{5}\right )} x^{2} +{\left (b^{5} c^{3} d^{2} - a b^{4} c^{2} d^{3} - a^{2} b^{3} c d^{4} + a^{3} b^{2} d^{5}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x + a)^2*(d*x + c)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.218228, size = 551, normalized size = 4.92 \[ \frac{a b^{3} c^{4} - a^{2} b^{2} c^{3} d + a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (b^{4} c^{4} - a b^{3} c^{3} d + a^{3} b c d^{3} - a^{4} d^{4}\right )} x +{\left (3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{2} +{\left (3 \, a^{2} b^{2} c^{2} d^{2} + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x\right )} \log \left (b x + a\right ) +{\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d +{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2}\right )} x^{2} +{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d - 3 \, a^{2} b^{2} c^{2} d^{2}\right )} x\right )} \log \left (d x + c\right )}{a b^{5} c^{4} d^{2} - 3 \, a^{2} b^{4} c^{3} d^{3} + 3 \, a^{3} b^{3} c^{2} d^{4} - a^{4} b^{2} c d^{5} +{\left (b^{6} c^{3} d^{3} - 3 \, a b^{5} c^{2} d^{4} + 3 \, a^{2} b^{4} c d^{5} - a^{3} b^{3} d^{6}\right )} x^{2} +{\left (b^{6} c^{4} d^{2} - 2 \, a b^{5} c^{3} d^{3} + 2 \, a^{3} b^{3} c d^{5} - a^{4} b^{2} d^{6}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x + a)^2*(d*x + c)^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 15.1727, size = 627, normalized size = 5.6 \[ \frac{a^{2} \left (a d - 3 b c\right ) \log{\left (x + \frac{\frac{a^{6} d^{5} \left (a d - 3 b c\right )}{b \left (a d - b c\right )^{3}} - \frac{4 a^{5} c d^{4} \left (a d - 3 b c\right )}{\left (a d - b c\right )^{3}} + \frac{6 a^{4} b c^{2} d^{3} \left (a d - 3 b c\right )}{\left (a d - b c\right )^{3}} - \frac{4 a^{3} b^{2} c^{3} d^{2} \left (a d - 3 b c\right )}{\left (a d - b c\right )^{3}} + a^{3} c d^{2} + \frac{a^{2} b^{3} c^{4} d \left (a d - 3 b c\right )}{\left (a d - b c\right )^{3}} - 6 a^{2} b c^{2} d + a b^{2} c^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + b^{3} c^{3}} \right )}}{b^{2} \left (a d - b c\right )^{3}} + \frac{c^{2} \left (3 a d - b c\right ) \log{\left (x + \frac{\frac{a^{4} b c^{2} d^{3} \left (3 a d - b c\right )}{\left (a d - b c\right )^{3}} - \frac{4 a^{3} b^{2} c^{3} d^{2} \left (3 a d - b c\right )}{\left (a d - b c\right )^{3}} + a^{3} c d^{2} + \frac{6 a^{2} b^{3} c^{4} d \left (3 a d - b c\right )}{\left (a d - b c\right )^{3}} - 6 a^{2} b c^{2} d - \frac{4 a b^{4} c^{5} \left (3 a d - b c\right )}{\left (a d - b c\right )^{3}} + a b^{2} c^{3} + \frac{b^{5} c^{6} \left (3 a d - b c\right )}{d \left (a d - b c\right )^{3}}}{a^{3} d^{3} - 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + b^{3} c^{3}} \right )}}{d^{2} \left (a d - b c\right )^{3}} + \frac{a^{3} c d^{2} + a b^{2} c^{3} + x \left (a^{3} d^{3} + b^{3} c^{3}\right )}{a^{3} b^{2} c d^{4} - 2 a^{2} b^{3} c^{2} d^{3} + a b^{4} c^{3} d^{2} + x^{2} \left (a^{2} b^{3} d^{5} - 2 a b^{4} c d^{4} + b^{5} c^{2} d^{3}\right ) + x \left (a^{3} b^{2} d^{5} - a^{2} b^{3} c d^{4} - a b^{4} c^{2} d^{3} + b^{5} c^{3} d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(b*x+a)**2/(d*x+c)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.299874, size = 273, normalized size = 2.44 \[ \frac{a^{3} b^{2}}{{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )}{\left (b x + a\right )}} + \frac{{\left (b^{2} c^{3} - 3 \, a b c^{2} d\right )}{\rm ln}\left ({\left | \frac{b c}{b x + a} - \frac{a d}{b x + a} + d \right |}\right )}{b^{4} c^{3} d^{2} - 3 \, a b^{3} c^{2} d^{3} + 3 \, a^{2} b^{2} c d^{4} - a^{3} b d^{5}} - \frac{b c^{3}}{{\left (b c - a d\right )}^{3}{\left (\frac{b c}{b x + a} - \frac{a d}{b x + a} + d\right )} d} - \frac{{\rm ln}\left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x + a)^2*(d*x + c)^2),x, algorithm="giac")
[Out]