Optimal. Leaf size=169 \[ -\frac{a^4}{2 b^2 (a+b x)^2 (b c-a d)^3}+\frac{a^3 (4 b c-a d)}{b^2 (a+b x) (b c-a d)^4}+\frac{6 a^2 c^2 \log (a+b x)}{(b c-a d)^5}-\frac{6 a^2 c^2 \log (c+d x)}{(b c-a d)^5}+\frac{c^4}{2 d^2 (c+d x)^2 (b c-a d)^3}-\frac{c^3 (b c-4 a d)}{d^2 (c+d x) (b c-a d)^4} \]
[Out]
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Rubi [A] time = 0.402205, antiderivative size = 169, 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 \[ -\frac{a^4}{2 b^2 (a+b x)^2 (b c-a d)^3}+\frac{a^3 (4 b c-a d)}{b^2 (a+b x) (b c-a d)^4}+\frac{6 a^2 c^2 \log (a+b x)}{(b c-a d)^5}-\frac{6 a^2 c^2 \log (c+d x)}{(b c-a d)^5}+\frac{c^4}{2 d^2 (c+d x)^2 (b c-a d)^3}-\frac{c^3 (b c-4 a d)}{d^2 (c+d x) (b c-a d)^4} \]
Antiderivative was successfully verified.
[In] Int[x^4/((a + b*x)^3*(c + d*x)^3),x]
[Out]
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Rubi in Sympy [A] time = 97.3552, size = 151, normalized size = 0.89 \[ \frac{a^{4}}{2 b^{2} \left (a + b x\right )^{2} \left (a d - b c\right )^{3}} - \frac{a^{3} \left (a d - 4 b c\right )}{b^{2} \left (a + b x\right ) \left (a d - b c\right )^{4}} - \frac{6 a^{2} c^{2} \log{\left (a + b x \right )}}{\left (a d - b c\right )^{5}} + \frac{6 a^{2} c^{2} \log{\left (c + d x \right )}}{\left (a d - b c\right )^{5}} - \frac{c^{4}}{2 d^{2} \left (c + d x\right )^{2} \left (a d - b c\right )^{3}} + \frac{c^{3} \left (4 a d - b c\right )}{d^{2} \left (c + d x\right ) \left (a d - b c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(b*x+a)**3/(d*x+c)**3,x)
[Out]
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Mathematica [A] time = 0.438368, size = 171, normalized size = 1.01 \[ -\frac{a^4}{2 b^2 (a+b x)^2 (b c-a d)^3}+\frac{6 a^2 c^2 \log (a+b x)}{(b c-a d)^5}-\frac{6 a^2 c^2 \log (c+d x)}{(b c-a d)^5}+\frac{4 a^3 b c-a^4 d}{b^2 (a+b x) (b c-a d)^4}-\frac{c^4}{2 d^2 (c+d x)^2 (a d-b c)^3}-\frac{c^3 (b c-4 a d)}{d^2 (c+d x) (a d-b c)^4} \]
Antiderivative was successfully verified.
[In] Integrate[x^4/((a + b*x)^3*(c + d*x)^3),x]
[Out]
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Maple [A] time = 0.02, size = 204, normalized size = 1.2 \[ -{\frac{{c}^{4}}{2\,{d}^{2} \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) ^{2}}}+6\,{\frac{{a}^{2}{c}^{2}\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{5}}}+4\,{\frac{{c}^{3}a}{ \left ( ad-bc \right ) ^{4}d \left ( dx+c \right ) }}-{\frac{{c}^{4}b}{ \left ( ad-bc \right ) ^{4}{d}^{2} \left ( dx+c \right ) }}-{\frac{d{a}^{4}}{ \left ( ad-bc \right ) ^{4}{b}^{2} \left ( bx+a \right ) }}+4\,{\frac{{a}^{3}c}{ \left ( ad-bc \right ) ^{4}b \left ( bx+a \right ) }}+{\frac{{a}^{4}}{2\,{b}^{2} \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) ^{2}}}-6\,{\frac{{a}^{2}{c}^{2}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{5}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(b*x+a)^3/(d*x+c)^3,x)
[Out]
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Maxima [A] time = 1.39971, size = 999, normalized size = 5.91 \[ \frac{6 \, a^{2} c^{2} \log \left (b x + a\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} - \frac{6 \, a^{2} c^{2} \log \left (d x + c\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} - \frac{a^{2} b^{3} c^{5} - 7 \, a^{3} b^{2} c^{4} d - 7 \, a^{4} b c^{3} d^{2} + a^{5} c^{2} d^{3} + 2 \,{\left (b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} - 4 \, a^{3} b^{2} c d^{4} + a^{4} b d^{5}\right )} x^{3} +{\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d - 16 \, a^{2} b^{3} c^{3} d^{2} - 16 \, a^{3} b^{2} c^{2} d^{3} - 3 \, a^{4} b c d^{4} + a^{5} d^{5}\right )} x^{2} + 2 \,{\left (a b^{4} c^{5} - 6 \, a^{2} b^{3} c^{4} d - 8 \, a^{3} b^{2} c^{3} d^{2} - 6 \, a^{4} b c^{2} d^{3} + a^{5} c d^{4}\right )} x}{2 \,{\left (a^{2} b^{6} c^{6} d^{2} - 4 \, a^{3} b^{5} c^{5} d^{3} + 6 \, a^{4} b^{4} c^{4} d^{4} - 4 \, a^{5} b^{3} c^{3} d^{5} + a^{6} b^{2} c^{2} d^{6} +{\left (b^{8} c^{4} d^{4} - 4 \, a b^{7} c^{3} d^{5} + 6 \, a^{2} b^{6} c^{2} d^{6} - 4 \, a^{3} b^{5} c d^{7} + a^{4} b^{4} d^{8}\right )} x^{4} + 2 \,{\left (b^{8} c^{5} d^{3} - 3 \, a b^{7} c^{4} d^{4} + 2 \, a^{2} b^{6} c^{3} d^{5} + 2 \, a^{3} b^{5} c^{2} d^{6} - 3 \, a^{4} b^{4} c d^{7} + a^{5} b^{3} d^{8}\right )} x^{3} +{\left (b^{8} c^{6} d^{2} - 9 \, a^{2} b^{6} c^{4} d^{4} + 16 \, a^{3} b^{5} c^{3} d^{5} - 9 \, a^{4} b^{4} c^{2} d^{6} + a^{6} b^{2} d^{8}\right )} x^{2} + 2 \,{\left (a b^{7} c^{6} d^{2} - 3 \, a^{2} b^{6} c^{5} d^{3} + 2 \, a^{3} b^{5} c^{4} d^{4} + 2 \, a^{4} b^{4} c^{3} d^{5} - 3 \, a^{5} b^{3} c^{2} d^{6} + a^{6} b^{2} c d^{7}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/((b*x + a)^3*(d*x + c)^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.230256, size = 1330, normalized size = 7.87 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/((b*x + a)^3*(d*x + c)^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 23.2544, size = 1046, normalized size = 6.19 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(b*x+a)**3/(d*x+c)**3,x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/((b*x + a)^3*(d*x + c)^3),x, algorithm="giac")
[Out]