Optimal. Leaf size=180 \[ \frac{\left (a^2 d^2+4 a b c d+b^2 c^2\right ) \log (a+b x)}{(b c-a d)^5}-\frac{\left (a^2 d^2+4 a b c d+b^2 c^2\right ) \log (c+d x)}{(b c-a d)^5}-\frac{a^2}{2 (a+b x)^2 (b c-a d)^3}+\frac{c^2}{2 (c+d x)^2 (b c-a d)^3}+\frac{a (a d+2 b c)}{(a+b x) (b c-a d)^4}+\frac{c (2 a d+b c)}{(c+d x) (b c-a d)^4} \]
[Out]
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Rubi [A] time = 0.366659, antiderivative size = 180, 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 \[ \frac{\left (a^2 d^2+4 a b c d+b^2 c^2\right ) \log (a+b x)}{(b c-a d)^5}-\frac{\left (a^2 d^2+4 a b c d+b^2 c^2\right ) \log (c+d x)}{(b c-a d)^5}-\frac{a^2}{2 (a+b x)^2 (b c-a d)^3}+\frac{c^2}{2 (c+d x)^2 (b c-a d)^3}+\frac{a (a d+2 b c)}{(a+b x) (b c-a d)^4}+\frac{c (2 a d+b c)}{(c+d x) (b c-a d)^4} \]
Antiderivative was successfully verified.
[In] Int[x^2/((a + b*x)^3*(c + d*x)^3),x]
[Out]
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Rubi in Sympy [A] time = 86.8179, size = 162, normalized size = 0.9 \[ \frac{a^{2}}{2 \left (a + b x\right )^{2} \left (a d - b c\right )^{3}} + \frac{a \left (a d + 2 b c\right )}{\left (a + b x\right ) \left (a d - b c\right )^{4}} - \frac{c^{2}}{2 \left (c + d x\right )^{2} \left (a d - b c\right )^{3}} + \frac{c \left (2 a d + b c\right )}{\left (c + d x\right ) \left (a d - b c\right )^{4}} - \frac{\left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right ) \log{\left (a + b x \right )}}{\left (a d - b c\right )^{5}} + \frac{\left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right ) \log{\left (c + d x \right )}}{\left (a d - b c\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(b*x+a)**3/(d*x+c)**3,x)
[Out]
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Mathematica [A] time = 0.26788, size = 168, normalized size = 0.93 \[ \frac{2 \left (a^2 d^2+4 a b c d+b^2 c^2\right ) \log (a+b x)-2 \left (a^2 d^2+4 a b c d+b^2 c^2\right ) \log (c+d x)-\frac{a^2 (b c-a d)^2}{(a+b x)^2}+\frac{c^2 (b c-a d)^2}{(c+d x)^2}+\frac{2 a (a d+2 b c) (b c-a d)}{a+b x}+\frac{2 c (2 a d+b c) (b c-a d)}{c+d x}}{2 (b c-a d)^5} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/((a + b*x)^3*(c + d*x)^3),x]
[Out]
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Maple [A] time = 0.022, size = 272, normalized size = 1.5 \[ -{\frac{{c}^{2}}{2\, \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) ^{2}}}+{\frac{\ln \left ( dx+c \right ){a}^{2}{d}^{2}}{ \left ( ad-bc \right ) ^{5}}}+4\,{\frac{\ln \left ( dx+c \right ) abcd}{ \left ( ad-bc \right ) ^{5}}}+{\frac{\ln \left ( dx+c \right ){b}^{2}{c}^{2}}{ \left ( ad-bc \right ) ^{5}}}+2\,{\frac{acd}{ \left ( ad-bc \right ) ^{4} \left ( dx+c \right ) }}+{\frac{{c}^{2}b}{ \left ( ad-bc \right ) ^{4} \left ( dx+c \right ) }}+{\frac{{a}^{2}}{2\, \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) ^{2}}}-{\frac{\ln \left ( bx+a \right ){a}^{2}{d}^{2}}{ \left ( ad-bc \right ) ^{5}}}-4\,{\frac{\ln \left ( bx+a \right ) abcd}{ \left ( ad-bc \right ) ^{5}}}-{\frac{\ln \left ( bx+a \right ){b}^{2}{c}^{2}}{ \left ( ad-bc \right ) ^{5}}}+{\frac{{a}^{2}d}{ \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) }}+2\,{\frac{abc}{ \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(b*x+a)^3/(d*x+c)^3,x)
[Out]
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Maxima [A] time = 1.41306, size = 872, normalized size = 4.84 \[ \frac{{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} \log \left (b x + a\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} - \frac{{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} \log \left (d x + c\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} + \frac{6 \, a^{2} b c^{3} + 6 \, a^{3} c^{2} d + 2 \,{\left (b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{3} + 3 \,{\left (b^{3} c^{3} + 5 \, a b^{2} c^{2} d + 5 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2} + 2 \,{\left (5 \, a b^{2} c^{3} + 8 \, a^{2} b c^{2} d + 5 \, a^{3} c d^{2}\right )} x}{2 \,{\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} +{\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \,{\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} +{\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \,{\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x + a)^3*(d*x + c)^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.236636, size = 1337, normalized size = 7.43 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((b*x + a)^3*(d*x + c)^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 19.1782, size = 1299, normalized size = 7.22 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(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^2/((b*x + a)^3*(d*x + c)^3),x, algorithm="giac")
[Out]