Optimal. Leaf size=143 \[ \frac{6 b^2 d^2 \log (a+b x)}{(b c-a d)^5}-\frac{6 b^2 d^2 \log (c+d x)}{(b c-a d)^5}+\frac{3 b d (a d+b c+2 b d x)}{(b c-a d)^4 \left (x (a d+b c)+a c+b d x^2\right )}-\frac{a d+b c+2 b d x}{2 (b c-a d)^2 \left (x (a d+b c)+a c+b d x^2\right )^2} \]
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Rubi [A] time = 0.0486094, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {614, 616, 31} \[ \frac{6 b^2 d^2 \log (a+b x)}{(b c-a d)^5}-\frac{6 b^2 d^2 \log (c+d x)}{(b c-a d)^5}+\frac{3 b d (a d+b c+2 b d x)}{(b c-a d)^4 \left (x (a d+b c)+a c+b d x^2\right )}-\frac{a d+b c+2 b d x}{2 (b c-a d)^2 \left (x (a d+b c)+a c+b d x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 614
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx &=-\frac{b c+a d+2 b d x}{2 (b c-a d)^2 \left (a c+(b c+a d) x+b d x^2\right )^2}-\frac{(3 b d) \int \frac{1}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx}{(b c-a d)^2}\\ &=-\frac{b c+a d+2 b d x}{2 (b c-a d)^2 \left (a c+(b c+a d) x+b d x^2\right )^2}+\frac{3 b d (b c+a d+2 b d x)}{(b c-a d)^4 \left (a c+(b c+a d) x+b d x^2\right )}+\frac{\left (6 b^2 d^2\right ) \int \frac{1}{a c+(b c+a d) x+b d x^2} \, dx}{(b c-a d)^4}\\ &=-\frac{b c+a d+2 b d x}{2 (b c-a d)^2 \left (a c+(b c+a d) x+b d x^2\right )^2}+\frac{3 b d (b c+a d+2 b d x)}{(b c-a d)^4 \left (a c+(b c+a d) x+b d x^2\right )}-\frac{\left (6 b^3 d^3\right ) \int \frac{1}{b c+b d x} \, dx}{(b c-a d)^5}+\frac{\left (6 b^3 d^3\right ) \int \frac{1}{a d+b d x} \, dx}{(b c-a d)^5}\\ &=-\frac{b c+a d+2 b d x}{2 (b c-a d)^2 \left (a c+(b c+a d) x+b d x^2\right )^2}+\frac{3 b d (b c+a d+2 b d x)}{(b c-a d)^4 \left (a c+(b c+a d) x+b d x^2\right )}+\frac{6 b^2 d^2 \log (a+b x)}{(b c-a d)^5}-\frac{6 b^2 d^2 \log (c+d x)}{(b c-a d)^5}\\ \end{align*}
Mathematica [A] time = 0.109183, size = 128, normalized size = 0.9 \[ \frac{\frac{6 b^2 d (b c-a d)}{a+b x}-\frac{b^2 (b c-a d)^2}{(a+b x)^2}+12 b^2 d^2 \log (a+b x)+\frac{6 b d^2 (b c-a d)}{c+d x}+\frac{d^2 (b c-a d)^2}{(c+d x)^2}-12 b^2 d^2 \log (c+d x)}{2 (b c-a d)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 140, normalized size = 1. \begin{align*} -{\frac{{d}^{2}}{2\, \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) ^{2}}}+6\,{\frac{{b}^{2}{d}^{2}\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{5}}}+3\,{\frac{b{d}^{2}}{ \left ( ad-bc \right ) ^{4} \left ( dx+c \right ) }}+{\frac{{b}^{2}}{2\, \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) ^{2}}}-6\,{\frac{{b}^{2}{d}^{2}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{5}}}+3\,{\frac{{b}^{2}d}{ \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.15268, size = 802, normalized size = 5.61 \begin{align*} \frac{6 \, b^{2} d^{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 \, b^{2} d^{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{12 \, b^{3} d^{3} x^{3} - b^{3} c^{3} + 7 \, a b^{2} c^{2} d + 7 \, a^{2} b c d^{2} - a^{3} d^{3} + 18 \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 4 \,{\left (b^{3} c^{2} d + 7 \, a b^{2} c d^{2} + a^{2} b d^{3}\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.75451, size = 1488, normalized size = 10.41 \begin{align*} -\frac{b^{4} c^{4} - 8 \, a b^{3} c^{3} d + 8 \, a^{3} b c d^{3} - a^{4} d^{4} - 12 \,{\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{3} - 18 \,{\left (b^{4} c^{2} d^{2} - a^{2} b^{2} d^{4}\right )} x^{2} - 4 \,{\left (b^{4} c^{3} d + 6 \, a b^{3} c^{2} d^{2} - 6 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x - 12 \,{\left (b^{4} d^{4} x^{4} + a^{2} b^{2} c^{2} d^{2} + 2 \,{\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{3} +{\left (b^{4} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} + 2 \,{\left (a b^{3} c^{2} d^{2} + a^{2} b^{2} c d^{3}\right )} x\right )} \log \left (b x + a\right ) + 12 \,{\left (b^{4} d^{4} x^{4} + a^{2} b^{2} c^{2} d^{2} + 2 \,{\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{3} +{\left (b^{4} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} + 2 \,{\left (a b^{3} c^{2} d^{2} + a^{2} b^{2} c d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (a^{2} b^{5} c^{7} - 5 \, a^{3} b^{4} c^{6} d + 10 \, a^{4} b^{3} c^{5} d^{2} - 10 \, a^{5} b^{2} c^{4} d^{3} + 5 \, a^{6} b c^{3} d^{4} - a^{7} c^{2} d^{5} +{\left (b^{7} c^{5} d^{2} - 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} - 10 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{4} + 2 \,{\left (b^{7} c^{6} d - 4 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} - 5 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} x^{3} +{\left (b^{7} c^{7} - a b^{6} c^{6} d - 9 \, a^{2} b^{5} c^{5} d^{2} + 25 \, a^{3} b^{4} c^{4} d^{3} - 25 \, a^{4} b^{3} c^{3} d^{4} + 9 \, a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} - a^{7} d^{7}\right )} x^{2} + 2 \,{\left (a b^{6} c^{7} - 4 \, a^{2} b^{5} c^{6} d + 5 \, a^{3} b^{4} c^{5} d^{2} - 5 \, a^{5} b^{2} c^{3} d^{4} + 4 \, a^{6} b c^{2} d^{5} - a^{7} c d^{6}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.79196, size = 881, normalized size = 6.16 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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