Optimal. Leaf size=82 \[ \frac{d^2 \log (a+b x)}{(b c-a d)^3}-\frac{d^2 \log (c+d x)}{(b c-a d)^3}+\frac{d}{(a+b x) (b c-a d)^2}-\frac{1}{2 (a+b x)^2 (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0454462, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {44} \[ \frac{d^2 \log (a+b x)}{(b c-a d)^3}-\frac{d^2 \log (c+d x)}{(b c-a d)^3}+\frac{d}{(a+b x) (b c-a d)^2}-\frac{1}{2 (a+b x)^2 (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^3 (c+d x)} \, dx &=\int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx\\ &=-\frac{1}{2 (b c-a d) (a+b x)^2}+\frac{d}{(b c-a d)^2 (a+b x)}+\frac{d^2 \log (a+b x)}{(b c-a d)^3}-\frac{d^2 \log (c+d x)}{(b c-a d)^3}\\ \end{align*}
Mathematica [A] time = 0.0639444, size = 67, normalized size = 0.82 \[ \frac{\frac{(b c-a d) (3 a d-b c+2 b d x)}{(a+b x)^2}+2 d^2 \log (a+b x)-2 d^2 \log (c+d x)}{2 (b c-a d)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 81, normalized size = 1. \begin{align*}{\frac{{d}^{2}\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{3}}}+{\frac{1}{ \left ( 2\,ad-2\,bc \right ) \left ( bx+a \right ) ^{2}}}+{\frac{d}{ \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) }}-{\frac{{d}^{2}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.0096, size = 273, normalized size = 3.33 \begin{align*} \frac{d^{2} \log \left (b x + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac{d^{2} \log \left (d x + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac{2 \, b d x - b c + 3 \, a d}{2 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.83099, size = 491, normalized size = 5.99 \begin{align*} -\frac{b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2} - 2 \,{\left (b^{2} c d - a b d^{2}\right )} x - 2 \,{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right ) + 2 \,{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (d x + c\right )}{2 \,{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3} +{\left (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}\right )} x^{2} + 2 \,{\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 1.21214, size = 381, normalized size = 4.65 \begin{align*} \frac{d^{2} \log{\left (x + \frac{- \frac{a^{4} d^{6}}{\left (a d - b c\right )^{3}} + \frac{4 a^{3} b c d^{5}}{\left (a d - b c\right )^{3}} - \frac{6 a^{2} b^{2} c^{2} d^{4}}{\left (a d - b c\right )^{3}} + \frac{4 a b^{3} c^{3} d^{3}}{\left (a d - b c\right )^{3}} + a d^{3} - \frac{b^{4} c^{4} d^{2}}{\left (a d - b c\right )^{3}} + b c d^{2}}{2 b d^{3}} \right )}}{\left (a d - b c\right )^{3}} - \frac{d^{2} \log{\left (x + \frac{\frac{a^{4} d^{6}}{\left (a d - b c\right )^{3}} - \frac{4 a^{3} b c d^{5}}{\left (a d - b c\right )^{3}} + \frac{6 a^{2} b^{2} c^{2} d^{4}}{\left (a d - b c\right )^{3}} - \frac{4 a b^{3} c^{3} d^{3}}{\left (a d - b c\right )^{3}} + a d^{3} + \frac{b^{4} c^{4} d^{2}}{\left (a d - b c\right )^{3}} + b c d^{2}}{2 b d^{3}} \right )}}{\left (a d - b c\right )^{3}} + \frac{3 a d - b c + 2 b d x}{2 a^{4} d^{2} - 4 a^{3} b c d + 2 a^{2} b^{2} c^{2} + x^{2} \left (2 a^{2} b^{2} d^{2} - 4 a b^{3} c d + 2 b^{4} c^{2}\right ) + x \left (4 a^{3} b d^{2} - 8 a^{2} b^{2} c d + 4 a b^{3} c^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.06785, size = 223, normalized size = 2.72 \begin{align*} \frac{b d^{2} \log \left ({\left | b x + a \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac{d^{3} \log \left ({\left | d x + c \right |}\right )}{b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}} - \frac{b^{2} c^{2} - 4 \, a b c d + 3 \, a^{2} d^{2} - 2 \,{\left (b^{2} c d - a b d^{2}\right )} x}{2 \,{\left (b c - a d\right )}^{3}{\left (b x + a\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]