Optimal. Leaf size=81 \[ -\frac {b}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d)^2 (c+d x)}-\frac {2 b d \log (a+b x)}{(b c-a d)^3}+\frac {2 b d \log (c+d x)}{(b c-a d)^3} \]
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Rubi [A]
time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, 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 = {46}
\begin {gather*} -\frac {b}{(a+b x) (b c-a d)^2}-\frac {d}{(c+d x) (b c-a d)^2}-\frac {2 b d \log (a+b x)}{(b c-a d)^3}+\frac {2 b d \log (c+d x)}{(b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rubi steps
\begin {align*} \int \frac {1}{(a+b x)^2 (c+d x)^2} \, dx &=\int \left (\frac {b^2}{(b c-a d)^2 (a+b x)^2}-\frac {2 b^2 d}{(b c-a d)^3 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)^2}+\frac {2 b d^2}{(b c-a d)^3 (c+d x)}\right ) \, dx\\ &=-\frac {b}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d)^2 (c+d x)}-\frac {2 b d \log (a+b x)}{(b c-a d)^3}+\frac {2 b d \log (c+d x)}{(b c-a d)^3}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 66, normalized size = 0.81 \begin {gather*} \frac {\frac {b (-b c+a d)}{a+b x}+\frac {d (-b c+a d)}{c+d x}-2 b d \log (a+b x)+2 b d \log (c+d x)}{(b c-a d)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 82, normalized size = 1.01
method | result | size |
default | \(-\frac {d}{\left (a d -b c \right )^{2} \left (d x +c \right )}-\frac {2 d b \ln \left (d x +c \right )}{\left (a d -b c \right )^{3}}-\frac {b}{\left (a d -b c \right )^{2} \left (b x +a \right )}+\frac {2 d b \ln \left (b x +a \right )}{\left (a d -b c \right )^{3}}\) | \(82\) |
risch | \(\frac {-\frac {2 b d x}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}-\frac {a d +b c}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}}{\left (b x +a \right ) \left (d x +c \right )}-\frac {2 b d \ln \left (d x +c \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}+\frac {2 b d \ln \left (-b x -a \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}\) | \(177\) |
norman | \(\frac {\frac {-a b \,d^{2}-b^{2} c d}{d b \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {2 b d x}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}}{\left (b x +a \right ) \left (d x +c \right )}+\frac {2 b d \ln \left (b x +a \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}-\frac {2 b d \ln \left (d x +c \right )}{a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}\) | \(187\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 208 vs.
\(2 (81) = 162\).
time = 0.29, size = 208, normalized size = 2.57 \begin {gather*} -\frac {2 \, b d \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 {2 \, b d \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 + a d}{a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 241 vs.
\(2 (81) = 162\).
time = 0.96, size = 241, normalized size = 2.98 \begin {gather*} -\frac {b^{2} c^{2} - a^{2} d^{2} + 2 \, {\left (b^{2} c d - a b d^{2}\right )} x + 2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right ) - 2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (d x + c\right )}{a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} + {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{2} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 406 vs.
\(2 (70) = 140\).
time = 0.60, size = 406, normalized size = 5.01 \begin {gather*} - \frac {2 b d \log {\left (x + \frac {- \frac {2 a^{4} b d^{5}}{\left (a d - b c\right )^{3}} + \frac {8 a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} - \frac {12 a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} + \frac {8 a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + 2 a b d^{2} - \frac {2 b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + 2 b^{2} c d}{4 b^{2} d^{2}} \right )}}{\left (a d - b c\right )^{3}} + \frac {2 b d \log {\left (x + \frac {\frac {2 a^{4} b d^{5}}{\left (a d - b c\right )^{3}} - \frac {8 a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} + \frac {12 a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} - \frac {8 a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + 2 a b d^{2} + \frac {2 b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + 2 b^{2} c d}{4 b^{2} d^{2}} \right )}}{\left (a d - b c\right )^{3}} + \frac {- a d - b c - 2 b d x}{a^{3} c d^{2} - 2 a^{2} b c^{2} d + a b^{2} c^{3} + x^{2} \left (a^{2} b d^{3} - 2 a b^{2} c d^{2} + b^{3} c^{2} d\right ) + x \left (a^{3} d^{3} - a^{2} b c d^{2} - a b^{2} c^{2} d + b^{3} c^{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.11, size = 153, normalized size = 1.89 \begin {gather*} \frac {2 \, b^{2} d \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \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 {b^{3}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} {\left (b x + a\right )}} + \frac {b d^{2}}{{\left (b c - a d\right )}^{3} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.33, size = 74, normalized size = 0.91 \begin {gather*} \frac {1}{\left (a\,d-b\,c\right )\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {2\,d}{{\left (a\,d-b\,c\right )}^2\,\left (c+d\,x\right )}-\frac {2\,b\,d\,\ln \left (\frac {c+d\,x}{a+b\,x}\right )}{{\left (a\,d-b\,c\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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