Optimal. Leaf size=77 \[ \frac {c^2}{d^2 (b c-a d) (c+d x)}+\frac {a^2 \log (a+b x)}{b (b c-a d)^2}+\frac {c (b c-2 a d) \log (c+d x)}{d^2 (b c-a d)^2} \]
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Rubi [A]
time = 0.04, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {90}
\begin {gather*} \frac {a^2 \log (a+b x)}{b (b c-a d)^2}+\frac {c^2}{d^2 (c+d x) (b c-a d)}+\frac {c (b c-2 a d) \log (c+d x)}{d^2 (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rubi steps
\begin {align*} \int \frac {x^2}{(a+b x) (c+d x)^2} \, dx &=\int \left (\frac {a^2}{(b c-a d)^2 (a+b x)}+\frac {c^2}{d (-b c+a d) (c+d x)^2}+\frac {c (b c-2 a d)}{d (-b c+a d)^2 (c+d x)}\right ) \, dx\\ &=\frac {c^2}{d^2 (b c-a d) (c+d x)}+\frac {a^2 \log (a+b x)}{b (b c-a d)^2}+\frac {c (b c-2 a d) \log (c+d x)}{d^2 (b c-a d)^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 77, normalized size = 1.00 \begin {gather*} \frac {a^2 d^2 (c+d x) \log (a+b x)+b c (c (b c-a d)+(b c-2 a d) (c+d x) \log (c+d x))}{b d^2 (b c-a d)^2 (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 81, normalized size = 1.05
method | result | size |
default | \(\frac {a^{2} \ln \left (b x +a \right )}{\left (a d -b c \right )^{2} b}-\frac {c^{2}}{d^{2} \left (a d -b c \right ) \left (d x +c \right )}-\frac {c \left (2 a d -b c \right ) \ln \left (d x +c \right )}{\left (a d -b c \right )^{2} d^{2}}\) | \(81\) |
norman | \(-\frac {c^{2}}{d^{2} \left (a d -b c \right ) \left (d x +c \right )}+\frac {a^{2} \ln \left (b x +a \right )}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b}-\frac {c \left (2 a d -b c \right ) \ln \left (d x +c \right )}{d^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(107\) |
risch | \(-\frac {c^{2}}{d^{2} \left (a d -b c \right ) \left (d x +c \right )}-\frac {2 c \ln \left (d x +c \right ) a}{d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {c^{2} \ln \left (d x +c \right ) b}{d^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {a^{2} \ln \left (-b x -a \right )}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b}\) | \(139\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 120, normalized size = 1.56 \begin {gather*} \frac {a^{2} \log \left (b x + a\right )}{b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}} + \frac {c^{2}}{b c^{2} d^{2} - a c d^{3} + {\left (b c d^{3} - a d^{4}\right )} x} + \frac {{\left (b c^{2} - 2 \, a c d\right )} \log \left (d x + c\right )}{b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.03, size = 148, normalized size = 1.92 \begin {gather*} \frac {b^{2} c^{3} - a b c^{2} d + {\left (a^{2} d^{3} x + a^{2} c d^{2}\right )} \log \left (b x + a\right ) + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + {\left (b^{2} c^{2} d - 2 \, a b c d^{2}\right )} x\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{2} - 2 \, a b^{2} c^{2} d^{3} + a^{2} b c d^{4} + {\left (b^{3} c^{2} d^{3} - 2 \, a b^{2} c d^{4} + a^{2} b d^{5}\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. 333 vs.
\(2 (66) = 132\).
time = 1.12, size = 333, normalized size = 4.32 \begin {gather*} \frac {a^{2} \log {\left (x + \frac {\frac {a^{5} d^{4}}{b \left (a d - b c\right )^{2}} - \frac {3 a^{4} c d^{3}}{\left (a d - b c\right )^{2}} + \frac {3 a^{3} b c^{2} d^{2}}{\left (a d - b c\right )^{2}} - \frac {a^{2} b^{2} c^{3} d}{\left (a d - b c\right )^{2}} + 3 a^{2} c d - a b c^{2}}{a^{2} d^{2} + 2 a b c d - b^{2} c^{2}} \right )}}{b \left (a d - b c\right )^{2}} - \frac {c^{2}}{a c d^{3} - b c^{2} d^{2} + x \left (a d^{4} - b c d^{3}\right )} - \frac {c \left (2 a d - b c\right ) \log {\left (x + \frac {- \frac {a^{3} c d^{2} \cdot \left (2 a d - b c\right )}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b c^{2} d \left (2 a d - b c\right )}{\left (a d - b c\right )^{2}} + 3 a^{2} c d - \frac {3 a b^{2} c^{3} \cdot \left (2 a d - b c\right )}{\left (a d - b c\right )^{2}} - a b c^{2} + \frac {b^{3} c^{4} \cdot \left (2 a d - b c\right )}{d \left (a d - b c\right )^{2}}}{a^{2} d^{2} + 2 a b c d - b^{2} c^{2}} \right )}}{d^{2} \left (a d - b c\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.83, size = 114, normalized size = 1.48 \begin {gather*} \frac {a^{2} d \log \left ({\left | b - \frac {b c}{d x + c} + \frac {a d}{d x + c} \right |}\right )}{b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}} + \frac {c^{2} d}{{\left (b c d^{3} - a d^{4}\right )} {\left (d x + c\right )}} - \frac {\log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{b d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.22, size = 80, normalized size = 1.04 \begin {gather*} \frac {a^2\,\ln \left (a+b\,x\right )}{b\,{\left (a\,d-b\,c\right )}^2}-\frac {c^2}{d^2\,\left (a\,d-b\,c\right )\,\left (c+d\,x\right )}-\frac {c\,\ln \left (c+d\,x\right )\,\left (2\,a\,d-b\,c\right )}{d^2\,{\left (a\,d-b\,c\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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