Optimal. Leaf size=58 \[ -\left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right ) \log (a+b x)+\frac {c^2 \log (x)}{a^2}+\frac {(b c-a d)^2}{a b^2 (a+b x)} \]
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Rubi [A] time = 0.04, antiderivative size = 58, 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 = {88} \[ -\left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right ) \log (a+b x)+\frac {c^2 \log (x)}{a^2}+\frac {(b c-a d)^2}{a b^2 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 88
Rubi steps
\begin {align*} \int \frac {(c+d x)^2}{x (a+b x)^2} \, dx &=\int \left (\frac {c^2}{a^2 x}-\frac {(-b c+a d)^2}{a b (a+b x)^2}+\frac {-b^2 c^2+a^2 d^2}{a^2 b (a+b x)}\right ) \, dx\\ &=\frac {(b c-a d)^2}{a b^2 (a+b x)}+\frac {c^2 \log (x)}{a^2}-\left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right ) \log (a+b x)\\ \end {align*}
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Mathematica [A] time = 0.06, size = 60, normalized size = 1.03 \[ \frac {\frac {(a d-b c) ((a+b x) (a d+b c) \log (a+b x)+a (a d-b c))}{b^2 (a+b x)}+c^2 \log (x)}{a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 107, normalized size = 1.84 \[ \frac {a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} - {\left (a b^{2} c^{2} - a^{3} d^{2} + {\left (b^{3} c^{2} - a^{2} b d^{2}\right )} x\right )} \log \left (b x + a\right ) + {\left (b^{3} c^{2} x + a b^{2} c^{2}\right )} \log \relax (x)}{a^{2} b^{3} x + a^{3} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 108, normalized size = 1.86 \[ -b {\left (\frac {d^{2} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{3}} - \frac {c^{2} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{2} b} - \frac {\frac {b^{3} c^{2}}{b x + a} - \frac {2 \, a b^{2} c d}{b x + a} + \frac {a^{2} b d^{2}}{b x + a}}{a b^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 81, normalized size = 1.40 \[ \frac {a \,d^{2}}{\left (b x +a \right ) b^{2}}+\frac {c^{2}}{\left (b x +a \right ) a}+\frac {c^{2} \ln \relax (x )}{a^{2}}-\frac {c^{2} \ln \left (b x +a \right )}{a^{2}}-\frac {2 c d}{\left (b x +a \right ) b}+\frac {d^{2} \ln \left (b x +a \right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 78, normalized size = 1.34 \[ \frac {c^{2} \log \relax (x)}{a^{2}} + \frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{a b^{3} x + a^{2} b^{2}} - \frac {{\left (b^{2} c^{2} - a^{2} d^{2}\right )} \log \left (b x + a\right )}{a^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 69, normalized size = 1.19 \[ \frac {c^2\,\ln \relax (x)}{a^2}-\ln \left (a+b\,x\right )\,\left (\frac {c^2}{a^2}-\frac {d^2}{b^2}\right )+\frac {a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}{a\,b^2\,\left (a+b\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.88, size = 107, normalized size = 1.84 \[ \frac {a^{2} d^{2} - 2 a b c d + b^{2} c^{2}}{a^{2} b^{2} + a b^{3} x} + \frac {c^{2} \log {\relax (x )}}{a^{2}} + \frac {\left (a d - b c\right ) \left (a d + b c\right ) \log {\left (x + \frac {- a b c^{2} + \frac {a \left (a d - b c\right ) \left (a d + b c\right )}{b}}{a^{2} d^{2} - 2 b^{2} c^{2}} \right )}}{a^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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