Optimal. Leaf size=45 \[ -\frac {a (b c-a d) \log (a+b x)}{b^3}+\frac {x (b c-a d)}{b^2}+\frac {d x^2}{2 b} \]
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Rubi [A] time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {77} \[ \frac {x (b c-a d)}{b^2}-\frac {a (b c-a d) \log (a+b x)}{b^3}+\frac {d x^2}{2 b} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int \frac {x (c+d x)}{a+b x} \, dx &=\int \left (\frac {b c-a d}{b^2}+\frac {d x}{b}+\frac {a (-b c+a d)}{b^2 (a+b x)}\right ) \, dx\\ &=\frac {(b c-a d) x}{b^2}+\frac {d x^2}{2 b}-\frac {a (b c-a d) \log (a+b x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 41, normalized size = 0.91 \[ \frac {b x (-2 a d+2 b c+b d x)+2 a (a d-b c) \log (a+b x)}{2 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 47, normalized size = 1.04 \[ \frac {b^{2} d x^{2} + 2 \, {\left (b^{2} c - a b d\right )} x - 2 \, {\left (a b c - a^{2} d\right )} \log \left (b x + a\right )}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.97, size = 46, normalized size = 1.02 \[ \frac {b d x^{2} + 2 \, b c x - 2 \, a d x}{2 \, b^{2}} - \frac {{\left (a b c - a^{2} d\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 52, normalized size = 1.16 \[ \frac {d \,x^{2}}{2 b}+\frac {a^{2} d \ln \left (b x +a \right )}{b^{3}}-\frac {a c \ln \left (b x +a \right )}{b^{2}}-\frac {a d x}{b^{2}}+\frac {c x}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.94, size = 46, normalized size = 1.02 \[ \frac {b d x^{2} + 2 \, {\left (b c - a d\right )} x}{2 \, b^{2}} - \frac {{\left (a b c - a^{2} d\right )} \log \left (b x + a\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 46, normalized size = 1.02 \[ x\,\left (\frac {c}{b}-\frac {a\,d}{b^2}\right )+\frac {d\,x^2}{2\,b}+\frac {\ln \left (a+b\,x\right )\,\left (a^2\,d-a\,b\,c\right )}{b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 37, normalized size = 0.82 \[ \frac {a \left (a d - b c\right ) \log {\left (a + b x \right )}}{b^{3}} + x \left (- \frac {a d}{b^{2}} + \frac {c}{b}\right ) + \frac {d x^{2}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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