Optimal. Leaf size=38 \[ -\frac{b c-a d}{3 b^2 (a+b x)^3}-\frac{d}{2 b^2 (a+b x)^2} \]
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Rubi [A] time = 0.0242644, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {24, 43} \[ -\frac{b c-a d}{3 b^2 (a+b x)^3}-\frac{d}{2 b^2 (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 24
Rule 43
Rubi steps
\begin{align*} \int \frac{a c+(b c+a d) x+b d x^2}{(a+b x)^5} \, dx &=\frac{\int \frac{b^2 c+b^2 d x}{(a+b x)^4} \, dx}{b^2}\\ &=\frac{\int \left (\frac{b (b c-a d)}{(a+b x)^4}+\frac{b d}{(a+b x)^3}\right ) \, dx}{b^2}\\ &=-\frac{b c-a d}{3 b^2 (a+b x)^3}-\frac{d}{2 b^2 (a+b x)^2}\\ \end{align*}
Mathematica [A] time = 0.0087894, size = 27, normalized size = 0.71 \[ -\frac{a d+2 b c+3 b d x}{6 b^2 (a+b x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 35, normalized size = 0.9 \begin{align*} -{\frac{d}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{-ad+bc}{3\,{b}^{2} \left ( bx+a \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08868, size = 68, normalized size = 1.79 \begin{align*} -\frac{3 \, b d x + 2 \, b c + a d}{6 \,{\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78756, size = 105, normalized size = 2.76 \begin{align*} -\frac{3 \, b d x + 2 \, b c + a d}{6 \,{\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.57405, size = 53, normalized size = 1.39 \begin{align*} - \frac{a d + 2 b c + 3 b d x}{6 a^{3} b^{2} + 18 a^{2} b^{3} x + 18 a b^{4} x^{2} + 6 b^{5} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14854, size = 55, normalized size = 1.45 \begin{align*} -\frac{c}{3 \,{\left (b x + a\right )}^{3} b} - \frac{d}{2 \,{\left (b x + a\right )}^{2} b^{2}} + \frac{a d}{3 \,{\left (b x + a\right )}^{3} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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