Optimal. Leaf size=59 \[ \frac{2 a^2}{3 b c^7 (a-b x)^6}-\frac{4 a}{5 b c^7 (a-b x)^5}+\frac{1}{4 b c^7 (a-b x)^4} \]
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Rubi [A] time = 0.0290148, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {43} \[ \frac{2 a^2}{3 b c^7 (a-b x)^6}-\frac{4 a}{5 b c^7 (a-b x)^5}+\frac{1}{4 b c^7 (a-b x)^4} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{(a c-b c x)^7} \, dx &=\int \left (\frac{4 a^2}{c^7 (a-b x)^7}-\frac{4 a}{c^7 (a-b x)^6}+\frac{1}{c^7 (a-b x)^5}\right ) \, dx\\ &=\frac{2 a^2}{3 b c^7 (a-b x)^6}-\frac{4 a}{5 b c^7 (a-b x)^5}+\frac{1}{4 b c^7 (a-b x)^4}\\ \end{align*}
Mathematica [A] time = 0.0148413, size = 37, normalized size = 0.63 \[ \frac{7 a^2+18 a b x+15 b^2 x^2}{60 b c^7 (a-b x)^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 52, normalized size = 0.9 \begin{align*}{\frac{1}{{c}^{7}} \left ({\frac{1}{4\,b \left ( bx-a \right ) ^{4}}}+{\frac{4\,a}{5\,b \left ( bx-a \right ) ^{5}}}+{\frac{2\,{a}^{2}}{3\,b \left ( bx-a \right ) ^{6}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05498, size = 146, normalized size = 2.47 \begin{align*} \frac{15 \, b^{2} x^{2} + 18 \, a b x + 7 \, a^{2}}{60 \,{\left (b^{7} c^{7} x^{6} - 6 \, a b^{6} c^{7} x^{5} + 15 \, a^{2} b^{5} c^{7} x^{4} - 20 \, a^{3} b^{4} c^{7} x^{3} + 15 \, a^{4} b^{3} c^{7} x^{2} - 6 \, a^{5} b^{2} c^{7} x + a^{6} b c^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45674, size = 220, normalized size = 3.73 \begin{align*} \frac{15 \, b^{2} x^{2} + 18 \, a b x + 7 \, a^{2}}{60 \,{\left (b^{7} c^{7} x^{6} - 6 \, a b^{6} c^{7} x^{5} + 15 \, a^{2} b^{5} c^{7} x^{4} - 20 \, a^{3} b^{4} c^{7} x^{3} + 15 \, a^{4} b^{3} c^{7} x^{2} - 6 \, a^{5} b^{2} c^{7} x + a^{6} b c^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.759735, size = 114, normalized size = 1.93 \begin{align*} \frac{7 a^{2} + 18 a b x + 15 b^{2} x^{2}}{60 a^{6} b c^{7} - 360 a^{5} b^{2} c^{7} x + 900 a^{4} b^{3} c^{7} x^{2} - 1200 a^{3} b^{4} c^{7} x^{3} + 900 a^{2} b^{5} c^{7} x^{4} - 360 a b^{6} c^{7} x^{5} + 60 b^{7} c^{7} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06035, size = 49, normalized size = 0.83 \begin{align*} \frac{15 \, b^{2} x^{2} + 18 \, a b x + 7 \, a^{2}}{60 \,{\left (b x - a\right )}^{6} b c^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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