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 = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.44, size = 108, normalized size = 1.83 \[ \frac {15 \, b^{2} x^{2} + 18 \, a b x + 7 \, a^{2}}{60 \, {\left (b^{7} c^{7} x^{6} - 6 \, a b^{6} c^{7} x^{5} + 15 \, a^{2} b^{5} c^{7} x^{4} - 20 \, a^{3} b^{4} c^{7} x^{3} + 15 \, a^{4} b^{3} c^{7} x^{2} - 6 \, a^{5} b^{2} c^{7} x + a^{6} b c^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.93, size = 36, normalized size = 0.61 \[ \frac {15 \, b^{2} x^{2} + 18 \, a b x + 7 \, a^{2}}{60 \, {\left (b x - a\right )}^{6} b c^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 52, normalized size = 0.88 \[ \frac {\frac {2 a^{2}}{3 \left (b x -a \right )^{6} b}+\frac {4 a}{5 \left (b x -a \right )^{5} b}+\frac {1}{4 \left (b x -a \right )^{4} b}}{c^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 108, normalized size = 1.83 \[ \frac {15 \, b^{2} x^{2} + 18 \, a b x + 7 \, a^{2}}{60 \, {\left (b^{7} c^{7} x^{6} - 6 \, a b^{6} c^{7} x^{5} + 15 \, a^{2} b^{5} c^{7} x^{4} - 20 \, a^{3} b^{4} c^{7} x^{3} + 15 \, a^{4} b^{3} c^{7} x^{2} - 6 \, a^{5} b^{2} c^{7} x + a^{6} b c^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 104, normalized size = 1.76 \[ \frac {\frac {3\,a\,x}{10}+\frac {b\,x^2}{4}+\frac {7\,a^2}{60\,b}}{a^6\,c^7-6\,a^5\,b\,c^7\,x+15\,a^4\,b^2\,c^7\,x^2-20\,a^3\,b^3\,c^7\,x^3+15\,a^2\,b^4\,c^7\,x^4-6\,a\,b^5\,c^7\,x^5+b^6\,c^7\,x^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.60, size = 117, normalized size = 1.98 \[ - \frac {- 7 a^{2} - 18 a b x - 15 b^{2} x^{2}}{60 a^{6} b c^{7} - 360 a^{5} b^{2} c^{7} x + 900 a^{4} b^{3} c^{7} x^{2} - 1200 a^{3} b^{4} c^{7} x^{3} + 900 a^{2} b^{5} c^{7} x^{4} - 360 a b^{6} c^{7} x^{5} + 60 b^{7} c^{7} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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