Optimal. Leaf size=70 \[ 49 a^3 x+\frac {23}{2} a^2 b x^2+\frac {7}{3} a b^2 x^3+\frac {b^3 x^4}{4}+\frac {32 a^5}{b (a-b x)}+\frac {80 a^4 \log (a-b x)}{b} \]
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Rubi [A]
time = 0.04, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {641, 45}
\begin {gather*} \frac {32 a^5}{b (a-b x)}+\frac {80 a^4 \log (a-b x)}{b}+49 a^3 x+\frac {23}{2} a^2 b x^2+\frac {7}{3} a b^2 x^3+\frac {b^3 x^4}{4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 641
Rubi steps
\begin {align*} \int \frac {(a+b x)^7}{\left (a^2-b^2 x^2\right )^2} \, dx &=\int \frac {(a+b x)^5}{(a-b x)^2} \, dx\\ &=\int \left (49 a^3+23 a^2 b x+7 a b^2 x^2+b^3 x^3+\frac {32 a^5}{(a-b x)^2}-\frac {80 a^4}{a-b x}\right ) \, dx\\ &=49 a^3 x+\frac {23}{2} a^2 b x^2+\frac {7}{3} a b^2 x^3+\frac {b^3 x^4}{4}+\frac {32 a^5}{b (a-b x)}+\frac {80 a^4 \log (a-b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 71, normalized size = 1.01 \begin {gather*} 49 a^3 x+\frac {23}{2} a^2 b x^2+\frac {7}{3} a b^2 x^3+\frac {b^3 x^4}{4}-\frac {32 a^5}{b (-a+b x)}+\frac {80 a^4 \log (a-b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.45, size = 65, normalized size = 0.93
method | result | size |
default | \(49 a^{3} x +\frac {23 a^{2} b \,x^{2}}{2}+\frac {7 a \,b^{2} x^{3}}{3}+\frac {b^{3} x^{4}}{4}+\frac {32 a^{5}}{b \left (-b x +a \right )}+\frac {80 a^{4} \ln \left (-b x +a \right )}{b}\) | \(65\) |
risch | \(49 a^{3} x +\frac {23 a^{2} b \,x^{2}}{2}+\frac {7 a \,b^{2} x^{3}}{3}+\frac {b^{3} x^{4}}{4}+\frac {32 a^{5}}{b \left (-b x +a \right )}+\frac {80 a^{4} \ln \left (-b x +a \right )}{b}\) | \(65\) |
norman | \(\frac {81 a^{5} x -\frac {b^{5} x^{6}}{4}-\frac {7 a \,b^{4} x^{5}}{3}-\frac {45 a^{2} b^{3} x^{4}}{4}+\frac {87 a^{6}}{2 b}-\frac {140 a^{3} b^{2} x^{3}}{3}}{-b^{2} x^{2}+a^{2}}+\frac {80 a^{4} \ln \left (-b x +a \right )}{b}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 66, normalized size = 0.94 \begin {gather*} \frac {1}{4} \, b^{3} x^{4} + \frac {7}{3} \, a b^{2} x^{3} + \frac {23}{2} \, a^{2} b x^{2} - \frac {32 \, a^{5}}{b^{2} x - a b} + 49 \, a^{3} x + \frac {80 \, a^{4} \log \left (b x - a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.44, size = 88, normalized size = 1.26 \begin {gather*} \frac {3 \, b^{5} x^{5} + 25 \, a b^{4} x^{4} + 110 \, a^{2} b^{3} x^{3} + 450 \, a^{3} b^{2} x^{2} - 588 \, a^{4} b x - 384 \, a^{5} + 960 \, {\left (a^{4} b x - a^{5}\right )} \log \left (b x - a\right )}{12 \, {\left (b^{2} x - a b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 65, normalized size = 0.93 \begin {gather*} - \frac {32 a^{5}}{- a b + b^{2} x} + \frac {80 a^{4} \log {\left (- a + b x \right )}}{b} + 49 a^{3} x + \frac {23 a^{2} b x^{2}}{2} + \frac {7 a b^{2} x^{3}}{3} + \frac {b^{3} x^{4}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.13, size = 78, normalized size = 1.11 \begin {gather*} \frac {80 \, a^{4} \log \left ({\left | b x - a \right |}\right )}{b} - \frac {32 \, a^{5}}{{\left (b x - a\right )} b} + \frac {3 \, b^{11} x^{4} + 28 \, a b^{10} x^{3} + 138 \, a^{2} b^{9} x^{2} + 588 \, a^{3} b^{8} x}{12 \, b^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 64, normalized size = 0.91 \begin {gather*} 49\,a^3\,x+\frac {b^3\,x^4}{4}+\frac {32\,a^5}{b\,\left (a-b\,x\right )}+\frac {80\,a^4\,\ln \left (a-b\,x\right )}{b}+\frac {23\,a^2\,b\,x^2}{2}+\frac {7\,a\,b^2\,x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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