Optimal. Leaf size=52 \[ \frac{x^4}{60 a^3 (a+b x)^4}+\frac{x^4}{15 a^2 (a+b x)^5}+\frac{x^4}{6 a (a+b x)^6} \]
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Rubi [A] time = 0.0293148, antiderivative size = 64, normalized size of antiderivative = 1.23, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{a^3}{6 b^4 (a+b x)^6}-\frac{3 a^2}{5 b^4 (a+b x)^5}+\frac{3 a}{4 b^4 (a+b x)^4}-\frac{1}{3 b^4 (a+b x)^3} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{(a+b x)^7} \, dx &=\int \left (-\frac{a^3}{b^3 (a+b x)^7}+\frac{3 a^2}{b^3 (a+b x)^6}-\frac{3 a}{b^3 (a+b x)^5}+\frac{1}{b^3 (a+b x)^4}\right ) \, dx\\ &=\frac{a^3}{6 b^4 (a+b x)^6}-\frac{3 a^2}{5 b^4 (a+b x)^5}+\frac{3 a}{4 b^4 (a+b x)^4}-\frac{1}{3 b^4 (a+b x)^3}\\ \end{align*}
Mathematica [A] time = 0.0182702, size = 42, normalized size = 0.81 \[ -\frac{6 a^2 b x+a^3+15 a b^2 x^2+20 b^3 x^3}{60 b^4 (a+b x)^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 57, normalized size = 1.1 \begin{align*}{\frac{{a}^{3}}{6\,{b}^{4} \left ( bx+a \right ) ^{6}}}-{\frac{3\,{a}^{2}}{5\,{b}^{4} \left ( bx+a \right ) ^{5}}}+{\frac{3\,a}{4\,{b}^{4} \left ( bx+a \right ) ^{4}}}-{\frac{1}{3\,{b}^{4} \left ( bx+a \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.04443, size = 132, normalized size = 2.54 \begin{align*} -\frac{20 \, b^{3} x^{3} + 15 \, a b^{2} x^{2} + 6 \, a^{2} b x + a^{3}}{60 \,{\left (b^{10} x^{6} + 6 \, a b^{9} x^{5} + 15 \, a^{2} b^{8} x^{4} + 20 \, a^{3} b^{7} x^{3} + 15 \, a^{4} b^{6} x^{2} + 6 \, a^{5} b^{5} x + a^{6} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.47325, size = 207, normalized size = 3.98 \begin{align*} -\frac{20 \, b^{3} x^{3} + 15 \, a b^{2} x^{2} + 6 \, a^{2} b x + a^{3}}{60 \,{\left (b^{10} x^{6} + 6 \, a b^{9} x^{5} + 15 \, a^{2} b^{8} x^{4} + 20 \, a^{3} b^{7} x^{3} + 15 \, a^{4} b^{6} x^{2} + 6 \, a^{5} b^{5} x + a^{6} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.873424, size = 104, normalized size = 2. \begin{align*} - \frac{a^{3} + 6 a^{2} b x + 15 a b^{2} x^{2} + 20 b^{3} x^{3}}{60 a^{6} b^{4} + 360 a^{5} b^{5} x + 900 a^{4} b^{6} x^{2} + 1200 a^{3} b^{7} x^{3} + 900 a^{2} b^{8} x^{4} + 360 a b^{9} x^{5} + 60 b^{10} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21901, size = 54, normalized size = 1.04 \begin{align*} -\frac{20 \, b^{3} x^{3} + 15 \, a b^{2} x^{2} + 6 \, a^{2} b x + a^{3}}{60 \,{\left (b x + a\right )}^{6} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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