Optimal. Leaf size=92 \[ -\frac{3 d^2 (b c-a d)}{5 b^4 (a+b x)^5}-\frac{d (b c-a d)^2}{2 b^4 (a+b x)^6}-\frac{(b c-a d)^3}{7 b^4 (a+b x)^7}-\frac{d^3}{4 b^4 (a+b x)^4} \]
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Rubi [A] time = 0.0576906, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {626, 43} \[ -\frac{3 d^2 (b c-a d)}{5 b^4 (a+b x)^5}-\frac{d (b c-a d)^2}{2 b^4 (a+b x)^6}-\frac{(b c-a d)^3}{7 b^4 (a+b x)^7}-\frac{d^3}{4 b^4 (a+b x)^4} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^{11}} \, dx &=\int \frac{(c+d x)^3}{(a+b x)^8} \, dx\\ &=\int \left (\frac{(b c-a d)^3}{b^3 (a+b x)^8}+\frac{3 d (b c-a d)^2}{b^3 (a+b x)^7}+\frac{3 d^2 (b c-a d)}{b^3 (a+b x)^6}+\frac{d^3}{b^3 (a+b x)^5}\right ) \, dx\\ &=-\frac{(b c-a d)^3}{7 b^4 (a+b x)^7}-\frac{d (b c-a d)^2}{2 b^4 (a+b x)^6}-\frac{3 d^2 (b c-a d)}{5 b^4 (a+b x)^5}-\frac{d^3}{4 b^4 (a+b x)^4}\\ \end{align*}
Mathematica [A] time = 0.03055, size = 97, normalized size = 1.05 \[ -\frac{a^2 b d^2 (4 c+7 d x)+a^3 d^3+a b^2 d \left (10 c^2+28 c d x+21 d^2 x^2\right )+b^3 \left (70 c^2 d x+20 c^3+84 c d^2 x^2+35 d^3 x^3\right )}{140 b^4 (a+b x)^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 122, normalized size = 1.3 \begin{align*}{\frac{3\,{d}^{2} \left ( ad-bc \right ) }{5\,{b}^{4} \left ( bx+a \right ) ^{5}}}-{\frac{{d}^{3}}{4\,{b}^{4} \left ( bx+a \right ) ^{4}}}-{\frac{-{a}^{3}{d}^{3}+3\,cb{a}^{2}{d}^{2}-3\,a{c}^{2}d{b}^{2}+{c}^{3}{b}^{3}}{7\,{b}^{4} \left ( bx+a \right ) ^{7}}}-{\frac{d \left ({a}^{2}{d}^{2}-2\,cabd+{b}^{2}{c}^{2} \right ) }{2\,{b}^{4} \left ( bx+a \right ) ^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.11019, size = 246, normalized size = 2.67 \begin{align*} -\frac{35 \, b^{3} d^{3} x^{3} + 20 \, b^{3} c^{3} + 10 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} + a^{3} d^{3} + 21 \,{\left (4 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 7 \,{\left (10 \, b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{140 \,{\left (b^{11} x^{7} + 7 \, a b^{10} x^{6} + 21 \, a^{2} b^{9} x^{5} + 35 \, a^{3} b^{8} x^{4} + 35 \, a^{4} b^{7} x^{3} + 21 \, a^{5} b^{6} x^{2} + 7 \, a^{6} b^{5} x + a^{7} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.52746, size = 382, normalized size = 4.15 \begin{align*} -\frac{35 \, b^{3} d^{3} x^{3} + 20 \, b^{3} c^{3} + 10 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} + a^{3} d^{3} + 21 \,{\left (4 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 7 \,{\left (10 \, b^{3} c^{2} d + 4 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{140 \,{\left (b^{11} x^{7} + 7 \, a b^{10} x^{6} + 21 \, a^{2} b^{9} x^{5} + 35 \, a^{3} b^{8} x^{4} + 35 \, a^{4} b^{7} x^{3} + 21 \, a^{5} b^{6} x^{2} + 7 \, a^{6} b^{5} x + a^{7} b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 13.7032, size = 194, normalized size = 2.11 \begin{align*} - \frac{a^{3} d^{3} + 4 a^{2} b c d^{2} + 10 a b^{2} c^{2} d + 20 b^{3} c^{3} + 35 b^{3} d^{3} x^{3} + x^{2} \left (21 a b^{2} d^{3} + 84 b^{3} c d^{2}\right ) + x \left (7 a^{2} b d^{3} + 28 a b^{2} c d^{2} + 70 b^{3} c^{2} d\right )}{140 a^{7} b^{4} + 980 a^{6} b^{5} x + 2940 a^{5} b^{6} x^{2} + 4900 a^{4} b^{7} x^{3} + 4900 a^{3} b^{8} x^{4} + 2940 a^{2} b^{9} x^{5} + 980 a b^{10} x^{6} + 140 b^{11} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26302, size = 154, normalized size = 1.67 \begin{align*} -\frac{35 \, b^{3} d^{3} x^{3} + 84 \, b^{3} c d^{2} x^{2} + 21 \, a b^{2} d^{3} x^{2} + 70 \, b^{3} c^{2} d x + 28 \, a b^{2} c d^{2} x + 7 \, a^{2} b d^{3} x + 20 \, b^{3} c^{3} + 10 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} + a^{3} d^{3}}{140 \,{\left (b x + a\right )}^{7} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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