Optimal. Leaf size=155 \[ -\frac{10 a^3 b^2 (3 a B+4 A b)}{3 x^{3/2}}-\frac{10 a^2 b^3 (4 a B+3 A b)}{\sqrt{x}}-\frac{2 a^5 (a B+6 A b)}{7 x^{7/2}}-\frac{6 a^4 b (2 a B+5 A b)}{5 x^{5/2}}-\frac{2 a^6 A}{9 x^{9/2}}+\frac{2}{3} b^5 x^{3/2} (6 a B+A b)+6 a b^4 \sqrt{x} (5 a B+2 A b)+\frac{2}{5} b^6 B x^{5/2} \]
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Rubi [A] time = 0.080033, antiderivative size = 155, 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 = {27, 76} \[ -\frac{10 a^3 b^2 (3 a B+4 A b)}{3 x^{3/2}}-\frac{10 a^2 b^3 (4 a B+3 A b)}{\sqrt{x}}-\frac{2 a^5 (a B+6 A b)}{7 x^{7/2}}-\frac{6 a^4 b (2 a B+5 A b)}{5 x^{5/2}}-\frac{2 a^6 A}{9 x^{9/2}}+\frac{2}{3} b^5 x^{3/2} (6 a B+A b)+6 a b^4 \sqrt{x} (5 a B+2 A b)+\frac{2}{5} b^6 B x^{5/2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{11/2}} \, dx &=\int \frac{(a+b x)^6 (A+B x)}{x^{11/2}} \, dx\\ &=\int \left (\frac{a^6 A}{x^{11/2}}+\frac{a^5 (6 A b+a B)}{x^{9/2}}+\frac{3 a^4 b (5 A b+2 a B)}{x^{7/2}}+\frac{5 a^3 b^2 (4 A b+3 a B)}{x^{5/2}}+\frac{5 a^2 b^3 (3 A b+4 a B)}{x^{3/2}}+\frac{3 a b^4 (2 A b+5 a B)}{\sqrt{x}}+b^5 (A b+6 a B) \sqrt{x}+b^6 B x^{3/2}\right ) \, dx\\ &=-\frac{2 a^6 A}{9 x^{9/2}}-\frac{2 a^5 (6 A b+a B)}{7 x^{7/2}}-\frac{6 a^4 b (5 A b+2 a B)}{5 x^{5/2}}-\frac{10 a^3 b^2 (4 A b+3 a B)}{3 x^{3/2}}-\frac{10 a^2 b^3 (3 A b+4 a B)}{\sqrt{x}}+6 a b^4 (2 A b+5 a B) \sqrt{x}+\frac{2}{3} b^5 (A b+6 a B) x^{3/2}+\frac{2}{5} b^6 B x^{5/2}\\ \end{align*}
Mathematica [A] time = 0.0398844, size = 123, normalized size = 0.79 \[ -\frac{2 \left (315 a^4 b^2 x^2 (3 A+5 B x)+2100 a^3 b^3 x^3 (A+3 B x)+4725 a^2 b^4 x^4 (A-B x)+54 a^5 b x (5 A+7 B x)+5 a^6 (7 A+9 B x)-630 a b^5 x^5 (3 A+B x)-21 b^6 x^6 (5 A+3 B x)\right )}{315 x^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 148, normalized size = 1. \begin{align*} -{\frac{-126\,B{b}^{6}{x}^{7}-210\,A{b}^{6}{x}^{6}-1260\,B{x}^{6}a{b}^{5}-3780\,aA{b}^{5}{x}^{5}-9450\,B{x}^{5}{a}^{2}{b}^{4}+9450\,{a}^{2}A{b}^{4}{x}^{4}+12600\,B{x}^{4}{a}^{3}{b}^{3}+4200\,{a}^{3}A{b}^{3}{x}^{3}+3150\,B{x}^{3}{a}^{4}{b}^{2}+1890\,{a}^{4}A{b}^{2}{x}^{2}+756\,B{x}^{2}{a}^{5}b+540\,{a}^{5}Abx+90\,B{a}^{6}x+70\,A{a}^{6}}{315}{x}^{-{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03024, size = 200, normalized size = 1.29 \begin{align*} \frac{2}{5} \, B b^{6} x^{\frac{5}{2}} + \frac{2}{3} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac{3}{2}} + 6 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} \sqrt{x} - \frac{2 \,{\left (35 \, A a^{6} + 1575 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 525 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 189 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 45 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{315 \, x^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61765, size = 333, normalized size = 2.15 \begin{align*} \frac{2 \,{\left (63 \, B b^{6} x^{7} - 35 \, A a^{6} + 105 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 945 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} - 1575 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} - 525 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 189 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 45 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{315 \, x^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 13.1712, size = 204, normalized size = 1.32 \begin{align*} - \frac{2 A a^{6}}{9 x^{\frac{9}{2}}} - \frac{12 A a^{5} b}{7 x^{\frac{7}{2}}} - \frac{6 A a^{4} b^{2}}{x^{\frac{5}{2}}} - \frac{40 A a^{3} b^{3}}{3 x^{\frac{3}{2}}} - \frac{30 A a^{2} b^{4}}{\sqrt{x}} + 12 A a b^{5} \sqrt{x} + \frac{2 A b^{6} x^{\frac{3}{2}}}{3} - \frac{2 B a^{6}}{7 x^{\frac{7}{2}}} - \frac{12 B a^{5} b}{5 x^{\frac{5}{2}}} - \frac{10 B a^{4} b^{2}}{x^{\frac{3}{2}}} - \frac{40 B a^{3} b^{3}}{\sqrt{x}} + 30 B a^{2} b^{4} \sqrt{x} + 4 B a b^{5} x^{\frac{3}{2}} + \frac{2 B b^{6} x^{\frac{5}{2}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16344, size = 200, normalized size = 1.29 \begin{align*} \frac{2}{5} \, B b^{6} x^{\frac{5}{2}} + 4 \, B a b^{5} x^{\frac{3}{2}} + \frac{2}{3} \, A b^{6} x^{\frac{3}{2}} + 30 \, B a^{2} b^{4} \sqrt{x} + 12 \, A a b^{5} \sqrt{x} - \frac{2 \,{\left (6300 \, B a^{3} b^{3} x^{4} + 4725 \, A a^{2} b^{4} x^{4} + 1575 \, B a^{4} b^{2} x^{3} + 2100 \, A a^{3} b^{3} x^{3} + 378 \, B a^{5} b x^{2} + 945 \, A a^{4} b^{2} x^{2} + 45 \, B a^{6} x + 270 \, A a^{5} b x + 35 \, A a^{6}\right )}}{315 \, x^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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