Optimal. Leaf size=155 \[ \frac{10}{3} a^2 b^3 x^{3/2} (4 a B+3 A b)+10 a^3 b^2 \sqrt{x} (3 a B+4 A b)-\frac{2 a^5 (a B+6 A b)}{3 x^{3/2}}-\frac{6 a^4 b (2 a B+5 A b)}{\sqrt{x}}-\frac{2 a^6 A}{5 x^{5/2}}+\frac{6}{5} a b^4 x^{5/2} (5 a B+2 A b)+\frac{2}{7} b^5 x^{7/2} (6 a B+A b)+\frac{2}{9} b^6 B x^{9/2} \]
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Rubi [A] time = 0.0830776, 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} \[ \frac{10}{3} a^2 b^3 x^{3/2} (4 a B+3 A b)+10 a^3 b^2 \sqrt{x} (3 a B+4 A b)-\frac{2 a^5 (a B+6 A b)}{3 x^{3/2}}-\frac{6 a^4 b (2 a B+5 A b)}{\sqrt{x}}-\frac{2 a^6 A}{5 x^{5/2}}+\frac{6}{5} a b^4 x^{5/2} (5 a B+2 A b)+\frac{2}{7} b^5 x^{7/2} (6 a B+A b)+\frac{2}{9} b^6 B x^{9/2} \]
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^{7/2}} \, dx &=\int \frac{(a+b x)^6 (A+B x)}{x^{7/2}} \, dx\\ &=\int \left (\frac{a^6 A}{x^{7/2}}+\frac{a^5 (6 A b+a B)}{x^{5/2}}+\frac{3 a^4 b (5 A b+2 a B)}{x^{3/2}}+\frac{5 a^3 b^2 (4 A b+3 a B)}{\sqrt{x}}+5 a^2 b^3 (3 A b+4 a B) \sqrt{x}+3 a b^4 (2 A b+5 a B) x^{3/2}+b^5 (A b+6 a B) x^{5/2}+b^6 B x^{7/2}\right ) \, dx\\ &=-\frac{2 a^6 A}{5 x^{5/2}}-\frac{2 a^5 (6 A b+a B)}{3 x^{3/2}}-\frac{6 a^4 b (5 A b+2 a B)}{\sqrt{x}}+10 a^3 b^2 (4 A b+3 a B) \sqrt{x}+\frac{10}{3} a^2 b^3 (3 A b+4 a B) x^{3/2}+\frac{6}{5} a b^4 (2 A b+5 a B) x^{5/2}+\frac{2}{7} b^5 (A b+6 a B) x^{7/2}+\frac{2}{9} b^6 B x^{9/2}\\ \end{align*}
Mathematica [A] time = 0.036669, size = 124, normalized size = 0.8 \[ \frac{2 \left (4725 a^4 b^2 x^2 (B x-A)+2100 a^3 b^3 x^3 (3 A+B x)+315 a^2 b^4 x^4 (5 A+3 B x)-630 a^5 b x (A+3 B x)-21 a^6 (3 A+5 B x)+54 a b^5 x^5 (7 A+5 B x)+5 b^6 x^6 (9 A+7 B x)\right )}{315 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 148, normalized size = 1. \begin{align*} -{\frac{-70\,B{b}^{6}{x}^{7}-90\,A{b}^{6}{x}^{6}-540\,B{x}^{6}a{b}^{5}-756\,aA{b}^{5}{x}^{5}-1890\,B{x}^{5}{a}^{2}{b}^{4}-3150\,{a}^{2}A{b}^{4}{x}^{4}-4200\,B{x}^{4}{a}^{3}{b}^{3}-12600\,{a}^{3}A{b}^{3}{x}^{3}-9450\,B{x}^{3}{a}^{4}{b}^{2}+9450\,{a}^{4}A{b}^{2}{x}^{2}+3780\,B{x}^{2}{a}^{5}b+1260\,{a}^{5}Abx+210\,B{a}^{6}x+126\,A{a}^{6}}{315}{x}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03157, size = 200, normalized size = 1.29 \begin{align*} \frac{2}{9} \, B b^{6} x^{\frac{9}{2}} + \frac{2}{7} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac{7}{2}} + \frac{6}{5} \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{\frac{5}{2}} + \frac{10}{3} \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{\frac{3}{2}} + 10 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} \sqrt{x} - \frac{2 \,{\left (3 \, A a^{6} + 45 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 5 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55768, size = 333, normalized size = 2.15 \begin{align*} \frac{2 \,{\left (35 \, B b^{6} x^{7} - 63 \, A a^{6} + 45 \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 189 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} + 525 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 1575 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 945 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 105 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{315 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.30525, size = 204, normalized size = 1.32 \begin{align*} - \frac{2 A a^{6}}{5 x^{\frac{5}{2}}} - \frac{4 A a^{5} b}{x^{\frac{3}{2}}} - \frac{30 A a^{4} b^{2}}{\sqrt{x}} + 40 A a^{3} b^{3} \sqrt{x} + 10 A a^{2} b^{4} x^{\frac{3}{2}} + \frac{12 A a b^{5} x^{\frac{5}{2}}}{5} + \frac{2 A b^{6} x^{\frac{7}{2}}}{7} - \frac{2 B a^{6}}{3 x^{\frac{3}{2}}} - \frac{12 B a^{5} b}{\sqrt{x}} + 30 B a^{4} b^{2} \sqrt{x} + \frac{40 B a^{3} b^{3} x^{\frac{3}{2}}}{3} + 6 B a^{2} b^{4} x^{\frac{5}{2}} + \frac{12 B a b^{5} x^{\frac{7}{2}}}{7} + \frac{2 B b^{6} x^{\frac{9}{2}}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15211, size = 200, normalized size = 1.29 \begin{align*} \frac{2}{9} \, B b^{6} x^{\frac{9}{2}} + \frac{12}{7} \, B a b^{5} x^{\frac{7}{2}} + \frac{2}{7} \, A b^{6} x^{\frac{7}{2}} + 6 \, B a^{2} b^{4} x^{\frac{5}{2}} + \frac{12}{5} \, A a b^{5} x^{\frac{5}{2}} + \frac{40}{3} \, B a^{3} b^{3} x^{\frac{3}{2}} + 10 \, A a^{2} b^{4} x^{\frac{3}{2}} + 30 \, B a^{4} b^{2} \sqrt{x} + 40 \, A a^{3} b^{3} \sqrt{x} - \frac{2 \,{\left (90 \, B a^{5} b x^{2} + 225 \, A a^{4} b^{2} x^{2} + 5 \, B a^{6} x + 30 \, A a^{5} b x + 3 \, A a^{6}\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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