Optimal. Leaf size=122 \[ \frac{2 a^2 (a+b x)^{5/2} (3 A b-4 a B)}{5 b^5}-\frac{2 a^3 (a+b x)^{3/2} (A b-a B)}{3 b^5}+\frac{2 (a+b x)^{9/2} (A b-4 a B)}{9 b^5}-\frac{6 a (a+b x)^{7/2} (A b-2 a B)}{7 b^5}+\frac{2 B (a+b x)^{11/2}}{11 b^5} \]
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Rubi [A] time = 0.0566592, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {77} \[ \frac{2 a^2 (a+b x)^{5/2} (3 A b-4 a B)}{5 b^5}-\frac{2 a^3 (a+b x)^{3/2} (A b-a B)}{3 b^5}+\frac{2 (a+b x)^{9/2} (A b-4 a B)}{9 b^5}-\frac{6 a (a+b x)^{7/2} (A b-2 a B)}{7 b^5}+\frac{2 B (a+b x)^{11/2}}{11 b^5} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int x^3 \sqrt{a+b x} (A+B x) \, dx &=\int \left (\frac{a^3 (-A b+a B) \sqrt{a+b x}}{b^4}-\frac{a^2 (-3 A b+4 a B) (a+b x)^{3/2}}{b^4}+\frac{3 a (-A b+2 a B) (a+b x)^{5/2}}{b^4}+\frac{(A b-4 a B) (a+b x)^{7/2}}{b^4}+\frac{B (a+b x)^{9/2}}{b^4}\right ) \, dx\\ &=-\frac{2 a^3 (A b-a B) (a+b x)^{3/2}}{3 b^5}+\frac{2 a^2 (3 A b-4 a B) (a+b x)^{5/2}}{5 b^5}-\frac{6 a (A b-2 a B) (a+b x)^{7/2}}{7 b^5}+\frac{2 (A b-4 a B) (a+b x)^{9/2}}{9 b^5}+\frac{2 B (a+b x)^{11/2}}{11 b^5}\\ \end{align*}
Mathematica [A] time = 0.0631749, size = 87, normalized size = 0.71 \[ \frac{2 (a+b x)^{3/2} \left (24 a^2 b^2 x (11 A+10 B x)-16 a^3 b (11 A+12 B x)+128 a^4 B-10 a b^3 x^2 (33 A+28 B x)+35 b^4 x^3 (11 A+9 B x)\right )}{3465 b^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 95, normalized size = 0.8 \begin{align*} -{\frac{-630\,B{x}^{4}{b}^{4}-770\,A{b}^{4}{x}^{3}+560\,Ba{b}^{3}{x}^{3}+660\,Aa{b}^{3}{x}^{2}-480\,B{a}^{2}{b}^{2}{x}^{2}-528\,A{a}^{2}{b}^{2}x+384\,B{a}^{3}bx+352\,A{a}^{3}b-256\,B{a}^{4}}{3465\,{b}^{5}} \left ( bx+a \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06399, size = 135, normalized size = 1.11 \begin{align*} \frac{2 \,{\left (315 \,{\left (b x + a\right )}^{\frac{11}{2}} B - 385 \,{\left (4 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{9}{2}} + 1485 \,{\left (2 \, B a^{2} - A a b\right )}{\left (b x + a\right )}^{\frac{7}{2}} - 693 \,{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )}{\left (b x + a\right )}^{\frac{5}{2}} + 1155 \,{\left (B a^{4} - A a^{3} b\right )}{\left (b x + a\right )}^{\frac{3}{2}}\right )}}{3465 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.28348, size = 274, normalized size = 2.25 \begin{align*} \frac{2 \,{\left (315 \, B b^{5} x^{5} + 128 \, B a^{5} - 176 \, A a^{4} b + 35 \,{\left (B a b^{4} + 11 \, A b^{5}\right )} x^{4} - 5 \,{\left (8 \, B a^{2} b^{3} - 11 \, A a b^{4}\right )} x^{3} + 6 \,{\left (8 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{2} - 8 \,{\left (8 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x\right )} \sqrt{b x + a}}{3465 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.1667, size = 121, normalized size = 0.99 \begin{align*} \frac{2 \left (\frac{B \left (a + b x\right )^{\frac{11}{2}}}{11 b} + \frac{\left (a + b x\right )^{\frac{9}{2}} \left (A b - 4 B a\right )}{9 b} + \frac{\left (a + b x\right )^{\frac{7}{2}} \left (- 3 A a b + 6 B a^{2}\right )}{7 b} + \frac{\left (a + b x\right )^{\frac{5}{2}} \left (3 A a^{2} b - 4 B a^{3}\right )}{5 b} + \frac{\left (a + b x\right )^{\frac{3}{2}} \left (- A a^{3} b + B a^{4}\right )}{3 b}\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17867, size = 158, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (\frac{11 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3}\right )} A}{b^{3}} + \frac{{\left (315 \,{\left (b x + a\right )}^{\frac{11}{2}} - 1540 \,{\left (b x + a\right )}^{\frac{9}{2}} a + 2970 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{3} + 1155 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4}\right )} B}{b^{4}}\right )}}{3465 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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