Optimal. Leaf size=122 \[ -\frac {2 a^3 (a+b x)^{3/2} (A b-a B)}{3 b^5}+\frac {2 a^2 (a+b x)^{5/2} (3 A b-4 a B)}{5 b^5}+\frac {2 (a+b x)^{9/2} (A b-4 a B)}{9 b^5}-\frac {6 a (a+b x)^{7/2} (A b-2 a B)}{7 b^5}+\frac {2 B (a+b x)^{11/2}}{11 b^5} \]
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Rubi [A] time = 0.06, antiderivative size = 122, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.09, size = 87, normalized size = 0.71 \[ \frac {2 (a+b x)^{3/2} \left (128 a^4 B-16 a^3 b (11 A+12 B x)+24 a^2 b^2 x (11 A+10 B x)-10 a b^3 x^2 (33 A+28 B x)+35 b^4 x^3 (11 A+9 B x)\right )}{3465 b^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 119, normalized size = 0.98 \[ \frac {2 \, {\left (315 \, B b^{5} x^{5} + 128 \, B a^{5} - 176 \, A a^{4} b + 35 \, {\left (B a b^{4} + 11 \, A b^{5}\right )} x^{4} - 5 \, {\left (8 \, B a^{2} b^{3} - 11 \, A a b^{4}\right )} x^{3} + 6 \, {\left (8 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{2} - 8 \, {\left (8 \, B a^{4} b - 11 \, A a^{3} b^{2}\right )} x\right )} \sqrt {b x + a}}{3465 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.25, size = 256, normalized size = 2.10 \[ \frac {2 \, {\left (\frac {99 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} A a}{b^{3}} + \frac {11 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} B a}{b^{4}} + \frac {11 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} A}{b^{3}} + \frac {5 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b x + a} a^{5}\right )} B}{b^{4}}\right )}}{3465 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 95, normalized size = 0.78 \[ -\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-315 B \,x^{4} b^{4}-385 A \,b^{4} x^{3}+280 B a \,b^{3} x^{3}+330 A a \,b^{3} x^{2}-240 B \,a^{2} b^{2} x^{2}-264 A \,a^{2} b^{2} x +192 B \,a^{3} b x +176 A \,a^{3} b -128 B \,a^{4}\right )}{3465 b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 100, normalized size = 0.82 \[ \frac {2 \, {\left (315 \, {\left (b x + a\right )}^{\frac {11}{2}} B - 385 \, {\left (4 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {9}{2}} + 1485 \, {\left (2 \, B a^{2} - A a b\right )} {\left (b x + a\right )}^{\frac {7}{2}} - 693 \, {\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {5}{2}} + 1155 \, {\left (B a^{4} - A a^{3} b\right )} {\left (b x + a\right )}^{\frac {3}{2}}\right )}}{3465 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 111, normalized size = 0.91 \[ \frac {\left (12\,B\,a^2-6\,A\,a\,b\right )\,{\left (a+b\,x\right )}^{7/2}}{7\,b^5}+\frac {2\,B\,{\left (a+b\,x\right )}^{11/2}}{11\,b^5}+\frac {\left (2\,A\,b-8\,B\,a\right )\,{\left (a+b\,x\right )}^{9/2}}{9\,b^5}+\frac {\left (2\,B\,a^4-2\,A\,a^3\,b\right )\,{\left (a+b\,x\right )}^{3/2}}{3\,b^5}-\frac {\left (8\,B\,a^3-6\,A\,a^2\,b\right )\,{\left (a+b\,x\right )}^{5/2}}{5\,b^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.32, size = 121, normalized size = 0.99 \[ \frac {2 \left (\frac {B \left (a + b x\right )^{\frac {11}{2}}}{11 b} + \frac {\left (a + b x\right )^{\frac {9}{2}} \left (A b - 4 B a\right )}{9 b} + \frac {\left (a + b x\right )^{\frac {7}{2}} \left (- 3 A a b + 6 B a^{2}\right )}{7 b} + \frac {\left (a + b x\right )^{\frac {5}{2}} \left (3 A a^{2} b - 4 B a^{3}\right )}{5 b} + \frac {\left (a + b x\right )^{\frac {3}{2}} \left (- A a^{3} b + B a^{4}\right )}{3 b}\right )}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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