Optimal. Leaf size=81 \[ -\frac {2 a^3 A}{3 x^{3/2}}-\frac {2 a^2 (a B+3 A b)}{\sqrt {x}}+\frac {2}{3} b^2 x^{3/2} (3 a B+A b)+6 a b \sqrt {x} (a B+A b)+\frac {2}{5} b^3 B x^{5/2} \]
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Rubi [A] time = 0.04, antiderivative size = 81, 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 = {76} \[ -\frac {2 a^2 (a B+3 A b)}{\sqrt {x}}-\frac {2 a^3 A}{3 x^{3/2}}+\frac {2}{3} b^2 x^{3/2} (3 a B+A b)+6 a b \sqrt {x} (a B+A b)+\frac {2}{5} b^3 B x^{5/2} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int \frac {(a+b x)^3 (A+B x)}{x^{5/2}} \, dx &=\int \left (\frac {a^3 A}{x^{5/2}}+\frac {a^2 (3 A b+a B)}{x^{3/2}}+\frac {3 a b (A b+a B)}{\sqrt {x}}+b^2 (A b+3 a B) \sqrt {x}+b^3 B x^{3/2}\right ) \, dx\\ &=-\frac {2 a^3 A}{3 x^{3/2}}-\frac {2 a^2 (3 A b+a B)}{\sqrt {x}}+6 a b (A b+a B) \sqrt {x}+\frac {2}{3} b^2 (A b+3 a B) x^{3/2}+\frac {2}{5} b^3 B x^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 66, normalized size = 0.81 \[ \frac {2 \left (-5 a^3 (A+3 B x)+45 a^2 b x (B x-A)+15 a b^2 x^2 (3 A+B x)+b^3 x^3 (5 A+3 B x)\right )}{15 x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 73, normalized size = 0.90 \[ \frac {2 \, {\left (3 \, B b^{3} x^{4} - 5 \, A a^{3} + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 45 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} - 15 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{15 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.28, size = 75, normalized size = 0.93 \[ \frac {2}{5} \, B b^{3} x^{\frac {5}{2}} + 2 \, B a b^{2} x^{\frac {3}{2}} + \frac {2}{3} \, A b^{3} x^{\frac {3}{2}} + 6 \, B a^{2} b \sqrt {x} + 6 \, A a b^{2} \sqrt {x} - \frac {2 \, {\left (3 \, B a^{3} x + 9 \, A a^{2} b x + A a^{3}\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 76, normalized size = 0.94 \[ -\frac {2 \left (-3 B \,b^{3} x^{4}-5 A \,b^{3} x^{3}-15 B a \,b^{2} x^{3}-45 A a \,b^{2} x^{2}-45 B \,a^{2} b \,x^{2}+45 A \,a^{2} b x +15 B \,a^{3} x +5 a^{3} A \right )}{15 x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 73, normalized size = 0.90 \[ \frac {2}{5} \, B b^{3} x^{\frac {5}{2}} + \frac {2}{3} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {3}{2}} + 6 \, {\left (B a^{2} b + A a b^{2}\right )} \sqrt {x} - \frac {2 \, {\left (A a^{3} + 3 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 70, normalized size = 0.86 \[ x^{3/2}\,\left (\frac {2\,A\,b^3}{3}+2\,B\,a\,b^2\right )-\frac {x\,\left (2\,B\,a^3+6\,A\,b\,a^2\right )+\frac {2\,A\,a^3}{3}}{x^{3/2}}+\frac {2\,B\,b^3\,x^{5/2}}{5}+6\,a\,b\,\sqrt {x}\,\left (A\,b+B\,a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.47, size = 105, normalized size = 1.30 \[ - \frac {2 A a^{3}}{3 x^{\frac {3}{2}}} - \frac {6 A a^{2} b}{\sqrt {x}} + 6 A a b^{2} \sqrt {x} + \frac {2 A b^{3} x^{\frac {3}{2}}}{3} - \frac {2 B a^{3}}{\sqrt {x}} + 6 B a^{2} b \sqrt {x} + 2 B a b^{2} x^{\frac {3}{2}} + \frac {2 B b^{3} x^{\frac {5}{2}}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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