Optimal. Leaf size=79 \[ -\frac {2 a^3 A}{\sqrt {x}}+2 a^2 \sqrt {x} (a B+3 A b)+\frac {2}{5} b^2 x^{5/2} (3 a B+A b)+2 a b x^{3/2} (a B+A b)+\frac {2}{7} b^3 B x^{7/2} \]
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Rubi [A] time = 0.04, antiderivative size = 79, 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} \[ 2 a^2 \sqrt {x} (a B+3 A b)-\frac {2 a^3 A}{\sqrt {x}}+\frac {2}{5} b^2 x^{5/2} (3 a B+A b)+2 a b x^{3/2} (a B+A b)+\frac {2}{7} b^3 B x^{7/2} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int \frac {(a+b x)^3 (A+B x)}{x^{3/2}} \, dx &=\int \left (\frac {a^3 A}{x^{3/2}}+\frac {a^2 (3 A b+a B)}{\sqrt {x}}+3 a b (A b+a B) \sqrt {x}+b^2 (A b+3 a B) x^{3/2}+b^3 B x^{5/2}\right ) \, dx\\ &=-\frac {2 a^3 A}{\sqrt {x}}+2 a^2 (3 A b+a B) \sqrt {x}+2 a b (A b+a B) x^{3/2}+\frac {2}{5} b^2 (A b+3 a B) x^{5/2}+\frac {2}{7} b^3 B x^{7/2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 67, normalized size = 0.85 \[ \frac {2 \left (-35 a^3 (A-B x)+35 a^2 b x (3 A+B x)+7 a b^2 x^2 (5 A+3 B x)+b^3 x^3 (7 A+5 B x)\right )}{35 \sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 73, normalized size = 0.92 \[ \frac {2 \, {\left (5 \, B b^{3} x^{4} - 35 \, A a^{3} + 7 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 35 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} + 35 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{35 \, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.25, size = 77, normalized size = 0.97 \[ \frac {2}{7} \, B b^{3} x^{\frac {7}{2}} + \frac {6}{5} \, B a b^{2} x^{\frac {5}{2}} + \frac {2}{5} \, A b^{3} x^{\frac {5}{2}} + 2 \, B a^{2} b x^{\frac {3}{2}} + 2 \, A a b^{2} x^{\frac {3}{2}} + 2 \, B a^{3} \sqrt {x} + 6 \, A a^{2} b \sqrt {x} - \frac {2 \, A a^{3}}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 76, normalized size = 0.96 \[ -\frac {2 \left (-5 B \,b^{3} x^{4}-7 A \,b^{3} x^{3}-21 B a \,b^{2} x^{3}-35 A a \,b^{2} x^{2}-35 B \,a^{2} b \,x^{2}-105 A \,a^{2} b x -35 B \,a^{3} x +35 a^{3} A \right )}{35 \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.83, size = 73, normalized size = 0.92 \[ \frac {2}{7} \, B b^{3} x^{\frac {7}{2}} - \frac {2 \, A a^{3}}{\sqrt {x}} + \frac {2}{5} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {5}{2}} + 2 \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {3}{2}} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 69, normalized size = 0.87 \[ \sqrt {x}\,\left (2\,B\,a^3+6\,A\,b\,a^2\right )+x^{5/2}\,\left (\frac {2\,A\,b^3}{5}+\frac {6\,B\,a\,b^2}{5}\right )-\frac {2\,A\,a^3}{\sqrt {x}}+\frac {2\,B\,b^3\,x^{7/2}}{7}+2\,a\,b\,x^{3/2}\,\left (A\,b+B\,a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.49, size = 105, normalized size = 1.33 \[ - \frac {2 A a^{3}}{\sqrt {x}} + 6 A a^{2} b \sqrt {x} + 2 A a b^{2} x^{\frac {3}{2}} + \frac {2 A b^{3} x^{\frac {5}{2}}}{5} + 2 B a^{3} \sqrt {x} + 2 B a^{2} b x^{\frac {3}{2}} + \frac {6 B a b^{2} x^{\frac {5}{2}}}{5} + \frac {2 B b^{3} x^{\frac {7}{2}}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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