Optimal. Leaf size=59 \[ -\frac {2 a^2 A}{\sqrt {x}}+\frac {2}{3} b x^{3/2} (2 a B+A b)+2 a \sqrt {x} (a B+2 A b)+\frac {2}{5} b^2 B x^{5/2} \]
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Rubi [A] time = 0.03, antiderivative size = 59, 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} \[ -\frac {2 a^2 A}{\sqrt {x}}+\frac {2}{3} b x^{3/2} (2 a B+A b)+2 a \sqrt {x} (a B+2 A b)+\frac {2}{5} b^2 B x^{5/2} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int \frac {(a+b x)^2 (A+B x)}{x^{3/2}} \, dx &=\int \left (\frac {a^2 A}{x^{3/2}}+\frac {a (2 A b+a B)}{\sqrt {x}}+b (A b+2 a B) \sqrt {x}+b^2 B x^{3/2}\right ) \, dx\\ &=-\frac {2 a^2 A}{\sqrt {x}}+2 a (2 A b+a B) \sqrt {x}+\frac {2}{3} b (A b+2 a B) x^{3/2}+\frac {2}{5} b^2 B x^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 49, normalized size = 0.83 \[ \frac {-30 a^2 (A-B x)+20 a b x (3 A+B x)+2 b^2 x^2 (5 A+3 B x)}{15 \sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 51, normalized size = 0.86 \[ \frac {2 \, {\left (3 \, B b^{2} x^{3} - 15 \, A a^{2} + 5 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 15 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )}}{15 \, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.28, size = 53, normalized size = 0.90 \[ \frac {2}{5} \, B b^{2} x^{\frac {5}{2}} + \frac {4}{3} \, B a b x^{\frac {3}{2}} + \frac {2}{3} \, A b^{2} x^{\frac {3}{2}} + 2 \, B a^{2} \sqrt {x} + 4 \, A a b \sqrt {x} - \frac {2 \, A a^{2}}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 52, normalized size = 0.88 \[ -\frac {2 \left (-3 B \,b^{2} x^{3}-5 A \,b^{2} x^{2}-10 B a b \,x^{2}-30 A a b x -15 B \,a^{2} x +15 a^{2} A \right )}{15 \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 51, normalized size = 0.86 \[ \frac {2}{5} \, B b^{2} x^{\frac {5}{2}} - \frac {2 \, A a^{2}}{\sqrt {x}} + \frac {2}{3} \, {\left (2 \, B a b + A b^{2}\right )} x^{\frac {3}{2}} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 51, normalized size = 0.86 \[ \sqrt {x}\,\left (2\,B\,a^2+4\,A\,b\,a\right )+x^{3/2}\,\left (\frac {2\,A\,b^2}{3}+\frac {4\,B\,a\,b}{3}\right )-\frac {2\,A\,a^2}{\sqrt {x}}+\frac {2\,B\,b^2\,x^{5/2}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.17, size = 75, normalized size = 1.27 \[ - \frac {2 A a^{2}}{\sqrt {x}} + 4 A a b \sqrt {x} + \frac {2 A b^{2} x^{\frac {3}{2}}}{3} + 2 B a^{2} \sqrt {x} + \frac {4 B a b x^{\frac {3}{2}}}{3} + \frac {2 B b^{2} x^{\frac {5}{2}}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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