Optimal. Leaf size=59 \[ -\frac {2 a^2 A}{3 x^{3/2}}-\frac {2 a (a B+2 A b)}{\sqrt {x}}+2 b \sqrt {x} (2 a B+A b)+\frac {2}{3} b^2 B x^{3/2} \]
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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} \[ -\frac {2 a^2 A}{3 x^{3/2}}-\frac {2 a (a B+2 A b)}{\sqrt {x}}+2 b \sqrt {x} (2 a B+A b)+\frac {2}{3} b^2 B x^{3/2} \]
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int \frac {(a+b x)^2 (A+B x)}{x^{5/2}} \, dx &=\int \left (\frac {a^2 A}{x^{5/2}}+\frac {a (2 A b+a B)}{x^{3/2}}+\frac {b (A b+2 a B)}{\sqrt {x}}+b^2 B \sqrt {x}\right ) \, dx\\ &=-\frac {2 a^2 A}{3 x^{3/2}}-\frac {2 a (2 A b+a B)}{\sqrt {x}}+2 b (A b+2 a B) \sqrt {x}+\frac {2}{3} b^2 B x^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 47, normalized size = 0.80 \[ \frac {2 \left (-\left (a^2 (A+3 B x)\right )+6 a b x (B x-A)+b^2 x^2 (3 A+B x)\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.18, size = 50, normalized size = 0.85 \[ \frac {2 \, {\left (B b^{2} x^{3} - A a^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} - 3 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.30, size = 51, normalized size = 0.86 \[ \frac {2}{3} \, B b^{2} x^{\frac {3}{2}} + 4 \, B a b \sqrt {x} + 2 \, A b^{2} \sqrt {x} - \frac {2 \, {\left (3 \, B a^{2} x + 6 \, A a b x + A a^{2}\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 51, normalized size = 0.86 \[ -\frac {2 \left (-B \,b^{2} x^{3}-3 A \,b^{2} x^{2}-6 B a b \,x^{2}+6 A a b x +3 B \,a^{2} x +a^{2} A \right )}{3 x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.80, size = 51, normalized size = 0.86 \[ \frac {2}{3} \, B b^{2} x^{\frac {3}{2}} + 2 \, {\left (2 \, B a b + A b^{2}\right )} \sqrt {x} - \frac {2 \, {\left (A a^{2} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 51, normalized size = 0.86 \[ -\frac {6\,B\,a^2\,x+2\,A\,a^2-12\,B\,a\,b\,x^2+12\,A\,a\,b\,x-2\,B\,b^2\,x^3-6\,A\,b^2\,x^2}{3\,x^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.88, size = 73, normalized size = 1.24 \[ - \frac {2 A a^{2}}{3 x^{\frac {3}{2}}} - \frac {4 A a b}{\sqrt {x}} + 2 A b^{2} \sqrt {x} - \frac {2 B a^{2}}{\sqrt {x}} + 4 B a b \sqrt {x} + \frac {2 B b^{2} x^{\frac {3}{2}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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