Optimal. Leaf size=59 \[ -\frac {2 a^2 A}{5 x^{5/2}}-\frac {2 a (a B+2 A b)}{3 x^{3/2}}-\frac {2 b (2 a B+A b)}{\sqrt {x}}+2 b^2 B \sqrt {x} \]
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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} \begin {gather*} -\frac {2 a^2 A}{5 x^{5/2}}-\frac {2 a (a B+2 A b)}{3 x^{3/2}}-\frac {2 b (2 a B+A b)}{\sqrt {x}}+2 b^2 B \sqrt {x} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rubi steps
\begin {align*} \int \frac {(a+b x)^2 (A+B x)}{x^{7/2}} \, dx &=\int \left (\frac {a^2 A}{x^{7/2}}+\frac {a (2 A b+a B)}{x^{5/2}}+\frac {b (A b+2 a B)}{x^{3/2}}+\frac {b^2 B}{\sqrt {x}}\right ) \, dx\\ &=-\frac {2 a^2 A}{5 x^{5/2}}-\frac {2 a (2 A b+a B)}{3 x^{3/2}}-\frac {2 b (A b+2 a B)}{\sqrt {x}}+2 b^2 B \sqrt {x}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 47, normalized size = 0.80 \begin {gather*} -\frac {2 \left (a^2 (3 A+5 B x)+10 a b x (A+3 B x)+15 b^2 x^2 (A-B x)\right )}{15 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 55, normalized size = 0.93 \begin {gather*} \frac {2 \left (-3 a^2 A-5 a^2 B x-10 a A b x-30 a b B x^2-15 A b^2 x^2+15 b^2 B x^3\right )}{15 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 51, normalized size = 0.86 \begin {gather*} \frac {2 \, {\left (15 \, B b^{2} x^{3} - 3 \, A a^{2} - 15 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} - 5 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.26, size = 52, normalized size = 0.88 \begin {gather*} 2 \, B b^{2} \sqrt {x} - \frac {2 \, {\left (30 \, B a b x^{2} + 15 \, A b^{2} x^{2} + 5 \, B a^{2} x + 10 \, A a b x + 3 \, A a^{2}\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}
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 \begin {gather*} -\frac {2 \left (-15 B \,b^{2} x^{3}+15 A \,b^{2} x^{2}+30 B a b \,x^{2}+10 A a b x +5 B \,a^{2} x +3 a^{2} A \right )}{15 x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 52, normalized size = 0.88 \begin {gather*} 2 \, B b^{2} \sqrt {x} - \frac {2 \, {\left (3 \, A a^{2} + 15 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 5 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 52, normalized size = 0.88 \begin {gather*} 2\,B\,b^2\,\sqrt {x}-\frac {x^2\,\left (2\,A\,b^2+4\,B\,a\,b\right )+\frac {2\,A\,a^2}{5}+x\,\left (\frac {2\,B\,a^2}{3}+\frac {4\,A\,b\,a}{3}\right )}{x^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.73, size = 75, normalized size = 1.27 \begin {gather*} - \frac {2 A a^{2}}{5 x^{\frac {5}{2}}} - \frac {4 A a b}{3 x^{\frac {3}{2}}} - \frac {2 A b^{2}}{\sqrt {x}} - \frac {2 B a^{2}}{3 x^{\frac {3}{2}}} - \frac {4 B a b}{\sqrt {x}} + 2 B b^{2} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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