Optimal. Leaf size=61 \[ -\frac {2 a^2 A}{7 x^{7/2}}-\frac {2 a (a B+2 A b)}{5 x^{5/2}}-\frac {2 b (2 a B+A b)}{3 x^{3/2}}-\frac {2 b^2 B}{\sqrt {x}} \]
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Rubi [A] time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {27, 76} \[ -\frac {2 a^2 A}{7 x^{7/2}}-\frac {2 a (a B+2 A b)}{5 x^{5/2}}-\frac {2 b (2 a B+A b)}{3 x^{3/2}}-\frac {2 b^2 B}{\sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )}{x^{9/2}} \, dx &=\int \frac {(a+b x)^2 (A+B x)}{x^{9/2}} \, dx\\ &=\int \left (\frac {a^2 A}{x^{9/2}}+\frac {a (2 A b+a B)}{x^{7/2}}+\frac {b (A b+2 a B)}{x^{5/2}}+\frac {b^2 B}{x^{3/2}}\right ) \, dx\\ &=-\frac {2 a^2 A}{7 x^{7/2}}-\frac {2 a (2 A b+a B)}{5 x^{5/2}}-\frac {2 b (A b+2 a B)}{3 x^{3/2}}-\frac {2 b^2 B}{\sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 50, normalized size = 0.82 \[ -\frac {2 \left (3 a^2 (5 A+7 B x)+14 a b x (3 A+5 B x)+35 b^2 x^2 (A+3 B x)\right )}{105 x^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 51, normalized size = 0.84 \[ -\frac {2 \, {\left (105 \, B b^{2} x^{3} + 15 \, A a^{2} + 35 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 21 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )}}{105 \, x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 51, normalized size = 0.84 \[ -\frac {2 \, {\left (105 \, B b^{2} x^{3} + 70 \, B a b x^{2} + 35 \, A b^{2} x^{2} + 21 \, B a^{2} x + 42 \, A a b x + 15 \, A a^{2}\right )}}{105 \, x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 52, normalized size = 0.85 \[ -\frac {2 \left (105 B \,b^{2} x^{3}+35 A \,b^{2} x^{2}+70 B a b \,x^{2}+42 A a b x +21 B \,a^{2} x +15 A \,a^{2}\right )}{105 x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 51, normalized size = 0.84 \[ -\frac {2 \, {\left (105 \, B b^{2} x^{3} + 15 \, A a^{2} + 35 \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + 21 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )}}{105 \, x^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 51, normalized size = 0.84 \[ -\frac {x^2\,\left (\frac {2\,A\,b^2}{3}+\frac {4\,B\,a\,b}{3}\right )+\frac {2\,A\,a^2}{7}+x\,\left (\frac {2\,B\,a^2}{5}+\frac {4\,A\,b\,a}{5}\right )+2\,B\,b^2\,x^3}{x^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.07, size = 80, normalized size = 1.31 \[ - \frac {2 A a^{2}}{7 x^{\frac {7}{2}}} - \frac {4 A a b}{5 x^{\frac {5}{2}}} - \frac {2 A b^{2}}{3 x^{\frac {3}{2}}} - \frac {2 B a^{2}}{5 x^{\frac {5}{2}}} - \frac {4 B a b}{3 x^{\frac {3}{2}}} - \frac {2 B b^{2}}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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