Optimal. Leaf size=84 \[ -\frac {4 b (a+b x)^{5/2} (4 A b-9 a B)}{315 a^3 x^{5/2}}+\frac {2 (a+b x)^{5/2} (4 A b-9 a B)}{63 a^2 x^{7/2}}-\frac {2 A (a+b x)^{5/2}}{9 a x^{9/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {78, 45, 37} \begin {gather*} -\frac {4 b (a+b x)^{5/2} (4 A b-9 a B)}{315 a^3 x^{5/2}}+\frac {2 (a+b x)^{5/2} (4 A b-9 a B)}{63 a^2 x^{7/2}}-\frac {2 A (a+b x)^{5/2}}{9 a x^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2} (A+B x)}{x^{11/2}} \, dx &=-\frac {2 A (a+b x)^{5/2}}{9 a x^{9/2}}+\frac {\left (2 \left (-2 A b+\frac {9 a B}{2}\right )\right ) \int \frac {(a+b x)^{3/2}}{x^{9/2}} \, dx}{9 a}\\ &=-\frac {2 A (a+b x)^{5/2}}{9 a x^{9/2}}+\frac {2 (4 A b-9 a B) (a+b x)^{5/2}}{63 a^2 x^{7/2}}+\frac {(2 b (4 A b-9 a B)) \int \frac {(a+b x)^{3/2}}{x^{7/2}} \, dx}{63 a^2}\\ &=-\frac {2 A (a+b x)^{5/2}}{9 a x^{9/2}}+\frac {2 (4 A b-9 a B) (a+b x)^{5/2}}{63 a^2 x^{7/2}}-\frac {4 b (4 A b-9 a B) (a+b x)^{5/2}}{315 a^3 x^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 57, normalized size = 0.68 \begin {gather*} -\frac {2 (a+b x)^{5/2} \left (5 a^2 (7 A+9 B x)-2 a b x (10 A+9 B x)+8 A b^2 x^2\right )}{315 a^3 x^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.30, size = 106, normalized size = 1.26 \begin {gather*} \frac {2 \sqrt {a+b x} \left (-35 a^4 A-45 a^4 B x-50 a^3 A b x-72 a^3 b B x^2-3 a^2 A b^2 x^2-9 a^2 b^2 B x^3+4 a A b^3 x^3+18 a b^3 B x^4-8 A b^4 x^4\right )}{315 a^3 x^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.20, size = 100, normalized size = 1.19 \begin {gather*} -\frac {2 \, {\left (35 \, A a^{4} - 2 \, {\left (9 \, B a b^{3} - 4 \, A b^{4}\right )} x^{4} + {\left (9 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{3} + 3 \, {\left (24 \, B a^{3} b + A a^{2} b^{2}\right )} x^{2} + 5 \, {\left (9 \, B a^{4} + 10 \, A a^{3} b\right )} x\right )} \sqrt {b x + a}}{315 \, a^{3} x^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.01, size = 110, normalized size = 1.31 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {5}{2}} {\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (9 \, B a^{2} b^{8} - 4 \, A a b^{9}\right )} {\left (b x + a\right )}}{a^{4}} - \frac {9 \, {\left (9 \, B a^{3} b^{8} - 4 \, A a^{2} b^{9}\right )}}{a^{4}}\right )} + \frac {63 \, {\left (B a^{4} b^{8} - A a^{3} b^{9}\right )}}{a^{4}}\right )} b}{315 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {9}{2}} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 53, normalized size = 0.63 \begin {gather*} -\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (8 A \,b^{2} x^{2}-18 B a b \,x^{2}-20 A a b x +45 B \,a^{2} x +35 A \,a^{2}\right )}{315 a^{3} x^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.87, size = 222, normalized size = 2.64 \begin {gather*} \frac {4 \, \sqrt {b x^{2} + a x} B b^{3}}{35 \, a^{2} x} - \frac {16 \, \sqrt {b x^{2} + a x} A b^{4}}{315 \, a^{3} x} - \frac {2 \, \sqrt {b x^{2} + a x} B b^{2}}{35 \, a x^{2}} + \frac {8 \, \sqrt {b x^{2} + a x} A b^{3}}{315 \, a^{2} x^{2}} + \frac {3 \, \sqrt {b x^{2} + a x} B b}{70 \, x^{3}} - \frac {2 \, \sqrt {b x^{2} + a x} A b^{2}}{105 \, a x^{3}} + \frac {3 \, \sqrt {b x^{2} + a x} B a}{14 \, x^{4}} + \frac {\sqrt {b x^{2} + a x} A b}{63 \, x^{4}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B}{2 \, x^{5}} + \frac {\sqrt {b x^{2} + a x} A a}{9 \, x^{5}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A}{3 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.79, size = 96, normalized size = 1.14 \begin {gather*} -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A\,a}{9}+\frac {x\,\left (90\,B\,a^4+100\,A\,b\,a^3\right )}{315\,a^3}+\frac {x^4\,\left (16\,A\,b^4-36\,B\,a\,b^3\right )}{315\,a^3}-\frac {2\,b^2\,x^3\,\left (4\,A\,b-9\,B\,a\right )}{315\,a^2}+\frac {2\,b\,x^2\,\left (A\,b+24\,B\,a\right )}{105\,a}\right )}{x^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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