Optimal. Leaf size=117 \[ \frac {16 b^2 (a+b x)^{3/2} (2 A b-3 a B)}{315 a^4 x^{3/2}}-\frac {8 b (a+b x)^{3/2} (2 A b-3 a B)}{105 a^3 x^{5/2}}+\frac {2 (a+b x)^{3/2} (2 A b-3 a B)}{21 a^2 x^{7/2}}-\frac {2 A (a+b x)^{3/2}}{9 a x^{9/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {78, 45, 37} \[ \frac {16 b^2 (a+b x)^{3/2} (2 A b-3 a B)}{315 a^4 x^{3/2}}-\frac {8 b (a+b x)^{3/2} (2 A b-3 a B)}{105 a^3 x^{5/2}}+\frac {2 (a+b x)^{3/2} (2 A b-3 a B)}{21 a^2 x^{7/2}}-\frac {2 A (a+b x)^{3/2}}{9 a x^{9/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b x} (A+B x)}{x^{11/2}} \, dx &=-\frac {2 A (a+b x)^{3/2}}{9 a x^{9/2}}+\frac {\left (2 \left (-3 A b+\frac {9 a B}{2}\right )\right ) \int \frac {\sqrt {a+b x}}{x^{9/2}} \, dx}{9 a}\\ &=-\frac {2 A (a+b x)^{3/2}}{9 a x^{9/2}}+\frac {2 (2 A b-3 a B) (a+b x)^{3/2}}{21 a^2 x^{7/2}}+\frac {(4 b (2 A b-3 a B)) \int \frac {\sqrt {a+b x}}{x^{7/2}} \, dx}{21 a^2}\\ &=-\frac {2 A (a+b x)^{3/2}}{9 a x^{9/2}}+\frac {2 (2 A b-3 a B) (a+b x)^{3/2}}{21 a^2 x^{7/2}}-\frac {8 b (2 A b-3 a B) (a+b x)^{3/2}}{105 a^3 x^{5/2}}-\frac {\left (8 b^2 (2 A b-3 a B)\right ) \int \frac {\sqrt {a+b x}}{x^{5/2}} \, dx}{105 a^3}\\ &=-\frac {2 A (a+b x)^{3/2}}{9 a x^{9/2}}+\frac {2 (2 A b-3 a B) (a+b x)^{3/2}}{21 a^2 x^{7/2}}-\frac {8 b (2 A b-3 a B) (a+b x)^{3/2}}{105 a^3 x^{5/2}}+\frac {16 b^2 (2 A b-3 a B) (a+b x)^{3/2}}{315 a^4 x^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 73, normalized size = 0.62 \[ -\frac {2 (a+b x)^{3/2} \left (5 a^3 (7 A+9 B x)-6 a^2 b x (5 A+6 B x)+24 a b^2 x^2 (A+B x)-16 A b^3 x^3\right )}{315 a^4 x^{9/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 101, normalized size = 0.86 \[ -\frac {2 \, {\left (35 \, A a^{4} + 8 \, {\left (3 \, B a b^{3} - 2 \, A b^{4}\right )} x^{4} - 4 \, {\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3} + 3 \, {\left (3 \, B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{2} + 5 \, {\left (9 \, B a^{4} + A a^{3} b\right )} x\right )} \sqrt {b x + a}}{315 \, a^{4} x^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.30, size = 137, normalized size = 1.17 \[ -\frac {2 \, {\left ({\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (3 \, B a b^{8} - 2 \, A b^{9}\right )} {\left (b x + a\right )}}{a^{4}} - \frac {9 \, {\left (3 \, B a^{2} b^{8} - 2 \, A a b^{9}\right )}}{a^{4}}\right )} + \frac {63 \, {\left (3 \, B a^{3} b^{8} - 2 \, A a^{2} b^{9}\right )}}{a^{4}}\right )} - \frac {105 \, {\left (B a^{4} b^{8} - A a^{3} b^{9}\right )}}{a^{4}}\right )} {\left (b x + a\right )}^{\frac {3}{2}} b}{315 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {9}{2}} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 77, normalized size = 0.66 \[ -\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-16 A \,b^{3} x^{3}+24 B a \,b^{2} x^{3}+24 A a \,b^{2} x^{2}-36 B \,a^{2} b \,x^{2}-30 A \,a^{2} b x +45 B \,a^{3} x +35 A \,a^{3}\right )}{315 a^{4} x^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.93, size = 192, normalized size = 1.64 \[ -\frac {16 \, \sqrt {b x^{2} + a x} B b^{3}}{105 \, a^{3} x} + \frac {32 \, \sqrt {b x^{2} + a x} A b^{4}}{315 \, a^{4} x} + \frac {8 \, \sqrt {b x^{2} + a x} B b^{2}}{105 \, a^{2} x^{2}} - \frac {16 \, \sqrt {b x^{2} + a x} A b^{3}}{315 \, a^{3} x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} B b}{35 \, a x^{3}} + \frac {4 \, \sqrt {b x^{2} + a x} A b^{2}}{105 \, a^{2} x^{3}} - \frac {2 \, \sqrt {b x^{2} + a x} B}{7 \, x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} A b}{63 \, a x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} A}{9 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.74, size = 96, normalized size = 0.82 \[ -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{9}+\frac {x\,\left (90\,B\,a^4+10\,A\,b\,a^3\right )}{315\,a^4}-\frac {x^4\,\left (32\,A\,b^4-48\,B\,a\,b^3\right )}{315\,a^4}+\frac {8\,b^2\,x^3\,\left (2\,A\,b-3\,B\,a\right )}{315\,a^3}-\frac {2\,b\,x^2\,\left (2\,A\,b-3\,B\,a\right )}{105\,a^2}\right )}{x^{9/2}} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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