Optimal. Leaf size=150 \[ -\frac {32 b^3 (a+b x)^{3/2} (8 A b-11 a B)}{3465 a^5 x^{3/2}}+\frac {16 b^2 (a+b x)^{3/2} (8 A b-11 a B)}{1155 a^4 x^{5/2}}-\frac {4 b (a+b x)^{3/2} (8 A b-11 a B)}{231 a^3 x^{7/2}}+\frac {2 (a+b x)^{3/2} (8 A b-11 a B)}{99 a^2 x^{9/2}}-\frac {2 A (a+b x)^{3/2}}{11 a x^{11/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {32 b^3 (a+b x)^{3/2} (8 A b-11 a B)}{3465 a^5 x^{3/2}}+\frac {16 b^2 (a+b x)^{3/2} (8 A b-11 a B)}{1155 a^4 x^{5/2}}-\frac {4 b (a+b x)^{3/2} (8 A b-11 a B)}{231 a^3 x^{7/2}}+\frac {2 (a+b x)^{3/2} (8 A b-11 a B)}{99 a^2 x^{9/2}}-\frac {2 A (a+b x)^{3/2}}{11 a x^{11/2}} \end {gather*}
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^{13/2}} \, dx &=-\frac {2 A (a+b x)^{3/2}}{11 a x^{11/2}}+\frac {\left (2 \left (-4 A b+\frac {11 a B}{2}\right )\right ) \int \frac {\sqrt {a+b x}}{x^{11/2}} \, dx}{11 a}\\ &=-\frac {2 A (a+b x)^{3/2}}{11 a x^{11/2}}+\frac {2 (8 A b-11 a B) (a+b x)^{3/2}}{99 a^2 x^{9/2}}+\frac {(2 b (8 A b-11 a B)) \int \frac {\sqrt {a+b x}}{x^{9/2}} \, dx}{33 a^2}\\ &=-\frac {2 A (a+b x)^{3/2}}{11 a x^{11/2}}+\frac {2 (8 A b-11 a B) (a+b x)^{3/2}}{99 a^2 x^{9/2}}-\frac {4 b (8 A b-11 a B) (a+b x)^{3/2}}{231 a^3 x^{7/2}}-\frac {\left (8 b^2 (8 A b-11 a B)\right ) \int \frac {\sqrt {a+b x}}{x^{7/2}} \, dx}{231 a^3}\\ &=-\frac {2 A (a+b x)^{3/2}}{11 a x^{11/2}}+\frac {2 (8 A b-11 a B) (a+b x)^{3/2}}{99 a^2 x^{9/2}}-\frac {4 b (8 A b-11 a B) (a+b x)^{3/2}}{231 a^3 x^{7/2}}+\frac {16 b^2 (8 A b-11 a B) (a+b x)^{3/2}}{1155 a^4 x^{5/2}}+\frac {\left (16 b^3 (8 A b-11 a B)\right ) \int \frac {\sqrt {a+b x}}{x^{5/2}} \, dx}{1155 a^4}\\ &=-\frac {2 A (a+b x)^{3/2}}{11 a x^{11/2}}+\frac {2 (8 A b-11 a B) (a+b x)^{3/2}}{99 a^2 x^{9/2}}-\frac {4 b (8 A b-11 a B) (a+b x)^{3/2}}{231 a^3 x^{7/2}}+\frac {16 b^2 (8 A b-11 a B) (a+b x)^{3/2}}{1155 a^4 x^{5/2}}-\frac {32 b^3 (8 A b-11 a B) (a+b x)^{3/2}}{3465 a^5 x^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 95, normalized size = 0.63 \begin {gather*} -\frac {2 (a+b x)^{3/2} \left (35 a^4 (9 A+11 B x)-10 a^3 b x (28 A+33 B x)+24 a^2 b^2 x^2 (10 A+11 B x)-16 a b^3 x^3 (12 A+11 B x)+128 A b^4 x^4\right )}{3465 a^5 x^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.24, size = 130, normalized size = 0.87 \begin {gather*} -\frac {2 \sqrt {a+b x} \left (315 a^5 A+385 a^5 B x+35 a^4 A b x+55 a^4 b B x^2-40 a^3 A b^2 x^2-66 a^3 b^2 B x^3+48 a^2 A b^3 x^3+88 a^2 b^3 B x^4-64 a A b^4 x^4-176 a b^4 B x^5+128 A b^5 x^5\right )}{3465 a^5 x^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.44, size = 125, normalized size = 0.83 \begin {gather*} -\frac {2 \, {\left (315 \, A a^{5} - 16 \, {\left (11 \, B a b^{4} - 8 \, A b^{5}\right )} x^{5} + 8 \, {\left (11 \, B a^{2} b^{3} - 8 \, A a b^{4}\right )} x^{4} - 6 \, {\left (11 \, B a^{3} b^{2} - 8 \, A a^{2} b^{3}\right )} x^{3} + 5 \, {\left (11 \, B a^{4} b - 8 \, A a^{3} b^{2}\right )} x^{2} + 35 \, {\left (11 \, B a^{5} + A a^{4} b\right )} x\right )} \sqrt {b x + a}}{3465 \, a^{5} x^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.12, size = 169, normalized size = 1.13 \begin {gather*} \frac {2 \, {\left ({\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (11 \, B a b^{10} - 8 \, A b^{11}\right )} {\left (b x + a\right )}}{a^{5}} - \frac {11 \, {\left (11 \, B a^{2} b^{10} - 8 \, A a b^{11}\right )}}{a^{5}}\right )} + \frac {99 \, {\left (11 \, B a^{3} b^{10} - 8 \, A a^{2} b^{11}\right )}}{a^{5}}\right )} - \frac {231 \, {\left (11 \, B a^{4} b^{10} - 8 \, A a^{3} b^{11}\right )}}{a^{5}}\right )} {\left (b x + a\right )} + \frac {1155 \, {\left (B a^{5} b^{10} - A a^{4} b^{11}\right )}}{a^{5}}\right )} {\left (b x + a\right )}^{\frac {3}{2}} b}{3465 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {11}{2}} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 101, normalized size = 0.67 \begin {gather*} -\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (128 A \,b^{4} x^{4}-176 B a \,b^{3} x^{4}-192 A a \,b^{3} x^{3}+264 B \,a^{2} b^{2} x^{3}+240 A \,a^{2} b^{2} x^{2}-330 B \,a^{3} b \,x^{2}-280 A \,a^{3} b x +385 B \,a^{4} x +315 A \,a^{4}\right )}{3465 a^{5} x^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.94, size = 238, normalized size = 1.59 \begin {gather*} \frac {32 \, \sqrt {b x^{2} + a x} B b^{4}}{315 \, a^{4} x} - \frac {256 \, \sqrt {b x^{2} + a x} A b^{5}}{3465 \, a^{5} x} - \frac {16 \, \sqrt {b x^{2} + a x} B b^{3}}{315 \, a^{3} x^{2}} + \frac {128 \, \sqrt {b x^{2} + a x} A b^{4}}{3465 \, a^{4} x^{2}} + \frac {4 \, \sqrt {b x^{2} + a x} B b^{2}}{105 \, a^{2} x^{3}} - \frac {32 \, \sqrt {b x^{2} + a x} A b^{3}}{1155 \, a^{3} x^{3}} - \frac {2 \, \sqrt {b x^{2} + a x} B b}{63 \, a x^{4}} + \frac {16 \, \sqrt {b x^{2} + a x} A b^{2}}{693 \, a^{2} x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} B}{9 \, x^{5}} - \frac {2 \, \sqrt {b x^{2} + a x} A b}{99 \, a x^{5}} - \frac {2 \, \sqrt {b x^{2} + a x} A}{11 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.78, size = 116, normalized size = 0.77 \begin {gather*} -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A}{11}+\frac {x\,\left (770\,B\,a^5+70\,A\,b\,a^4\right )}{3465\,a^5}+\frac {x^5\,\left (256\,A\,b^5-352\,B\,a\,b^4\right )}{3465\,a^5}+\frac {4\,b^2\,x^3\,\left (8\,A\,b-11\,B\,a\right )}{1155\,a^3}-\frac {16\,b^3\,x^4\,\left (8\,A\,b-11\,B\,a\right )}{3465\,a^4}-\frac {2\,b\,x^2\,\left (8\,A\,b-11\,B\,a\right )}{693\,a^2}\right )}{x^{11/2}} \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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