Optimal. Leaf size=84 \[ -\frac {4 b (a+b x)^{7/2} (4 A b-11 a B)}{693 a^3 x^{7/2}}+\frac {2 (a+b x)^{7/2} (4 A b-11 a B)}{99 a^2 x^{9/2}}-\frac {2 A (a+b x)^{7/2}}{11 a x^{11/2}} \]
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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} \[ -\frac {4 b (a+b x)^{7/2} (4 A b-11 a B)}{693 a^3 x^{7/2}}+\frac {2 (a+b x)^{7/2} (4 A b-11 a B)}{99 a^2 x^{9/2}}-\frac {2 A (a+b x)^{7/2}}{11 a x^{11/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 78
Rubi steps
\begin {align*} \int \frac {(a+b x)^{5/2} (A+B x)}{x^{13/2}} \, dx &=-\frac {2 A (a+b x)^{7/2}}{11 a x^{11/2}}+\frac {\left (2 \left (-2 A b+\frac {11 a B}{2}\right )\right ) \int \frac {(a+b x)^{5/2}}{x^{11/2}} \, dx}{11 a}\\ &=-\frac {2 A (a+b x)^{7/2}}{11 a x^{11/2}}+\frac {2 (4 A b-11 a B) (a+b x)^{7/2}}{99 a^2 x^{9/2}}+\frac {(2 b (4 A b-11 a B)) \int \frac {(a+b x)^{5/2}}{x^{9/2}} \, dx}{99 a^2}\\ &=-\frac {2 A (a+b x)^{7/2}}{11 a x^{11/2}}+\frac {2 (4 A b-11 a B) (a+b x)^{7/2}}{99 a^2 x^{9/2}}-\frac {4 b (4 A b-11 a B) (a+b x)^{7/2}}{693 a^3 x^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 57, normalized size = 0.68 \[ -\frac {2 (a+b x)^{7/2} \left (7 a^2 (9 A+11 B x)-2 a b x (14 A+11 B x)+8 A b^2 x^2\right )}{693 a^3 x^{11/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 123, normalized size = 1.46 \[ -\frac {2 \, {\left (63 \, A a^{5} - 2 \, {\left (11 \, B a b^{4} - 4 \, A b^{5}\right )} x^{5} + {\left (11 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} x^{4} + 3 \, {\left (55 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + {\left (209 \, B a^{4} b + 113 \, A a^{3} b^{2}\right )} x^{2} + 7 \, {\left (11 \, B a^{5} + 23 \, A a^{4} b\right )} x\right )} \sqrt {b x + a}}{693 \, a^{3} x^{\frac {11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.52, size = 112, normalized size = 1.33 \[ \frac {2 \, {\left (b x + a\right )}^{\frac {7}{2}} {\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (11 \, B a^{3} b^{10} - 4 \, A a^{2} b^{11}\right )} {\left (b x + a\right )}}{a^{5}} - \frac {11 \, {\left (11 \, B a^{4} b^{10} - 4 \, A a^{3} b^{11}\right )}}{a^{5}}\right )} + \frac {99 \, {\left (B a^{5} b^{10} - A a^{4} b^{11}\right )}}{a^{5}}\right )} b}{693 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {11}{2}} {\left | b \right |}} \]
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 \[ -\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (8 A \,b^{2} x^{2}-22 B a b \,x^{2}-28 A a b x +77 B \,a^{2} x +63 A \,a^{2}\right )}{693 a^{3} x^{\frac {11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.94, size = 304, normalized size = 3.62 \[ \frac {4 \, \sqrt {b x^{2} + a x} B b^{4}}{63 \, a^{2} x} - \frac {16 \, \sqrt {b x^{2} + a x} A b^{5}}{693 \, a^{3} x} - \frac {2 \, \sqrt {b x^{2} + a x} B b^{3}}{63 \, a x^{2}} + \frac {8 \, \sqrt {b x^{2} + a x} A b^{4}}{693 \, a^{2} x^{2}} + \frac {\sqrt {b x^{2} + a x} B b^{2}}{42 \, x^{3}} - \frac {2 \, \sqrt {b x^{2} + a x} A b^{3}}{231 \, a x^{3}} - \frac {5 \, \sqrt {b x^{2} + a x} B a b}{252 \, x^{4}} + \frac {5 \, \sqrt {b x^{2} + a x} A b^{2}}{693 \, x^{4}} - \frac {5 \, \sqrt {b x^{2} + a x} B a^{2}}{36 \, x^{5}} - \frac {5 \, \sqrt {b x^{2} + a x} A a b}{792 \, x^{5}} + \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a}{12 \, x^{6}} - \frac {5 \, \sqrt {b x^{2} + a x} A a^{2}}{88 \, x^{6}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} B}{2 \, x^{7}} + \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a}{24 \, x^{7}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} A}{3 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 115, normalized size = 1.37 \[ -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A\,a^2}{11}+\frac {x\,\left (154\,B\,a^5+322\,A\,b\,a^4\right )}{693\,a^3}+\frac {x^5\,\left (16\,A\,b^5-44\,B\,a\,b^4\right )}{693\,a^3}+\frac {2\,b\,x^2\,\left (113\,A\,b+209\,B\,a\right )}{693}-\frac {2\,b^3\,x^4\,\left (4\,A\,b-11\,B\,a\right )}{693\,a^2}+\frac {2\,b^2\,x^3\,\left (A\,b+55\,B\,a\right )}{231\,a}\right )}{x^{11/2}} \]
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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