Optimal. Leaf size=117 \[ -\frac {2 A (a+b x)^{5/2}}{11 a x^{11/2}}+\frac {2 (6 A b-11 a B) (a+b x)^{5/2}}{99 a^2 x^{9/2}}-\frac {8 b (6 A b-11 a B) (a+b x)^{5/2}}{693 a^3 x^{7/2}}+\frac {16 b^2 (6 A b-11 a B) (a+b x)^{5/2}}{3465 a^4 x^{5/2}} \]
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Rubi [A]
time = 0.03, 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 = {79, 47, 37}
\begin {gather*} \frac {16 b^2 (a+b x)^{5/2} (6 A b-11 a B)}{3465 a^4 x^{5/2}}-\frac {8 b (a+b x)^{5/2} (6 A b-11 a B)}{693 a^3 x^{7/2}}+\frac {2 (a+b x)^{5/2} (6 A b-11 a B)}{99 a^2 x^{9/2}}-\frac {2 A (a+b x)^{5/2}}{11 a x^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 79
Rubi steps
\begin {align*} \int \frac {(a+b x)^{3/2} (A+B x)}{x^{13/2}} \, dx &=-\frac {2 A (a+b x)^{5/2}}{11 a x^{11/2}}+\frac {\left (2 \left (-3 A b+\frac {11 a B}{2}\right )\right ) \int \frac {(a+b x)^{3/2}}{x^{11/2}} \, dx}{11 a}\\ &=-\frac {2 A (a+b x)^{5/2}}{11 a x^{11/2}}+\frac {2 (6 A b-11 a B) (a+b x)^{5/2}}{99 a^2 x^{9/2}}+\frac {(4 b (6 A b-11 a B)) \int \frac {(a+b x)^{3/2}}{x^{9/2}} \, dx}{99 a^2}\\ &=-\frac {2 A (a+b x)^{5/2}}{11 a x^{11/2}}+\frac {2 (6 A b-11 a B) (a+b x)^{5/2}}{99 a^2 x^{9/2}}-\frac {8 b (6 A b-11 a B) (a+b x)^{5/2}}{693 a^3 x^{7/2}}-\frac {\left (8 b^2 (6 A b-11 a B)\right ) \int \frac {(a+b x)^{3/2}}{x^{7/2}} \, dx}{693 a^3}\\ &=-\frac {2 A (a+b x)^{5/2}}{11 a x^{11/2}}+\frac {2 (6 A b-11 a B) (a+b x)^{5/2}}{99 a^2 x^{9/2}}-\frac {8 b (6 A b-11 a B) (a+b x)^{5/2}}{693 a^3 x^{7/2}}+\frac {16 b^2 (6 A b-11 a B) (a+b x)^{5/2}}{3465 a^4 x^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 76, normalized size = 0.65 \begin {gather*} -\frac {2 (a+b x)^{5/2} \left (-48 A b^3 x^3+35 a^3 (9 A+11 B x)+8 a b^2 x^2 (15 A+11 B x)-10 a^2 b x (21 A+22 B x)\right )}{3465 a^4 x^{11/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 101, normalized size = 0.86
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {5}{2}} \left (-48 A \,b^{3} x^{3}+88 B a \,b^{2} x^{3}+120 a A \,b^{2} x^{2}-220 B \,a^{2} b \,x^{2}-210 a^{2} A b x +385 a^{3} B x +315 a^{3} A \right )}{3465 x^{\frac {11}{2}} a^{4}}\) | \(77\) |
default | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-48 A \,b^{4} x^{4}+88 B a \,b^{3} x^{4}+72 A a \,b^{3} x^{3}-132 B \,a^{2} b^{2} x^{3}-90 A \,a^{2} b^{2} x^{2}+165 B \,a^{3} b \,x^{2}+105 A \,a^{3} b x +385 B \,a^{4} x +315 A \,a^{4}\right )}{3465 x^{\frac {11}{2}} a^{4}}\) | \(101\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (-48 A \,b^{5} x^{5}+88 B a \,b^{4} x^{5}+24 a A \,b^{4} x^{4}-44 B \,a^{2} b^{3} x^{4}-18 a^{2} A \,b^{3} x^{3}+33 B \,a^{3} b^{2} x^{3}+15 a^{3} A \,b^{2} x^{2}+550 B \,a^{4} b \,x^{2}+420 a^{4} A b x +385 a^{5} B x +315 a^{5} A \right )}{3465 x^{\frac {11}{2}} a^{4}}\) | \(125\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 268 vs.
\(2 (93) = 186\).
time = 0.28, size = 268, normalized size = 2.29 \begin {gather*} -\frac {16 \, \sqrt {b x^{2} + a x} B b^{4}}{315 \, a^{3} x} + \frac {32 \, \sqrt {b x^{2} + a x} A b^{5}}{1155 \, a^{4} x} + \frac {8 \, \sqrt {b x^{2} + a x} B b^{3}}{315 \, a^{2} x^{2}} - \frac {16 \, \sqrt {b x^{2} + a x} A b^{4}}{1155 \, a^{3} x^{2}} - \frac {2 \, \sqrt {b x^{2} + a x} B b^{2}}{105 \, a x^{3}} + \frac {4 \, \sqrt {b x^{2} + a x} A b^{3}}{385 \, a^{2} x^{3}} + \frac {\sqrt {b x^{2} + a x} B b}{63 \, x^{4}} - \frac {2 \, \sqrt {b x^{2} + a x} A b^{2}}{231 \, a x^{4}} + \frac {\sqrt {b x^{2} + a x} B a}{9 \, x^{5}} + \frac {\sqrt {b x^{2} + a x} A b}{132 \, x^{5}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} B}{3 \, x^{6}} + \frac {3 \, \sqrt {b x^{2} + a x} A a}{44 \, x^{6}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} A}{4 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.56, size = 126, normalized size = 1.08 \begin {gather*} -\frac {2 \, {\left (315 \, A a^{5} + 8 \, {\left (11 \, B a b^{4} - 6 \, A b^{5}\right )} x^{5} - 4 \, {\left (11 \, B a^{2} b^{3} - 6 \, A a b^{4}\right )} x^{4} + 3 \, {\left (11 \, B a^{3} b^{2} - 6 \, A a^{2} b^{3}\right )} x^{3} + 5 \, {\left (110 \, B a^{4} b + 3 \, A a^{3} b^{2}\right )} x^{2} + 35 \, {\left (11 \, B a^{5} + 12 \, A a^{4} b\right )} x\right )} \sqrt {b x + a}}{3465 \, a^{4} x^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [A]
time = 1.75, size = 142, normalized size = 1.21 \begin {gather*} -\frac {2 \, {\left ({\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (11 \, B a^{2} b^{10} - 6 \, A a b^{11}\right )} {\left (b x + a\right )}}{a^{5}} - \frac {11 \, {\left (11 \, B a^{3} b^{10} - 6 \, A a^{2} b^{11}\right )}}{a^{5}}\right )} + \frac {99 \, {\left (11 \, B a^{4} b^{10} - 6 \, A a^{3} b^{11}\right )}}{a^{5}}\right )} - \frac {693 \, {\left (B a^{5} b^{10} - A a^{4} b^{11}\right )}}{a^{5}}\right )} {\left (b x + a\right )}^{\frac {5}{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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Mupad [B]
time = 0.85, size = 117, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {a+b\,x}\,\left (\frac {2\,A\,a}{11}+\frac {x\,\left (770\,B\,a^5+840\,A\,b\,a^4\right )}{3465\,a^4}-\frac {x^5\,\left (96\,A\,b^5-176\,B\,a\,b^4\right )}{3465\,a^4}-\frac {2\,b^2\,x^3\,\left (6\,A\,b-11\,B\,a\right )}{1155\,a^2}+\frac {8\,b^3\,x^4\,\left (6\,A\,b-11\,B\,a\right )}{3465\,a^3}+\frac {2\,b\,x^2\,\left (3\,A\,b+110\,B\,a\right )}{693\,a}\right )}{x^{11/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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