Optimal. Leaf size=100 \[ -\frac {32 b^3 \left (a+\frac {b}{x}\right )^{7/2} x^{7/2}}{3003 a^4}+\frac {16 b^2 \left (a+\frac {b}{x}\right )^{7/2} x^{9/2}}{429 a^3}-\frac {12 b \left (a+\frac {b}{x}\right )^{7/2} x^{11/2}}{143 a^2}+\frac {2 \left (a+\frac {b}{x}\right )^{7/2} x^{13/2}}{13 a} \]
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Rubi [A]
time = 0.02, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {277, 270}
\begin {gather*} -\frac {32 b^3 x^{7/2} \left (a+\frac {b}{x}\right )^{7/2}}{3003 a^4}+\frac {16 b^2 x^{9/2} \left (a+\frac {b}{x}\right )^{7/2}}{429 a^3}-\frac {12 b x^{11/2} \left (a+\frac {b}{x}\right )^{7/2}}{143 a^2}+\frac {2 x^{13/2} \left (a+\frac {b}{x}\right )^{7/2}}{13 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 277
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x}\right )^{5/2} x^{11/2} \, dx &=\frac {2 \left (a+\frac {b}{x}\right )^{7/2} x^{13/2}}{13 a}-\frac {(6 b) \int \left (a+\frac {b}{x}\right )^{5/2} x^{9/2} \, dx}{13 a}\\ &=-\frac {12 b \left (a+\frac {b}{x}\right )^{7/2} x^{11/2}}{143 a^2}+\frac {2 \left (a+\frac {b}{x}\right )^{7/2} x^{13/2}}{13 a}+\frac {\left (24 b^2\right ) \int \left (a+\frac {b}{x}\right )^{5/2} x^{7/2} \, dx}{143 a^2}\\ &=\frac {16 b^2 \left (a+\frac {b}{x}\right )^{7/2} x^{9/2}}{429 a^3}-\frac {12 b \left (a+\frac {b}{x}\right )^{7/2} x^{11/2}}{143 a^2}+\frac {2 \left (a+\frac {b}{x}\right )^{7/2} x^{13/2}}{13 a}-\frac {\left (16 b^3\right ) \int \left (a+\frac {b}{x}\right )^{5/2} x^{5/2} \, dx}{429 a^3}\\ &=-\frac {32 b^3 \left (a+\frac {b}{x}\right )^{7/2} x^{7/2}}{3003 a^4}+\frac {16 b^2 \left (a+\frac {b}{x}\right )^{7/2} x^{9/2}}{429 a^3}-\frac {12 b \left (a+\frac {b}{x}\right )^{7/2} x^{11/2}}{143 a^2}+\frac {2 \left (a+\frac {b}{x}\right )^{7/2} x^{13/2}}{13 a}\\ \end {align*}
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Mathematica [A]
time = 5.84, size = 60, normalized size = 0.60 \begin {gather*} \frac {2 \sqrt {a+\frac {b}{x}} \sqrt {x} (b+a x)^3 \left (-16 b^3+56 a b^2 x-126 a^2 b x^2+231 a^3 x^3\right )}{3003 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 57, normalized size = 0.57
method | result | size |
gosper | \(\frac {2 \left (a x +b \right ) \left (231 a^{3} x^{3}-126 a^{2} b \,x^{2}+56 a \,b^{2} x -16 b^{3}\right ) x^{\frac {5}{2}} \left (\frac {a x +b}{x}\right )^{\frac {5}{2}}}{3003 a^{4}}\) | \(55\) |
default | \(\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (a x +b \right )^{3} \left (231 a^{3} x^{3}-126 a^{2} b \,x^{2}+56 a \,b^{2} x -16 b^{3}\right )}{3003 a^{4}}\) | \(57\) |
risch | \(\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (231 x^{6} a^{6}+567 a^{5} b \,x^{5}+371 a^{4} b^{2} x^{4}+5 a^{3} b^{3} x^{3}-6 a^{2} b^{4} x^{2}+8 b^{5} x a -16 b^{6}\right )}{3003 a^{4}}\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 69, normalized size = 0.69 \begin {gather*} \frac {2 \, {\left (231 \, {\left (a + \frac {b}{x}\right )}^{\frac {13}{2}} x^{\frac {13}{2}} - 819 \, {\left (a + \frac {b}{x}\right )}^{\frac {11}{2}} b x^{\frac {11}{2}} + 1001 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{2}} b^{2} x^{\frac {9}{2}} - 429 \, {\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} b^{3} x^{\frac {7}{2}}\right )}}{3003 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 82, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (231 \, a^{6} x^{6} + 567 \, a^{5} b x^{5} + 371 \, a^{4} b^{2} x^{4} + 5 \, a^{3} b^{3} x^{3} - 6 \, a^{2} b^{4} x^{2} + 8 \, a b^{5} x - 16 \, b^{6}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{3003 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 229 vs.
\(2 (76) = 152\).
time = 1.08, size = 229, normalized size = 2.29 \begin {gather*} \frac {2}{315} \, b^{2} {\left (\frac {16 \, b^{\frac {9}{2}}}{a^{4}} + \frac {35 \, {\left (a x + b\right )}^{\frac {9}{2}} - 135 \, {\left (a x + b\right )}^{\frac {7}{2}} b + 189 \, {\left (a x + b\right )}^{\frac {5}{2}} b^{2} - 105 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{3}}{a^{4}}\right )} \mathrm {sgn}\left (x\right ) - \frac {4}{3465} \, a b {\left (\frac {128 \, b^{\frac {11}{2}}}{a^{5}} - \frac {315 \, {\left (a x + b\right )}^{\frac {11}{2}} - 1540 \, {\left (a x + b\right )}^{\frac {9}{2}} b + 2970 \, {\left (a x + b\right )}^{\frac {7}{2}} b^{2} - 2772 \, {\left (a x + b\right )}^{\frac {5}{2}} b^{3} + 1155 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{4}}{a^{5}}\right )} \mathrm {sgn}\left (x\right ) + \frac {2}{9009} \, a^{2} {\left (\frac {256 \, b^{\frac {13}{2}}}{a^{6}} + \frac {693 \, {\left (a x + b\right )}^{\frac {13}{2}} - 4095 \, {\left (a x + b\right )}^{\frac {11}{2}} b + 10010 \, {\left (a x + b\right )}^{\frac {9}{2}} b^{2} - 12870 \, {\left (a x + b\right )}^{\frac {7}{2}} b^{3} + 9009 \, {\left (a x + b\right )}^{\frac {5}{2}} b^{4} - 3003 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{5}}{a^{6}}\right )} \mathrm {sgn}\left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.54, size = 78, normalized size = 0.78 \begin {gather*} \sqrt {a+\frac {b}{x}}\,\left (\frac {2\,a^2\,x^{13/2}}{13}+\frac {106\,b^2\,x^{9/2}}{429}+\frac {10\,b^3\,x^{7/2}}{3003\,a}-\frac {4\,b^4\,x^{5/2}}{1001\,a^2}+\frac {16\,b^5\,x^{3/2}}{3003\,a^3}-\frac {32\,b^6\,\sqrt {x}}{3003\,a^4}+\frac {54\,a\,b\,x^{11/2}}{143}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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