Optimal. Leaf size=110 \[ -\frac {63 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{4 a^{11/2}}+\frac {63 b^2 \sqrt {x}}{4 a^5}-\frac {21 b x^{3/2}}{4 a^4}+\frac {63 x^{5/2}}{20 a^3}-\frac {9 x^{7/2}}{4 a^2 (a x+b)}-\frac {x^{9/2}}{2 a (a x+b)^2} \]
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Rubi [A] time = 0.04, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {263, 47, 50, 63, 205} \[ \frac {63 b^2 \sqrt {x}}{4 a^5}-\frac {63 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{4 a^{11/2}}-\frac {9 x^{7/2}}{4 a^2 (a x+b)}-\frac {21 b x^{3/2}}{4 a^4}+\frac {63 x^{5/2}}{20 a^3}-\frac {x^{9/2}}{2 a (a x+b)^2} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 205
Rule 263
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{\left (a+\frac {b}{x}\right )^3} \, dx &=\int \frac {x^{9/2}}{(b+a x)^3} \, dx\\ &=-\frac {x^{9/2}}{2 a (b+a x)^2}+\frac {9 \int \frac {x^{7/2}}{(b+a x)^2} \, dx}{4 a}\\ &=-\frac {x^{9/2}}{2 a (b+a x)^2}-\frac {9 x^{7/2}}{4 a^2 (b+a x)}+\frac {63 \int \frac {x^{5/2}}{b+a x} \, dx}{8 a^2}\\ &=\frac {63 x^{5/2}}{20 a^3}-\frac {x^{9/2}}{2 a (b+a x)^2}-\frac {9 x^{7/2}}{4 a^2 (b+a x)}-\frac {(63 b) \int \frac {x^{3/2}}{b+a x} \, dx}{8 a^3}\\ &=-\frac {21 b x^{3/2}}{4 a^4}+\frac {63 x^{5/2}}{20 a^3}-\frac {x^{9/2}}{2 a (b+a x)^2}-\frac {9 x^{7/2}}{4 a^2 (b+a x)}+\frac {\left (63 b^2\right ) \int \frac {\sqrt {x}}{b+a x} \, dx}{8 a^4}\\ &=\frac {63 b^2 \sqrt {x}}{4 a^5}-\frac {21 b x^{3/2}}{4 a^4}+\frac {63 x^{5/2}}{20 a^3}-\frac {x^{9/2}}{2 a (b+a x)^2}-\frac {9 x^{7/2}}{4 a^2 (b+a x)}-\frac {\left (63 b^3\right ) \int \frac {1}{\sqrt {x} (b+a x)} \, dx}{8 a^5}\\ &=\frac {63 b^2 \sqrt {x}}{4 a^5}-\frac {21 b x^{3/2}}{4 a^4}+\frac {63 x^{5/2}}{20 a^3}-\frac {x^{9/2}}{2 a (b+a x)^2}-\frac {9 x^{7/2}}{4 a^2 (b+a x)}-\frac {\left (63 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {x}\right )}{4 a^5}\\ &=\frac {63 b^2 \sqrt {x}}{4 a^5}-\frac {21 b x^{3/2}}{4 a^4}+\frac {63 x^{5/2}}{20 a^3}-\frac {x^{9/2}}{2 a (b+a x)^2}-\frac {9 x^{7/2}}{4 a^2 (b+a x)}-\frac {63 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{4 a^{11/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 27, normalized size = 0.25 \[ \frac {2 x^{11/2} \, _2F_1\left (3,\frac {11}{2};\frac {13}{2};-\frac {a x}{b}\right )}{11 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 254, normalized size = 2.31 \[ \left [\frac {315 \, {\left (a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {a x - 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - b}{a x + b}\right ) + 2 \, {\left (8 \, a^{4} x^{4} - 24 \, a^{3} b x^{3} + 168 \, a^{2} b^{2} x^{2} + 525 \, a b^{3} x + 315 \, b^{4}\right )} \sqrt {x}}{40 \, {\left (a^{7} x^{2} + 2 \, a^{6} b x + a^{5} b^{2}\right )}}, -\frac {315 \, {\left (a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {x} \sqrt {\frac {b}{a}}}{b}\right ) - {\left (8 \, a^{4} x^{4} - 24 \, a^{3} b x^{3} + 168 \, a^{2} b^{2} x^{2} + 525 \, a b^{3} x + 315 \, b^{4}\right )} \sqrt {x}}{20 \, {\left (a^{7} x^{2} + 2 \, a^{6} b x + a^{5} b^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 88, normalized size = 0.80 \[ -\frac {63 \, b^{3} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{5}} + \frac {17 \, a b^{3} x^{\frac {3}{2}} + 15 \, b^{4} \sqrt {x}}{4 \, {\left (a x + b\right )}^{2} a^{5}} + \frac {2 \, {\left (a^{12} x^{\frac {5}{2}} - 5 \, a^{11} b x^{\frac {3}{2}} + 30 \, a^{10} b^{2} \sqrt {x}\right )}}{5 \, a^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 90, normalized size = 0.82 \[ \frac {17 b^{3} x^{\frac {3}{2}}}{4 \left (a x +b \right )^{2} a^{4}}+\frac {2 x^{\frac {5}{2}}}{5 a^{3}}+\frac {15 b^{4} \sqrt {x}}{4 \left (a x +b \right )^{2} a^{5}}-\frac {63 b^{3} \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}\, a^{5}}-\frac {2 b \,x^{\frac {3}{2}}}{a^{4}}+\frac {12 b^{2} \sqrt {x}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.36, size = 99, normalized size = 0.90 \[ \frac {8 \, a^{4} - \frac {24 \, a^{3} b}{x} + \frac {168 \, a^{2} b^{2}}{x^{2}} + \frac {525 \, a b^{3}}{x^{3}} + \frac {315 \, b^{4}}{x^{4}}}{20 \, {\left (\frac {a^{7}}{x^{\frac {5}{2}}} + \frac {2 \, a^{6} b}{x^{\frac {7}{2}}} + \frac {a^{5} b^{2}}{x^{\frac {9}{2}}}\right )}} + \frac {63 \, b^{3} \arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{4 \, \sqrt {a b} a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 91, normalized size = 0.83 \[ \frac {\frac {15\,b^4\,\sqrt {x}}{4}+\frac {17\,a\,b^3\,x^{3/2}}{4}}{a^7\,x^2+2\,a^6\,b\,x+a^5\,b^2}+\frac {2\,x^{5/2}}{5\,a^3}-\frac {2\,b\,x^{3/2}}{a^4}+\frac {12\,b^2\,\sqrt {x}}{a^5}-\frac {63\,b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{4\,a^{11/2}} \]
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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