Optimal. Leaf size=70 \[ \frac {3 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{4 a^{5/2} \sqrt {b}}-\frac {3 \sqrt {x}}{4 a^2 (a x+b)}-\frac {x^{3/2}}{2 a (a x+b)^2} \]
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Rubi [A] time = 0.02, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {263, 47, 63, 205} \[ -\frac {3 \sqrt {x}}{4 a^2 (a x+b)}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{4 a^{5/2} \sqrt {b}}-\frac {x^{3/2}}{2 a (a x+b)^2} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 205
Rule 263
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^{3/2}} \, dx &=\int \frac {x^{3/2}}{(b+a x)^3} \, dx\\ &=-\frac {x^{3/2}}{2 a (b+a x)^2}+\frac {3 \int \frac {\sqrt {x}}{(b+a x)^2} \, dx}{4 a}\\ &=-\frac {x^{3/2}}{2 a (b+a x)^2}-\frac {3 \sqrt {x}}{4 a^2 (b+a x)}+\frac {3 \int \frac {1}{\sqrt {x} (b+a x)} \, dx}{8 a^2}\\ &=-\frac {x^{3/2}}{2 a (b+a x)^2}-\frac {3 \sqrt {x}}{4 a^2 (b+a x)}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {x}\right )}{4 a^2}\\ &=-\frac {x^{3/2}}{2 a (b+a x)^2}-\frac {3 \sqrt {x}}{4 a^2 (b+a x)}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{4 a^{5/2} \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 59, normalized size = 0.84 \[ \frac {3 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{4 a^{5/2} \sqrt {b}}-\frac {\sqrt {x} (5 a x+3 b)}{4 a^2 (a x+b)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 185, normalized size = 2.64 \[ \left [-\frac {3 \, {\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt {-a b} \log \left (\frac {a x - b - 2 \, \sqrt {-a b} \sqrt {x}}{a x + b}\right ) + 2 \, {\left (5 \, a^{2} b x + 3 \, a b^{2}\right )} \sqrt {x}}{8 \, {\left (a^{5} b x^{2} + 2 \, a^{4} b^{2} x + a^{3} b^{3}\right )}}, -\frac {3 \, {\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{a \sqrt {x}}\right ) + {\left (5 \, a^{2} b x + 3 \, a b^{2}\right )} \sqrt {x}}{4 \, {\left (a^{5} b x^{2} + 2 \, a^{4} b^{2} x + a^{3} b^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 47, normalized size = 0.67 \[ \frac {3 \, \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} a^{2}} - \frac {5 \, a x^{\frac {3}{2}} + 3 \, b \sqrt {x}}{4 \, {\left (a x + b\right )}^{2} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 50, normalized size = 0.71 \[ \frac {3 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}\, a^{2}}+\frac {-\frac {5 x^{\frac {3}{2}}}{4 a}-\frac {3 b \sqrt {x}}{4 a^{2}}}{\left (a x +b \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.33, size = 62, normalized size = 0.89 \[ -\frac {\frac {5 \, a}{\sqrt {x}} + \frac {3 \, b}{x^{\frac {3}{2}}}}{4 \, {\left (a^{4} + \frac {2 \, a^{3} b}{x} + \frac {a^{2} b^{2}}{x^{2}}\right )}} - \frac {3 \, \arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{4 \, \sqrt {a b} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 58, normalized size = 0.83 \[ \frac {3\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{4\,a^{5/2}\,\sqrt {b}}-\frac {\frac {5\,x^{3/2}}{4\,a}+\frac {3\,b\,\sqrt {x}}{4\,a^2}}{a^2\,x^2+2\,a\,b\,x+b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 110.84, size = 726, normalized size = 10.37 \[ \begin {cases} \tilde {\infty } x^{\frac {5}{2}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {5}{2}}}{5 b^{3}} & \text {for}\: a = 0 \\- \frac {2}{a^{3} \sqrt {x}} & \text {for}\: b = 0 \\- \frac {10 i a^{2} \sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {1}{a}}}{8 i a^{5} \sqrt {b} x^{2} \sqrt {\frac {1}{a}} + 16 i a^{4} b^{\frac {3}{2}} x \sqrt {\frac {1}{a}} + 8 i a^{3} b^{\frac {5}{2}} \sqrt {\frac {1}{a}}} + \frac {3 a^{2} x^{2} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{8 i a^{5} \sqrt {b} x^{2} \sqrt {\frac {1}{a}} + 16 i a^{4} b^{\frac {3}{2}} x \sqrt {\frac {1}{a}} + 8 i a^{3} b^{\frac {5}{2}} \sqrt {\frac {1}{a}}} - \frac {3 a^{2} x^{2} \log {\left (i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{8 i a^{5} \sqrt {b} x^{2} \sqrt {\frac {1}{a}} + 16 i a^{4} b^{\frac {3}{2}} x \sqrt {\frac {1}{a}} + 8 i a^{3} b^{\frac {5}{2}} \sqrt {\frac {1}{a}}} - \frac {6 i a b^{\frac {3}{2}} \sqrt {x} \sqrt {\frac {1}{a}}}{8 i a^{5} \sqrt {b} x^{2} \sqrt {\frac {1}{a}} + 16 i a^{4} b^{\frac {3}{2}} x \sqrt {\frac {1}{a}} + 8 i a^{3} b^{\frac {5}{2}} \sqrt {\frac {1}{a}}} + \frac {6 a b x \log {\left (- i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{8 i a^{5} \sqrt {b} x^{2} \sqrt {\frac {1}{a}} + 16 i a^{4} b^{\frac {3}{2}} x \sqrt {\frac {1}{a}} + 8 i a^{3} b^{\frac {5}{2}} \sqrt {\frac {1}{a}}} - \frac {6 a b x \log {\left (i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{8 i a^{5} \sqrt {b} x^{2} \sqrt {\frac {1}{a}} + 16 i a^{4} b^{\frac {3}{2}} x \sqrt {\frac {1}{a}} + 8 i a^{3} b^{\frac {5}{2}} \sqrt {\frac {1}{a}}} + \frac {3 b^{2} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{8 i a^{5} \sqrt {b} x^{2} \sqrt {\frac {1}{a}} + 16 i a^{4} b^{\frac {3}{2}} x \sqrt {\frac {1}{a}} + 8 i a^{3} b^{\frac {5}{2}} \sqrt {\frac {1}{a}}} - \frac {3 b^{2} \log {\left (i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{8 i a^{5} \sqrt {b} x^{2} \sqrt {\frac {1}{a}} + 16 i a^{4} b^{\frac {3}{2}} x \sqrt {\frac {1}{a}} + 8 i a^{3} b^{\frac {5}{2}} \sqrt {\frac {1}{a}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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