Optimal. Leaf size=82 \[ -\frac {15 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{4 b^{7/2}}+\frac {5}{4 b^2 \sqrt {x} (a x+b)}+\frac {1}{2 b \sqrt {x} (a x+b)^2}-\frac {15}{4 b^3 \sqrt {x}} \]
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Rubi [A] time = 0.03, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {263, 51, 63, 205} \[ \frac {5}{4 b^2 \sqrt {x} (a x+b)}-\frac {15 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{4 b^{7/2}}+\frac {1}{2 b \sqrt {x} (a x+b)^2}-\frac {15}{4 b^3 \sqrt {x}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 205
Rule 263
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x^{9/2}} \, dx &=\int \frac {1}{x^{3/2} (b+a x)^3} \, dx\\ &=\frac {1}{2 b \sqrt {x} (b+a x)^2}+\frac {5 \int \frac {1}{x^{3/2} (b+a x)^2} \, dx}{4 b}\\ &=\frac {1}{2 b \sqrt {x} (b+a x)^2}+\frac {5}{4 b^2 \sqrt {x} (b+a x)}+\frac {15 \int \frac {1}{x^{3/2} (b+a x)} \, dx}{8 b^2}\\ &=-\frac {15}{4 b^3 \sqrt {x}}+\frac {1}{2 b \sqrt {x} (b+a x)^2}+\frac {5}{4 b^2 \sqrt {x} (b+a x)}-\frac {(15 a) \int \frac {1}{\sqrt {x} (b+a x)} \, dx}{8 b^3}\\ &=-\frac {15}{4 b^3 \sqrt {x}}+\frac {1}{2 b \sqrt {x} (b+a x)^2}+\frac {5}{4 b^2 \sqrt {x} (b+a x)}-\frac {(15 a) \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {x}\right )}{4 b^3}\\ &=-\frac {15}{4 b^3 \sqrt {x}}+\frac {1}{2 b \sqrt {x} (b+a x)^2}+\frac {5}{4 b^2 \sqrt {x} (b+a x)}-\frac {15 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{4 b^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 25, normalized size = 0.30 \[ -\frac {2 \, _2F_1\left (-\frac {1}{2},3;\frac {1}{2};-\frac {a x}{b}\right )}{b^3 \sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 214, normalized size = 2.61 \[ \left [\frac {15 \, {\left (a^{2} x^{3} + 2 \, a b x^{2} + b^{2} x\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {a x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - b}{a x + b}\right ) - 2 \, {\left (15 \, a^{2} x^{2} + 25 \, a b x + 8 \, b^{2}\right )} \sqrt {x}}{8 \, {\left (a^{2} b^{3} x^{3} + 2 \, a b^{4} x^{2} + b^{5} x\right )}}, \frac {15 \, {\left (a^{2} x^{3} + 2 \, a b x^{2} + b^{2} x\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {\frac {a}{b}}}{a \sqrt {x}}\right ) - {\left (15 \, a^{2} x^{2} + 25 \, a b x + 8 \, b^{2}\right )} \sqrt {x}}{4 \, {\left (a^{2} b^{3} x^{3} + 2 \, a b^{4} x^{2} + b^{5} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 59, normalized size = 0.72 \[ -\frac {15 \, a \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{4 \, \sqrt {a b} b^{3}} - \frac {2}{b^{3} \sqrt {x}} - \frac {7 \, a^{2} x^{\frac {3}{2}} + 9 \, a b \sqrt {x}}{4 \, {\left (a x + b\right )}^{2} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 66, normalized size = 0.80 \[ -\frac {7 a^{2} x^{\frac {3}{2}}}{4 \left (a x +b \right )^{2} b^{3}}-\frac {9 a \sqrt {x}}{4 \left (a x +b \right )^{2} b^{2}}-\frac {15 a \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}\, b^{3}}-\frac {2}{b^{3} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.42, size = 75, normalized size = 0.91 \[ -\frac {\frac {7 \, a^{2}}{\sqrt {x}} + \frac {9 \, a b}{x^{\frac {3}{2}}}}{4 \, {\left (a^{2} b^{3} + \frac {2 \, a b^{4}}{x} + \frac {b^{5}}{x^{2}}\right )}} + \frac {15 \, a \arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{4 \, \sqrt {a b} b^{3}} - \frac {2}{b^{3} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.13, size = 70, normalized size = 0.85 \[ -\frac {\frac {2}{b}+\frac {15\,a^2\,x^2}{4\,b^3}+\frac {25\,a\,x}{4\,b^2}}{a^2\,x^{5/2}+b^2\,\sqrt {x}+2\,a\,b\,x^{3/2}}-\frac {15\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{4\,b^{7/2}} \]
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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