Optimal. Leaf size=56 \[ -\frac {3 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{b^{5/2}}+\frac {1}{b \sqrt {x} (a x+b)}-\frac {3}{b^2 \sqrt {x}} \]
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Rubi [A] time = 0.02, antiderivative size = 56, 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, 51, 63, 205} \[ -\frac {3 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{b^{5/2}}+\frac {1}{b \sqrt {x} (a x+b)}-\frac {3}{b^2 \sqrt {x}} \]
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 )^2 x^{7/2}} \, dx &=\int \frac {1}{x^{3/2} (b+a x)^2} \, dx\\ &=\frac {1}{b \sqrt {x} (b+a x)}+\frac {3 \int \frac {1}{x^{3/2} (b+a x)} \, dx}{2 b}\\ &=-\frac {3}{b^2 \sqrt {x}}+\frac {1}{b \sqrt {x} (b+a x)}-\frac {(3 a) \int \frac {1}{\sqrt {x} (b+a x)} \, dx}{2 b^2}\\ &=-\frac {3}{b^2 \sqrt {x}}+\frac {1}{b \sqrt {x} (b+a x)}-\frac {(3 a) \operatorname {Subst}\left (\int \frac {1}{b+a x^2} \, dx,x,\sqrt {x}\right )}{b^2}\\ &=-\frac {3}{b^2 \sqrt {x}}+\frac {1}{b \sqrt {x} (b+a x)}-\frac {3 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 25, normalized size = 0.45 \[ -\frac {2 \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};-\frac {a x}{b}\right )}{b^2 \sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 147, normalized size = 2.62 \[ \left [\frac {3 \, {\left (a x^{2} + b x\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {a x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - b}{a x + b}\right ) - 2 \, {\left (3 \, a x + 2 \, b\right )} \sqrt {x}}{2 \, {\left (a b^{2} x^{2} + b^{3} x\right )}}, \frac {3 \, {\left (a x^{2} + b x\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {\frac {a}{b}}}{a \sqrt {x}}\right ) - {\left (3 \, a x + 2 \, b\right )} \sqrt {x}}{a b^{2} x^{2} + b^{3} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 49, normalized size = 0.88 \[ -\frac {3 \, a \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} - \frac {3 \, a x + 2 \, b}{{\left (a x^{\frac {3}{2}} + b \sqrt {x}\right )} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 48, normalized size = 0.86 \[ -\frac {3 a \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, b^{2}}-\frac {a \sqrt {x}}{\left (a x +b \right ) b^{2}}-\frac {2}{b^{2} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.35, size = 52, normalized size = 0.93 \[ -\frac {a}{{\left (a b^{2} + \frac {b^{3}}{x}\right )} \sqrt {x}} + \frac {3 \, a \arctan \left (\frac {b}{\sqrt {a b} \sqrt {x}}\right )}{\sqrt {a b} b^{2}} - \frac {2}{b^{2} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.13, size = 48, normalized size = 0.86 \[ -\frac {\frac {2}{b}+\frac {3\,a\,x}{b^2}}{a\,x^{3/2}+b\,\sqrt {x}}-\frac {3\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{b^{5/2}} \]
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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