Optimal. Leaf size=57 \[ \frac {3 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a-b x)} \]
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Rubi [A] time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {51, 63, 208} \[ \frac {3 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a-b x)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{x^{3/2} (-a+b x)^2} \, dx &=\frac {1}{a \sqrt {x} (a-b x)}-\frac {3 \int \frac {1}{x^{3/2} (-a+b x)} \, dx}{2 a}\\ &=-\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a-b x)}-\frac {(3 b) \int \frac {1}{\sqrt {x} (-a+b x)} \, dx}{2 a^2}\\ &=-\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a-b x)}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{-a+b x^2} \, dx,x,\sqrt {x}\right )}{a^2}\\ &=-\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a-b x)}+\frac {3 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 24, normalized size = 0.42 \[ -\frac {2 \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {b x}{a}\right )}{a^2 \sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 151, normalized size = 2.65 \[ \left [\frac {3 \, {\left (b x^{2} - a x\right )} \sqrt {\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {\frac {b}{a}} + a}{b x - a}\right ) - 2 \, {\left (3 \, b x - 2 \, a\right )} \sqrt {x}}{2 \, {\left (a^{2} b x^{2} - a^{3} x\right )}}, -\frac {3 \, {\left (b x^{2} - a x\right )} \sqrt {-\frac {b}{a}} \arctan \left (\frac {a \sqrt {-\frac {b}{a}}}{b \sqrt {x}}\right ) + {\left (3 \, b x - 2 \, a\right )} \sqrt {x}}{a^{2} b x^{2} - a^{3} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 52, normalized size = 0.91 \[ -\frac {3 \, b \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{\sqrt {-a b} a^{2}} - \frac {3 \, b x - 2 \, a}{{\left (b x^{\frac {3}{2}} - a \sqrt {x}\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 49, normalized size = 0.86 \[ -\frac {2 \left (-\frac {3 \arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}+\frac {\sqrt {x}}{2 b x -2 a}\right ) b}{a^{2}}-\frac {2}{a^{2} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 69, normalized size = 1.21 \[ -\frac {3 \, b x - 2 \, a}{a^{2} b x^{\frac {3}{2}} - a^{3} \sqrt {x}} - \frac {3 \, b \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 49, normalized size = 0.86 \[ \frac {3\,\sqrt {b}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {\frac {2}{a}-\frac {3\,b\,x}{a^2}}{a\,\sqrt {x}-b\,x^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.54, size = 403, normalized size = 7.07 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{a^{2} \sqrt {x}} & \text {for}\: b = 0 \\- \frac {2}{5 b^{2} x^{\frac {5}{2}}} & \text {for}\: a = 0 \\- \frac {4 a^{\frac {3}{2}} \sqrt {\frac {1}{b}}}{2 a^{\frac {7}{2}} \sqrt {x} \sqrt {\frac {1}{b}} - 2 a^{\frac {5}{2}} b x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} + \frac {6 \sqrt {a} b x \sqrt {\frac {1}{b}}}{2 a^{\frac {7}{2}} \sqrt {x} \sqrt {\frac {1}{b}} - 2 a^{\frac {5}{2}} b x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} - \frac {3 a \sqrt {x} \log {\left (- \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 a^{\frac {7}{2}} \sqrt {x} \sqrt {\frac {1}{b}} - 2 a^{\frac {5}{2}} b x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} + \frac {3 a \sqrt {x} \log {\left (\sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 a^{\frac {7}{2}} \sqrt {x} \sqrt {\frac {1}{b}} - 2 a^{\frac {5}{2}} b x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} + \frac {3 b x^{\frac {3}{2}} \log {\left (- \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 a^{\frac {7}{2}} \sqrt {x} \sqrt {\frac {1}{b}} - 2 a^{\frac {5}{2}} b x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} - \frac {3 b x^{\frac {3}{2}} \log {\left (\sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 a^{\frac {7}{2}} \sqrt {x} \sqrt {\frac {1}{b}} - 2 a^{\frac {5}{2}} b x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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