Optimal. Leaf size=72 \[ -\frac {3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b}}-\frac {3 \sqrt {x}}{4 a^2 (a-b x)}-\frac {\sqrt {x}}{2 a (a-b x)^2} \]
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Rubi [A] time = 0.02, antiderivative size = 72, 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} \begin {gather*} -\frac {3 \sqrt {x}}{4 a^2 (a-b x)}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b}}-\frac {\sqrt {x}}{2 a (a-b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} (-a+b x)^3} \, dx &=-\frac {\sqrt {x}}{2 a (a-b x)^2}-\frac {3 \int \frac {1}{\sqrt {x} (-a+b x)^2} \, dx}{4 a}\\ &=-\frac {\sqrt {x}}{2 a (a-b x)^2}-\frac {3 \sqrt {x}}{4 a^2 (a-b x)}+\frac {3 \int \frac {1}{\sqrt {x} (-a+b x)} \, dx}{8 a^2}\\ &=-\frac {\sqrt {x}}{2 a (a-b x)^2}-\frac {3 \sqrt {x}}{4 a^2 (a-b x)}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-a+b x^2} \, dx,x,\sqrt {x}\right )}{4 a^2}\\ &=-\frac {\sqrt {x}}{2 a (a-b x)^2}-\frac {3 \sqrt {x}}{4 a^2 (a-b x)}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 24, normalized size = 0.33 \begin {gather*} -\frac {2 \sqrt {x} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};\frac {b x}{a}\right )}{a^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 64, normalized size = 0.89 \begin {gather*} \frac {3 b x^{3/2}-5 a \sqrt {x}}{4 a^2 (a-b x)^2}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{5/2} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 185, normalized size = 2.57 \begin {gather*} \left [\frac {3 \, {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt {a b} \log \left (\frac {b x + a - 2 \, \sqrt {a b} \sqrt {x}}{b x - a}\right ) + 2 \, {\left (3 \, a b^{2} x - 5 \, a^{2} b\right )} \sqrt {x}}{8 \, {\left (a^{3} b^{3} x^{2} - 2 \, a^{4} b^{2} x + a^{5} b\right )}}, \frac {3 \, {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt {-a b} \arctan \left (\frac {\sqrt {-a b}}{b \sqrt {x}}\right ) + {\left (3 \, a b^{2} x - 5 \, a^{2} b\right )} \sqrt {x}}{4 \, {\left (a^{3} b^{3} x^{2} - 2 \, a^{4} b^{2} x + a^{5} b\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.15, size = 51, normalized size = 0.71 \begin {gather*} \frac {3 \, \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{4 \, \sqrt {-a b} a^{2}} + \frac {3 \, b x^{\frac {3}{2}} - 5 \, a \sqrt {x}}{4 \, {\left (b x - a\right )}^{2} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 63, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {x}}{2 \left (b x -a \right )^{2} a}-\frac {3 \left (\frac {\arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a}-\frac {\sqrt {x}}{2 \left (b x -a \right ) a}\right )}{2 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.88, size = 77, normalized size = 1.07 \begin {gather*} \frac {3 \, b x^{\frac {3}{2}} - 5 \, a \sqrt {x}}{4 \, {\left (a^{2} b^{2} x^{2} - 2 \, a^{3} b x + a^{4}\right )}} + \frac {3 \, \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 58, normalized size = 0.81 \begin {gather*} -\frac {\frac {5\,\sqrt {x}}{4\,a}-\frac {3\,b\,x^{3/2}}{4\,a^2}}{a^2-2\,a\,b\,x+b^2\,x^2}-\frac {3\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,a^{5/2}\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 25.67, size = 660, normalized size = 9.17 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2 \sqrt {x}}{a^{3}} & \text {for}\: b = 0 \\- \frac {2}{5 b^{3} x^{\frac {5}{2}}} & \text {for}\: a = 0 \\- \frac {10 a^{\frac {3}{2}} b \sqrt {x} \sqrt {\frac {1}{b}}}{8 a^{\frac {9}{2}} b \sqrt {\frac {1}{b}} - 16 a^{\frac {7}{2}} b^{2} x \sqrt {\frac {1}{b}} + 8 a^{\frac {5}{2}} b^{3} x^{2} \sqrt {\frac {1}{b}}} + \frac {6 \sqrt {a} b^{2} x^{\frac {3}{2}} \sqrt {\frac {1}{b}}}{8 a^{\frac {9}{2}} b \sqrt {\frac {1}{b}} - 16 a^{\frac {7}{2}} b^{2} x \sqrt {\frac {1}{b}} + 8 a^{\frac {5}{2}} b^{3} x^{2} \sqrt {\frac {1}{b}}} + \frac {3 a^{2} \log {\left (- \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 a^{\frac {9}{2}} b \sqrt {\frac {1}{b}} - 16 a^{\frac {7}{2}} b^{2} x \sqrt {\frac {1}{b}} + 8 a^{\frac {5}{2}} b^{3} x^{2} \sqrt {\frac {1}{b}}} - \frac {3 a^{2} \log {\left (\sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 a^{\frac {9}{2}} b \sqrt {\frac {1}{b}} - 16 a^{\frac {7}{2}} b^{2} x \sqrt {\frac {1}{b}} + 8 a^{\frac {5}{2}} b^{3} x^{2} \sqrt {\frac {1}{b}}} - \frac {6 a b x \log {\left (- \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 a^{\frac {9}{2}} b \sqrt {\frac {1}{b}} - 16 a^{\frac {7}{2}} b^{2} x \sqrt {\frac {1}{b}} + 8 a^{\frac {5}{2}} b^{3} x^{2} \sqrt {\frac {1}{b}}} + \frac {6 a b x \log {\left (\sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 a^{\frac {9}{2}} b \sqrt {\frac {1}{b}} - 16 a^{\frac {7}{2}} b^{2} x \sqrt {\frac {1}{b}} + 8 a^{\frac {5}{2}} b^{3} x^{2} \sqrt {\frac {1}{b}}} + \frac {3 b^{2} x^{2} \log {\left (- \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 a^{\frac {9}{2}} b \sqrt {\frac {1}{b}} - 16 a^{\frac {7}{2}} b^{2} x \sqrt {\frac {1}{b}} + 8 a^{\frac {5}{2}} b^{3} x^{2} \sqrt {\frac {1}{b}}} - \frac {3 b^{2} x^{2} \log {\left (\sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{8 a^{\frac {9}{2}} b \sqrt {\frac {1}{b}} - 16 a^{\frac {7}{2}} b^{2} x \sqrt {\frac {1}{b}} + 8 a^{\frac {5}{2}} b^{3} x^{2} \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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