Optimal. Leaf size=57 \[ -\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a-b x)}+\frac {3 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {44, 53, 65, 214}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 53
Rule 65
Rule 214
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) \text {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 [A]
time = 0.06, size = 55, normalized size = 0.96 \begin {gather*} \frac {-2 a+3 b x}{a^2 \sqrt {x} (a-b x)}+\frac {3 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 7.25, size = 340, normalized size = 5.96 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\text {DirectedInfinity}\left [\frac {1}{x^{\frac {5}{2}}}\right ],a\text {==}0\text {\&\&}b\text {==}0\right \},\left \{\frac {-2}{a^2 \sqrt {x}},b\text {==}0\right \},\left \{\frac {-2}{5 b^2 x^{\frac {5}{2}}},a\text {==}0\right \}\right \},\frac {-4 a \sqrt {\frac {a}{b}}}{2 a^3 \sqrt {x} \sqrt {\frac {a}{b}}-2 a^2 b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}}-\frac {3 a \sqrt {x} \text {Log}\left [\sqrt {x}-\sqrt {\frac {a}{b}}\right ]}{2 a^3 \sqrt {x} \sqrt {\frac {a}{b}}-2 a^2 b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}}+\frac {3 a \sqrt {x} \text {Log}\left [\sqrt {x}+\sqrt {\frac {a}{b}}\right ]}{2 a^3 \sqrt {x} \sqrt {\frac {a}{b}}-2 a^2 b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}}+\frac {6 b x \sqrt {\frac {a}{b}}}{2 a^3 \sqrt {x} \sqrt {\frac {a}{b}}-2 a^2 b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}}-\frac {3 b x^{\frac {3}{2}} \text {Log}\left [\sqrt {x}+\sqrt {\frac {a}{b}}\right ]}{2 a^3 \sqrt {x} \sqrt {\frac {a}{b}}-2 a^2 b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}}+\frac {3 b x^{\frac {3}{2}} \text {Log}\left [\sqrt {x}-\sqrt {\frac {a}{b}}\right ]}{2 a^3 \sqrt {x} \sqrt {\frac {a}{b}}-2 a^2 b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.13, size = 48, normalized size = 0.84
method | result | size |
derivativedivides | \(-\frac {2}{a^{2} \sqrt {x}}+\frac {2 b \left (\frac {\sqrt {x}}{-2 b x +2 a}+\frac {3 \arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}\) | \(48\) |
default | \(-\frac {2}{a^{2} \sqrt {x}}+\frac {2 b \left (\frac {\sqrt {x}}{-2 b x +2 a}+\frac {3 \arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}\) | \(48\) |
risch | \(-\frac {2}{a^{2} \sqrt {x}}-\frac {b \left (\frac {\sqrt {x}}{b x -a}-\frac {3 \arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{a^{2}}\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 69, normalized size = 1.21 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 151, normalized size = 2.65 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 7.22, size = 354, normalized size = 6.21 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{5 b^{2} x^{\frac {5}{2}}} & \text {for}\: a = 0 \\- \frac {2}{a^{2} \sqrt {x}} & \text {for}\: b = 0 \\- \frac {3 a \sqrt {x} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{2 a^{3} \sqrt {x} \sqrt {\frac {a}{b}} - 2 a^{2} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}} + \frac {3 a \sqrt {x} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{2 a^{3} \sqrt {x} \sqrt {\frac {a}{b}} - 2 a^{2} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}} - \frac {4 a \sqrt {\frac {a}{b}}}{2 a^{3} \sqrt {x} \sqrt {\frac {a}{b}} - 2 a^{2} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}} + \frac {3 b x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{2 a^{3} \sqrt {x} \sqrt {\frac {a}{b}} - 2 a^{2} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}} - \frac {3 b x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{2 a^{3} \sqrt {x} \sqrt {\frac {a}{b}} - 2 a^{2} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}} + \frac {6 b x \sqrt {\frac {a}{b}}}{2 a^{3} \sqrt {x} \sqrt {\frac {a}{b}} - 2 a^{2} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 74, normalized size = 1.30 \begin {gather*} 2 \left (-\frac {3 x b-2 a}{2 a^{2} \left (\sqrt {x} x b-\sqrt {x} a\right )}-\frac {3 b \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{2 a^{2} \sqrt {-a b}}\right ) \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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