Optimal. Leaf size=53 \[ -\frac {2 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {2 b}{a^2 \sqrt {x}}+\frac {2}{3 a x^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 53, 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} \[ -\frac {2 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {2 b}{a^2 \sqrt {x}}+\frac {2}{3 a x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{x^{5/2} (-a+b x)} \, dx &=\frac {2}{3 a x^{3/2}}+\frac {b \int \frac {1}{x^{3/2} (-a+b x)} \, dx}{a}\\ &=\frac {2}{3 a x^{3/2}}+\frac {2 b}{a^2 \sqrt {x}}+\frac {b^2 \int \frac {1}{\sqrt {x} (-a+b x)} \, dx}{a^2}\\ &=\frac {2}{3 a x^{3/2}}+\frac {2 b}{a^2 \sqrt {x}}+\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-a+b x^2} \, dx,x,\sqrt {x}\right )}{a^2}\\ &=\frac {2}{3 a x^{3/2}}+\frac {2 b}{a^2 \sqrt {x}}-\frac {2 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 26, normalized size = 0.49 \[ \frac {2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {b x}{a}\right )}{3 a x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 113, normalized size = 2.13 \[ \left [\frac {3 \, b x^{2} \sqrt {\frac {b}{a}} \log \left (\frac {b x - 2 \, a \sqrt {x} \sqrt {\frac {b}{a}} + a}{b x - a}\right ) + 2 \, {\left (3 \, b x + a\right )} \sqrt {x}}{3 \, a^{2} x^{2}}, \frac {2 \, {\left (3 \, b x^{2} \sqrt {-\frac {b}{a}} \arctan \left (\frac {a \sqrt {-\frac {b}{a}}}{b \sqrt {x}}\right ) + {\left (3 \, b x + a\right )} \sqrt {x}\right )}}{3 \, a^{2} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 41, normalized size = 0.77 \[ \frac {2 \, b^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{\sqrt {-a b} a^{2}} + \frac {2 \, {\left (3 \, b x + a\right )}}{3 \, a^{2} x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 43, normalized size = 0.81 \[ -\frac {2 b^{2} \arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{2}}+\frac {2 b}{a^{2} \sqrt {x}}+\frac {2}{3 a \,x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 55, normalized size = 1.04 \[ \frac {b^{2} \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {2 \, {\left (3 \, b x + a\right )}}{3 \, a^{2} x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 37, normalized size = 0.70 \[ \frac {\frac {2}{3\,a}+\frac {2\,b\,x}{a^2}}{x^{3/2}}-\frac {2\,b^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.67, size = 112, normalized size = 2.11 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2}{3 a x^{\frac {3}{2}}} & \text {for}\: b = 0 \\- \frac {2}{5 b x^{\frac {5}{2}}} & \text {for}\: a = 0 \\\frac {2}{3 a x^{\frac {3}{2}}} + \frac {2 b}{a^{2} \sqrt {x}} + \frac {b \log {\left (- \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{a^{\frac {5}{2}} \sqrt {\frac {1}{b}}} - \frac {b \log {\left (\sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{a^{\frac {5}{2}} \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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