Optimal. Leaf size=106 \[ -\frac {a^{3/2} e^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{b^{7/2} c}+\frac {a^{3/2} e^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{b^{7/2} c}-\frac {2 e (e x)^{3/2}}{3 b^2 c} \]
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Rubi [A] time = 0.08, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {73, 321, 329, 298, 205, 208} \[ -\frac {a^{3/2} e^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{b^{7/2} c}+\frac {a^{3/2} e^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{b^{7/2} c}-\frac {2 e (e x)^{3/2}}{3 b^2 c} \]
Antiderivative was successfully verified.
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Rule 73
Rule 205
Rule 208
Rule 298
Rule 321
Rule 329
Rubi steps
\begin {align*} \int \frac {(e x)^{5/2}}{(a+b x) (a c-b c x)} \, dx &=\int \frac {(e x)^{5/2}}{a^2 c-b^2 c x^2} \, dx\\ &=-\frac {2 e (e x)^{3/2}}{3 b^2 c}+\frac {\left (a^2 e^2\right ) \int \frac {\sqrt {e x}}{a^2 c-b^2 c x^2} \, dx}{b^2}\\ &=-\frac {2 e (e x)^{3/2}}{3 b^2 c}+\frac {\left (2 a^2 e\right ) \operatorname {Subst}\left (\int \frac {x^2}{a^2 c-\frac {b^2 c x^4}{e^2}} \, dx,x,\sqrt {e x}\right )}{b^2}\\ &=-\frac {2 e (e x)^{3/2}}{3 b^2 c}+\frac {\left (a^2 e^3\right ) \operatorname {Subst}\left (\int \frac {1}{a e-b x^2} \, dx,x,\sqrt {e x}\right )}{b^3 c}-\frac {\left (a^2 e^3\right ) \operatorname {Subst}\left (\int \frac {1}{a e+b x^2} \, dx,x,\sqrt {e x}\right )}{b^3 c}\\ &=-\frac {2 e (e x)^{3/2}}{3 b^2 c}-\frac {a^{3/2} e^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{b^{7/2} c}+\frac {a^{3/2} e^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{b^{7/2} c}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 85, normalized size = 0.80 \[ -\frac {(e x)^{5/2} \left (3 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )-3 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )+2 b^{3/2} x^{3/2}\right )}{3 b^{7/2} c x^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 216, normalized size = 2.04 \[ \left [-\frac {4 \, \sqrt {e x} b e^{2} x + 6 \, a \sqrt {\frac {a e}{b}} e^{2} \arctan \left (\frac {\sqrt {e x} b \sqrt {\frac {a e}{b}}}{a e}\right ) - 3 \, a \sqrt {\frac {a e}{b}} e^{2} \log \left (\frac {b e x + 2 \, \sqrt {e x} b \sqrt {\frac {a e}{b}} + a e}{b x - a}\right )}{6 \, b^{3} c}, -\frac {4 \, \sqrt {e x} b e^{2} x + 6 \, a \sqrt {-\frac {a e}{b}} e^{2} \arctan \left (\frac {\sqrt {e x} b \sqrt {-\frac {a e}{b}}}{a e}\right ) - 3 \, a \sqrt {-\frac {a e}{b}} e^{2} \log \left (\frac {b e x - 2 \, \sqrt {e x} b \sqrt {-\frac {a e}{b}} - a e}{b x + a}\right )}{6 \, b^{3} c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 84, normalized size = 0.79 \[ -\frac {1}{3} \, {\left (\frac {3 \, a^{2} \arctan \left (\frac {b \sqrt {x} e^{\frac {1}{2}}}{\sqrt {-a b e}}\right ) e}{\sqrt {-a b e} b^{3} c} + \frac {2 \, x^{\frac {3}{2}} e^{\frac {1}{2}}}{b^{2} c} + \frac {3 \, a^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) e^{\frac {1}{2}}}{\sqrt {a b} b^{3} c}\right )} e^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 83, normalized size = 0.78 \[ \frac {a^{2} e^{3} \arctanh \left (\frac {\sqrt {e x}\, b}{\sqrt {a b e}}\right )}{\sqrt {a b e}\, b^{3} c}-\frac {a^{2} e^{3} \arctan \left (\frac {\sqrt {e x}\, b}{\sqrt {a b e}}\right )}{\sqrt {a b e}\, b^{3} c}-\frac {2 \left (e x \right )^{\frac {3}{2}} e}{3 b^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.38, size = 110, normalized size = 1.04 \[ -\frac {\frac {6 \, a^{2} e^{4} \arctan \left (\frac {\sqrt {e x} b}{\sqrt {a b e}}\right )}{\sqrt {a b e} b^{3} c} + \frac {3 \, a^{2} e^{4} \log \left (\frac {\sqrt {e x} b - \sqrt {a b e}}{\sqrt {e x} b + \sqrt {a b e}}\right )}{\sqrt {a b e} b^{3} c} + \frac {4 \, \left (e x\right )^{\frac {3}{2}} e^{2}}{b^{2} c}}{6 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 74, normalized size = 0.70 \[ \frac {a^{3/2}\,e^{5/2}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {e\,x}}{\sqrt {a}\,\sqrt {e}}\right )}{b^{7/2}\,c}-\frac {a^{3/2}\,e^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {e\,x}}{\sqrt {a}\,\sqrt {e}}\right )}{b^{7/2}\,c}-\frac {2\,e\,{\left (e\,x\right )}^{3/2}}{3\,b^2\,c} \]
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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