Optimal. Leaf size=103 \[ -\frac {2 e \sqrt {e x}}{b^2 c}+\frac {\sqrt {a} e^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{b^{5/2} c}+\frac {\sqrt {a} e^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{b^{5/2} c} \]
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Rubi [A]
time = 0.05, antiderivative size = 103, 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 = {74, 327, 335,
218, 214, 211} \begin {gather*} \frac {\sqrt {a} e^{3/2} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{b^{5/2} c}+\frac {\sqrt {a} e^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{b^{5/2} c}-\frac {2 e \sqrt {e x}}{b^2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 74
Rule 211
Rule 214
Rule 218
Rule 327
Rule 335
Rubi steps
\begin {align*} \int \frac {(e x)^{3/2}}{(a+b x) (a c-b c x)} \, dx &=\int \frac {(e x)^{3/2}}{a^2 c-b^2 c x^2} \, dx\\ &=-\frac {2 e \sqrt {e x}}{b^2 c}+\frac {\left (a^2 e^2\right ) \int \frac {1}{\sqrt {e x} \left (a^2 c-b^2 c x^2\right )} \, dx}{b^2}\\ &=-\frac {2 e \sqrt {e x}}{b^2 c}+\frac {\left (2 a^2 e\right ) \text {Subst}\left (\int \frac {1}{a^2 c-\frac {b^2 c x^4}{e^2}} \, dx,x,\sqrt {e x}\right )}{b^2}\\ &=-\frac {2 e \sqrt {e x}}{b^2 c}+\frac {\left (a e^2\right ) \text {Subst}\left (\int \frac {1}{a e-b x^2} \, dx,x,\sqrt {e x}\right )}{b^2 c}+\frac {\left (a e^2\right ) \text {Subst}\left (\int \frac {1}{a e+b x^2} \, dx,x,\sqrt {e x}\right )}{b^2 c}\\ &=-\frac {2 e \sqrt {e x}}{b^2 c}+\frac {\sqrt {a} e^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{b^{5/2} c}+\frac {\sqrt {a} e^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{b^{5/2} c}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 80, normalized size = 0.78 \begin {gather*} \frac {(e x)^{3/2} \left (-2 \sqrt {b} \sqrt {x}+\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )+\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right )}{b^{5/2} c x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 71, normalized size = 0.69
method | result | size |
derivativedivides | \(-\frac {2 e \left (\frac {\sqrt {e x}}{b^{2}}-\frac {a e \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 b^{2} \sqrt {a e b}}-\frac {a e \arctanh \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 b^{2} \sqrt {a e b}}\right )}{c}\) | \(71\) |
default | \(-\frac {2 e \left (\frac {\sqrt {e x}}{b^{2}}-\frac {a e \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 b^{2} \sqrt {a e b}}-\frac {a e \arctanh \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{2 b^{2} \sqrt {a e b}}\right )}{c}\) | \(71\) |
risch | \(-\frac {2 x \,e^{2}}{b^{2} \sqrt {e x}\, c}+\frac {\left (\frac {a \arctan \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{b^{2} \sqrt {a e b}}+\frac {a \arctanh \left (\frac {b \sqrt {e x}}{\sqrt {a e b}}\right )}{b^{2} \sqrt {a e b}}\right ) e^{2}}{c}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 83, normalized size = 0.81 \begin {gather*} \frac {1}{2} \, {\left (\frac {2 \, a \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2} c} - \frac {a \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{\sqrt {a b} b^{2} c} - \frac {4 \, \sqrt {x}}{b^{2} c}\right )} e^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.74, size = 170, normalized size = 1.65 \begin {gather*} \left [\frac {2 \, \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) e^{\frac {3}{2}} + \sqrt {\frac {a}{b}} e^{\frac {3}{2}} \log \left (\frac {b x + 2 \, b \sqrt {x} \sqrt {\frac {a}{b}} + a}{b x - a}\right ) - 4 \, \sqrt {x} e^{\frac {3}{2}}}{2 \, b^{2} c}, -\frac {2 \, \sqrt {-\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {-\frac {a}{b}}}{a}\right ) e^{\frac {3}{2}} - \sqrt {-\frac {a}{b}} e^{\frac {3}{2}} \log \left (\frac {b x + 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) + 4 \, \sqrt {x} e^{\frac {3}{2}}}{2 \, b^{2} c}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 207 vs.
\(2 (95) = 190\).
time = 1.46, size = 207, normalized size = 2.01 \begin {gather*} \begin {cases} \frac {\sqrt {a} e^{\frac {3}{2}} \operatorname {acoth}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{b^{\frac {5}{2}} c} - \frac {\sqrt {a} e^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{b^{\frac {5}{2}} c} - \frac {2 e^{\frac {3}{2}} \sqrt {x}}{b^{2} c} - \frac {e^{\frac {3}{2}} x^{\frac {3}{2}}}{3 a b c} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {\sqrt {a} e^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{b^{\frac {5}{2}} c} + \frac {\sqrt {a} e^{\frac {3}{2}} \operatorname {atanh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{b^{\frac {5}{2}} c} - \frac {2 e^{\frac {3}{2}} \sqrt {x}}{b^{2} c} - \frac {e^{\frac {3}{2}} x^{\frac {3}{2}}}{3 a b c} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.92, size = 66, normalized size = 0.64 \begin {gather*} {\left (\frac {a \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2} c} - \frac {a \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{\sqrt {-a b} b^{2} c} - \frac {2 \, \sqrt {x}}{b^{2} c}\right )} e^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 73, normalized size = 0.71 \begin {gather*} \frac {\sqrt {a}\,e^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {e\,x}}{\sqrt {a}\,\sqrt {e}}\right )}{b^{5/2}\,c}-\frac {2\,e\,\sqrt {e\,x}}{b^2\,c}+\frac {\sqrt {a}\,e^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {e\,x}}{\sqrt {a}\,\sqrt {e}}\right )}{b^{5/2}\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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