Optimal. Leaf size=87 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{a^{3/2} \sqrt {b} c \sqrt {e}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{a^{3/2} \sqrt {b} c \sqrt {e}} \]
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Rubi [A] time = 0.05, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {73, 329, 212, 208, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{a^{3/2} \sqrt {b} c \sqrt {e}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{a^{3/2} \sqrt {b} c \sqrt {e}} \]
Antiderivative was successfully verified.
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Rule 73
Rule 205
Rule 208
Rule 212
Rule 329
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {e x} (a+b x) (a c-b c x)} \, dx &=\int \frac {1}{\sqrt {e x} \left (a^2 c-b^2 c x^2\right )} \, dx\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{a^2 c-\frac {b^2 c x^4}{e^2}} \, dx,x,\sqrt {e x}\right )}{e}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{a e-b x^2} \, dx,x,\sqrt {e x}\right )}{a c}+\frac {\operatorname {Subst}\left (\int \frac {1}{a e+b x^2} \, dx,x,\sqrt {e x}\right )}{a c}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{a^{3/2} \sqrt {b} c \sqrt {e}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {e x}}{\sqrt {a} \sqrt {e}}\right )}{a^{3/2} \sqrt {b} c \sqrt {e}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 61, normalized size = 0.70 \[ \frac {\sqrt {x} \left (\tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )+\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )\right )}{a^{3/2} \sqrt {b} c \sqrt {e x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 173, normalized size = 1.99 \[ \left [-\frac {2 \, \sqrt {a b e} \arctan \left (\frac {\sqrt {a b e} \sqrt {e x}}{b e x}\right ) - \sqrt {a b e} \log \left (\frac {b e x + a e + 2 \, \sqrt {a b e} \sqrt {e x}}{b x - a}\right )}{2 \, a^{2} b c e}, -\frac {2 \, \sqrt {-a b e} \arctan \left (\frac {\sqrt {-a b e} \sqrt {e x}}{b e x}\right ) + \sqrt {-a b e} \log \left (\frac {b e x - a e - 2 \, \sqrt {-a b e} \sqrt {e x}}{b x + a}\right )}{2 \, a^{2} b c e}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 58, normalized size = 0.67 \[ \frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) e^{\left (-\frac {1}{2}\right )}}{\sqrt {a b} a c} - \frac {\arctan \left (\frac {b \sqrt {x} e^{\frac {1}{2}}}{\sqrt {-a b e}}\right )}{\sqrt {-a b e} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 56, normalized size = 0.64 \[ \frac {\arctanh \left (\frac {\sqrt {e x}\, b}{\sqrt {a b e}}\right )}{\sqrt {a b e}\, a c}+\frac {\arctan \left (\frac {\sqrt {e x}\, b}{\sqrt {a b e}}\right )}{\sqrt {a b e}\, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.50, size = 84, normalized size = 0.97 \[ \frac {\frac {2 \, e \arctan \left (\frac {\sqrt {e x} b}{\sqrt {a b e}}\right )}{\sqrt {a b e} a c} - \frac {e \log \left (\frac {\sqrt {e x} b - \sqrt {a b e}}{\sqrt {e x} b + \sqrt {a b e}}\right )}{\sqrt {a b e} a c}}{2 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 46, normalized size = 0.53 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {e\,x}}{\sqrt {a}\,\sqrt {e}}\right )+\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {e\,x}}{\sqrt {a}\,\sqrt {e}}\right )}{a^{3/2}\,\sqrt {b}\,c\,\sqrt {e}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.66, size = 291, normalized size = 3.34 \[ \begin {cases} \frac {1}{a b c \sqrt {e} \sqrt {x}} + \frac {\operatorname {acoth}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{a^{\frac {3}{2}} \sqrt {b} c \sqrt {e}} + \frac {\operatorname {atan}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{a^{\frac {3}{2}} \sqrt {b} c \sqrt {e}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\\frac {1 - i}{2 a b c \sqrt {e} \sqrt {x}} + \frac {i \left (1 - i\right )}{2 a b c \sqrt {e} \sqrt {x}} + \frac {\left (1 - i\right ) \operatorname {atan}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{2 a^{\frac {3}{2}} \sqrt {b} c \sqrt {e}} + \frac {i \left (1 - i\right ) \operatorname {atan}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{2 a^{\frac {3}{2}} \sqrt {b} c \sqrt {e}} + \frac {\left (1 - i\right ) \operatorname {atanh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{2 a^{\frac {3}{2}} \sqrt {b} c \sqrt {e}} + \frac {i \left (1 - i\right ) \operatorname {atanh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{2 a^{\frac {3}{2}} \sqrt {b} c \sqrt {e}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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