Optimal. Leaf size=62 \[ \frac {2 \sqrt {d+e x}}{b}-\frac {2 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {27, 50, 63, 208} \begin {gather*} \frac {2 \sqrt {d+e x}}{b}-\frac {2 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 50
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {(a+b x) \sqrt {d+e x}}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac {\sqrt {d+e x}}{a+b x} \, dx\\ &=\frac {2 \sqrt {d+e x}}{b}+\frac {(b d-a e) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{b}\\ &=\frac {2 \sqrt {d+e x}}{b}+\frac {(2 (b d-a e)) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{b e}\\ &=\frac {2 \sqrt {d+e x}}{b}-\frac {2 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 62, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {d+e x}}{b}-\frac {2 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 72, normalized size = 1.16 \begin {gather*} \frac {2 \sqrt {a e-b d} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{b^{3/2}}+\frac {2 \sqrt {d+e x}}{b} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 143, normalized size = 2.31 \begin {gather*} \left [\frac {\sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) + 2 \, \sqrt {e x + d}}{b}, -\frac {2 \, {\left (\sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - \sqrt {e x + d}\right )}}{b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 67, normalized size = 1.08 \begin {gather*} \frac {2 \, {\left (b d - a e\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b} + \frac {2 \, \sqrt {x e + d}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 92, normalized size = 1.48 \begin {gather*} -\frac {2 a e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}\, b}+\frac {2 d \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}}+\frac {2 \sqrt {e x +d}}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 50, normalized size = 0.81 \begin {gather*} \frac {2\,\sqrt {d+e\,x}}{b}-\frac {2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )\,\sqrt {a\,e-b\,d}}{b^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 58.92, size = 61, normalized size = 0.98 \begin {gather*} \frac {2 \left (\frac {e \sqrt {d + e x}}{b} - \frac {e \left (a e - b d\right ) \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e - b d}{b}}} \right )}}{b^{2} \sqrt {\frac {a e - b d}{b}}}\right )}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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