Optimal. Leaf size=70 \[ -\frac {e \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {b d-a e}}-\frac {\sqrt {d+e x}}{b (a+b x)} \]
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Rubi [A] time = 0.03, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {27, 47, 63, 208} \begin {gather*} -\frac {e \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {b d-a e}}-\frac {\sqrt {d+e x}}{b (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 47
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x}}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac {\sqrt {d+e x}}{(a+b x)^2} \, dx\\ &=-\frac {\sqrt {d+e x}}{b (a+b x)}+\frac {e \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 b}\\ &=-\frac {\sqrt {d+e x}}{b (a+b x)}+\frac {\operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{b}\\ &=-\frac {\sqrt {d+e x}}{b (a+b x)}-\frac {e \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {b d-a e}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 69, normalized size = 0.99 \begin {gather*} \frac {e \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {a e-b d}}\right )}{b^{3/2} \sqrt {a e-b d}}-\frac {\sqrt {d+e x}}{b (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.23, size = 91, normalized size = 1.30 \begin {gather*} -\frac {e \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{b^{3/2} \sqrt {a e-b d}}-\frac {e \sqrt {d+e x}}{b (a e+b (d+e x)-b d)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 232, normalized size = 3.31 \begin {gather*} \left [\frac {\sqrt {b^{2} d - a b e} {\left (b e x + a e\right )} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) - 2 \, {\left (b^{2} d - a b e\right )} \sqrt {e x + d}}{2 \, {\left (a b^{3} d - a^{2} b^{2} e + {\left (b^{4} d - a b^{3} e\right )} x\right )}}, \frac {\sqrt {-b^{2} d + a b e} {\left (b e x + a e\right )} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) - {\left (b^{2} d - a b e\right )} \sqrt {e x + d}}{a b^{3} d - a^{2} b^{2} e + {\left (b^{4} d - a b^{3} e\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 80, normalized size = 1.14 \begin {gather*} \frac {\arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e}{\sqrt {-b^{2} d + a b e} b} - \frac {\sqrt {x e + d} e}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 64, normalized size = 0.91 \begin {gather*} \frac {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 {\sqrt {e x +d}\, e}{\left (b e x +a e \right ) 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 = 0.07, size = 61, normalized size = 0.87 \begin {gather*} \frac {e\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{b^{3/2}\,\sqrt {a\,e-b\,d}}-\frac {e\,\sqrt {d+e\,x}}{e\,x\,b^2+a\,e\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 39.85, size = 573, normalized size = 8.19 \begin {gather*} - \frac {2 a e^{2} \sqrt {d + e x}}{2 a^{2} b e^{2} - 2 a b^{2} d e + 2 a b^{2} e^{2} x - 2 b^{3} d e x} + \frac {a e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (- a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b} - \frac {a e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{2 b} - \frac {d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (- a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{2} + \frac {d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} \log {\left (a^{2} e^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} - 2 a b d e \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + b^{2} d^{2} \sqrt {- \frac {1}{b \left (a e - b d\right )^{3}}} + \sqrt {d + e x} \right )}}{2} + \frac {2 d e \sqrt {d + e x}}{2 a^{2} e^{2} - 2 a b d e + 2 a b e^{2} x - 2 b^{2} d e x} + \frac {2 e \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e}{b} - d}} \right )}}{b^{2} \sqrt {\frac {a e}{b} - d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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