Optimal. Leaf size=112 \[ \frac {2 (a+b x) \sqrt {d+e x}}{b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (a+b x) \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.05, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {646, 50, 63, 208} \[ \frac {2 (a+b x) \sqrt {d+e x}}{b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (a+b x) \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rule 646
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx &=\frac {\left (a b+b^2 x\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (a+b x) \sqrt {d+e x}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (\left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (a+b x) \sqrt {d+e x}}{b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{b^2 e \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 (a+b x) \sqrt {d+e x}}{b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 \sqrt {b d-a e} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 81, normalized size = 0.72 \[ \frac {2 (a+b x) \left (\sqrt {b} \sqrt {d+e x}-\sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )\right )}{b^{3/2} \sqrt {(a+b x)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 143, normalized size = 1.28 \[ \left [\frac {\sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) + 2 \, \sqrt {e x + d}}{b}, -\frac {2 \, {\left (\sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - \sqrt {e x + d}\right )}}{b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 85, normalized size = 0.76 \[ \frac {2 \, {\left (b d \mathrm {sgn}\left (b x + a\right ) - a e \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b} + \frac {2 \, \sqrt {x e + d} \mathrm {sgn}\left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 104, normalized size = 0.93 \[ \frac {2 \left (b x +a \right ) \left (-a e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+b d \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )+\sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\right )}{\sqrt {\left (b x +a \right )^{2}}\, \sqrt {\left (a e -b d \right ) b}\, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e x + d}}{\sqrt {{\left (b x + a\right )}^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {d+e\,x}}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.56, size = 95, normalized size = 0.85 \[ \sqrt {- \frac {a e}{b^{3}} + \frac {d}{b^{2}}} \log {\left (- b \sqrt {- \frac {a e}{b^{3}} + \frac {d}{b^{2}}} + \sqrt {d + e x} \right )} - \sqrt {- \frac {a e}{b^{3}} + \frac {d}{b^{2}}} \log {\left (b \sqrt {- \frac {a e}{b^{3}} + \frac {d}{b^{2}}} + \sqrt {d + e x} \right )} + \frac {2 \sqrt {d + e x}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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