Optimal. Leaf size=124 \[ \frac {2 B (a+b x) \sqrt {d+e x}}{b e \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (a+b x) (A b-a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} \sqrt {b d-a e}} \]
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Rubi [A] time = 0.08, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {770, 80, 63, 208} \begin {gather*} \frac {2 B (a+b x) \sqrt {d+e x}}{b e \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (a+b x) (A b-a B) \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} \sqrt {b d-a e}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 208
Rule 770
Rubi steps
\begin {align*} \int \frac {A+B x}{\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 {A+B x}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 B (a+b x) \sqrt {d+e x}}{b e \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (\frac {1}{2} A b^2 e-\frac {1}{2} a b 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 e \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 B (a+b x) \sqrt {d+e x}}{b e \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (4 \left (\frac {1}{2} A b^2 e-\frac {1}{2} a b 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^2 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=\frac {2 B (a+b x) \sqrt {d+e x}}{b e \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (A b-a B) (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {b d-a e} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 111, normalized size = 0.90 \begin {gather*} \frac {2 (a+b x) \left (e (a B-A b) \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )+\sqrt {b} B \sqrt {d+e x} (b d-a e)\right )}{b^{3/2} e \sqrt {(a+b x)^2} (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 22.48, size = 116, normalized size = 0.94 \begin {gather*} \frac {(-a e-b e x) \left (\frac {2 (A b-a B) \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 {2 B \sqrt {d+e x}}{b e}\right )}{e \sqrt {\frac {(a e+b e x)^2}{e^2}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 209, normalized size = 1.69 \begin {gather*} \left [-\frac {\sqrt {b^{2} d - a b e} {\left (B a - A b\right )} e \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 b^{2} d - B a b e\right )} \sqrt {e x + d}}{b^{3} d e - a b^{2} e^{2}}, -\frac {2 \, {\left (\sqrt {-b^{2} d + a b e} {\left (B a - A b\right )} e \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) - {\left (B b^{2} d - B a b e\right )} \sqrt {e x + d}\right )}}{b^{3} d e - a b^{2} e^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 87, normalized size = 0.70 \begin {gather*} \frac {2 \, \sqrt {x e + d} B e^{\left (-1\right )} \mathrm {sgn}\left (b x + a\right )}{b} - \frac {2 \, {\left (B a \mathrm {sgn}\left (b x + a\right ) - A b \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 110, normalized size = 0.89 \begin {gather*} \frac {2 \left (b x +a \right ) \left (A b e \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )-B a e \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}\, B \right )}{\sqrt {\left (b x +a \right )^{2}}\, \sqrt {\left (a e -b d \right ) b}\, b e} \end {gather*}
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 \begin {gather*} \int \frac {B x + A}{\sqrt {{\left (b x + a\right )}^{2}} \sqrt {e x + d}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {A+B\,x}{\sqrt {{\left (a+b\,x\right )}^2}\,\sqrt {d+e\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{\sqrt {d + e x} \sqrt {\left (a + b x\right )^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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