Optimal. Leaf size=168 \[ -\frac {e \sqrt {d+e x}}{4 b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^2 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{3/2} (b d-a e)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {660, 43, 44, 65,
214} \begin {gather*} -\frac {e \sqrt {d+e x}}{4 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {\sqrt {d+e x}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^2 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 44
Rule 65
Rule 214
Rule 660
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {\sqrt {d+e x}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (e \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^2 \sqrt {d+e x}} \, dx}{4 \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {e \sqrt {d+e x}}{4 b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (e^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{8 b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {e \sqrt {d+e x}}{4 b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (e \left (a b+b^2 x\right )\right ) \text {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 )}{4 b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}\\ &=-\frac {e \sqrt {d+e x}}{4 b (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^2 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{3/2} (b d-a e)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 121, normalized size = 0.72 \begin {gather*} -\frac {\sqrt {b} \sqrt {-b d+a e} \sqrt {d+e x} (a e-b (2 d+e x))-e^2 (a+b x)^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{4 b^{3/2} (-b d+a e)^{3/2} (a+b x) \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.68, size = 200, normalized size = 1.19
method | result | size |
default | \(\frac {\left (\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) b^{2} e^{2} x^{2}+2 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a b \,e^{2} x +\left (e x +d \right )^{\frac {3}{2}} \sqrt {b \left (a e -b d \right )}\, b +\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a^{2} e^{2}-\sqrt {b \left (a e -b d \right )}\, \sqrt {e x +d}\, a e +\sqrt {e x +d}\, \sqrt {b \left (a e -b d \right )}\, b d \right ) \left (b x +a \right )}{4 \sqrt {b \left (a e -b d \right )}\, \left (a e -b d \right ) b \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(200\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.30, size = 451, normalized size = 2.68 \begin {gather*} \left [-\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {b^{2} d - a b e} e^{2} \log \left (\frac {2 \, b d + {\left (b x - a\right )} e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {x e + d}}{b x + a}\right ) + 2 \, {\left (2 \, b^{3} d^{2} - {\left (a b^{2} x - a^{2} b\right )} e^{2} + {\left (b^{3} d x - 3 \, a b^{2} d\right )} e\right )} \sqrt {x e + d}}{8 \, {\left (b^{6} d^{2} x^{2} + 2 \, a b^{5} d^{2} x + a^{2} b^{4} d^{2} + {\left (a^{2} b^{4} x^{2} + 2 \, a^{3} b^{3} x + a^{4} b^{2}\right )} e^{2} - 2 \, {\left (a b^{5} d x^{2} + 2 \, a^{2} b^{4} d x + a^{3} b^{3} d\right )} e\right )}}, -\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {x e + d}}{b x e + b d}\right ) e^{2} + {\left (2 \, b^{3} d^{2} - {\left (a b^{2} x - a^{2} b\right )} e^{2} + {\left (b^{3} d x - 3 \, a b^{2} d\right )} e\right )} \sqrt {x e + d}}{4 \, {\left (b^{6} d^{2} x^{2} + 2 \, a b^{5} d^{2} x + a^{2} b^{4} d^{2} + {\left (a^{2} b^{4} x^{2} + 2 \, a^{3} b^{3} x + a^{4} b^{2}\right )} e^{2} - 2 \, {\left (a b^{5} d x^{2} + 2 \, a^{2} b^{4} d x + a^{3} b^{3} d\right )} e\right )}}\right ] \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 {\sqrt {d + e x}}{\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.84, size = 156, normalized size = 0.93 \begin {gather*} -\frac {\arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right ) e^{2}}{4 \, {\left (b^{2} d \mathrm {sgn}\left (b x + a\right ) - a b e \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} - \frac {{\left (x e + d\right )}^{\frac {3}{2}} b e^{2} + \sqrt {x e + d} b d e^{2} - \sqrt {x e + d} a e^{3}}{4 \, {\left (b^{2} d \mathrm {sgn}\left (b x + a\right ) - a b e \mathrm {sgn}\left (b x + a\right )\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \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 {\sqrt {d+e\,x}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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