Optimal. Leaf size=112 \[ \frac {2 (a+b x) \sqrt {d+e x}}{b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 \sqrt {b d-a e} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, 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 = {660, 52, 65,
214} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 65
Rule 214
Rule 660
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 ) \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 )}{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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 81, normalized size = 0.72 \begin {gather*} \frac {2 (a+b x) \left (\sqrt {b} \sqrt {d+e x}-\sqrt {-b d+a e} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{b^{3/2} \sqrt {(a+b x)^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.58, size = 104, normalized size = 0.93
method | result | size |
default | \(\frac {2 \left (b x +a \right ) \left (-\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a e +\arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) b d +\sqrt {e x +d}\, \sqrt {b \left (a e -b d \right )}\right )}{\sqrt {\left (b x +a \right )^{2}}\, b \sqrt {b \left (a e -b d \right )}}\) | \(104\) |
risch | \(\frac {2 \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{b \left (b x +a \right )}+\frac {\left (-\frac {2 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) a e}{b \sqrt {b \left (a e -b d \right )}}+\frac {2 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {b \left (a e -b d \right )}}\right ) d}{\sqrt {b \left (a e -b d \right )}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b x +a}\) | \(126\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.40, size = 154, normalized size = 1.38 \begin {gather*} \left [\frac {\sqrt {\frac {b d - a e}{b}} \log \left (\frac {2 \, b d - 2 \, \sqrt {x e + d} b \sqrt {\frac {b d - a e}{b}} + {\left (b x - a\right )} e}{b x + a}\right ) + 2 \, \sqrt {x e + d}}{b}, -\frac {2 \, {\left (\sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {x e + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - \sqrt {x e + d}\right )}}{b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 2.14, size = 95, normalized size = 0.85 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.21, size = 85, normalized size = 0.76 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {d+e\,x}}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________