Optimal. Leaf size=116 \[ \frac {2 (b e g-c (e f+d g)) \sqrt {f+g x}}{e^2 g^2}+\frac {2 c (f+g x)^{3/2}}{3 e g^2}-\frac {2 \left (c d^2-b d e+a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{5/2} \sqrt {e f-d g}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {911, 1167, 214}
\begin {gather*} -\frac {2 \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{5/2} \sqrt {e f-d g}}+\frac {2 \sqrt {f+g x} (b e g-c (d g+e f))}{e^2 g^2}+\frac {2 c (f+g x)^{3/2}}{3 e g^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 214
Rule 911
Rule 1167
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{(d+e x) \sqrt {f+g x}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {\frac {c f^2-b f g+a g^2}{g^2}-\frac {(2 c f-b g) x^2}{g^2}+\frac {c x^4}{g^2}}{\frac {-e f+d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{g}\\ &=\frac {2 \text {Subst}\left (\int \left (\frac {b e g-c (e f+d g)}{e^2 g}+\frac {c x^2}{e g}+\frac {c d^2-b d e+a e^2}{e^2 \left (d-\frac {e f}{g}+\frac {e x^2}{g}\right )}\right ) \, dx,x,\sqrt {f+g x}\right )}{g}\\ &=\frac {2 (b e g-c (e f+d g)) \sqrt {f+g x}}{e^2 g^2}+\frac {2 c (f+g x)^{3/2}}{3 e g^2}+\frac {\left (2 \left (c d^2-b d e+a e^2\right )\right ) \text {Subst}\left (\int \frac {1}{d-\frac {e f}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{e^2 g}\\ &=\frac {2 (b e g-c (e f+d g)) \sqrt {f+g x}}{e^2 g^2}+\frac {2 c (f+g x)^{3/2}}{3 e g^2}-\frac {2 \left (c d^2-b d e+a e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{5/2} \sqrt {e f-d g}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.16, size = 104, normalized size = 0.90 \begin {gather*} \frac {2 \sqrt {f+g x} (3 b e g+c (-2 e f-3 d g+e g x))}{3 e^2 g^2}+\frac {2 \left (c d^2+e (-b d+a e)\right ) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {-e f+d g}}\right )}{e^{5/2} \sqrt {-e f+d g}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.11, size = 115, normalized size = 0.99
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (\frac {c \left (g x +f \right )^{\frac {3}{2}} e}{3}+b e g \sqrt {g x +f}-c d g \sqrt {g x +f}-c e f \sqrt {g x +f}\right )}{e^{2}}+\frac {2 g^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{e^{2} \sqrt {\left (d g -e f \right ) e}}}{g^{2}}\) | \(115\) |
default | \(\frac {\frac {2 \left (\frac {c \left (g x +f \right )^{\frac {3}{2}} e}{3}+b e g \sqrt {g x +f}-c d g \sqrt {g x +f}-c e f \sqrt {g x +f}\right )}{e^{2}}+\frac {2 g^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{e^{2} \sqrt {\left (d g -e f \right ) e}}}{g^{2}}\) | \(115\) |
risch | \(\frac {2 \left (c e g x +3 b e g -3 d g c -2 c e f \right ) \sqrt {g x +f}}{3 g^{2} e^{2}}+\frac {2 \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right ) a}{\sqrt {\left (d g -e f \right ) e}}-\frac {2 \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right ) b d}{e \sqrt {\left (d g -e f \right ) e}}+\frac {2 \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right ) c \,d^{2}}{e^{2} \sqrt {\left (d g -e f \right ) e}}\) | \(159\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.42, size = 334, normalized size = 2.88 \begin {gather*} \left [-\frac {3 \, {\left (c d^{2} g^{2} - b d g^{2} e + a g^{2} e^{2}\right )} \sqrt {-d g e + f e^{2}} \log \left (-\frac {d g - {\left (g x + 2 \, f\right )} e + 2 \, \sqrt {-d g e + f e^{2}} \sqrt {g x + f}}{x e + d}\right ) + 2 \, {\left (3 \, c d^{2} g^{2} e + {\left (c f g x - 2 \, c f^{2} + 3 \, b f g\right )} e^{3} - {\left (c d g^{2} x + c d f g + 3 \, b d g^{2}\right )} e^{2}\right )} \sqrt {g x + f}}{3 \, {\left (d g^{3} e^{3} - f g^{2} e^{4}\right )}}, -\frac {2 \, {\left (3 \, {\left (c d^{2} g^{2} - b d g^{2} e + a g^{2} e^{2}\right )} \sqrt {d g e - f e^{2}} \arctan \left (-\frac {\sqrt {d g e - f e^{2}} \sqrt {g x + f}}{d g - f e}\right ) + {\left (3 \, c d^{2} g^{2} e + {\left (c f g x - 2 \, c f^{2} + 3 \, b f g\right )} e^{3} - {\left (c d g^{2} x + c d f g + 3 \, b d g^{2}\right )} e^{2}\right )} \sqrt {g x + f}\right )}}{3 \, {\left (d g^{3} e^{3} - f g^{2} e^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 14.42, size = 112, normalized size = 0.97 \begin {gather*} \frac {2 c \left (f + g x\right )^{\frac {3}{2}}}{3 e g^{2}} - \frac {2 \left (a e^{2} - b d e + c d^{2}\right ) \operatorname {atan}{\left (\frac {1}{\sqrt {\frac {e}{d g - e f}} \sqrt {f + g x}} \right )}}{e^{2} \sqrt {\frac {e}{d g - e f}} \left (d g - e f\right )} + \frac {2 \sqrt {f + g x} \left (b e g - c d g - c e f\right )}{e^{2} g^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 5.44, size = 128, normalized size = 1.10 \begin {gather*} \frac {2 \, {\left (c d^{2} - b d e + a e^{2}\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {d g e - f e^{2}}}\right ) e^{\left (-2\right )}}{\sqrt {d g e - f e^{2}}} - \frac {2 \, {\left (3 \, \sqrt {g x + f} c d g^{5} e - {\left (g x + f\right )}^{\frac {3}{2}} c g^{4} e^{2} + 3 \, \sqrt {g x + f} c f g^{4} e^{2} - 3 \, \sqrt {g x + f} b g^{5} e^{2}\right )} e^{\left (-3\right )}}{3 \, g^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.14, size = 117, normalized size = 1.01 \begin {gather*} \sqrt {f+g\,x}\,\left (\frac {2\,b\,g-4\,c\,f}{e\,g^2}-\frac {2\,c\,\left (d\,g^3-e\,f\,g^2\right )}{e^2\,g^4}\right )+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {e}\,\sqrt {f+g\,x}}{\sqrt {d\,g-e\,f}}\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{e^{5/2}\,\sqrt {d\,g-e\,f}}+\frac {2\,c\,{\left (f+g\,x\right )}^{3/2}}{3\,e\,g^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________