Optimal. Leaf size=140 \[ \frac {2 c \sqrt {f+g x}}{e^2 g}-\frac {\left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{(e f-d g) (d+e x)}+\frac {(c d (4 e f-3 d g)-e (2 b e f-b d g-a e g)) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{5/2} (e f-d g)^{3/2}} \]
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Rubi [A]
time = 0.18, antiderivative size = 144, normalized size of antiderivative = 1.03, number of steps
used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {911, 1171, 396,
214} \begin {gather*} -\frac {\sqrt {f+g x} \left (a e^2-b d e+c d^2\right )}{e^2 (d+e x) (e f-d g)}+\frac {\tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) (c d (4 e f-3 d g)-e (-a e g-b d g+2 b e f))}{e^{5/2} (e f-d g)^{3/2}}+\frac {2 c \sqrt {f+g x}}{e^2 g} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 396
Rule 911
Rule 1171
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{(d+e x)^2 \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}}{\left (\frac {-e f+d g}{g}+\frac {e x^2}{g}\right )^2} \, dx,x,\sqrt {f+g x}\right )}{g}\\ &=-\frac {\left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{(e f-d g) (d+e x)}+\frac {\text {Subst}\left (\int \frac {-a+\frac {c d^2}{e^2}-\frac {b d}{e}-\frac {2 c f^2}{g^2}+\frac {2 b f}{g}+\frac {2 c (e f-d g) x^2}{e g^2}}{\frac {-e f+d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{e f-d g}\\ &=\frac {2 c \sqrt {f+g x}}{e^2 g}-\frac {\left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{(e f-d g) (d+e x)}-\frac {(c d (4 e f-3 d g)-e (2 b e f-b d g-a e g)) \text {Subst}\left (\int \frac {1}{\frac {-e f+d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{e^2 g (e f-d g)}\\ &=\frac {2 c \sqrt {f+g x}}{e^2 g}-\frac {\left (a+\frac {d (c d-b e)}{e^2}\right ) \sqrt {f+g x}}{(e f-d g) (d+e x)}+\frac {(c d (4 e f-3 d g)-e (2 b e f-b d g-a e g)) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{5/2} (e f-d g)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.48, size = 150, normalized size = 1.07 \begin {gather*} \frac {\sqrt {f+g x} \left (e (b d-a e) g+c \left (-3 d^2 g+2 e^2 f x+2 d e (f-g x)\right )\right )}{e^2 g (e f-d g) (d+e x)}-\frac {(c d (-4 e f+3 d g)+e (2 b e f-b d g-a e g)) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {-e f+d g}}\right )}{e^{5/2} (-e f+d g)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 156, normalized size = 1.11
method | result | size |
derivativedivides | \(\frac {\frac {2 c \sqrt {g x +f}}{e^{2}}+\frac {2 g \left (\frac {g \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {g x +f}}{2 \left (d g -e f \right ) \left (e \left (g x +f \right )+d g -e f \right )}+\frac {\left (a \,e^{2} g +b d e g -2 b \,e^{2} f -3 c \,d^{2} g +4 c d e f \right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{2 \left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}}\right )}{e^{2}}}{g}\) | \(156\) |
default | \(\frac {\frac {2 c \sqrt {g x +f}}{e^{2}}+\frac {2 g \left (\frac {g \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {g x +f}}{2 \left (d g -e f \right ) \left (e \left (g x +f \right )+d g -e f \right )}+\frac {\left (a \,e^{2} g +b d e g -2 b \,e^{2} f -3 c \,d^{2} g +4 c d e f \right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{2 \left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}}\right )}{e^{2}}}{g}\) | \(156\) |
risch | \(\frac {2 c \sqrt {g x +f}}{e^{2} g}+\frac {g \sqrt {g x +f}\, a}{\left (d g -e f \right ) \left (e g x +d g \right )}-\frac {g \sqrt {g x +f}\, b d}{e \left (d g -e f \right ) \left (e g x +d g \right )}+\frac {g \sqrt {g x +f}\, c \,d^{2}}{e^{2} \left (d g -e f \right ) \left (e g x +d g \right )}+\frac {\arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right ) a g}{\left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}}+\frac {\arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right ) b d g}{e \left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}}-\frac {2 \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right ) b f}{\left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}}-\frac {3 \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right ) c \,d^{2} g}{e^{2} \left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}}+\frac {4 \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right ) c d f}{e \left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}}\) | \(371\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 300 vs.
\(2 (129) = 258\).
time = 1.11, size = 617, normalized size = 4.41 \begin {gather*} \left [-\frac {{\left (3 \, c d^{3} g^{2} + {\left (2 \, b f g - a g^{2}\right )} x e^{3} + {\left (2 \, b d f g - a d g^{2} - {\left (4 \, c d f g + b d g^{2}\right )} x\right )} e^{2} + {\left (3 \, c d^{2} g^{2} x - 4 \, c d^{2} f g - b d^{2} g^{2}\right )} e\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^{3} g^{2} e + {\left (2 \, c f^{2} x - a f g\right )} e^{4} - {\left (4 \, c d f g x - 2 \, c d f^{2} - b d f g - a d g^{2}\right )} e^{3} + {\left (2 \, c d^{2} g^{2} x - 5 \, c d^{2} f g - b d^{2} g^{2}\right )} e^{2}\right )} \sqrt {g x + f}}{2 \, {\left (d^{3} g^{3} e^{3} + f^{2} g x e^{6} - {\left (2 \, d f g^{2} x - d f^{2} g\right )} e^{5} + {\left (d^{2} g^{3} x - 2 \, d^{2} f g^{2}\right )} e^{4}\right )}}, \frac {{\left (3 \, c d^{3} g^{2} + {\left (2 \, b f g - a g^{2}\right )} x e^{3} + {\left (2 \, b d f g - a d g^{2} - {\left (4 \, c d f g + b d g^{2}\right )} x\right )} e^{2} + {\left (3 \, c d^{2} g^{2} x - 4 \, c d^{2} f g - b d^{2} g^{2}\right )} e\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^{3} g^{2} e + {\left (2 \, c f^{2} x - a f g\right )} e^{4} - {\left (4 \, c d f g x - 2 \, c d f^{2} - b d f g - a d g^{2}\right )} e^{3} + {\left (2 \, c d^{2} g^{2} x - 5 \, c d^{2} f g - b d^{2} g^{2}\right )} e^{2}\right )} \sqrt {g x + f}}{d^{3} g^{3} e^{3} + f^{2} g x e^{6} - {\left (2 \, d f g^{2} x - d f^{2} g\right )} e^{5} + {\left (d^{2} g^{3} x - 2 \, d^{2} f g^{2}\right )} e^{4}}\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 {a + b x + c x^{2}}{\left (d + e x\right )^{2} \sqrt {f + g x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.11, size = 175, normalized size = 1.25 \begin {gather*} \frac {2 \, \sqrt {g x + f} c e^{\left (-2\right )}}{g} - \frac {{\left (3 \, c d^{2} g - 4 \, c d f e - b d g e + 2 \, b f e^{2} - a g e^{2}\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {d g e - f e^{2}}}\right )}{{\left (d g e^{2} - f e^{3}\right )} \sqrt {d g e - f e^{2}}} + \frac {\sqrt {g x + f} c d^{2} g - \sqrt {g x + f} b d g e + \sqrt {g x + f} a g e^{2}}{{\left (d g e^{2} - f e^{3}\right )} {\left (d g + {\left (g x + f\right )} e - f e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.23, size = 146, normalized size = 1.04 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {e}\,\sqrt {f+g\,x}}{\sqrt {d\,g-e\,f}}\right )\,\left (a\,e^2\,g-2\,b\,e^2\,f-3\,c\,d^2\,g+b\,d\,e\,g+4\,c\,d\,e\,f\right )}{e^{5/2}\,{\left (d\,g-e\,f\right )}^{3/2}}+\frac {\sqrt {f+g\,x}\,\left (c\,g\,d^2-b\,g\,d\,e+a\,g\,e^2\right )}{\left (d\,g-e\,f\right )\,\left (e^3\,\left (f+g\,x\right )-e^3\,f+d\,e^2\,g\right )}+\frac {2\,c\,\sqrt {f+g\,x}}{e^2\,g} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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