Optimal. Leaf size=122 \[ \frac {2 c \sqrt {f+g x}}{e^2 g}-\frac {\left (a+\frac {c d^2}{e^2}\right ) \sqrt {f+g x}}{(e f-d g) (d+e x)}+\frac {\left (a e^2 g+c d (4 e f-3 d g)\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{5/2} (e f-d g)^{3/2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 126, normalized size of antiderivative = 1.03, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {912, 1171, 396,
214} \begin {gather*} -\frac {\sqrt {f+g x} \left (a e^2+c d^2\right )}{e^2 (d+e x) (e f-d g)}+\frac {\left (a e^2 g+c d (4 e f-3 d g)\right ) \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}}+\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 912
Rule 1171
Rubi steps
\begin {align*} \int \frac {a+c x^2}{(d+e x)^2 \sqrt {f+g x}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {\frac {c f^2+a g^2}{g^2}-\frac {2 c f 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 {c d^2}{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 {2 c f^2}{g^2}+\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 {c d^2}{e^2}\right ) \sqrt {f+g x}}{(e f-d g) (d+e x)}-\frac {\left (a+\frac {c d (4 e f-3 d g)}{e^2 g}\right ) \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 f-d g}\\ &=\frac {2 c \sqrt {f+g x}}{e^2 g}-\frac {\left (a+\frac {c d^2}{e^2}\right ) \sqrt {f+g x}}{(e f-d g) (d+e x)}+\frac {\left (a e^2 g+c d (4 e f-3 d g)\right ) \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 = 0.43, size = 133, normalized size = 1.09 \begin {gather*} \frac {\sqrt {f+g x} \left (-a e^2 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 {\left (a e^2 g+c d (4 e f-3 d g)\right ) \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 = 0.09, size = 139, normalized size = 1.14
method | result | size |
derivativedivides | \(\frac {\frac {2 c \sqrt {g x +f}}{e^{2}}+\frac {2 g \left (\frac {g \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {g x +f}}{2 \left (d g -e f \right ) \left (e \left (g x +f \right )+d g -e f \right )}+\frac {\left (a \,e^{2} g -3 c \,d^{2} g +4 c d e f \right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{2 \left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}}\right )}{e^{2}}}{g}\) | \(139\) |
default | \(\frac {\frac {2 c \sqrt {g x +f}}{e^{2}}+\frac {2 g \left (\frac {g \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {g x +f}}{2 \left (d g -e f \right ) \left (e \left (g x +f \right )+d g -e f \right )}+\frac {\left (a \,e^{2} g -3 c \,d^{2} g +4 c d e f \right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{2 \left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}}\right )}{e^{2}}}{g}\) | \(139\) |
risch | \(\frac {2 c \sqrt {g x +f}}{e^{2} g}+\frac {g \sqrt {g x +f}\, a}{\left (d g -e f \right ) \left (e g x +d g \right )}+\frac {g \sqrt {g x +f}\, c \,d^{2}}{e^{2} \left (d g -e f \right ) \left (e g x +d g \right )}+\frac {\arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right ) a g}{\left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}}-\frac {3 \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right ) c \,d^{2} g}{e^{2} \left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}}+\frac {4 \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right ) c d f}{e \left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}}\) | \(237\) |
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. 254 vs.
\(2 (109) = 218\).
time = 2.49, size = 525, normalized size = 4.30 \begin {gather*} \left [-\frac {{\left (3 \, c d^{3} g^{2} - a g^{2} x e^{3} - {\left (4 \, c d f g x + a d g^{2}\right )} e^{2} + {\left (3 \, c d^{2} g^{2} x - 4 \, c d^{2} f g\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} - a d g^{2}\right )} e^{3} + {\left (2 \, c d^{2} g^{2} x - 5 \, c d^{2} f g\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} - a g^{2} x e^{3} - {\left (4 \, c d f g x + a d g^{2}\right )} e^{2} + {\left (3 \, c d^{2} g^{2} x - 4 \, c d^{2} f g\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} - a d g^{2}\right )} e^{3} + {\left (2 \, c d^{2} g^{2} x - 5 \, c d^{2} f g\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 + 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 = 1.07, size = 148, normalized size = 1.21 \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 - 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} 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 = 2.68, size = 128, normalized size = 1.05 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {e}\,\sqrt {f+g\,x}}{\sqrt {d\,g-e\,f}}\right )\,\left (-3\,c\,g\,d^2+4\,c\,f\,d\,e+a\,g\,e^2\right )}{e^{5/2}\,{\left (d\,g-e\,f\right )}^{3/2}}+\frac {\sqrt {f+g\,x}\,\left (c\,g\,d^2+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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