Optimal. Leaf size=116 \[ -\frac {2 \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{5/2} \sqrt {e f-d g}}+\frac {2 \sqrt {f+g x} (b e g-c (d g+e f))}{e^2 g^2}+\frac {2 c (f+g x)^{3/2}}{3 e g^2} \]
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Rubi [A] time = 0.17, 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 = {897, 1153, 208} \[ -\frac {2 \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{5/2} \sqrt {e f-d g}}+\frac {2 \sqrt {f+g x} (b e g-c (d g+e f))}{e^2 g^2}+\frac {2 c (f+g x)^{3/2}}{3 e g^2} \]
Antiderivative was successfully verified.
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Rule 208
Rule 897
Rule 1153
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{(d+e x) \sqrt {f+g x}} \, dx &=\frac {2 \operatorname {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 \operatorname {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 ) \operatorname {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*}
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Mathematica [A] time = 0.19, size = 118, normalized size = 1.02 \[ \frac {2 \left (-\frac {g^2 \left (c d^2-e (b d-a e)\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{5/2} \sqrt {e f-d g}}+\frac {\sqrt {f+g x} (b e g-c (d g+e f))}{e^2}+\frac {c (f+g x)^{3/2}}{3 e}\right )}{g^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 341, normalized size = 2.94 \[ \left [\frac {3 \, {\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {e^{2} f - d e g} g^{2} \log \left (\frac {e g x + 2 \, e f - d g - 2 \, \sqrt {e^{2} f - d e g} \sqrt {g x + f}}{e x + d}\right ) - 2 \, {\left (2 \, c e^{3} f^{2} + {\left (c d e^{2} - 3 \, b e^{3}\right )} f g - 3 \, {\left (c d^{2} e - b d e^{2}\right )} g^{2} - {\left (c e^{3} f g - c d e^{2} g^{2}\right )} x\right )} \sqrt {g x + f}}{3 \, {\left (e^{4} f g^{2} - d e^{3} g^{3}\right )}}, \frac {2 \, {\left (3 \, {\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {-e^{2} f + d e g} g^{2} \arctan \left (\frac {\sqrt {-e^{2} f + d e g} \sqrt {g x + f}}{e g x + e f}\right ) - {\left (2 \, c e^{3} f^{2} + {\left (c d e^{2} - 3 \, b e^{3}\right )} f g - 3 \, {\left (c d^{2} e - b d e^{2}\right )} g^{2} - {\left (c e^{3} f g - c d e^{2} g^{2}\right )} x\right )} \sqrt {g x + f}\right )}}{3 \, {\left (e^{4} f g^{2} - d e^{3} g^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 128, normalized size = 1.10 \[ \frac {2 \, {\left (c d^{2} - b d e + a e^{2}\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {d g e - f e^{2}}}\right ) e^{\left (-2\right )}}{\sqrt {d g e - f e^{2}}} - \frac {2 \, {\left (3 \, \sqrt {g x + f} c d g^{5} e - {\left (g x + f\right )}^{\frac {3}{2}} c g^{4} e^{2} + 3 \, \sqrt {g x + f} c f g^{4} e^{2} - 3 \, \sqrt {g x + f} b g^{5} e^{2}\right )} e^{\left (-3\right )}}{3 \, g^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 189, normalized size = 1.63 \[ \frac {2 a \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\sqrt {\left (d g -e f \right ) e}}-\frac {2 b d \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\sqrt {\left (d g -e f \right ) e}\, e}+\frac {2 c \,d^{2} \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\sqrt {\left (d g -e f \right ) e}\, e^{2}}+\frac {2 \sqrt {g x +f}\, b}{e g}-\frac {2 \sqrt {g x +f}\, c d}{e^{2} g}-\frac {2 \sqrt {g x +f}\, c f}{e \,g^{2}}+\frac {2 \left (g x +f \right )^{\frac {3}{2}} c}{3 e \,g^{2}} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 117, normalized size = 1.01 \[ \sqrt {f+g\,x}\,\left (\frac {2\,b\,g-4\,c\,f}{e\,g^2}-\frac {2\,c\,\left (d\,g^3-e\,f\,g^2\right )}{e^2\,g^4}\right )+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {e}\,\sqrt {f+g\,x}}{\sqrt {d\,g-e\,f}}\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{e^{5/2}\,\sqrt {d\,g-e\,f}}+\frac {2\,c\,{\left (f+g\,x\right )}^{3/2}}{3\,e\,g^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 37.63, size = 112, normalized size = 0.97 \[ \frac {2 c \left (f + g x\right )^{\frac {3}{2}}}{3 e g^{2}} - \frac {2 \left (a e^{2} - b d e + c d^{2}\right ) \operatorname {atan}{\left (\frac {1}{\sqrt {\frac {e}{d g - e f}} \sqrt {f + g x}} \right )}}{e^{2} \sqrt {\frac {e}{d g - e f}} \left (d g - e f\right )} + \frac {2 \sqrt {f + g x} \left (b e g - c d g - c e f\right )}{e^{2} g^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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