Optimal. Leaf size=104 \[ -\frac {2 \left (a e^2+c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{5/2} \sqrt {e f-d g}}-\frac {2 c \sqrt {f+g x} (d g+e f)}{e^2 g^2}+\frac {2 c (f+g x)^{3/2}}{3 e g^2} \]
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Rubi [A] time = 0.12, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {898, 1153, 208} \[ -\frac {2 \left (a e^2+c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{5/2} \sqrt {e f-d g}}-\frac {2 c \sqrt {f+g x} (d g+e f)}{e^2 g^2}+\frac {2 c (f+g x)^{3/2}}{3 e g^2} \]
Antiderivative was successfully verified.
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Rule 208
Rule 898
Rule 1153
Rubi steps
\begin {align*} \int \frac {a+c x^2}{(d+e x) \sqrt {f+g x}} \, dx &=\frac {2 \operatorname {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}}{\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 {c (e f+d g)}{e^2 g}+\frac {c x^2}{e g}+\frac {c d^2+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 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 (a+\frac {c d^2}{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 )}{g}\\ &=-\frac {2 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+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 = 0.16, size = 92, normalized size = 0.88 \[ \frac {2 c \sqrt {f+g x} (-3 d g-2 e f+e g x)}{3 e^2 g^2}-\frac {2 \left (a e^2+c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{5/2} \sqrt {e f-d g}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 297, normalized size = 2.86 \[ \left [\frac {3 \, {\left (c d^{2} + a e^{2}\right )} \sqrt {e^{2} f - d e g} g^{2} \log \left (\frac {e g x + 2 \, e f - d g - 2 \, \sqrt {e^{2} f - d e g} \sqrt {g x + f}}{e x + d}\right ) - 2 \, {\left (2 \, c e^{3} f^{2} + c d e^{2} f g - 3 \, c d^{2} e g^{2} - {\left (c e^{3} f g - c d e^{2} g^{2}\right )} x\right )} \sqrt {g x + f}}{3 \, {\left (e^{4} f g^{2} - d e^{3} g^{3}\right )}}, \frac {2 \, {\left (3 \, {\left (c d^{2} + a e^{2}\right )} \sqrt {-e^{2} f + d e g} g^{2} \arctan \left (\frac {\sqrt {-e^{2} f + d e g} \sqrt {g x + f}}{e g x + e f}\right ) - {\left (2 \, c e^{3} f^{2} + c d e^{2} f g - 3 \, c d^{2} e g^{2} - {\left (c e^{3} f g - c d e^{2} g^{2}\right )} x\right )} \sqrt {g x + f}\right )}}{3 \, {\left (e^{4} f g^{2} - d e^{3} g^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 107, normalized size = 1.03 \[ \frac {2 \, {\left (c d^{2} + a e^{2}\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {d g e - f e^{2}}}\right ) e^{\left (-2\right )}}{\sqrt {d g e - f e^{2}}} - \frac {2 \, {\left (3 \, \sqrt {g x + f} c d g^{5} e - {\left (g x + f\right )}^{\frac {3}{2}} c g^{4} e^{2} + 3 \, \sqrt {g x + f} c f g^{4} e^{2}\right )} e^{\left (-3\right )}}{3 \, g^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 132, normalized size = 1.27 \[ \frac {2 a \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\sqrt {\left (d g -e f \right ) e}}+\frac {2 c \,d^{2} \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\sqrt {\left (d g -e f \right ) e}\, e^{2}}-\frac {2 \sqrt {g x +f}\, c d}{e^{2} g}-\frac {2 \sqrt {g x +f}\, c f}{e \,g^{2}}+\frac {2 \left (g x +f \right )^{\frac {3}{2}} c}{3 e \,g^{2}} \]
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 = 0.11, size = 107, normalized size = 1.03 \[ \frac {2\,\mathrm {atan}\left (\frac {\sqrt {e}\,\sqrt {f+g\,x}}{\sqrt {d\,g-e\,f}}\right )\,\left (c\,d^2+a\,e^2\right )}{e^{5/2}\,\sqrt {d\,g-e\,f}}-\sqrt {f+g\,x}\,\left (\frac {2\,c\,\left (d\,g^3-e\,f\,g^2\right )}{e^2\,g^4}+\frac {4\,c\,f}{e\,g^2}\right )+\frac {2\,c\,{\left (f+g\,x\right )}^{3/2}}{3\,e\,g^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 48.66, size = 100, normalized size = 0.96 \[ \frac {2 c \left (f + g x\right )^{\frac {3}{2}}}{3 e g^{2}} - \frac {2 c \sqrt {f + g x} \left (d g + e f\right )}{e^{2} g^{2}} - \frac {2 \left (a e^{2} + c d^{2}\right ) \operatorname {atan}{\left (\frac {1}{\sqrt {\frac {e}{d g - e f}} \sqrt {f + g x}} \right )}}{e^{2} \sqrt {\frac {e}{d g - e f}} \left (d g - e f\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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