Optimal. Leaf size=122 \[ -\frac {\sqrt {f+g x} \left (a+\frac {c d^2}{e^2}\right )}{(d+e x) (e f-d g)}+\frac {\left (a e^2 g+c d (4 e f-3 d g)\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{5/2} (e f-d g)^{3/2}}+\frac {2 c \sqrt {f+g x}}{e^2 g} \]
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Rubi [A] time = 0.20, antiderivative size = 122, normalized size of antiderivative = 1.00, 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 = {898, 1157, 388, 208} \[ -\frac {\sqrt {f+g x} \left (a+\frac {c d^2}{e^2}\right )}{(d+e x) (e f-d g)}+\frac {\left (a e^2 g+c d (4 e f-3 d g)\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{5/2} (e f-d g)^{3/2}}+\frac {2 c \sqrt {f+g x}}{e^2 g} \]
Antiderivative was successfully verified.
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Rule 208
Rule 388
Rule 898
Rule 1157
Rubi steps
\begin {align*} \int \frac {a+c x^2}{(d+e x)^2 \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}}{\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 {\operatorname {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 ) \operatorname {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 = 0.29, size = 171, normalized size = 1.40 \[ \frac {\frac {\left (a e^2+c d^2\right ) \left (\sqrt {e} \sqrt {f+g x} (d g-e f)+g (d+e x) \sqrt {d g-e f} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {d g-e f}}\right )\right )}{(d+e x) (e f-d g)^2}+\frac {4 c d \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}+\frac {2 c \sqrt {e} \sqrt {f+g x}}{g}}{e^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 539, normalized size = 4.42 \[ \left [-\frac {{\left (4 \, c d^{2} e f g - {\left (3 \, c d^{3} - a d e^{2}\right )} g^{2} + {\left (4 \, c d e^{2} f g - {\left (3 \, c d^{2} e - a e^{3}\right )} g^{2}\right )} x\right )} \sqrt {e^{2} f - d e g} \log \left (\frac {e g x + 2 \, e f - d g - 2 \, \sqrt {e^{2} f - d e g} \sqrt {g x + f}}{e x + d}\right ) - 2 \, {\left (2 \, c d e^{3} f^{2} - {\left (5 \, c d^{2} e^{2} + a e^{4}\right )} f g + {\left (3 \, c d^{3} e + a d e^{3}\right )} g^{2} + 2 \, {\left (c e^{4} f^{2} - 2 \, c d e^{3} f g + c d^{2} e^{2} g^{2}\right )} x\right )} \sqrt {g x + f}}{2 \, {\left (d e^{5} f^{2} g - 2 \, d^{2} e^{4} f g^{2} + d^{3} e^{3} g^{3} + {\left (e^{6} f^{2} g - 2 \, d e^{5} f g^{2} + d^{2} e^{4} g^{3}\right )} x\right )}}, -\frac {{\left (4 \, c d^{2} e f g - {\left (3 \, c d^{3} - a d e^{2}\right )} g^{2} + {\left (4 \, c d e^{2} f g - {\left (3 \, c d^{2} e - a e^{3}\right )} g^{2}\right )} x\right )} \sqrt {-e^{2} f + d e g} \arctan \left (\frac {\sqrt {-e^{2} f + d e g} \sqrt {g x + f}}{e g x + e f}\right ) - {\left (2 \, c d e^{3} f^{2} - {\left (5 \, c d^{2} e^{2} + a e^{4}\right )} f g + {\left (3 \, c d^{3} e + a d e^{3}\right )} g^{2} + 2 \, {\left (c e^{4} f^{2} - 2 \, c d e^{3} f g + c d^{2} e^{2} g^{2}\right )} x\right )} \sqrt {g x + f}}{d e^{5} f^{2} g - 2 \, d^{2} e^{4} f g^{2} + d^{3} e^{3} g^{3} + {\left (e^{6} f^{2} g - 2 \, d e^{5} f g^{2} + d^{2} e^{4} g^{3}\right )} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 148, normalized size = 1.21 \[ \frac {2 \, \sqrt {g x + f} c e^{\left (-2\right )}}{g} - \frac {{\left (3 \, c d^{2} g - 4 \, c d f e - a g e^{2}\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {d g e - f e^{2}}}\right )}{{\left (d g e^{2} - f e^{3}\right )} \sqrt {d g e - f e^{2}}} + \frac {\sqrt {g x + f} c d^{2} g + \sqrt {g x + f} a g e^{2}}{{\left (d g e^{2} - f e^{3}\right )} {\left (d g + {\left (g x + f\right )} e - f e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 237, normalized size = 1.94 \[ \frac {a g \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}}-\frac {3 c \,d^{2} g \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}\, e^{2}}+\frac {4 c d f \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}\, e}+\frac {\sqrt {g x +f}\, a g}{\left (d g -e f \right ) \left (e g x +d g \right )}+\frac {\sqrt {g x +f}\, c \,d^{2} g}{\left (d g -e f \right ) \left (e g x +d g \right ) e^{2}}+\frac {2 \sqrt {g x +f}\, c}{e^{2} g} \]
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 = 2.68, size = 128, normalized size = 1.05 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {e}\,\sqrt {f+g\,x}}{\sqrt {d\,g-e\,f}}\right )\,\left (-3\,c\,g\,d^2+4\,c\,f\,d\,e+a\,g\,e^2\right )}{e^{5/2}\,{\left (d\,g-e\,f\right )}^{3/2}}+\frac {\sqrt {f+g\,x}\,\left (c\,g\,d^2+a\,g\,e^2\right )}{\left (d\,g-e\,f\right )\,\left (e^3\,\left (f+g\,x\right )-e^3\,f+d\,e^2\,g\right )}+\frac {2\,c\,\sqrt {f+g\,x}}{e^2\,g} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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