Optimal. Leaf size=140 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) (c d (4 e f-3 d g)-e (-a e g-b d g+2 b e f))}{e^{5/2} (e f-d g)^{3/2}}-\frac {\sqrt {f+g x} \left (a+\frac {d (c d-b e)}{e^2}\right )}{(d+e x) (e f-d g)}+\frac {2 c \sqrt {f+g x}}{e^2 g} \]
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Rubi [A] time = 0.29, antiderivative size = 140, normalized size of antiderivative = 1.00, 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 = {897, 1157, 388, 208} \[ -\frac {\sqrt {f+g x} \left (a+\frac {d (c d-b e)}{e^2}\right )}{(d+e x) (e f-d g)}+\frac {\tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) (c d (4 e f-3 d g)-e (-a e g-b d g+2 b e f))}{e^{5/2} (e f-d g)^{3/2}}+\frac {2 c \sqrt {f+g x}}{e^2 g} \]
Antiderivative was successfully verified.
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Rule 208
Rule 388
Rule 897
Rule 1157
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{(d+e x)^2 \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}}{\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 {\operatorname {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)) \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^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 = 0.56, size = 150, normalized size = 1.07 \[ \frac {\sqrt {f+g x} \left (e g (b d-a e)+c \left (-3 d^2 g+2 d e (f-g x)+2 e^2 f x\right )\right )}{e^2 g (d+e x) (e f-d g)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) (e (-a e g-b d g+2 b e f)+c d (3 d g-4 e f))}{e^{5/2} (e f-d g)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.24, size = 637, normalized size = 4.55 \[ \left [-\frac {\sqrt {e^{2} f - d e g} {\left (2 \, {\left (2 \, c d^{2} e - b d e^{2}\right )} f g - {\left (3 \, c d^{3} - b d^{2} e - a d e^{2}\right )} g^{2} + {\left (2 \, {\left (2 \, c d e^{2} - b e^{3}\right )} f g - {\left (3 \, c d^{2} e - b d e^{2} - a e^{3}\right )} g^{2}\right )} x\right )} \log \left (\frac {e g x + 2 \, e f - d g - 2 \, \sqrt {e^{2} f - d e g} \sqrt {g x + f}}{e x + d}\right ) - 2 \, {\left (2 \, c d e^{3} f^{2} - {\left (5 \, c d^{2} e^{2} - b d e^{3} + a e^{4}\right )} f g + {\left (3 \, c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )} g^{2} + 2 \, {\left (c e^{4} f^{2} - 2 \, c d e^{3} f g + c d^{2} e^{2} g^{2}\right )} x\right )} \sqrt {g x + f}}{2 \, {\left (d e^{5} f^{2} g - 2 \, d^{2} e^{4} f g^{2} + d^{3} e^{3} g^{3} + {\left (e^{6} f^{2} g - 2 \, d e^{5} f g^{2} + d^{2} e^{4} g^{3}\right )} x\right )}}, -\frac {\sqrt {-e^{2} f + d e g} {\left (2 \, {\left (2 \, c d^{2} e - b d e^{2}\right )} f g - {\left (3 \, c d^{3} - b d^{2} e - a d e^{2}\right )} g^{2} + {\left (2 \, {\left (2 \, c d e^{2} - b e^{3}\right )} f g - {\left (3 \, c d^{2} e - b d e^{2} - a e^{3}\right )} g^{2}\right )} x\right )} \arctan \left (\frac {\sqrt {-e^{2} f + d e g} \sqrt {g x + f}}{e g x + e f}\right ) - {\left (2 \, c d e^{3} f^{2} - {\left (5 \, c d^{2} e^{2} - b d e^{3} + a e^{4}\right )} f g + {\left (3 \, c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )} g^{2} + 2 \, {\left (c e^{4} f^{2} - 2 \, c d e^{3} f g + c d^{2} e^{2} g^{2}\right )} x\right )} \sqrt {g x + f}}{d e^{5} f^{2} g - 2 \, d^{2} e^{4} f g^{2} + d^{3} e^{3} g^{3} + {\left (e^{6} f^{2} g - 2 \, d e^{5} f g^{2} + d^{2} e^{4} g^{3}\right )} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 175, normalized size = 1.25 \[ \frac {2 \, \sqrt {g x + f} c e^{\left (-2\right )}}{g} - \frac {{\left (3 \, c d^{2} g - 4 \, c d f e - b d g e + 2 \, b f e^{2} - a g e^{2}\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {d g e - f e^{2}}}\right )}{{\left (d g e^{2} - f e^{3}\right )} \sqrt {d g e - f e^{2}}} + \frac {\sqrt {g x + f} c d^{2} g - \sqrt {g x + f} b d g e + \sqrt {g x + f} a g e^{2}}{{\left (d g e^{2} - f e^{3}\right )} {\left (d g + {\left (g x + f\right )} e - f e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 371, normalized size = 2.65 \[ \frac {a g \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}}+\frac {b d g \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}\, e}-\frac {2 b f \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}}-\frac {3 c \,d^{2} g \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}\, e^{2}}+\frac {4 c d f \arctan \left (\frac {\sqrt {g x +f}\, e}{\sqrt {\left (d g -e f \right ) e}}\right )}{\left (d g -e f \right ) \sqrt {\left (d g -e f \right ) e}\, e}+\frac {\sqrt {g x +f}\, a g}{\left (d g -e f \right ) \left (e g x +d g \right )}-\frac {\sqrt {g x +f}\, b d g}{\left (d g -e f \right ) \left (e g x +d g \right ) e}+\frac {\sqrt {g x +f}\, c \,d^{2} g}{\left (d g -e f \right ) \left (e g x +d g \right ) e^{2}}+\frac {2 \sqrt {g x +f}\, c}{e^{2} g} \]
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 = 0.23, size = 146, normalized size = 1.04 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {e}\,\sqrt {f+g\,x}}{\sqrt {d\,g-e\,f}}\right )\,\left (a\,e^2\,g-2\,b\,e^2\,f-3\,c\,d^2\,g+b\,d\,e\,g+4\,c\,d\,e\,f\right )}{e^{5/2}\,{\left (d\,g-e\,f\right )}^{3/2}}+\frac {\sqrt {f+g\,x}\,\left (c\,g\,d^2-b\,g\,d\,e+a\,g\,e^2\right )}{\left (d\,g-e\,f\right )\,\left (e^3\,\left (f+g\,x\right )-e^3\,f+d\,e^2\,g\right )}+\frac {2\,c\,\sqrt {f+g\,x}}{e^2\,g} \]
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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