Optimal. Leaf size=102 \[ \frac {\left (8 a f-b \left (\frac {4 d f}{e}+e\right )\right ) \tanh ^{-1}\left (\frac {e+2 f x}{2 \sqrt {f} \sqrt {d+e x+f x^2}}\right )}{8 f^{3/2}}+\frac {b x \sqrt {d+e x+f x^2}}{2 e}+\frac {b \sqrt {d+e x+f x^2}}{4 f} \]
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Rubi [A] time = 0.09, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1661, 640, 621, 206} \begin {gather*} \frac {\left (8 a f-b \left (\frac {4 d f}{e}+e\right )\right ) \tanh ^{-1}\left (\frac {e+2 f x}{2 \sqrt {f} \sqrt {d+e x+f x^2}}\right )}{8 f^{3/2}}+\frac {b x \sqrt {d+e x+f x^2}}{2 e}+\frac {b \sqrt {d+e x+f x^2}}{4 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 640
Rule 1661
Rubi steps
\begin {align*} \int \frac {a+b x+\frac {b f x^2}{e}}{\sqrt {d+e x+f x^2}} \, dx &=\frac {b x \sqrt {d+e x+f x^2}}{2 e}+\frac {\int \frac {\left (2 a-\frac {b d}{e}\right ) f+\frac {b f x}{2}}{\sqrt {d+e x+f x^2}} \, dx}{2 f}\\ &=\frac {b \sqrt {d+e x+f x^2}}{4 f}+\frac {b x \sqrt {d+e x+f x^2}}{2 e}+\frac {\left (-b e+8 a f-\frac {4 b d f}{e}\right ) \int \frac {1}{\sqrt {d+e x+f x^2}} \, dx}{8 f}\\ &=\frac {b \sqrt {d+e x+f x^2}}{4 f}+\frac {b x \sqrt {d+e x+f x^2}}{2 e}+\frac {\left (-b e+8 a f-\frac {4 b d f}{e}\right ) \operatorname {Subst}\left (\int \frac {1}{4 f-x^2} \, dx,x,\frac {e+2 f x}{\sqrt {d+e x+f x^2}}\right )}{4 f}\\ &=\frac {b \sqrt {d+e x+f x^2}}{4 f}+\frac {b x \sqrt {d+e x+f x^2}}{2 e}-\frac {\left (b e-8 a f+\frac {4 b d f}{e}\right ) \tanh ^{-1}\left (\frac {e+2 f x}{2 \sqrt {f} \sqrt {d+e x+f x^2}}\right )}{8 f^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 87, normalized size = 0.85 \begin {gather*} \frac {2 b \sqrt {f} (e+2 f x) \sqrt {d+x (e+f x)}-\left (b \left (4 d f+e^2\right )-8 a e f\right ) \tanh ^{-1}\left (\frac {e+2 f x}{2 \sqrt {f} \sqrt {d+x (e+f x)}}\right )}{8 e f^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.44, size = 96, normalized size = 0.94 \begin {gather*} \frac {\left (-8 a e f+4 b d f+b e^2\right ) \log \left (-2 e f^{3/2} \sqrt {d+e x+f x^2}+e^2 f+2 e f^2 x\right )}{8 e f^{3/2}}+\frac {b (e+2 f x) \sqrt {d+e x+f x^2}}{4 e f} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 205, normalized size = 2.01 \begin {gather*} \left [-\frac {{\left (b e^{2} + 4 \, {\left (b d - 2 \, a e\right )} f\right )} \sqrt {f} \log \left (-8 \, f^{2} x^{2} - 8 \, e f x - e^{2} - 4 \, \sqrt {f x^{2} + e x + d} {\left (2 \, f x + e\right )} \sqrt {f} - 4 \, d f\right ) - 4 \, {\left (2 \, b f^{2} x + b e f\right )} \sqrt {f x^{2} + e x + d}}{16 \, e f^{2}}, \frac {{\left (b e^{2} + 4 \, {\left (b d - 2 \, a e\right )} f\right )} \sqrt {-f} \arctan \left (\frac {\sqrt {f x^{2} + e x + d} {\left (2 \, f x + e\right )} \sqrt {-f}}{2 \, {\left (f^{2} x^{2} + e f x + d f\right )}}\right ) + 2 \, {\left (2 \, b f^{2} x + b e f\right )} \sqrt {f x^{2} + e x + d}}{8 \, e f^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 84, normalized size = 0.82 \begin {gather*} \frac {1}{4} \, \sqrt {f x^{2} + x e + d} {\left (2 \, b x e^{\left (-1\right )} + \frac {b}{f}\right )} + \frac {{\left (4 \, b d f - 8 \, a f e + b e^{2}\right )} e^{\left (-1\right )} \log \left ({\left | -2 \, {\left (\sqrt {f} x - \sqrt {f x^{2} + x e + d}\right )} \sqrt {f} - e \right |}\right )}{8 \, f^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 136, normalized size = 1.33 \begin {gather*} \frac {a \ln \left (\frac {f x +\frac {e}{2}}{\sqrt {f}}+\sqrt {f \,x^{2}+e x +d}\right )}{\sqrt {f}}-\frac {b d \ln \left (\frac {f x +\frac {e}{2}}{\sqrt {f}}+\sqrt {f \,x^{2}+e x +d}\right )}{2 e \sqrt {f}}-\frac {b e \ln \left (\frac {f x +\frac {e}{2}}{\sqrt {f}}+\sqrt {f \,x^{2}+e x +d}\right )}{8 f^{\frac {3}{2}}}+\frac {\sqrt {f \,x^{2}+e x +d}\, b x}{2 e}+\frac {\sqrt {f \,x^{2}+e x +d}\, b}{4 f} \end {gather*}
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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,x+\frac {b\,f\,x^2}{e}}{\sqrt {f\,x^2+e\,x+d}} \,d x \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*} \frac {\int \frac {a e}{\sqrt {d + e x + f x^{2}}}\, dx + \int \frac {b e x}{\sqrt {d + e x + f x^{2}}}\, dx + \int \frac {b f x^{2}}{\sqrt {d + e x + f x^{2}}}\, dx}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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