Optimal. Leaf size=102 \[ \frac {b \sqrt {d+e x+f x^2}}{4 f}+\frac {b x \sqrt {d+e x+f x^2}}{2 e}+\frac {\left (8 a f-b \left (e+\frac {4 d f}{e}\right )\right ) \tanh ^{-1}\left (\frac {e+2 f x}{2 \sqrt {f} \sqrt {d+e x+f x^2}}\right )}{8 f^{3/2}} \]
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Rubi [A]
time = 0.06, 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 = {1675, 654, 635,
212} \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 212
Rule 635
Rule 654
Rule 1675
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 ) \text {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 = 0.46, size = 87, normalized size = 0.85 \begin {gather*} \frac {2 b \sqrt {f} (e+2 f x) \sqrt {d+x (e+f x)}+\left (-8 a e f+b \left (e^2+4 d f\right )\right ) \log \left (e f \left (e+2 f x-2 \sqrt {f} \sqrt {d+x (e+f x)}\right )\right )}{8 e f^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(194\) vs.
\(2(84)=168\).
time = 0.14, size = 195, normalized size = 1.91
method | result | size |
risch | \(\frac {b \left (2 f x +e \right ) \sqrt {f \,x^{2}+e x +d}}{4 f e}+\frac {\frac {a e \ln \left (\frac {\frac {e}{2}+f x}{\sqrt {f}}+\sqrt {f \,x^{2}+e x +d}\right )}{\sqrt {f}}-\frac {\ln \left (\frac {\frac {e}{2}+f x}{\sqrt {f}}+\sqrt {f \,x^{2}+e x +d}\right ) b d}{2 \sqrt {f}}-\frac {\ln \left (\frac {\frac {e}{2}+f x}{\sqrt {f}}+\sqrt {f \,x^{2}+e x +d}\right ) e^{2} b}{8 f^{\frac {3}{2}}}}{e}\) | \(131\) |
default | \(\frac {b f \left (\frac {x \sqrt {f \,x^{2}+e x +d}}{2 f}-\frac {3 e \left (\frac {\sqrt {f \,x^{2}+e x +d}}{f}-\frac {e \ln \left (\frac {\frac {e}{2}+f x}{\sqrt {f}}+\sqrt {f \,x^{2}+e x +d}\right )}{2 f^{\frac {3}{2}}}\right )}{4 f}-\frac {d \ln \left (\frac {\frac {e}{2}+f x}{\sqrt {f}}+\sqrt {f \,x^{2}+e x +d}\right )}{2 f^{\frac {3}{2}}}\right )+e b \left (\frac {\sqrt {f \,x^{2}+e x +d}}{f}-\frac {e \ln \left (\frac {\frac {e}{2}+f x}{\sqrt {f}}+\sqrt {f \,x^{2}+e x +d}\right )}{2 f^{\frac {3}{2}}}\right )+\frac {a e \ln \left (\frac {\frac {e}{2}+f x}{\sqrt {f}}+\sqrt {f \,x^{2}+e x +d}\right )}{\sqrt {f}}}{e}\) | \(195\) |
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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Fricas [A]
time = 1.29, size = 208, normalized size = 2.04 \begin {gather*} \left [\frac {{\left ({\left (4 \, b d f - 8 \, a f e + b e^{2}\right )} \sqrt {f} \log \left (8 \, f^{2} x^{2} + 8 \, f x e - 4 \, \sqrt {f x^{2} + x e + d} {\left (2 \, f x + e\right )} \sqrt {f} + 4 \, d f + e^{2}\right ) + 4 \, {\left (2 \, b f^{2} x + b f e\right )} \sqrt {f x^{2} + x e + d}\right )} e^{\left (-1\right )}}{16 \, f^{2}}, \frac {{\left ({\left (4 \, b d f - 8 \, a f e + b e^{2}\right )} \sqrt {-f} \arctan \left (\frac {\sqrt {f x^{2} + x e + d} {\left (2 \, f x + e\right )} \sqrt {-f}}{2 \, {\left (f^{2} x^{2} + f x e + d f\right )}}\right ) + 2 \, {\left (2 \, b f^{2} x + b f e\right )} \sqrt {f x^{2} + x e + d}\right )} e^{\left (-1\right )}}{8 \, f^{2}}\right ] \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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Giac [A]
time = 4.74, 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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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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