Optimal. Leaf size=118 \[ \frac {f^2 \left (4 a e^2-b^2 f^2\right ) \tanh ^{-1}\left (\frac {b f^2+2 e^2 x}{2 e f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}\right )}{8 e^3}+\frac {f \left (b f^2+2 e^2 x\right ) \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}{4 e^2}+d x+\frac {e x^2}{2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {612, 621, 206} \[ \frac {f^2 \left (4 a e^2-b^2 f^2\right ) \tanh ^{-1}\left (\frac {b f^2+2 e^2 x}{2 e f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}\right )}{8 e^3}+\frac {f \left (b f^2+2 e^2 x\right ) \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}{4 e^2}+d x+\frac {e x^2}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 612
Rule 621
Rubi steps
\begin {align*} \int \left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right ) \, dx &=d x+\frac {e x^2}{2}+f \int \sqrt {a+b x+\frac {e^2 x^2}{f^2}} \, dx\\ &=d x+\frac {e x^2}{2}+\frac {f \left (b f^2+2 e^2 x\right ) \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}{4 e^2}+\frac {1}{8} \left (f \left (4 a-\frac {b^2 f^2}{e^2}\right )\right ) \int \frac {1}{\sqrt {a+b x+\frac {e^2 x^2}{f^2}}} \, dx\\ &=d x+\frac {e x^2}{2}+\frac {f \left (b f^2+2 e^2 x\right ) \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}{4 e^2}+\frac {1}{4} \left (f \left (4 a-\frac {b^2 f^2}{e^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {4 e^2}{f^2}-x^2} \, dx,x,\frac {b+\frac {2 e^2 x}{f^2}}{\sqrt {a+b x+\frac {e^2 x^2}{f^2}}}\right )\\ &=d x+\frac {e x^2}{2}+\frac {f \left (b f^2+2 e^2 x\right ) \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}{4 e^2}+\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \tanh ^{-1}\left (\frac {b f^2+2 e^2 x}{2 e f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}\right )}{8 e^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.21, size = 120, normalized size = 1.02 \[ \frac {1}{8} \left (\frac {\left (4 a e^2 f^2-b^2 f^4\right ) \log \left (2 e \left (f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+e x\right )+b f^2\right )}{e^3}+4 f x \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}+\frac {2 b f^3 \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}}{e^2}+8 d x+4 e x^2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.57, size = 123, normalized size = 1.04 \[ \frac {4 \, e^{4} x^{2} + 8 \, d e^{3} x + {\left (b^{2} f^{4} - 4 \, a e^{2} f^{2}\right )} \log \left (-b f^{2} - 2 \, e^{2} x + 2 \, e f \sqrt {\frac {b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) + 2 \, {\left (b e f^{3} + 2 \, e^{3} f x\right )} \sqrt {\frac {b f^{2} x + e^{2} x^{2} + a f^{2}}{f^{2}}}}{8 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.22, size = 111, normalized size = 0.94 \[ \frac {1}{2} \, x^{2} e + d x + \frac {{\left ({\left (b^{2} f^{4} - 4 \, a f^{2} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | -b f^{2} - 2 \, {\left (x e - \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}}\right )} e \right |}\right ) + 2 \, \sqrt {b f^{2} x + a f^{2} + x^{2} e^{2}} {\left (b f^{2} e^{\left (-2\right )} + 2 \, x\right )}\right )} {\left | f \right |}}{8 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 173, normalized size = 1.47 \[ -\frac {b^{2} f^{3} \ln \left (\frac {\frac {b}{2}+\frac {e^{2} x}{f^{2}}}{\sqrt {\frac {e^{2}}{f^{2}}}}+\sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\right )}{8 \sqrt {\frac {e^{2}}{f^{2}}}\, e^{2}}+\frac {a f \ln \left (\frac {\frac {b}{2}+\frac {e^{2} x}{f^{2}}}{\sqrt {\frac {e^{2}}{f^{2}}}}+\sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\right )}{2 \sqrt {\frac {e^{2}}{f^{2}}}}+\frac {e \,x^{2}}{2}+\frac {\sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\, b \,f^{3}}{4 e^{2}}+d x +\frac {\sqrt {b x +\frac {e^{2} x^{2}}{f^{2}}+a}\, f x}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int d+e\,x+f\,\sqrt {a+b\,x+\frac {e^2\,x^2}{f^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d + e x + f \sqrt {a + b x + \frac {e^{2} x^{2}}{f^{2}}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________