Optimal. Leaf size=233 \[ \frac {\left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^{3/2}}{3 e}-\frac {f^2 \left (4 a-\frac {b^2 f^2}{e^2}\right ) \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{4 \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}-\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{\sqrt {2 d e-b f^2}}\right )}{4 \sqrt {2} e^{5/2} \sqrt {2 d e-b f^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.23, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2141, 911,
1271, 1167, 214} \begin {gather*} -\frac {f^2 \left (4 a-\frac {b^2 f^2}{e^2}\right ) \sqrt {f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x}}{4 \left (2 e \left (f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}+e x\right )+b f^2\right )}-\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} \sqrt {f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x}}{\sqrt {2 d e-b f^2}}\right )}{4 \sqrt {2} e^{5/2} \sqrt {2 d e-b f^2}}+\frac {\left (f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}+d+e x\right )^{3/2}}{3 e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 214
Rule 911
Rule 1167
Rule 1271
Rule 2141
Rubi steps
\begin {align*} \int \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}} \, dx &=2 \text {Subst}\left (\int \frac {\sqrt {x} \left (d^2 e-(b d-a e) f^2-\left (2 d e-b f^2\right ) x+e x^2\right )}{\left (-2 d e+b f^2+2 e x\right )^2} \, dx,x,d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )\\ &=4 \text {Subst}\left (\int \frac {x^2 \left (d^2 e-(b d-a e) f^2+\left (-2 d e+b f^2\right ) x^2+e x^4\right )}{\left (-2 d e+b f^2+2 e x^2\right )^2} \, dx,x,\sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}\right )\\ &=-\frac {f^2 \left (4 a-\frac {b^2 f^2}{e^2}\right ) \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{4 \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}-\frac {\text {Subst}\left (\int \frac {-e f^2 \left (4 a e^2-b^2 f^2\right )+4 e^2 \left (2 d e-b f^2\right ) x^2-8 e^3 x^4}{-2 d e+b f^2+2 e x^2} \, dx,x,\sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}\right )}{4 e^3}\\ &=-\frac {f^2 \left (4 a-\frac {b^2 f^2}{e^2}\right ) \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{4 \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}-\frac {\text {Subst}\left (\int \left (-4 e^2 x^2-\frac {e f^2 \left (4 a e^2-b^2 f^2\right )}{-2 d e+b f^2+2 e x^2}\right ) \, dx,x,\sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}\right )}{4 e^3}\\ &=\frac {\left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^{3/2}}{3 e}-\frac {f^2 \left (4 a-\frac {b^2 f^2}{e^2}\right ) \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{4 \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}+\frac {1}{4} \left (f^2 \left (4 a-\frac {b^2 f^2}{e^2}\right )\right ) \text {Subst}\left (\int \frac {1}{-2 d e+b f^2+2 e x^2} \, dx,x,\sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}\right )\\ &=\frac {\left (d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}\right )^{3/2}}{3 e}-\frac {f^2 \left (4 a-\frac {b^2 f^2}{e^2}\right ) \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{4 \left (b f^2+2 e \left (e x+f \sqrt {a+\frac {x \left (b f^2+e^2 x\right )}{f^2}}\right )\right )}-\frac {f^2 \left (4 a e^2-b^2 f^2\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {e} \sqrt {d+e x+f \sqrt {a+b x+\frac {e^2 x^2}{f^2}}}}{\sqrt {2 d e-b f^2}}\right )}{4 \sqrt {2} e^{5/2} \sqrt {2 d e-b f^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.51, size = 327, normalized size = 1.40 \begin {gather*} \frac {\sqrt {d+e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}} \left (3 b^2 f^4+4 b e f^2 \left (d+3 e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )+4 e^2 \left (-a f^2+2 (d+2 e x) \left (e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )\right )\right )}{12 e^2 \left (b f^2+2 e \left (e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}\right )\right )}+\frac {a f^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e} \sqrt {d+e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}}}{\sqrt {-2 d e+b f^2}}\right )}{\sqrt {2} \sqrt {e} \sqrt {-2 d e+b f^2}}-\frac {b^2 f^4 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e} \sqrt {d+e x+f \sqrt {a+x \left (b+\frac {e^2 x}{f^2}\right )}}}{\sqrt {-2 d e+b f^2}}\right )}{4 e^{5/2} \sqrt {-4 d e+2 b f^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \sqrt {d +e x +f \sqrt {a +b x +\frac {e^{2} x^{2}}{f^{2}}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.50, size = 600, normalized size = 2.58 \begin {gather*} \left [\frac {3 \, {\left (b^{2} f^{4} - 4 \, a f^{2} e^{2}\right )} \sqrt {-2 \, b f^{2} e + 4 \, d e^{2}} \log \left (b^{2} f^{4} - 4 \, b d f^{2} e - 8 \, d x e^{3} + 4 \, {\left (b f^{2} x + a f^{2}\right )} e^{2} - 2 \, {\left (2 \, \sqrt {-2 \, b f^{2} e + 4 \, d e^{2}} f \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}} e - \sqrt {-2 \, b f^{2} e + 4 \, d e^{2}} {\left (b f^{2} + 2 \, x e^{2}\right )}\right )} \sqrt {x e + f \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}} + d} - 4 \, {\left (b f^{3} e - 2 \, d f e^{2}\right )} \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}}\right ) + 4 \, {\left (3 \, b^{2} f^{4} e - 2 \, b d f^{2} e^{2} - 20 \, d x e^{4} + 2 \, {\left (5 \, b f^{2} x - 4 \, d^{2}\right )} e^{3} - 2 \, {\left (b f^{3} e^{2} - 2 \, d f e^{3}\right )} \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}}\right )} \sqrt {x e + f \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}} + d}}{48 \, {\left (b f^{2} e^{3} - 2 \, d e^{4}\right )}}, \frac {3 \, {\left (b^{2} f^{4} - 4 \, a f^{2} e^{2}\right )} \sqrt {2 \, b f^{2} e - 4 \, d e^{2}} \arctan \left (-\frac {\sqrt {2 \, b f^{2} e - 4 \, d e^{2}} \sqrt {x e + f \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}} + d}}{b f^{2} - 2 \, d e}\right ) + 2 \, {\left (3 \, b^{2} f^{4} e - 2 \, b d f^{2} e^{2} - 20 \, d x e^{4} + 2 \, {\left (5 \, b f^{2} x - 4 \, d^{2}\right )} e^{3} - 2 \, {\left (b f^{3} e^{2} - 2 \, d f e^{3}\right )} \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}}\right )} \sqrt {x e + f \sqrt {\frac {b f^{2} x + a f^{2} + x^{2} e^{2}}{f^{2}}} + d}}{24 \, {\left (b f^{2} e^{3} - 2 \, d e^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {d + e x + f \sqrt {a + b x + \frac {e^{2} x^{2}}{f^{2}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sqrt {d+e\,x+f\,\sqrt {a+b\,x+\frac {e^2\,x^2}{f^2}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________