3.293 \(\int \left (d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )^3 \, dx\)

Optimal. Leaf size=175 \[ -\frac{a d^3 f^2}{2 e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}+\frac{3 a d^2 f^2 \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{2 e}+\frac{\left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^4}{8 e}+\frac{a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^2}{4 e}+\frac{a d f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{e} \]

[Out]

-(a*d^3*f^2)/(2*e*(e*x + f*Sqrt[a + (e^2*x^2)/f^2])) + (a*d*f^2*(e*x + f*Sqrt[a
+ (e^2*x^2)/f^2]))/e + (a*f^2*(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^2)/(4*e) + (
d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^4/(8*e) + (3*a*d^2*f^2*Log[e*x + f*Sqrt[a +
(e^2*x^2)/f^2]])/(2*e)

_______________________________________________________________________________________

Rubi [A]  time = 0.283324, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ -\frac{a d^3 f^2}{2 e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}+\frac{3 a d^2 f^2 \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{2 e}+\frac{\left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^4}{8 e}+\frac{a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^2}{4 e}+\frac{a d f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{e} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^3,x]

[Out]

-(a*d^3*f^2)/(2*e*(e*x + f*Sqrt[a + (e^2*x^2)/f^2])) + (a*d*f^2*(e*x + f*Sqrt[a
+ (e^2*x^2)/f^2]))/e + (a*f^2*(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^2)/(4*e) + (
d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^4/(8*e) + (3*a*d^2*f^2*Log[e*x + f*Sqrt[a +
(e^2*x^2)/f^2]])/(2*e)

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x + f \sqrt{a + \frac{e^{2} x^{2}}{f^{2}}}\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d+e*x+f*(a+e**2*x**2/f**2)**(1/2))**3,x)

[Out]

Integral((d + e*x + f*sqrt(a + e**2*x**2/f**2))**3, x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.598262, size = 143, normalized size = 0.82 \[ \frac{1}{2} \left (\frac{3 a d^2 f^2 \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{e}+\frac{\sqrt{a+\frac{e^2 x^2}{f^2}} \left (2 a f^3 (2 d+e x)+e f x \left (3 d^2+4 d e x+2 e^2 x^2\right )\right )}{e}+3 e x^2 \left (a f^2+d^2\right )+2 d x \left (3 a f^2+d^2\right )+4 d e^2 x^3+2 e^3 x^4\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^3,x]

[Out]

(2*d*(d^2 + 3*a*f^2)*x + 3*e*(d^2 + a*f^2)*x^2 + 4*d*e^2*x^3 + 2*e^3*x^4 + (Sqrt
[a + (e^2*x^2)/f^2]*(2*a*f^3*(2*d + e*x) + e*f*x*(3*d^2 + 4*d*e*x + 2*e^2*x^2)))
/e + (3*a*d^2*f^2*Log[e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/e)/2

_______________________________________________________________________________________

Maple [A]  time = 0.018, size = 175, normalized size = 1. \[{f}^{3}x \left ( a+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}} \right ) ^{{\frac{3}{2}}}+{e}^{3}{x}^{4}+2\,{x}^{3}{e}^{2}d+{\frac{3\,{f}^{2}ae{x}^{2}}{2}}+3\,{f}^{2}adx+{\frac{3\,f{d}^{2}x}{2}\sqrt{a+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}}}+{\frac{3\,f{d}^{2}a}{2}\ln \left ({\frac{{e}^{2}x}{{f}^{2}}{\frac{1}{\sqrt{{\frac{{e}^{2}}{{f}^{2}}}}}}}+\sqrt{a+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ){\frac{1}{\sqrt{{\frac{{e}^{2}}{{f}^{2}}}}}}}+2\,{\frac{d{f}^{3}}{e} \left ({\frac{{e}^{2}{x}^{2}+a{f}^{2}}{{f}^{2}}} \right ) ^{3/2}}+{\frac{3\,{x}^{2}{d}^{2}e}{2}}+{d}^{3}x+{\frac{{d}^{4}}{4\,e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d+e*x+f*(a+e^2*x^2/f^2)^(1/2))^3,x)

[Out]

f^3*x*(a+e^2*x^2/f^2)^(3/2)+e^3*x^4+2*x^3*e^2*d+3/2*f^2*a*e*x^2+3*f^2*a*d*x+3/2*
f*d^2*x*(a+e^2*x^2/f^2)^(1/2)+3/2*f*d^2*a*ln(e^2*x/f^2/(1/f^2*e^2)^(1/2)+(a+e^2*
x^2/f^2)^(1/2))/(1/f^2*e^2)^(1/2)+2*d/e*f^3*((e^2*x^2+a*f^2)/f^2)^(3/2)+3/2*x^2*
d^2*e+d^3*x+1/4*d^4/e

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.291018, size = 217, normalized size = 1.24 \[ \frac{2 \, e^{4} x^{4} + 4 \, d e^{3} x^{3} - 3 \, a d^{2} f^{2} \log \left (-e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) + 3 \,{\left (a e^{2} f^{2} + d^{2} e^{2}\right )} x^{2} + 2 \,{\left (3 \, a d e f^{2} + d^{3} e\right )} x +{\left (2 \, e^{3} f x^{3} + 4 \, d e^{2} f x^{2} + 4 \, a d f^{3} +{\left (2 \, a e f^{3} + 3 \, d^{2} e f\right )} x\right )} \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}}}{2 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^3,x, algorithm="fricas")

[Out]

1/2*(2*e^4*x^4 + 4*d*e^3*x^3 - 3*a*d^2*f^2*log(-e*x + f*sqrt((e^2*x^2 + a*f^2)/f
^2)) + 3*(a*e^2*f^2 + d^2*e^2)*x^2 + 2*(3*a*d*e*f^2 + d^3*e)*x + (2*e^3*f*x^3 +
4*d*e^2*f*x^2 + 4*a*d*f^3 + (2*a*e*f^3 + 3*d^2*e*f)*x)*sqrt((e^2*x^2 + a*f^2)/f^
2))/e

_______________________________________________________________________________________

Sympy [A]  time = 25.4691, size = 279, normalized size = 1.59 \[ \frac{a^{\frac{3}{2}} f^{3} x \sqrt{1 + \frac{e^{2} x^{2}}{a f^{2}}}}{2} + \frac{a^{\frac{3}{2}} f^{3} x}{2 \sqrt{1 + \frac{e^{2} x^{2}}{a f^{2}}}} + \frac{3 \sqrt{a} d^{2} f x \sqrt{1 + \frac{e^{2} x^{2}}{a f^{2}}}}{2} + \frac{3 \sqrt{a} e^{2} f x^{3}}{2 \sqrt{1 + \frac{e^{2} x^{2}}{a f^{2}}}} + \frac{3 a d^{2} f^{2} \operatorname{asinh}{\left (\frac{e x}{\sqrt{a} f} \right )}}{2 e} + 3 a d f^{2} x + \frac{3 a e f^{2} x^{2}}{2} + d^{3} x + \frac{3 d^{2} e x^{2}}{2} + 2 d e^{2} x^{3} + 6 d e f \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: e^{2} = 0 \\\frac{f^{2} \left (a + \frac{e^{2} x^{2}}{f^{2}}\right )^{\frac{3}{2}}}{3 e^{2}} & \text{otherwise} \end{cases}\right ) + e^{3} x^{4} + \frac{e^{4} x^{5}}{\sqrt{a} f \sqrt{1 + \frac{e^{2} x^{2}}{a f^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d+e*x+f*(a+e**2*x**2/f**2)**(1/2))**3,x)

[Out]

a**(3/2)*f**3*x*sqrt(1 + e**2*x**2/(a*f**2))/2 + a**(3/2)*f**3*x/(2*sqrt(1 + e**
2*x**2/(a*f**2))) + 3*sqrt(a)*d**2*f*x*sqrt(1 + e**2*x**2/(a*f**2))/2 + 3*sqrt(a
)*e**2*f*x**3/(2*sqrt(1 + e**2*x**2/(a*f**2))) + 3*a*d**2*f**2*asinh(e*x/(sqrt(a
)*f))/(2*e) + 3*a*d*f**2*x + 3*a*e*f**2*x**2/2 + d**3*x + 3*d**2*e*x**2/2 + 2*d*
e**2*x**3 + 6*d*e*f*Piecewise((sqrt(a)*x**2/2, Eq(e**2, 0)), (f**2*(a + e**2*x**
2/f**2)**(3/2)/(3*e**2), True)) + e**3*x**4 + e**4*x**5/(sqrt(a)*f*sqrt(1 + e**2
*x**2/(a*f**2)))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.288999, size = 220, normalized size = 1.26 \[ -\frac{3}{2} \, a d^{2} f{\left | f \right |} e^{\left (-1\right )}{\rm ln}\left ({\left | -x e + \sqrt{a f^{2} + x^{2} e^{2}} \right |}\right ) + \frac{3}{2} \, a f^{2} x^{2} e + 3 \, a d f^{2} x + x^{4} e^{3} + 2 \, d x^{3} e^{2} + \frac{3}{2} \, d^{2} x^{2} e + d^{3} x + \frac{1}{2} \,{\left (4 \, a d f{\left | f \right |} e^{\left (-1\right )} +{\left (2 \,{\left (\frac{x{\left | f \right |} e^{2}}{f} + \frac{2 \, d{\left | f \right |} e}{f}\right )} x + \frac{{\left (2 \, a f^{6}{\left | f \right |} e^{4} + 3 \, d^{2} f^{4}{\left | f \right |} e^{4}\right )} e^{\left (-4\right )}}{f^{5}}\right )} x\right )} \sqrt{a f^{2} + x^{2} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + sqrt(e^2*x^2/f^2 + a)*f + d)^3,x, algorithm="giac")

[Out]

-3/2*a*d^2*f*abs(f)*e^(-1)*ln(abs(-x*e + sqrt(a*f^2 + x^2*e^2))) + 3/2*a*f^2*x^2
*e + 3*a*d*f^2*x + x^4*e^3 + 2*d*x^3*e^2 + 3/2*d^2*x^2*e + d^3*x + 1/2*(4*a*d*f*
abs(f)*e^(-1) + (2*(x*abs(f)*e^2/f + 2*d*abs(f)*e/f)*x + (2*a*f^6*abs(f)*e^4 + 3
*d^2*f^4*abs(f)*e^4)*e^(-4)/f^5)*x)*sqrt(a*f^2 + x^2*e^2)