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

Optimal. Leaf size=136 \[ -\frac{a d^2 f^2}{2 e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}+\frac{\left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^3}{6 e}+\frac{a d f^2 \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{e}+\frac{a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{2 e} \]

[Out]

-(a*d^2*f^2)/(2*e*(e*x + f*Sqrt[a + (e^2*x^2)/f^2])) + (a*f^2*(e*x + f*Sqrt[a +
(e^2*x^2)/f^2]))/(2*e) + (d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^3/(6*e) + (a*d*f^
2*Log[e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/e

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Rubi [A]  time = 0.228167, antiderivative size = 136, 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 \[ -\frac{a d^2 f^2}{2 e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}+\frac{\left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^3}{6 e}+\frac{a d f^2 \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{e}+\frac{a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{2 e} \]

Antiderivative was successfully verified.

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

[Out]

-(a*d^2*f^2)/(2*e*(e*x + f*Sqrt[a + (e^2*x^2)/f^2])) + (a*f^2*(e*x + f*Sqrt[a +
(e^2*x^2)/f^2]))/(2*e) + (d + e*x + f*Sqrt[a + (e^2*x^2)/f^2])^3/(6*e) + (a*d*f^
2*Log[e*x + f*Sqrt[a + (e^2*x^2)/f^2]])/e

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

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))**2,x)

[Out]

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

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Mathematica [A]  time = 0.393465, size = 102, normalized size = 0.75 \[ x \left (a f^2+d^2\right )+\frac{a d f^2 \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{e}+\frac{\sqrt{a+\frac{e^2 x^2}{f^2}} \left (2 a f^3+e f x (3 d+2 e x)\right )}{3 e}+d e x^2+\frac{2 e^2 x^3}{3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 126, normalized size = 0.9 \[{f}^{2}xa+{\frac{2\,{x}^{3}{e}^{2}}{3}}+fdx\sqrt{a+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}}+{adf\ln \left ({\frac{{e}^{2}x}{{f}^{2}}{\frac{1}{\sqrt{{\frac{{e}^{2}}{{f}^{2}}}}}}}+\sqrt{a+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ){\frac{1}{\sqrt{{\frac{{e}^{2}}{{f}^{2}}}}}}}+{\frac{2\,{f}^{3}}{3\,e} \left ({\frac{{e}^{2}{x}^{2}+a{f}^{2}}{{f}^{2}}} \right ) ^{{\frac{3}{2}}}}+{x}^{2}de+x{d}^{2}+{\frac{{d}^{3}}{3\,e}} \]

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))^2,x)

[Out]

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

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.292464, size = 154, normalized size = 1.13 \[ \frac{2 \, e^{3} x^{3} + 3 \, d e^{2} x^{2} - 3 \, a d f^{2} \log \left (-e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) + 3 \,{\left (a e f^{2} + d^{2} e\right )} x +{\left (2 \, e^{2} f x^{2} + 2 \, a f^{3} + 3 \, d e f x\right )} \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}}}{3 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.60567, size = 116, normalized size = 0.85 \[ \sqrt{a} d f x \sqrt{1 + \frac{e^{2} x^{2}}{a f^{2}}} + \frac{a d f^{2} \operatorname{asinh}{\left (\frac{e x}{\sqrt{a} f} \right )}}{e} + a f^{2} x + d^{2} x + d e x^{2} + \frac{2 e^{2} x^{3}}{3} + 2 e f \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: e^{2} = 0 \\\frac{f^{2} \left (a + \frac{e^{2} x^{2}}{f^{2}}\right )^{\frac{3}{2}}}{3 e^{2}} & \text{otherwise} \end{cases}\right ) \]

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))**2,x)

[Out]

sqrt(a)*d*f*x*sqrt(1 + e**2*x**2/(a*f**2)) + a*d*f**2*asinh(e*x/(sqrt(a)*f))/e +
a*f**2*x + d**2*x + d*e*x**2 + 2*e**2*x**3/3 + 2*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))

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GIAC/XCAS [A]  time = 0.28552, size = 139, normalized size = 1.02 \[ -a d f{\left | f \right |} e^{\left (-1\right )}{\rm ln}\left ({\left | -x e + \sqrt{a f^{2} + x^{2} e^{2}} \right |}\right ) + a f^{2} x + \frac{2}{3} \, x^{3} e^{2} + d x^{2} e + d^{2} x + \frac{1}{3} \,{\left (2 \, a f{\left | f \right |} e^{\left (-1\right )} +{\left (\frac{2 \, x{\left | f \right |} e}{f} + \frac{3 \, d{\left | f \right |}}{f}\right )} x\right )} \sqrt{a f^{2} + x^{2} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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