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

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 = 0.103329, 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, Rules used = {2117, 893} \[ -\frac{a d^2 f^2}{2 e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}+\frac{\left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^3}{6 e}+\frac{a d f^2 \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{e}+\frac{a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{2 e} \]

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

Rule 2117

Int[((g_.) + (h_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_))^(p_.), x_Symbol] :> Dist[1/(2*
e), Subst[Int[((g + h*x^n)^p*(d^2 + a*f^2 - 2*d*x + x^2))/(d - x)^2, x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /
; FreeQ[{a, c, d, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[p]

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

\begin{align*} \int \left (d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (d^2+a f^2-2 d x+x^2\right )}{(d-x)^2} \, dx,x,d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )}{2 e}\\ &=\frac{\operatorname{Subst}\left (\int \left (a f^2+\frac{a d^2 f^2}{(d-x)^2}-\frac{2 a d f^2}{d-x}+x^2\right ) \, dx,x,d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )}{2 e}\\ &=-\frac{a d^2 f^2}{2 e \left (e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )}+\frac{a f^2 \left (e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )}{2 e}+\frac{\left (d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )^3}{6 e}+\frac{a d f^2 \log \left (e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )}{e}\\ \end{align*}

Mathematica [A]  time = 0.201769, size = 128, normalized size = 0.94 \[ \frac{\frac{a d^2 f^2}{f \left (-\sqrt{a+\frac{e^2 x^2}{f^2}}\right )-e x}+\frac{1}{3} \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^3+2 a d f^2 \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )+a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{2 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 126, normalized size = 0.9 \begin{align*}{f}^{2}ax+{\frac{2\,{e}^{2}{x}^{3}}{3}}+{\frac{2\,{f}^{3}}{3\,e} \left ({\frac{{e}^{2}{x}^{2}+a{f}^{2}}{{f}^{2}}} \right ) ^{{\frac{3}{2}}}}+fdx\sqrt{a+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}}+{dfa\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}}}}}}}+e{x}^{2}d+x{d}^{2}+{\frac{{d}^{3}}{3\,e}} \end{align*}

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.01359, size = 239, normalized size = 1.76 \begin{align*} \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} \end{align*}

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, 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 = 3.70792, size = 116, normalized size = 0.85 \begin{align*} \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 ) \end{align*}

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), T
rue))

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Giac [A]  time = 1.17823, size = 139, normalized size = 1.02 \begin{align*} -a d f{\left | f \right |} e^{\left (-1\right )} \log \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}} \end{align*}

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, algorithm="giac")

[Out]

-a*d*f*abs(f)*e^(-1)*log(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)