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

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Rubi [A]  time = 0.10, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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 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]))

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]

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*}

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Mathematica [A]  time = 0.20, 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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fricas [A]  time = 0.42, size = 114, normalized size = 0.84 \[ \frac {2 \, e^{3} x^{3} + 3 \, d e^{2} x^{2} - 3 \, a d f^{2} \log \left (-e x + f \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) + 3 \, {\left (a e f^{2} + d^{2} e\right )} x + {\left (2 \, e^{2} f x^{2} + 2 \, a f^{3} + 3 \, d e f x\right )} \sqrt {\frac {e^{2} x^{2} + a f^{2}}{f^{2}}}}{3 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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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giac [A]  time = 0.52, size = 103, normalized size = 0.76 \[ -a d f {\left | f \right |} e^{\left (-1\right )} \log \left ({\left | -x e + \sqrt {a f^{2} + x^{2} e^{2}} \right |}\right ) + a f^{2} x + \frac {2}{3} \, x^{3} e^{2} + d x^{2} e + d^{2} x + \frac {1}{3} \, {\left (2 \, a f {\left | f \right |} e^{\left (-1\right )} + {\left (\frac {2 \, x {\left | f \right |} e}{f} + \frac {3 \, d {\left | f \right |}}{f}\right )} x\right )} \sqrt {a f^{2} + x^{2} e^{2}} \]

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)

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maple [A]  time = 0.01, size = 126, normalized size = 0.93 \[ \frac {2 e^{2} x^{3}}{3}+\frac {a d f \ln \left (\frac {e^{2} x}{\sqrt {\frac {e^{2}}{f^{2}}}\, f^{2}}+\sqrt {\frac {e^{2} x^{2}}{f^{2}}+a}\right )}{\sqrt {\frac {e^{2}}{f^{2}}}}+a \,f^{2} x +d e \,x^{2}+d^{2} x +\sqrt {\frac {e^{2} x^{2}}{f^{2}}+a}\, d f x +\frac {d^{3}}{3 e}+\frac {2 \left (\frac {e^{2} x^{2}+a \,f^{2}}{f^{2}}\right )^{\frac {3}{2}} f^{3}}{3 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [A]  time = 0.89, size = 99, normalized size = 0.73 \[ \frac {1}{3} \, e^{2} x^{3} + \frac {2 \, {\left (\frac {e^{2} x^{2}}{f^{2}} + a\right )}^{\frac {3}{2}} f^{3}}{3 \, e} + \frac {1}{3} \, {\left (\frac {e^{2} x^{3}}{f^{2}} + 3 \, a x\right )} f^{2} + d^{2} x + {\left (e x^{2} + {\left (\frac {a f \operatorname {arsinh}\left (\frac {e x}{\sqrt {a} f}\right )}{e} + \sqrt {\frac {e^{2} x^{2}}{f^{2}} + a} x\right )} f\right )} d \]

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]

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

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mupad [B]  time = 4.66, size = 210, normalized size = 1.54 \[ \left \{\begin {array}{cl} x\,{\left (d+\sqrt {a}\,f\right )}^2 & \text {\ if\ \ }e=0\\ x\,\left (d^2+a\,f^2\right )+\frac {2\,e^2\,x^3}{3}+d\,e\,x^2+\frac {2\,a\,f^3\,\sqrt {a+\frac {e^2\,x^2}{f^2}}}{e}-\frac {2\,f\,\sqrt {a+\frac {e^2\,x^2}{f^2}}\,\left (2\,a\,f^2-e^2\,x^2\right )}{3\,e}+d\,f\,x\,\sqrt {a+\frac {e^2\,x^2}{f^2}}+\frac {2\,a\,d\,f\,\ln \left (x\,\sqrt {\frac {e^2}{f^2}}+\sqrt {a+\frac {e^2\,x^2}{f^2}}\right )}{\sqrt {\frac {e^2}{f^2}}}-\frac {a\,d\,e^2\,\ln \left (2\,x\,\sqrt {\frac {e^2}{f^2}}+2\,\sqrt {a+\frac {e^2\,x^2}{f^2}}\right )}{f\,{\left (\frac {e^2}{f^2}\right )}^{3/2}} & \text {\ if\ \ }e\neq 0 \end {array}\right . \]

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]

piecewise(e == 0, x*(d + a^(1/2)*f)^2, e ~= 0, x*(a*f^2 + d^2) + (2*e^2*x^3)/3 + d*e*x^2 + (2*a*f^3*(a + (e^2*
x^2)/f^2)^(1/2))/e - (2*f*(a + (e^2*x^2)/f^2)^(1/2)*(2*a*f^2 - e^2*x^2))/(3*e) + d*f*x*(a + (e^2*x^2)/f^2)^(1/
2) + (2*a*d*f*log(x*(e^2/f^2)^(1/2) + (a + (e^2*x^2)/f^2)^(1/2)))/(e^2/f^2)^(1/2) - (a*d*e^2*log(2*x*(e^2/f^2)
^(1/2) + 2*(a + (e^2*x^2)/f^2)^(1/2)))/(f*(e^2/f^2)^(3/2)))

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sympy [A]  time = 4.48, 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), T
rue))

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