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

Optimal. Leaf size=239 \[ \frac{\left (d^2-a f^2\right )^3 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-3}}{8 e f^2 (3-n)}-\frac{3 \left (d^2-a f^2\right )^2 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-1}}{8 e f^2 (1-n)}-\frac{3 \left (d^2-a f^2\right ) \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{8 e f^2 (n+1)}+\frac{\left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+3}}{8 e f^2 (n+3)} \]

[Out]

((d^2 - a*f^2)^3*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(-3 + n))/(8*e*f^2*(3 - n)) - (3*(d^2 -
a*f^2)^2*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(-1 + n))/(8*e*f^2*(1 - n)) - (3*(d^2 - a*f^2)
*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(1 + n))/(8*e*f^2*(1 + n)) + (d + e*x + f*Sqrt[a + (2*d
*e*x)/f^2 + (e^2*x^2)/f^2])^(3 + n)/(8*e*f^2*(3 + n))

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Rubi [A]  time = 0.250062, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 54, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {2121, 12, 270} \[ \frac{\left (d^2-a f^2\right )^3 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-3}}{8 e f^2 (3-n)}-\frac{3 \left (d^2-a f^2\right )^2 \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n-1}}{8 e f^2 (1-n)}-\frac{3 \left (d^2-a f^2\right ) \left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+1}}{8 e f^2 (n+1)}+\frac{\left (f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}+d+e x\right )^{n+3}}{8 e f^2 (n+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d^2 - a*f^2)^3*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(-3 + n))/(8*e*f^2*(3 - n)) - (3*(d^2 -
a*f^2)^2*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(-1 + n))/(8*e*f^2*(1 - n)) - (3*(d^2 - a*f^2)
*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(1 + n))/(8*e*f^2*(1 + n)) + (d + e*x + f*Sqrt[a + (2*d
*e*x)/f^2 + (e^2*x^2)/f^2])^(3 + n)/(8*e*f^2*(3 + n))

Rule 2121

Int[((g_.) + (h_.)*(x_) + (i_.)*(x_)^2)^(m_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)
^2])^(n_.), x_Symbol] :> Dist[(2*(i/c)^m)/f^(2*m), Subst[Int[(x^n*(d^2*e - (b*d - a*e)*f^2 - (2*d*e - b*f^2)*x
+ e*x^2)^(2*m + 1))/(-2*d*e + b*f^2 + 2*e*x)^(2*(m + 1)), x], x, d + e*x + f*Sqrt[a + b*x + c*x^2]], x] /; Fr
eeQ[{a, b, c, d, e, f, g, h, i, n}, x] && EqQ[e^2 - c*f^2, 0] && EqQ[c*g - a*i, 0] && EqQ[c*h - b*i, 0] && Int
egerQ[2*m] && (IntegerQ[m] || GtQ[i/c, 0])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \left (a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}\right ) \left (d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )^n \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{x^{-4+n} \left (d^2 e-\left (-a e+\frac{2 d^2 e}{f^2}\right ) f^2+e x^2\right )^3}{16 e^4} \, dx,x,d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )}{f^2}\\ &=\frac{\operatorname{Subst}\left (\int x^{-4+n} \left (d^2 e-\left (-a e+\frac{2 d^2 e}{f^2}\right ) f^2+e x^2\right )^3 \, dx,x,d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )}{8 e^4 f^2}\\ &=\frac{\operatorname{Subst}\left (\int \left (-e^3 \left (d^2-a f^2\right )^3 x^{-4+n}+3 e^3 \left (d^2-a f^2\right )^2 x^{-2+n}-3 e^3 \left (d^2-a f^2\right ) x^n+e^3 x^{2+n}\right ) \, dx,x,d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )}{8 e^4 f^2}\\ &=\frac{\left (d^2-a f^2\right )^3 \left (d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )^{-3+n}}{8 e f^2 (3-n)}-\frac{3 \left (d^2-a f^2\right )^2 \left (d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )^{-1+n}}{8 e f^2 (1-n)}-\frac{3 \left (d^2-a f^2\right ) \left (d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )^{1+n}}{8 e f^2 (1+n)}+\frac{\left (d+e x+f \sqrt{a+\frac{2 d e x}{f^2}+\frac{e^2 x^2}{f^2}}\right )^{3+n}}{8 e f^2 (3+n)}\\ \end{align*}

Mathematica [A]  time = 0.576077, size = 186, normalized size = 0.78 \[ \frac{\left (f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right )^{n-3} \left (-\frac{3 \left (d^2-a f^2\right ) \left (f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right )^4}{n+1}+\frac{3 \left (d^2-a f^2\right )^2 \left (f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right )^2}{n-1}-\frac{\left (d^2-a f^2\right )^3}{n-3}+\frac{\left (f \sqrt{a+\frac{e x (2 d+e x)}{f^2}}+d+e x\right )^6}{n+3}\right )}{8 e f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^(-3 + n)*(-((d^2 - a*f^2)^3/(-3 + n)) + (3*(d^2 - a*f^2)^2*(d +
e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^2)/(-1 + n) - (3*(d^2 - a*f^2)*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x
))/f^2])^4)/(1 + n) + (d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^6/(3 + n)))/(8*e*f^2)

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Maple [F]  time = 0.007, size = 0, normalized size = 0. \begin{align*} \int \left ( a+2\,{\frac{dex}{{f}^{2}}}+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}} \right ) \left ( d+ex+f\sqrt{a+2\,{\frac{dex}{{f}^{2}}}+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+2*d*e*x/f^2+e^2*x^2/f^2)*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2))^n,x)

[Out]

int((a+2*d*e*x/f^2+e^2*x^2/f^2)*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2))^n,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\frac{e^{2} x^{2}}{f^{2}} + a + \frac{2 \, d e x}{f^{2}}\right )}{\left (e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a + \frac{2 \, d e x}{f^{2}}} f + d\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+2*d*e*x/f^2+e^2*x^2/f^2)*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2))^n,x, algorithm="maxima")

[Out]

integrate((e^2*x^2/f^2 + a + 2*d*e*x/f^2)*(e*x + sqrt(e^2*x^2/f^2 + a + 2*d*e*x/f^2)*f + d)^n, x)

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Fricas [A]  time = 1.15083, size = 482, normalized size = 2.02 \begin{align*} -\frac{{\left (3 \, a d f^{2} n^{2} - 9 \, a d f^{2} + 3 \,{\left (e^{3} n^{2} - e^{3}\right )} x^{3} + 6 \, d^{3} + 9 \,{\left (d e^{2} n^{2} - d e^{2}\right )} x^{2} - 3 \,{\left (3 \, a e f^{2} -{\left (a e f^{2} + 2 \, d^{2} e\right )} n^{2}\right )} x -{\left (a f^{3} n^{3} +{\left (e^{2} f n^{3} - e^{2} f n\right )} x^{2} -{\left (7 \, a f^{3} - 6 \, d^{2} f\right )} n + 2 \,{\left (d e f n^{3} - d e f n\right )} x\right )} \sqrt{\frac{e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}}\right )}{\left (e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2} + 2 \, d e x}{f^{2}}} + d\right )}^{n}}{e f^{2} n^{4} - 10 \, e f^{2} n^{2} + 9 \, e f^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+2*d*e*x/f^2+e^2*x^2/f^2)*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2))^n,x, algorithm="fricas")

[Out]

-(3*a*d*f^2*n^2 - 9*a*d*f^2 + 3*(e^3*n^2 - e^3)*x^3 + 6*d^3 + 9*(d*e^2*n^2 - d*e^2)*x^2 - 3*(3*a*e*f^2 - (a*e*
f^2 + 2*d^2*e)*n^2)*x - (a*f^3*n^3 + (e^2*f*n^3 - e^2*f*n)*x^2 - (7*a*f^3 - 6*d^2*f)*n + 2*(d*e*f*n^3 - d*e*f*
n)*x)*sqrt((e^2*x^2 + a*f^2 + 2*d*e*x)/f^2))*(e*x + f*sqrt((e^2*x^2 + a*f^2 + 2*d*e*x)/f^2) + d)^n/(e*f^2*n^4
- 10*e*f^2*n^2 + 9*e*f^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+2*d*e*x/f**2+e**2*x**2/f**2)*(d+e*x+f*(a+2*d*e*x/f**2+e**2*x**2/f**2)**(1/2))**n,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\frac{e^{2} x^{2}}{f^{2}} + a + \frac{2 \, d e x}{f^{2}}\right )}{\left (e x + \sqrt{\frac{e^{2} x^{2}}{f^{2}} + a + \frac{2 \, d e x}{f^{2}}} f + d\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+2*d*e*x/f^2+e^2*x^2/f^2)*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2))^n,x, algorithm="giac")

[Out]

integrate((e^2*x^2/f^2 + a + 2*d*e*x/f^2)*(e*x + sqrt(e^2*x^2/f^2 + a + 2*d*e*x/f^2)*f + d)^n, x)