3.6.8 \(\int (a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}) (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}})^n \, dx\) [508]

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

[Out]

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

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Rubi [A]
time = 0.18, antiderivative size = 239, normalized size of antiderivative = 1.00, 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 = {2146, 12, 276} \begin {gather*} \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)} \end {gather*}

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 12

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

Rule 276

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]

Rule 2146

Int[((g_.) + (h_.)*(x_) + (i_.)*(x_)^2)^(m_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)
^2])^(n_.), x_Symbol] :> Dist[(2/f^(2*m))*(i/c)^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])

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 \text {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 {\text {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 {\text {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*}

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Mathematica [A]
time = 1.59, size = 186, normalized size = 0.78 \begin {gather*} \frac {\left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^{-3+n} \left (-\frac {\left (d^2-a f^2\right )^3}{-3+n}+\frac {3 \left (d^2-a f^2\right )^2 \left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^2}{-1+n}-\frac {3 \left (d^2-a f^2\right ) \left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^4}{1+n}+\frac {\left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^6}{3+n}\right )}{8 e f^2} \end {gather*}

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 = 0.01, size = 0, normalized size = 0.00 \[\int \left (a +\frac {2 d e x}{f^{2}}+\frac {e^{2} x^{2}}{f^{2}}\right ) \left (d +e x +f \sqrt {a +\frac {2 d e x}{f^{2}}+\frac {e^{2} x^{2}}{f^{2}}}\right )^{n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [A]
time = 0.42, size = 224, normalized size = 0.94 \begin {gather*} -\frac {{\left (3 \, a d f^{2} n^{2} + 3 \, {\left (n^{2} - 1\right )} x^{3} e^{3} - 9 \, a d f^{2} + 9 \, {\left (d n^{2} - d\right )} x^{2} e^{2} + 6 \, d^{3} - 3 \, {\left (3 \, a f^{2} - {\left (a f^{2} + 2 \, d^{2}\right )} n^{2}\right )} x e - {\left (a f^{3} n^{3} + {\left (f n^{3} - f n\right )} x^{2} e^{2} + 2 \, {\left (d f n^{3} - d f n\right )} x e - {\left (7 \, a f^{3} - 6 \, d^{2} f\right )} n\right )} \sqrt {\frac {a f^{2} + x^{2} e^{2} + 2 \, d x e}{f^{2}}}\right )} {\left (x e + f \sqrt {\frac {a f^{2} + x^{2} e^{2} + 2 \, d x e}{f^{2}}} + d\right )}^{n} e^{\left (-1\right )}}{f^{2} n^{4} - 10 \, f^{2} n^{2} + 9 \, f^{2}} \end {gather*}

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 + 3*(n^2 - 1)*x^3*e^3 - 9*a*d*f^2 + 9*(d*n^2 - d)*x^2*e^2 + 6*d^3 - 3*(3*a*f^2 - (a*f^2 + 2*d^
2)*n^2)*x*e - (a*f^3*n^3 + (f*n^3 - f*n)*x^2*e^2 + 2*(d*f*n^3 - d*f*n)*x*e - (7*a*f^3 - 6*d^2*f)*n)*sqrt((a*f^
2 + x^2*e^2 + 2*d*x*e)/f^2))*(x*e + f*sqrt((a*f^2 + x^2*e^2 + 2*d*x*e)/f^2) + d)^n*e^(-1)/(f^2*n^4 - 10*f^2*n^
2 + 9*f^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int a f^{2} \left (d + e x + f \sqrt {a + \frac {2 d e x}{f^{2}} + \frac {e^{2} x^{2}}{f^{2}}}\right )^{n}\, dx + \int e^{2} x^{2} \left (d + e x + f \sqrt {a + \frac {2 d e x}{f^{2}} + \frac {e^{2} x^{2}}{f^{2}}}\right )^{n}\, dx + \int 2 d e x \left (d + e x + f \sqrt {a + \frac {2 d e x}{f^{2}} + \frac {e^{2} x^{2}}{f^{2}}}\right )^{n}\, dx}{f^{2}} \end {gather*}

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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (d+f\,\sqrt {a+\frac {e^2\,x^2}{f^2}+\frac {2\,d\,e\,x}{f^2}}+e\,x\right )}^n\,\left (a+\frac {e^2\,x^2}{f^2}+\frac {2\,d\,e\,x}{f^2}\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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