3.6.7 \(\int (a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2})^2 (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}})^n \, dx\) [507]

Optimal. Leaf size=365 \[ \frac {\left (d^2-a f^2\right )^5 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{-5+n}}{32 e f^4 (5-n)}-\frac {5 \left (d^2-a f^2\right )^4 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{-3+n}}{32 e f^4 (3-n)}+\frac {5 \left (d^2-a f^2\right )^3 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{-1+n}}{16 e f^4 (1-n)}+\frac {5 \left (d^2-a f^2\right )^2 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{1+n}}{16 e f^4 (1+n)}-\frac {5 \left (d^2-a f^2\right ) \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{3+n}}{32 e f^4 (3+n)}+\frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{5+n}}{32 e f^4 (5+n)} \]

[Out]

1/32*(-a*f^2+d^2)^5*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2))^(-5+n)/e/f^4/(5-n)-5/32*(-a*f^2+d^2)^4*(d+e*x+
f*(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2))^(-3+n)/e/f^4/(3-n)+5/16*(-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))^(-1+n)/e/f^4/(1-n)+5/16*(-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^4/(1+n)
-5/32*(-a*f^2+d^2)*(d+e*x+f*(a+2*d*e*x/f^2+e^2*x^2/f^2)^(1/2))^(3+n)/e/f^4/(3+n)+1/32*(d+e*x+f*(a+2*d*e*x/f^2+
e^2*x^2/f^2)^(1/2))^(5+n)/e/f^4/(5+n)

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Rubi [A]
time = 0.33, antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2146, 12, 276} \begin {gather*} \frac {\left (d^2-a f^2\right )^5 \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n-5}}{32 e f^4 (5-n)}-\frac {5 \left (d^2-a f^2\right )^4 \left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n-3}}{32 e f^4 (3-n)}+\frac {5 \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-1}}{16 e f^4 (1-n)}+\frac {5 \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}}{16 e f^4 (n+1)}-\frac {5 \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+3}}{32 e f^4 (n+3)}+\frac {\left (f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}+d+e x\right )^{n+5}}{32 e f^4 (n+5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2)^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)^5*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(-5 + n))/(32*e*f^4*(5 - n)) - (5*(d^2
- a*f^2)^4*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(-3 + n))/(32*e*f^4*(3 - n)) + (5*(d^2 - a*f^
2)^3*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(-1 + n))/(16*e*f^4*(1 - n)) + (5*(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))/(16*e*f^4*(1 + n)) - (5*(d^2 - a*f^2)*(d + e*x +
f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(3 + n))/(32*e*f^4*(3 + n)) + (d + e*x + f*Sqrt[a + (2*d*e*x)/f^2
+ (e^2*x^2)/f^2])^(5 + n)/(32*e*f^4*(5 + 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 )^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 &=\frac {2 \text {Subst}\left (\int \frac {x^{-6+n} \left (d^2 e-\left (-a e+\frac {2 d^2 e}{f^2}\right ) f^2+e x^2\right )^5}{64 e^6} \, dx,x,d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )}{f^4}\\ &=\frac {\text {Subst}\left (\int x^{-6+n} \left (d^2 e-\left (-a e+\frac {2 d^2 e}{f^2}\right ) f^2+e x^2\right )^5 \, dx,x,d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )}{32 e^6 f^4}\\ &=\frac {\text {Subst}\left (\int \left (-e^5 \left (d^2-a f^2\right )^5 x^{-6+n}+5 e^5 \left (d^2-a f^2\right )^4 x^{-4+n}-10 e^5 \left (d^2-a f^2\right )^3 x^{-2+n}+10 e^5 \left (d^2-a f^2\right )^2 x^n-5 e^5 \left (d^2-a f^2\right ) x^{2+n}+e^5 x^{4+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 )}{32 e^6 f^4}\\ &=\frac {\left (d^2-a f^2\right )^5 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{-5+n}}{32 e f^4 (5-n)}-\frac {5 \left (d^2-a f^2\right )^4 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{-3+n}}{32 e f^4 (3-n)}+\frac {5 \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 )^{-1+n}}{16 e f^4 (1-n)}+\frac {5 \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}}{16 e f^4 (1+n)}-\frac {5 \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 )^{3+n}}{32 e f^4 (3+n)}+\frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{5+n}}{32 e f^4 (5+n)}\\ \end {align*}

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Mathematica [A]
time = 6.92, size = 280, normalized size = 0.77 \begin {gather*} \frac {\left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^{-5+n} \left (-\frac {\left (d^2-a f^2\right )^5}{-5+n}+\frac {5 \left (d^2-a f^2\right )^4 \left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^2}{-3+n}-\frac {10 \left (d^2-a f^2\right )^3 \left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^4}{-1+n}+\frac {10 \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 )^6}{1+n}-\frac {5 \left (d^2-a f^2\right ) \left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^8}{3+n}+\frac {\left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^{10}}{5+n}\right )}{32 e f^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2)^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])^(-5 + n)*(-((d^2 - a*f^2)^5/(-5 + n)) + (5*(d^2 - a*f^2)^4*(d +
e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^2)/(-3 + n) - (10*(d^2 - a*f^2)^3*(d + e*x + f*Sqrt[a + (e*x*(2*d +
e*x))/f^2])^4)/(-1 + n) + (10*(d^2 - a*f^2)^2*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^6)/(1 + n) - (5*(d
^2 - a*f^2)*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^8)/(3 + n) + (d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))
/f^2])^10/(5 + n)))/(32*e*f^4)

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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \left (a +\frac {2 d e x}{f^{2}}+\frac {e^{2} x^{2}}{f^{2}}\right )^{2} \left (d +e x +f \sqrt {a +\frac {2 d e x}{f^{2}}+\frac {e^{2} x^{2}}{f^{2}}}\right )^{n}\, dx\]

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)^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)^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)^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)^2, x)

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Fricas [A]
time = 0.43, size = 570, normalized size = 1.56 \begin {gather*} -\frac {{\left (5 \, a^{2} d f^{4} n^{4} + 225 \, a^{2} d f^{4} + 5 \, {\left (n^{4} - 10 \, n^{2} + 9\right )} x^{5} e^{5} - 300 \, a d^{3} f^{2} + 25 \, {\left (d n^{4} - 10 \, d n^{2} + 9 \, d\right )} x^{4} e^{4} + 120 \, d^{5} + 10 \, {\left ({\left (a f^{2} + 4 \, d^{2}\right )} n^{4} + 15 \, a f^{2} - 2 \, {\left (8 \, a f^{2} + 17 \, d^{2}\right )} n^{2} + 30 \, d^{2}\right )} x^{3} e^{3} + 10 \, {\left ({\left (3 \, a d f^{2} + 2 \, d^{3}\right )} n^{4} + 45 \, a d f^{2} - 2 \, {\left (24 \, a d f^{2} + d^{3}\right )} n^{2}\right )} x^{2} e^{2} - 10 \, {\left (11 \, a^{2} d f^{4} - 6 \, a d^{3} f^{2}\right )} n^{2} + 5 \, {\left (45 \, a^{2} f^{4} + {\left (a^{2} f^{4} + 4 \, a d^{2} f^{2}\right )} n^{4} - 2 \, {\left (11 \, a^{2} f^{4} + 26 \, a d^{2} f^{2} - 12 \, d^{4}\right )} n^{2}\right )} x e - {\left (a^{2} f^{5} n^{5} + {\left (f n^{5} - 10 \, f n^{3} + 9 \, f n\right )} x^{4} e^{4} + 4 \, {\left (d f n^{5} - 10 \, d f n^{3} + 9 \, d f n\right )} x^{3} e^{3} - 10 \, {\left (3 \, a^{2} f^{5} - 2 \, a d^{2} f^{3}\right )} n^{3} + 2 \, {\left ({\left (a f^{3} + 2 \, d^{2} f\right )} n^{5} - 10 \, {\left (2 \, a f^{3} + d^{2} f\right )} n^{3} + {\left (19 \, a f^{3} + 8 \, d^{2} f\right )} n\right )} x^{2} e^{2} + 4 \, {\left (a d f^{3} n^{5} - 10 \, {\left (2 \, a d f^{3} - d^{3} f\right )} n^{3} + {\left (19 \, a d f^{3} - 10 \, d^{3} f\right )} n\right )} x e + {\left (149 \, a^{2} f^{5} - 260 \, a d^{2} f^{3} + 120 \, d^{4} 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^{4} n^{6} - 35 \, f^{4} n^{4} + 259 \, f^{4} n^{2} - 225 \, f^{4}} \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)^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]

-(5*a^2*d*f^4*n^4 + 225*a^2*d*f^4 + 5*(n^4 - 10*n^2 + 9)*x^5*e^5 - 300*a*d^3*f^2 + 25*(d*n^4 - 10*d*n^2 + 9*d)
*x^4*e^4 + 120*d^5 + 10*((a*f^2 + 4*d^2)*n^4 + 15*a*f^2 - 2*(8*a*f^2 + 17*d^2)*n^2 + 30*d^2)*x^3*e^3 + 10*((3*
a*d*f^2 + 2*d^3)*n^4 + 45*a*d*f^2 - 2*(24*a*d*f^2 + d^3)*n^2)*x^2*e^2 - 10*(11*a^2*d*f^4 - 6*a*d^3*f^2)*n^2 +
5*(45*a^2*f^4 + (a^2*f^4 + 4*a*d^2*f^2)*n^4 - 2*(11*a^2*f^4 + 26*a*d^2*f^2 - 12*d^4)*n^2)*x*e - (a^2*f^5*n^5 +
(f*n^5 - 10*f*n^3 + 9*f*n)*x^4*e^4 + 4*(d*f*n^5 - 10*d*f*n^3 + 9*d*f*n)*x^3*e^3 - 10*(3*a^2*f^5 - 2*a*d^2*f^3
)*n^3 + 2*((a*f^3 + 2*d^2*f)*n^5 - 10*(2*a*f^3 + d^2*f)*n^3 + (19*a*f^3 + 8*d^2*f)*n)*x^2*e^2 + 4*(a*d*f^3*n^5
- 10*(2*a*d*f^3 - d^3*f)*n^3 + (19*a*d*f^3 - 10*d^3*f)*n)*x*e + (149*a^2*f^5 - 260*a*d^2*f^3 + 120*d^4*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^4*n^6 -
35*f^4*n^4 + 259*f^4*n^2 - 225*f^4)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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)**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 = 0.00, 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)^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)^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 )}^2 \,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)^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)^2, x)

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