3.6.13 \(\int \frac {(d+e x+f \sqrt {\frac {a f^2+e x (2 d+e x)}{f^2}})^n}{a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}} \, dx\) [513]

Optimal. Leaf size=122 \[ -\frac {2 f^2 \left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^{1+n} \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};\frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^2}{d^2-a f^2}\right )}{e \left (d^2-a f^2\right ) (1+n)} \]

[Out]

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

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Rubi [A]
time = 0.33, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2152, 2146, 371} \begin {gather*} -\frac {2 f^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} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};\frac {\left (d+e x+f \sqrt {\frac {e^2 x^2}{f^2}+\frac {2 d e x}{f^2}+a}\right )^2}{d^2-a f^2}\right )}{e (n+1) \left (d^2-a f^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*f^2*(d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/
2, (d + e*x + f*Sqrt[a + (2*d*e*x)/f^2 + (e^2*x^2)/f^2])^2/(d^2 - a*f^2)])/(e*(d^2 - a*f^2)*(1 + n))

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 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])

Rule 2152

Int[((u_) + (f_.)*((j_.) + (k_.)*Sqrt[v_]))^(n_.)*(w_)^(m_.), x_Symbol] :> Int[ExpandToSum[w, x]^m*(ExpandToSu
m[u + f*j, x] + f*k*Sqrt[ExpandToSum[v, x]])^n, x] /; FreeQ[{f, j, k, m, n}, x] && LinearQ[u, x] && QuadraticQ
[{v, w}, x] &&  !(LinearMatchQ[u, x] && QuadraticMatchQ[{v, w}, x] && (EqQ[j, 0] || EqQ[f, 1])) && EqQ[Coeffic
ient[u, x, 1]^2 - Coefficient[v, x, 2]*f^2*k^2, 0]

Rubi steps

\begin {align*} \int \frac {\left (d+e x+f \sqrt {\frac {a f^2+e x (2 d+e x)}{f^2}}\right )^n}{a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}} \, dx &=\int \frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^n}{a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}} \, dx\\ &=\left (2 f^2\right ) \text {Subst}\left (\int \frac {x^n}{d^2 e-\left (-a e+\frac {2 d^2 e}{f^2}\right ) f^2+e x^2} \, dx,x,d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )\\ &=-\frac {2 f^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} \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};\frac {\left (d+e x+f \sqrt {a+\frac {2 d e x}{f^2}+\frac {e^2 x^2}{f^2}}\right )^2}{d^2-a f^2}\right )}{e \left (d^2-a f^2\right ) (1+n)}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 112, normalized size = 0.92 \begin {gather*} -\frac {2 f^2 \left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^{1+n} \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};\frac {\left (d+e x+f \sqrt {a+\frac {e x (2 d+e x)}{f^2}}\right )^2}{d^2-a f^2}\right )}{e \left (d^2-a f^2\right ) (1+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*f^2*(d + e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, (d +
e*x + f*Sqrt[a + (e*x*(2*d + e*x))/f^2])^2/(d^2 - a*f^2)])/(e*(d^2 - a*f^2)*(1 + n))

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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {\left (d +e x +f \sqrt {\frac {a \,f^{2}+e x \left (e x +2 d \right )}{f^{2}}}\right )^{n}}{a +\frac {2 d e x}{f^{2}}+\frac {e^{2} x^{2}}{f^{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

[Out]

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

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

[Out]

integrate((x*e + f*sqrt((a*f^2 + (x*e + 2*d)*x*e)/f^2) + 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 = -0.01 \begin {gather*} \int \frac {{\left (d+e\,x+f\,\sqrt {\frac {a\,f^2+e\,x\,\left (2\,d+e\,x\right )}{f^2}}\right )}^n}{a+\frac {e^2\,x^2}{f^2}+\frac {2\,d\,e\,x}{f^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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