3.3060 \(\int \frac{x^m}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}} \, dx\)

Optimal. Leaf size=230 \[ \frac{x^{m+1} \sqrt{\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d}-\sqrt{b^2 d-4 a c}\right )}+1} \sqrt{\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (\sqrt{b^2 d-4 a c}+b \sqrt{d}\right )}+1} F_1\left (-2 (m+1);\frac{1}{2},\frac{1}{2};-2 m-1;-\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d}-\sqrt{b^2 d-4 a c}\right )},-\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (\sqrt{d} b+\sqrt{b^2 d-4 a c}\right )}\right )}{(m+1) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}} \]

[Out]

(Sqrt[1 + (2*c*Sqrt[d/x])/(Sqrt[d]*(b*Sqrt[d] - Sqrt[-4*a*c + b^2*d]))]*Sqrt[1 + (2*c*Sqrt[d/x])/(Sqrt[d]*(b*S
qrt[d] + Sqrt[-4*a*c + b^2*d]))]*x^(1 + m)*AppellF1[-2*(1 + m), 1/2, 1/2, -1 - 2*m, (-2*c*Sqrt[d/x])/(Sqrt[d]*
(b*Sqrt[d] - Sqrt[-4*a*c + b^2*d])), (-2*c*Sqrt[d/x])/(Sqrt[d]*(b*Sqrt[d] + Sqrt[-4*a*c + b^2*d]))])/((1 + m)*
Sqrt[a + b*Sqrt[d/x] + c/x])

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Rubi [A]  time = 0.474719, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {1971, 1379, 759, 133} \[ \frac{x^{m+1} \sqrt{\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d}-\sqrt{b^2 d-4 a c}\right )}+1} \sqrt{\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (\sqrt{b^2 d-4 a c}+b \sqrt{d}\right )}+1} F_1\left (-2 (m+1);\frac{1}{2},\frac{1}{2};-2 m-1;-\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d}-\sqrt{b^2 d-4 a c}\right )},-\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (\sqrt{d} b+\sqrt{b^2 d-4 a c}\right )}\right )}{(m+1) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}} \]

Antiderivative was successfully verified.

[In]

Int[x^m/Sqrt[a + b*Sqrt[d/x] + c/x],x]

[Out]

(Sqrt[1 + (2*c*Sqrt[d/x])/(Sqrt[d]*(b*Sqrt[d] - Sqrt[-4*a*c + b^2*d]))]*Sqrt[1 + (2*c*Sqrt[d/x])/(Sqrt[d]*(b*S
qrt[d] + Sqrt[-4*a*c + b^2*d]))]*x^(1 + m)*AppellF1[-2*(1 + m), 1/2, 1/2, -1 - 2*m, (-2*c*Sqrt[d/x])/(Sqrt[d]*
(b*Sqrt[d] - Sqrt[-4*a*c + b^2*d])), (-2*c*Sqrt[d/x])/(Sqrt[d]*(b*Sqrt[d] + Sqrt[-4*a*c + b^2*d]))])/((1 + m)*
Sqrt[a + b*Sqrt[d/x] + c/x])

Rule 1971

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*((d_.)/(x_))^(n_) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> -Dist[d*(e*x)^m
*(d/x)^m, Subst[Int[(a + b*x^n + (c*x^(2*n))/d^(2*n))^p/x^(m + 2), x], x, d/x], x] /; FreeQ[{a, b, c, d, e, n,
p}, x] && EqQ[n2, -2*n] &&  !IntegerQ[m] && IntegerQ[2*n]

Rule 1379

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Dist
[k, Subst[Int[x^(k*(m + 1) - 1)*(a + b*x^(k*n) + c*x^(2*k*n))^p, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, m, p}
, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && FractionQ[n]

Rule 759

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c,
2]}, Dist[(a + b*x + c*x^2)^p/(e*(1 - (d + e*x)/(d - (e*(b - q))/(2*c)))^p*(1 - (d + e*x)/(d - (e*(b + q))/(2
*c)))^p), Subst[Int[x^m*Simp[1 - x/(d - (e*(b - q))/(2*c)), x]^p*Simp[1 - x/(d - (e*(b + q))/(2*c)), x]^p, x],
x, d + e*x], x]] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &
& NeQ[2*c*d - b*e, 0] &&  !IntegerQ[p]

Rule 133

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(c^n*e^p*(b*x)^(m +
1)*AppellF1[m + 1, -n, -p, m + 2, -((d*x)/c), -((f*x)/e)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

Rubi steps

\begin{align*} \int \frac{x^m}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}} \, dx &=-\left (\left (d \left (\frac{d}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int \frac{x^{-2-m}}{\sqrt{a+b \sqrt{x}+\frac{c x}{d}}} \, dx,x,\frac{d}{x}\right )\right )\\ &=-\left (\left (2 d \left (\frac{d}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int \frac{x^{-1+2 (-1-m)}}{\sqrt{a+b x+\frac{c x^2}{d}}} \, dx,x,\sqrt{\frac{d}{x}}\right )\right )\\ &=-\frac{\left (2 d \sqrt{1+\frac{2 c \sqrt{\frac{d}{x}}}{d \left (b-\frac{\sqrt{-4 a c+b^2 d}}{\sqrt{d}}\right )}} \sqrt{1+\frac{2 c \sqrt{\frac{d}{x}}}{d \left (b+\frac{\sqrt{-4 a c+b^2 d}}{\sqrt{d}}\right )}} \left (\frac{d}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int \frac{x^{-1+2 (-1-m)}}{\sqrt{1+\frac{2 c x}{\sqrt{d} \left (b \sqrt{d}-\sqrt{-4 a c+b^2 d}\right )}} \sqrt{1+\frac{2 c x}{\sqrt{d} \left (b \sqrt{d}+\sqrt{-4 a c+b^2 d}\right )}}} \, dx,x,\sqrt{\frac{d}{x}}\right )}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\\ &=\frac{\sqrt{1+\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d}-\sqrt{-4 a c+b^2 d}\right )}} \sqrt{1+\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d}+\sqrt{-4 a c+b^2 d}\right )}} x^{1+m} F_1\left (-2 (1+m);\frac{1}{2},\frac{1}{2};-1-2 m;-\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d}-\sqrt{-4 a c+b^2 d}\right )},-\frac{2 c \sqrt{\frac{d}{x}}}{\sqrt{d} \left (b \sqrt{d}+\sqrt{-4 a c+b^2 d}\right )}\right )}{(1+m) \sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}}\\ \end{align*}

Mathematica [F]  time = 0.229175, size = 0, normalized size = 0. \[ \int \frac{x^m}{\sqrt{a+b \sqrt{\frac{d}{x}}+\frac{c}{x}}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^m/Sqrt[a + b*Sqrt[d/x] + c/x],x]

[Out]

Integrate[x^m/Sqrt[a + b*Sqrt[d/x] + c/x], x]

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Maple [F]  time = 0.008, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m}{\frac{1}{\sqrt{a+{\frac{c}{x}}+b\sqrt{{\frac{d}{x}}}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/(a+c/x+b*(d/x)^(1/2))^(1/2),x)

[Out]

int(x^m/(a+c/x+b*(d/x)^(1/2))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{\sqrt{b \sqrt{\frac{d}{x}} + a + \frac{c}{x}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(a+c/x+b*(d/x)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(x^m/sqrt(b*sqrt(d/x) + a + c/x), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(a+c/x+b*(d/x)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{\sqrt{a + b \sqrt{\frac{d}{x}} + \frac{c}{x}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m/(a+c/x+b*(d/x)**(1/2))**(1/2),x)

[Out]

Integral(x**m/sqrt(a + b*sqrt(d/x) + c/x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{\sqrt{b \sqrt{\frac{d}{x}} + a + \frac{c}{x}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(a+c/x+b*(d/x)^(1/2))^(1/2),x, algorithm="giac")

[Out]

integrate(x^m/sqrt(b*sqrt(d/x) + a + c/x), x)