3.3052 \(\int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^m \, dx\)

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

[Out]

x^(1+m)*AppellF1(-2-2*m,-1/2,-1/2,-1-2*m,-2*c*(d/x)^(1/2)/d^(1/2)/(b*d^(1/2)-(b^2*d-4*a*c)^(1/2)),-2*c*(d/x)^(
1/2)/d^(1/2)/(b*d^(1/2)+(b^2*d-4*a*c)^(1/2)))*(a+c/x+b*(d/x)^(1/2))^(1/2)/(1+m)/(1+2*c*(d/x)^(1/2)/d^(1/2)/(b*
d^(1/2)-(b^2*d-4*a*c)^(1/2)))^(1/2)/(1+2*c*(d/x)^(1/2)/d^(1/2)/(b*d^(1/2)+(b^2*d-4*a*c)^(1/2)))^(1/2)

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Rubi [A]  time = 0.72, antiderivative size = 230, normalized size of antiderivative = 1.00, 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} \[ \frac {x^{m+1} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} F_1\left (-2 (m+1);-\frac {1}{2},-\frac {1}{2};-2 m-1;-\frac {2 c \sqrt {\frac {d}{x}}}{\sqrt {d} \left (b \sqrt {d}-\sqrt {b^2 d-4 a c}\right )},-\frac {2 c \sqrt {\frac {d}{x}}}{\sqrt {d} \left (\sqrt {d} b+\sqrt {b^2 d-4 a c}\right )}\right )}{(m+1) \sqrt {\frac {2 c \sqrt {\frac {d}{x}}}{\sqrt {d} \left (b \sqrt {d}-\sqrt {b^2 d-4 a c}\right )}+1} \sqrt {\frac {2 c \sqrt {\frac {d}{x}}}{\sqrt {d} \left (\sqrt {b^2 d-4 a c}+b \sqrt {d}\right )}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*Sqrt[d/x] + c/x]*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)*Sq
rt[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*Sqrt
[d] + Sqrt[-4*a*c + b^2*d]))])

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])

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 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 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]

Rubi steps

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

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Mathematica [F]  time = 0.18, size = 0, normalized size = 0.00 \[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^m \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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: TypeError >>  Error detected within library code:   alglogextint: unimplemented

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}} x^{m}\,{d x} \]

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(sqrt(b*sqrt(d/x) + a + c/x)*x^m, x)

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maple [F]  time = 0.22, size = 0, normalized size = 0.00 \[ \int \sqrt {a +\sqrt {\frac {d}{x}}\, b +\frac {c}{x}}\, x^{m}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}} x^{m}\,{d x} \]

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(sqrt(b*sqrt(d/x) + a + c/x)*x^m, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^m\,\sqrt {a+\frac {c}{x}+b\,\sqrt {\frac {d}{x}}} \,d x \]

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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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \sqrt {a + b \sqrt {\frac {d}{x}} + \frac {c}{x}}\, dx \]

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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