3.3000 \(\int \frac{(d x)^m}{\sqrt{a+b \sqrt{\frac{c}{x}}}} \, dx\)

Optimal. Leaf size=58 \[ \frac{4 x^{m+1} \sqrt{a+b \sqrt{\frac{c}{x}}} \, _2F_1\left (1,\frac{1}{2} (-4 m-3);\frac{3}{2};\frac{a+b \sqrt{\frac{c}{x}}}{a}\right )}{a} \]

[Out]

(4*Sqrt[a + b*Sqrt[c/x]]*x^(1 + m)*Hypergeometric2F1[1, (-3 - 4*m)/2, 3/2, (a + b*Sqrt[c/x])/a])/a

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Rubi [A]  time = 0.0876514, antiderivative size = 78, normalized size of antiderivative = 1.34, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {369, 343, 341, 339, 67, 65} \[ \frac{4 b^2 c (d x)^m \sqrt{a+b \sqrt{\frac{c}{x}}} \left (-\frac{b \sqrt{\frac{c}{x}}}{a}\right )^{2 m} \, _2F_1\left (\frac{1}{2},2 m+3;\frac{3}{2};\frac{\sqrt{\frac{c}{x}} b}{a}+1\right )}{a^3} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m/Sqrt[a + b*Sqrt[c/x]],x]

[Out]

(4*b^2*c*Sqrt[a + b*Sqrt[c/x]]*(-((b*Sqrt[c/x])/a))^(2*m)*(d*x)^m*Hypergeometric2F1[1/2, 3 + 2*m, 3/2, 1 + (b*
Sqrt[c/x])/a])/a^3

Rule 369

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> With[{k = Denominator[n]}, Su
bst[Int[(d*x)^m*(a + b*c^n*x^(n*q))^p, x], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b
, c, d, m, p, q}, x] && FractionQ[n]

Rule 343

Int[((c_)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracPart[m])/x^FracP
art[m], Int[x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && FractionQ[n]

Rule 341

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

Rule 339

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Dist[((c*x)^(m + 1)*(1/x)^(m + 1))/c, Subst
[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, m, p}, x] && ILtQ[n, 0] &&  !RationalQ[m]

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[((-((b*c)/d))^IntPart[m]*(b*x)^FracPart[m])/
(-((d*x)/c))^FracPart[m], Int[(-((d*x)/c))^m*(c + d*x)^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m]
&&  !IntegerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-(d/(b*c)), 0]

Rule 65

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)*Hypergeometric2F1[-m, n +
1, n + 2, 1 + (d*x)/c])/(d*(n + 1)*(-(d/(b*c)))^m), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Inte
gerQ[m] || GtQ[-(d/(b*c)), 0])

Rubi steps

\begin{align*} \int \frac{(d x)^m}{\sqrt{a+b \sqrt{\frac{c}{x}}}} \, dx &=\operatorname{Subst}\left (\int \frac{(d x)^m}{\sqrt{a+\frac{b \sqrt{c}}{\sqrt{x}}}} \, dx,\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\operatorname{Subst}\left (\left (x^{-m} (d x)^m\right ) \int \frac{x^m}{\sqrt{a+\frac{b \sqrt{c}}{\sqrt{x}}}} \, dx,\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\operatorname{Subst}\left (\left (2 x^{-m} (d x)^m\right ) \operatorname{Subst}\left (\int \frac{x^{-1+2 (1+m)}}{\sqrt{a+\frac{b \sqrt{c}}{x}}} \, dx,x,\sqrt{x}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\operatorname{Subst}\left (\left (2 x^{-m} (d x)^m\right ) \operatorname{Subst}\left (\int \frac{x^{-1-2 (1+m)}}{\sqrt{a+b \sqrt{c} x}} \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\operatorname{Subst}\left (\frac{\left (2 b^3 c^{3/2} \left (-\frac{b \sqrt{c}}{a \sqrt{x}}\right )^{2 m} (d x)^m\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{b \sqrt{c} x}{a}\right )^{-1-2 (1+m)}}{\sqrt{a+b \sqrt{c} x}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{a^3},\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\frac{4 b^2 c \sqrt{a+b \sqrt{\frac{c}{x}}} \left (-\frac{b \sqrt{\frac{c}{x}}}{a}\right )^{2 m} (d x)^m \, _2F_1\left (\frac{1}{2},3+2 m;\frac{3}{2};1+\frac{b \sqrt{\frac{c}{x}}}{a}\right )}{a^3}\\ \end{align*}

Mathematica [A]  time = 0.186254, size = 96, normalized size = 1.66 \[ \frac{4 b^2 c (d x)^m \left (1-\frac{a}{a+b \sqrt{\frac{c}{x}}}\right )^{2 m} \, _2F_1\left (2 m+\frac{5}{2},2 m+3;2 m+\frac{7}{2};\frac{a}{a+b \sqrt{\frac{c}{x}}}\right )}{(4 m+5) \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*b^2*c*(1 - a/(a + b*Sqrt[c/x]))^(2*m)*(d*x)^m*Hypergeometric2F1[5/2 + 2*m, 3 + 2*m, 7/2 + 2*m, a/(a + b*Sqr
t[c/x])])/((5 + 4*m)*(a + b*Sqrt[c/x])^(5/2))

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dx \right ) ^{m}{\frac{1}{\sqrt{a+b\sqrt{{\frac{c}{x}}}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x)^m/sqrt(b*sqrt(c/x) + a), 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((d*x)^m/(a+b*(c/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{\left (d x\right )^{m}}{\sqrt{a + b \sqrt{\frac{c}{x}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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