3.2996 \(\int \sqrt{a+b \sqrt{\frac{c}{x}}} (d x)^m \, dx\)

Optimal. Leaf size=60 \[ \frac{4 x^{m+1} \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2} \, _2F_1\left (1,\frac{1}{2} (-4 m-1);\frac{5}{2};\frac{a+b \sqrt{\frac{c}{x}}}{a}\right )}{3 a} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0904063, antiderivative size = 80, normalized size of antiderivative = 1.33, 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} \[ \frac{4 b^2 c (d x)^m \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2} \left (-\frac{b \sqrt{\frac{c}{x}}}{a}\right )^{2 m} \, _2F_1\left (\frac{3}{2},2 m+3;\frac{5}{2};\frac{\sqrt{\frac{c}{x}} b}{a}+1\right )}{3 a^3} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sqrt[c/x]]*(d*x)^m,x]

[Out]

(4*b^2*c*(a + b*Sqrt[c/x])^(3/2)*(-((b*Sqrt[c/x])/a))^(2*m)*(d*x)^m*Hypergeometric2F1[3/2, 3 + 2*m, 5/2, 1 + (
b*Sqrt[c/x])/a])/(3*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 \sqrt{a+b \sqrt{\frac{c}{x}}} (d x)^m \, dx &=\operatorname{Subst}\left (\int \sqrt{a+\frac{b \sqrt{c}}{\sqrt{x}}} (d x)^m \, dx,\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\operatorname{Subst}\left (\left (x^{-m} (d x)^m\right ) \int \sqrt{a+\frac{b \sqrt{c}}{\sqrt{x}}} x^m \, 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 \sqrt{a+\frac{b \sqrt{c}}{x}} x^{-1+2 (1+m)} \, 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 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 \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 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2} \left (-\frac{b \sqrt{\frac{c}{x}}}{a}\right )^{2 m} (d x)^m \, _2F_1\left (\frac{3}{2},3+2 m;\frac{5}{2};1+\frac{b \sqrt{\frac{c}{x}}}{a}\right )}{3 a^3}\\ \end{align*}

Mathematica [A]  time = 0.0694583, size = 78, normalized size = 1.3 \[ \frac{x (d x)^m \sqrt{a+b \sqrt{\frac{c}{x}}} \, _2F_1\left (-\frac{1}{2},-2 (m+1);-2 m-1;-\frac{b \sqrt{\frac{c}{x}}}{a}\right )}{(m+1) \sqrt{\frac{b \sqrt{\frac{c}{x}}}{a}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.048, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m}\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)

________________________________________________________________________________________

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

________________________________________________________________________________________

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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)

________________________________________________________________________________________

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