3.2920 \(\int (d x)^m \sqrt{a+b \sqrt{c x}} \, dx\)

Optimal. Leaf size=74 \[ -\frac{4 a (d x)^m \left (a+b \sqrt{c x}\right )^{3/2} \left (-\frac{b \sqrt{c x}}{a}\right )^{-2 m} \, _2F_1\left (\frac{3}{2},-2 m-1;\frac{5}{2};\frac{\sqrt{c x} b}{a}+1\right )}{3 b^2 c} \]

[Out]

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

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Rubi [A]  time = 0.0660279, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {367, 343, 341, 67, 65} \[ -\frac{4 a (d x)^m \left (a+b \sqrt{c x}\right )^{3/2} \left (-\frac{b \sqrt{c x}}{a}\right )^{-2 m} \, _2F_1\left (\frac{3}{2},-2 m-1;\frac{5}{2};\frac{\sqrt{c x} b}{a}+1\right )}{3 b^2 c} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*Sqrt[a + b*Sqrt[c*x]],x]

[Out]

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

Rule 367

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_)*(x_))^(n_))^(p_.), x_Symbol] :> Dist[1/c, Subst[Int[((d*x)/c)^m*(a
+ b*x^n)^p, x], x, c*x], x] /; FreeQ[{a, b, c, d, m, n, p}, x]

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

Mathematica [A]  time = 0.148216, size = 112, normalized size = 1.51 \[ \frac{4 (d x)^m \left (a+b \sqrt{c x}\right )^{3/2} \left (-\frac{b \sqrt{c x}}{a}\right )^{-2 m} \left (3 \left (a+b \sqrt{c x}\right ) \, _2F_1\left (\frac{5}{2},-2 m;\frac{7}{2};\frac{\sqrt{c x} b}{a}+1\right )-5 a \, _2F_1\left (\frac{3}{2},-2 m;\frac{5}{2};\frac{\sqrt{c x} b}{a}+1\right )\right )}{15 b^2 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(d*x)^m*(a + b*Sqrt[c*x])^(3/2)*(-5*a*Hypergeometric2F1[3/2, -2*m, 5/2, 1 + (b*Sqrt[c*x])/a] + 3*(a + b*Sqr
t[c*x])*Hypergeometric2F1[5/2, -2*m, 7/2, 1 + (b*Sqrt[c*x])/a]))/(15*b^2*c*(-((b*Sqrt[c*x])/a))^(2*m))

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m}\sqrt{a+b\sqrt{cx}}\, 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 \sqrt{\sqrt{c x} b + 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(sqrt(c*x)*b + a)*(d*x)^m, 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 \left (d x\right )^{m} \sqrt{a + b \sqrt{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 \sqrt{\sqrt{c x} b + 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(sqrt(c*x)*b + a)*(d*x)^m, x)