3.422 \(\int (c x)^m \sqrt{a x^j+b x^n} \, dx\)

Optimal. Leaf size=100 \[ \frac{2 x (c x)^m \sqrt{a x^j+b x^n} \, _2F_1\left (-\frac{1}{2},\frac{m+\frac{n}{2}+1}{j-n};\frac{2 m+n+2}{2 j-2 n}+1;-\frac{a x^{j-n}}{b}\right )}{(2 m+n+2) \sqrt{\frac{a x^{j-n}}{b}+1}} \]

[Out]

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

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Rubi [A]  time = 0.094578, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2032, 365, 364} \[ \frac{2 x (c x)^m \sqrt{a x^j+b x^n} \, _2F_1\left (-\frac{1}{2},\frac{m+\frac{n}{2}+1}{j-n};\frac{2 m+n+2}{2 j-2 n}+1;-\frac{a x^{j-n}}{b}\right )}{(2 m+n+2) \sqrt{\frac{a x^{j-n}}{b}+1}} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^m*Sqrt[a*x^j + b*x^n],x]

[Out]

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

Rule 2032

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

Rule 365

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

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int (c x)^m \sqrt{a x^j+b x^n} \, dx &=\frac{\left (x^{-m-\frac{n}{2}} (c x)^m \sqrt{a x^j+b x^n}\right ) \int x^{m+\frac{n}{2}} \sqrt{b+a x^{j-n}} \, dx}{\sqrt{b+a x^{j-n}}}\\ &=\frac{\left (x^{-m-\frac{n}{2}} (c x)^m \sqrt{a x^j+b x^n}\right ) \int x^{m+\frac{n}{2}} \sqrt{1+\frac{a x^{j-n}}{b}} \, dx}{\sqrt{1+\frac{a x^{j-n}}{b}}}\\ &=\frac{2 x (c x)^m \sqrt{a x^j+b x^n} \, _2F_1\left (-\frac{1}{2},\frac{1+m+\frac{n}{2}}{j-n};1+\frac{2+2 m+n}{2 j-2 n};-\frac{a x^{j-n}}{b}\right )}{(2+2 m+n) \sqrt{1+\frac{a x^{j-n}}{b}}}\\ \end{align*}

Mathematica [A]  time = 0.191334, size = 156, normalized size = 1.56 \[ \frac{2 x (c x)^m \left ((2 j+2 m-n+2) \left (a x^j+b x^n\right )-a (j-n) x^j \sqrt{\frac{a x^{j-n}}{b}+1} \, _2F_1\left (\frac{1}{2},\frac{2 j+2 m-n+2}{2 j-2 n};\frac{4 j+2 m-3 n+2}{2 j-2 n};-\frac{a x^{j-n}}{b}\right )\right )}{(2 m+n+2) (2 j+2 m-n+2) \sqrt{a x^j+b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.512, size = 0, normalized size = 0. \begin{align*} \int \left ( cx \right ) ^{m}\sqrt{a{x}^{j}+b{x}^{n}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*x^j + b*x^n)*(c*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((c*x)^m*(a*x^j+b*x^n)^(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 (c x\right )^{m} \sqrt{a x^{j} + b x^{n}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c*x)**m*sqrt(a*x**j + b*x**n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*x^j + b*x^n)*(c*x)^m, x)