3.3027 \(\int \frac{(a+b (c x^n)^{\frac{1}{n}})^p}{x^2} \, dx\)

Optimal. Leaf size=63 \[ \frac{b \left (c x^n\right )^{\frac{1}{n}} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+1} \, _2F_1\left (2,p+1;p+2;\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )}{a^2 (p+1) x} \]

[Out]

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

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Rubi [A]  time = 0.0248413, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {368, 65} \[ \frac{b \left (c x^n\right )^{\frac{1}{n}} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+1} \, _2F_1\left (2,p+1;p+2;\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )}{a^2 (p+1) x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*(c*x^n)^n^(-1))^p/x^2,x]

[Out]

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

Rule 368

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

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{\left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^p}{x^2} \, dx &=\frac{\left (c x^n\right )^{\frac{1}{n}} \operatorname{Subst}\left (\int \frac{(a+b x)^p}{x^2} \, dx,x,\left (c x^n\right )^{\frac{1}{n}}\right )}{x}\\ &=\frac{b \left (c x^n\right )^{\frac{1}{n}} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{1+p} \, _2F_1\left (2,1+p;2+p;1+\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}\right )}{a^2 (1+p) x}\\ \end{align*}

Mathematica [A]  time = 0.0111352, size = 63, normalized size = 1. \[ \frac{b \left (c x^n\right )^{\frac{1}{n}} \left (a+b \left (c x^n\right )^{\frac{1}{n}}\right )^{p+1} \, _2F_1\left (2,p+1;p+2;\frac{b \left (c x^n\right )^{\frac{1}{n}}}{a}+1\right )}{a^2 (p+1) x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*(c*x^n)^n^(-1))^p/x^2,x]

[Out]

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

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Maple [F]  time = 0.819, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\sqrt [n]{c{x}^{n}} \right ) ^{p}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{p}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((c*x^n)^(1/n)*b + a)^p/x^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{p}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c*x^n)^(1/n))^p/x^2,x, algorithm="fricas")

[Out]

integral(((c*x^n)^(1/n)*b + a)^p/x^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \left (c x^{n}\right )^{\frac{1}{n}}\right )^{p}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*(c*x**n)**(1/n))**p/x**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (\left (c x^{n}\right )^{\left (\frac{1}{n}\right )} b + a\right )}^{p}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((c*x^n)^(1/n)*b + a)^p/x^2, x)