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

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Rubi [A]  time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, 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 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])

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)]

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*}

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Mathematica [A]  time = 0.01, size = 63, normalized size = 1.00 \[ \frac {b \left (c x^n\right )^{\frac {1}{n}} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+1} \, _2F_1\left (2,p+1;p+2;\frac {b \left (c x^n\right )^{\frac {1}{n}}}{a}+1\right )}{a^2 (p+1) x} \]

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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fricas [F]  time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{p}}{x^{2}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{p}}{x^{2}}\,{d x} \]

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)

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maple [F]  time = 1.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \left (c \,x^{n}\right )^{\frac {1}{n}}+a \right )^{p}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{p}}{x^{2}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^p}{x^2} \,d x \]

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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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )^{p}}{x^{2}}\, dx \]

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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