3.2796 \(\int (c x)^{-3 n} (a+b x^n)^p \, dx\)

Optimal. Leaf size=56 \[ \frac {(c x)^{1-3 n} \left (a+b x^n\right )^{p+1} \, _2F_1\left (1,p+\frac {1}{n}-2;\frac {1}{n}-2;-\frac {b x^n}{a}\right )}{a c (1-3 n)} \]

[Out]

(c*x)^(1-3*n)*(a+b*x^n)^(1+p)*hypergeom([1, -2+1/n+p],[-2+1/n],-b*x^n/a)/a/c/(1-3*n)

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Rubi [A]  time = 0.03, antiderivative size = 66, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {365, 364} \[ \frac {(c x)^{1-3 n} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{n}-3,-p;\frac {1}{n}-2;-\frac {b x^n}{a}\right )}{c (1-3 n)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^p/(c*x)^(3*n),x]

[Out]

((c*x)^(1 - 3*n)*(a + b*x^n)^p*Hypergeometric2F1[-3 + n^(-1), -p, -2 + n^(-1), -((b*x^n)/a)])/(c*(1 - 3*n)*(1
+ (b*x^n)/a)^p)

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])

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])

Rubi steps

\begin {align*} \int (c x)^{-3 n} \left (a+b x^n\right )^p \, dx &=\left (\left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p}\right ) \int (c x)^{-3 n} \left (1+\frac {b x^n}{a}\right )^p \, dx\\ &=\frac {(c x)^{1-3 n} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \, _2F_1\left (-3+\frac {1}{n},-p;-2+\frac {1}{n};-\frac {b x^n}{a}\right )}{c (1-3 n)}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 63, normalized size = 1.12 \[ -\frac {x (c x)^{-3 n} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{n}-3,-p;\frac {1}{n}-2;-\frac {b x^n}{a}\right )}{3 n-1} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^n)^p/(c*x)^(3*n),x]

[Out]

-((x*(a + b*x^n)^p*Hypergeometric2F1[-3 + n^(-1), -p, -2 + n^(-1), -((b*x^n)/a)])/((-1 + 3*n)*(c*x)^(3*n)*(1 +
(b*x^n)/a)^p))

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fricas [F]  time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{n} + a\right )}^{p}}{\left (c x\right )^{3 \, n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^p/((c*x)^(3*n)),x, algorithm="fricas")

[Out]

integral((b*x^n + a)^p/(c*x)^(3*n), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{n} + a\right )}^{p}}{\left (c x\right )^{3 \, n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^p/((c*x)^(3*n)),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^p/(c*x)^(3*n), x)

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maple [F]  time = 0.22, size = 0, normalized size = 0.00 \[ \int \left (c x \right )^{-3 n} \left (b \,x^{n}+a \right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^n+a)^p/((c*x)^(3*n)),x)

[Out]

int((b*x^n+a)^p/((c*x)^(3*n)),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{n} + a\right )}^{p}}{\left (c x\right )^{3 \, n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^p/((c*x)^(3*n)),x, algorithm="maxima")

[Out]

integrate((b*x^n + a)^p/(c*x)^(3*n), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (a+b\,x^n\right )}^p}{{\left (c\,x\right )}^{3\,n}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^p/(c*x)^(3*n),x)

[Out]

int((a + b*x^n)^p/(c*x)^(3*n), x)

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sympy [C]  time = 7.63, size = 51, normalized size = 0.91 \[ \frac {a^{p} c^{- 3 n} x x^{- 3 n} \Gamma \left (-3 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, -3 + \frac {1}{n} \\ -2 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (-2 + \frac {1}{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**n)**p/((c*x)**(3*n)),x)

[Out]

a**p*c**(-3*n)*x*x**(-3*n)*gamma(-3 + 1/n)*hyper((-p, -3 + 1/n), (-2 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gam
ma(-2 + 1/n))

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