3.2926 \(\int (a+b (c x)^n)^p \, dx\)

Optimal. Leaf size=52 \[ x \left (a+b (c x)^n\right )^p \left (\frac {b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {b (c x)^n}{a}\right ) \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {247, 246, 245} \[ x \left (a+b (c x)^n\right )^p \left (\frac {b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {b (c x)^n}{a}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*(c*x)^n)^p,x]

[Out]

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

Rule 245

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

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^Fr
acPart[p], Int[(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 52, normalized size = 1.00 \[ x \left (a+b (c x)^n\right )^p \left (\frac {b (c x)^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {b (c x)^n}{a}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*(c*x)^n)^p,x]

[Out]

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

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fricas [F]  time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\left (c x\right )^{n} b + a\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(((c*x)^n*b + a)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((c*x)^n*b + a)^p, x)

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maple [F]  time = 0.26, size = 0, normalized size = 0.00 \[ \int \left (b \left (c x \right )^{n}+a \right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(((c*x)^n*b + a)^p, x)

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mupad [B]  time = 2.80, size = 53, normalized size = 1.02 \[ \frac {x\,{\left (a+b\,{\left (c\,x\right )}^n\right )}^p\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{n},-p;\ \frac {1}{n}+1;\ -\frac {b\,{\left (c\,x\right )}^n}{a}\right )}{{\left (\frac {b\,{\left (c\,x\right )}^n}{a}+1\right )}^p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*(c*x)^n)^p,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*(c*x)**n)**p, x)

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