3.3.79 \(\int x^{-1-3 n} (a x^2+b x^3)^n \, dx\) [279]

Optimal. Leaf size=54 \[ -\frac {x^{-3 n} \left (1+\frac {b x}{a}\right )^{-n} \left (a x^2+b x^3\right )^n \, _2F_1\left (-n,-n;1-n;-\frac {b x}{a}\right )}{n} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, 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 = {2057, 68, 66} \begin {gather*} -\frac {x^{-3 n} \left (\frac {b x}{a}+1\right )^{-n} \left (a x^2+b x^3\right )^n \, _2F_1\left (-n,-n;1-n;-\frac {b x}{a}\right )}{n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 3*n)*(a*x^2 + b*x^3)^n,x]

[Out]

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

Rule 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x)^(m + 1)/(b*(m + 1)))*Hypergeometr
ic2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))

Rule 68

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(
x/c))^FracPart[n]), Int[(b*x)^m*(1 + d*(x/c))^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-d/(b*c), 0] && ((RationalQ[m] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0])) |
|  !RationalQ[n])

Rule 2057

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[c^IntPart[m]*(c*x)^FracPa
rt[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]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 52, normalized size = 0.96 \begin {gather*} -\frac {x^{-3 n} \left (x^2 (a+b x)\right )^n \left (1+\frac {b x}{a}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac {b x}{a}\right )}{n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 3*n)*(a*x^2 + b*x^3)^n,x]

[Out]

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

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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int x^{-1-3 n} \left (b \,x^{3}+a \,x^{2}\right )^{n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-3*n)*(b*x^3+a*x^2)^n,x)

[Out]

int(x^(-1-3*n)*(b*x^3+a*x^2)^n,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-3*n)*(b*x^3+a*x^2)^n,x, algorithm="maxima")

[Out]

integrate((b*x^3 + a*x^2)^n*x^(-3*n - 1), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-3*n)*(b*x^3+a*x^2)^n,x, algorithm="fricas")

[Out]

integral((b*x^3 + a*x^2)^n*x^(-3*n - 1), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{- 3 n - 1} \left (x^{2} \left (a + b x\right )\right )^{n}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1-3*n)*(b*x**3+a*x**2)**n,x)

[Out]

Integral(x**(-3*n - 1)*(x**2*(a + b*x))**n, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-3*n)*(b*x^3+a*x^2)^n,x, algorithm="giac")

[Out]

integrate((b*x^3 + a*x^2)^n*x^(-3*n - 1), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (b\,x^3+a\,x^2\right )}^n}{x^{3\,n+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^2 + b*x^3)^n/x^(3*n + 1),x)

[Out]

int((a*x^2 + b*x^3)^n/x^(3*n + 1), x)

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