3.476 \(\int x^2 (c+d x+e x^2) (a+b x^3)^p \, dx\)

Optimal. Leaf size=107 \[ \frac{c \left (a+b x^3\right )^{p+1}}{3 b (p+1)}+\frac{d x^4 \left (a+b x^3\right )^{p+1} \, _2F_1\left (1,p+\frac{7}{3};\frac{7}{3};-\frac{b x^3}{a}\right )}{4 a}+\frac{e x^5 \left (a+b x^3\right )^{p+1} \, _2F_1\left (1,p+\frac{8}{3};\frac{8}{3};-\frac{b x^3}{a}\right )}{5 a} \]

[Out]

(c*(a + b*x^3)^(1 + p))/(3*b*(1 + p)) + (d*x^4*(a + b*x^3)^(1 + p)*Hypergeometric2F1[1, 7/3 + p, 7/3, -((b*x^3
)/a)])/(4*a) + (e*x^5*(a + b*x^3)^(1 + p)*Hypergeometric2F1[1, 8/3 + p, 8/3, -((b*x^3)/a)])/(5*a)

________________________________________________________________________________________

Rubi [A]  time = 0.106132, antiderivative size = 125, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {1893, 261, 365, 364} \[ \frac{c \left (a+b x^3\right )^{p+1}}{3 b (p+1)}+\frac{1}{4} d x^4 \left (a+b x^3\right )^p \left (\frac{b x^3}{a}+1\right )^{-p} \, _2F_1\left (\frac{4}{3},-p;\frac{7}{3};-\frac{b x^3}{a}\right )+\frac{1}{5} e x^5 \left (a+b x^3\right )^p \left (\frac{b x^3}{a}+1\right )^{-p} \, _2F_1\left (\frac{5}{3},-p;\frac{8}{3};-\frac{b x^3}{a}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2*(c + d*x + e*x^2)*(a + b*x^3)^p,x]

[Out]

(c*(a + b*x^3)^(1 + p))/(3*b*(1 + p)) + (d*x^4*(a + b*x^3)^p*Hypergeometric2F1[4/3, -p, 7/3, -((b*x^3)/a)])/(4
*(1 + (b*x^3)/a)^p) + (e*x^5*(a + b*x^3)^p*Hypergeometric2F1[5/3, -p, 8/3, -((b*x^3)/a)])/(5*(1 + (b*x^3)/a)^p
)

Rule 1893

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n])

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

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])

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])

Rubi steps

\begin{align*} \int x^2 \left (c+d x+e x^2\right ) \left (a+b x^3\right )^p \, dx &=\int \left (c x^2 \left (a+b x^3\right )^p+d x^3 \left (a+b x^3\right )^p+e x^4 \left (a+b x^3\right )^p\right ) \, dx\\ &=c \int x^2 \left (a+b x^3\right )^p \, dx+d \int x^3 \left (a+b x^3\right )^p \, dx+e \int x^4 \left (a+b x^3\right )^p \, dx\\ &=\frac{c \left (a+b x^3\right )^{1+p}}{3 b (1+p)}+\left (d \left (a+b x^3\right )^p \left (1+\frac{b x^3}{a}\right )^{-p}\right ) \int x^3 \left (1+\frac{b x^3}{a}\right )^p \, dx+\left (e \left (a+b x^3\right )^p \left (1+\frac{b x^3}{a}\right )^{-p}\right ) \int x^4 \left (1+\frac{b x^3}{a}\right )^p \, dx\\ &=\frac{c \left (a+b x^3\right )^{1+p}}{3 b (1+p)}+\frac{1}{4} d x^4 \left (a+b x^3\right )^p \left (1+\frac{b x^3}{a}\right )^{-p} \, _2F_1\left (\frac{4}{3},-p;\frac{7}{3};-\frac{b x^3}{a}\right )+\frac{1}{5} e x^5 \left (a+b x^3\right )^p \left (1+\frac{b x^3}{a}\right )^{-p} \, _2F_1\left (\frac{5}{3},-p;\frac{8}{3};-\frac{b x^3}{a}\right )\\ \end{align*}

Mathematica [A]  time = 0.0762696, size = 116, normalized size = 1.08 \[ \frac{\left (a+b x^3\right )^p \left (\frac{b x^3}{a}+1\right )^{-p} \left (20 c \left (a+b x^3\right ) \left (\frac{b x^3}{a}+1\right )^p+15 b d (p+1) x^4 \, _2F_1\left (\frac{4}{3},-p;\frac{7}{3};-\frac{b x^3}{a}\right )+12 b e (p+1) x^5 \, _2F_1\left (\frac{5}{3},-p;\frac{8}{3};-\frac{b x^3}{a}\right )\right )}{60 b (p+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(c + d*x + e*x^2)*(a + b*x^3)^p,x]

[Out]

((a + b*x^3)^p*(20*c*(a + b*x^3)*(1 + (b*x^3)/a)^p + 15*b*d*(1 + p)*x^4*Hypergeometric2F1[4/3, -p, 7/3, -((b*x
^3)/a)] + 12*b*e*(1 + p)*x^5*Hypergeometric2F1[5/3, -p, 8/3, -((b*x^3)/a)]))/(60*b*(1 + p)*(1 + (b*x^3)/a)^p)

________________________________________________________________________________________

Maple [F]  time = 0.234, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( e{x}^{2}+dx+c \right ) \left ( b{x}^{3}+a \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(e*x^2+d*x+c)*(b*x^3+a)^p,x)

[Out]

int(x^2*(e*x^2+d*x+c)*(b*x^3+a)^p,x)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(e*x^2+d*x+c)*(b*x^3+a)^p,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (e x^{4} + d x^{3} + c x^{2}\right )}{\left (b x^{3} + a\right )}^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(e*x^2+d*x+c)*(b*x^3+a)^p,x, algorithm="fricas")

[Out]

integral((e*x^4 + d*x^3 + c*x^2)*(b*x^3 + a)^p, x)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(e*x**2+d*x+c)*(b*x**3+a)**p,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{2} + d x + c\right )}{\left (b x^{3} + a\right )}^{p} x^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(e*x^2+d*x+c)*(b*x^3+a)^p,x, algorithm="giac")

[Out]

integrate((e*x^2 + d*x + c)*(b*x^3 + a)^p*x^2, x)