\(\int x (d+e x)^2 (a+b x^2)^p \, dx\) [394]

Optimal result

Integrand size = 18, antiderivative size = 113 \[ \int x (d+e x)^2 \left (a+b x^2\right )^p \, dx=\frac {\left (b d^2-a e^2\right ) \left (a+b x^2\right )^{1+p}}{2 b^2 (1+p)}+\frac {e^2 \left (a+b x^2\right )^{2+p}}{2 b^2 (2+p)}+\frac {2}{3} d e x^3 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},-\frac {b x^2}{a}\right ) \]

[Out]

1/2*(-a*e^2+b*d^2)*(b*x^2+a)^(p+1)/b^2/(p+1)+1/2*e^2*(b*x^2+a)^(2+p)/b^2/(2+p)+2/3*d*e*x^3*(b*x^2+a)^p*hyperge
om([3/2, -p],[5/2],-b*x^2/a)/((1+b*x^2/a)^p)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1666, 455, 45, 12, 372, 371} \[ \int x (d+e x)^2 \left (a+b x^2\right )^p \, dx=\frac {\left (b d^2-a e^2\right ) \left (a+b x^2\right )^{p+1}}{2 b^2 (p+1)}+\frac {e^2 \left (a+b x^2\right )^{p+2}}{2 b^2 (p+2)}+\frac {2}{3} d e x^3 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},-\frac {b x^2}{a}\right ) \]

[In]

Int[x*(d + e*x)^2*(a + b*x^2)^p,x]

[Out]

((b*d^2 - a*e^2)*(a + b*x^2)^(1 + p))/(2*b^2*(1 + p)) + (e^2*(a + b*x^2)^(2 + p))/(2*b^2*(2 + p)) + (2*d*e*x^3
*(a + b*x^2)^p*Hypergeometric2F1[3/2, -p, 5/2, -((b*x^2)/a)])/(3*(1 + (b*x^2)/a)^p)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

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 455

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 1666

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[x^m*Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2)^p, x] + Int[x^(m + 1)*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2)^p, x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2] && IGtQ[m, -2] &&  !
IntegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = \int 2 d e x^2 \left (a+b x^2\right )^p \, dx+\int x \left (a+b x^2\right )^p \left (d^2+e^2 x^2\right ) \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int (a+b x)^p \left (d^2+e^2 x\right ) \, dx,x,x^2\right )+(2 d e) \int x^2 \left (a+b x^2\right )^p \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {\left (b d^2-a e^2\right ) (a+b x)^p}{b}+\frac {e^2 (a+b x)^{1+p}}{b}\right ) \, dx,x,x^2\right )+\left (2 d e \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int x^2 \left (1+\frac {b x^2}{a}\right )^p \, dx \\ & = \frac {\left (b d^2-a e^2\right ) \left (a+b x^2\right )^{1+p}}{2 b^2 (1+p)}+\frac {e^2 \left (a+b x^2\right )^{2+p}}{2 b^2 (2+p)}+\frac {2}{3} d e x^3 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {3}{2},-p;\frac {5}{2};-\frac {b x^2}{a}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.63 \[ \int x (d+e x)^2 \left (a+b x^2\right )^p \, dx=\frac {\left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (3 b^2 x^2 \left (1+\frac {b x^2}{a}\right )^p \left (d^2 (2+p)+e^2 (1+p) x^2\right )-3 a^2 e^2 \left (-1+\left (1+\frac {b x^2}{a}\right )^p\right )+3 a b \left (e^2 p x^2 \left (1+\frac {b x^2}{a}\right )^p+d^2 (2+p) \left (-1+\left (1+\frac {b x^2}{a}\right )^p\right )\right )+4 b^2 d e \left (2+3 p+p^2\right ) x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},-\frac {b x^2}{a}\right )\right )}{6 b^2 (1+p) (2+p)} \]

[In]

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

[Out]

((a + b*x^2)^p*(3*b^2*x^2*(1 + (b*x^2)/a)^p*(d^2*(2 + p) + e^2*(1 + p)*x^2) - 3*a^2*e^2*(-1 + (1 + (b*x^2)/a)^
p) + 3*a*b*(e^2*p*x^2*(1 + (b*x^2)/a)^p + d^2*(2 + p)*(-1 + (1 + (b*x^2)/a)^p)) + 4*b^2*d*e*(2 + 3*p + p^2)*x^
3*Hypergeometric2F1[3/2, -p, 5/2, -((b*x^2)/a)]))/(6*b^2*(1 + p)*(2 + p)*(1 + (b*x^2)/a)^p)

Maple [F]

\[\int x \left (e x +d \right )^{2} \left (b \,x^{2}+a \right )^{p}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int x (d+e x)^2 \left (a+b x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (b x^{2} + a\right )}^{p} x \,d x } \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 4.84 (sec) , antiderivative size = 408, normalized size of antiderivative = 3.61 \[ \int x (d+e x)^2 \left (a+b x^2\right )^p \, dx=\frac {2 a^{p} d e x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, - p \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3} + d^{2} \left (\begin {cases} \frac {a^{p} x^{2}}{2} & \text {for}\: b = 0 \\\frac {\begin {cases} \frac {\left (a + b x^{2}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (a + b x^{2} \right )} & \text {otherwise} \end {cases}}{2 b} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} \frac {a^{p} x^{4}}{4} & \text {for}\: b = 0 \\\frac {a \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{2 a b^{2} + 2 b^{3} x^{2}} + \frac {a \log {\left (x + \sqrt {- \frac {a}{b}} \right )}}{2 a b^{2} + 2 b^{3} x^{2}} + \frac {a}{2 a b^{2} + 2 b^{3} x^{2}} + \frac {b x^{2} \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{2 a b^{2} + 2 b^{3} x^{2}} + \frac {b x^{2} \log {\left (x + \sqrt {- \frac {a}{b}} \right )}}{2 a b^{2} + 2 b^{3} x^{2}} & \text {for}\: p = -2 \\- \frac {a \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{2 b^{2}} - \frac {a \log {\left (x + \sqrt {- \frac {a}{b}} \right )}}{2 b^{2}} + \frac {x^{2}}{2 b} & \text {for}\: p = -1 \\- \frac {a^{2} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} + \frac {a b p x^{2} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} + \frac {b^{2} p x^{4} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} + \frac {b^{2} x^{4} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

2*a**p*d*e*x**3*hyper((3/2, -p), (5/2,), b*x**2*exp_polar(I*pi)/a)/3 + d**2*Piecewise((a**p*x**2/2, Eq(b, 0)),
(Piecewise(((a + b*x**2)**(p + 1)/(p + 1), Ne(p, -1)), (log(a + b*x**2), True))/(2*b), True)) + e**2*Piecewis
e((a**p*x**4/4, Eq(b, 0)), (a*log(x - sqrt(-a/b))/(2*a*b**2 + 2*b**3*x**2) + a*log(x + sqrt(-a/b))/(2*a*b**2 +
2*b**3*x**2) + a/(2*a*b**2 + 2*b**3*x**2) + b*x**2*log(x - sqrt(-a/b))/(2*a*b**2 + 2*b**3*x**2) + b*x**2*log(
x + sqrt(-a/b))/(2*a*b**2 + 2*b**3*x**2), Eq(p, -2)), (-a*log(x - sqrt(-a/b))/(2*b**2) - a*log(x + sqrt(-a/b))
/(2*b**2) + x**2/(2*b), Eq(p, -1)), (-a**2*(a + b*x**2)**p/(2*b**2*p**2 + 6*b**2*p + 4*b**2) + a*b*p*x**2*(a +
b*x**2)**p/(2*b**2*p**2 + 6*b**2*p + 4*b**2) + b**2*p*x**4*(a + b*x**2)**p/(2*b**2*p**2 + 6*b**2*p + 4*b**2)
+ b**2*x**4*(a + b*x**2)**p/(2*b**2*p**2 + 6*b**2*p + 4*b**2), True))

Maxima [F]

\[ \int x (d+e x)^2 \left (a+b x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (b x^{2} + a\right )}^{p} x \,d x } \]

[In]

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

[Out]

1/2*(b*x^2 + a)^(p + 1)*d^2/(b*(p + 1)) + integrate((e^2*x^3 + 2*d*e*x^2)*(b*x^2 + a)^p, x)

Giac [F]

\[ \int x (d+e x)^2 \left (a+b x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (b x^{2} + a\right )}^{p} x \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x (d+e x)^2 \left (a+b x^2\right )^p \, dx=\int x\,{\left (b\,x^2+a\right )}^p\,{\left (d+e\,x\right )}^2 \,d x \]

[In]

int(x*(a + b*x^2)^p*(d + e*x)^2,x)

[Out]

int(x*(a + b*x^2)^p*(d + e*x)^2, x)