\(\int x^4 (d+e x)^2 (a+b x^2)^p \, dx\) [391]

Optimal result

Integrand size = 20, antiderivative size = 177 \[ \int x^4 (d+e x)^2 \left (a+b x^2\right )^p \, dx=\frac {a^2 d e \left (a+b x^2\right )^{1+p}}{b^3 (1+p)}+\frac {e^2 x^5 \left (a+b x^2\right )^{1+p}}{b (7+2 p)}-\frac {2 a d e \left (a+b x^2\right )^{2+p}}{b^3 (2+p)}+\frac {d e \left (a+b x^2\right )^{3+p}}{b^3 (3+p)}-\frac {\left (5 a e^2-b d^2 (7+2 p)\right ) x^5 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-p,\frac {7}{2},-\frac {b x^2}{a}\right )}{5 b (7+2 p)} \]

[Out]

a^2*d*e*(b*x^2+a)^(p+1)/b^3/(p+1)+e^2*x^5*(b*x^2+a)^(p+1)/b/(7+2*p)-2*a*d*e*(b*x^2+a)^(2+p)/b^3/(2+p)+d*e*(b*x
^2+a)^(3+p)/b^3/(3+p)-1/5*(5*a*e^2-b*d^2*(7+2*p))*x^5*(b*x^2+a)^p*hypergeom([5/2, -p],[7/2],-b*x^2/a)/b/(7+2*p
)/((1+b*x^2/a)^p)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1666, 470, 372, 371, 12, 272, 45} \[ \int x^4 (d+e x)^2 \left (a+b x^2\right )^p \, dx=\frac {a^2 d e \left (a+b x^2\right )^{p+1}}{b^3 (p+1)}-\frac {2 a d e \left (a+b x^2\right )^{p+2}}{b^3 (p+2)}+\frac {d e \left (a+b x^2\right )^{p+3}}{b^3 (p+3)}+\frac {1}{5} x^5 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (d^2-\frac {5 a e^2}{2 b p+7 b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-p,\frac {7}{2},-\frac {b x^2}{a}\right )+\frac {e^2 x^5 \left (a+b x^2\right )^{p+1}}{b (2 p+7)} \]

[In]

Int[x^4*(d + e*x)^2*(a + b*x^2)^p,x]

[Out]

(a^2*d*e*(a + b*x^2)^(1 + p))/(b^3*(1 + p)) + (e^2*x^5*(a + b*x^2)^(1 + p))/(b*(7 + 2*p)) - (2*a*d*e*(a + b*x^
2)^(2 + p))/(b^3*(2 + p)) + (d*e*(a + b*x^2)^(3 + p))/(b^3*(3 + p)) + ((d^2 - (5*a*e^2)/(7*b + 2*b*p))*x^5*(a
+ b*x^2)^p*Hypergeometric2F1[5/2, -p, 7/2, -((b*x^2)/a)])/(5*(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 272

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

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 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 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^5 \left (a+b x^2\right )^p \, dx+\int x^4 \left (a+b x^2\right )^p \left (d^2+e^2 x^2\right ) \, dx \\ & = \frac {e^2 x^5 \left (a+b x^2\right )^{1+p}}{b (7+2 p)}+(2 d e) \int x^5 \left (a+b x^2\right )^p \, dx-\left (-d^2+\frac {5 a e^2}{7 b+2 b p}\right ) \int x^4 \left (a+b x^2\right )^p \, dx \\ & = \frac {e^2 x^5 \left (a+b x^2\right )^{1+p}}{b (7+2 p)}+(d e) \text {Subst}\left (\int x^2 (a+b x)^p \, dx,x,x^2\right )-\left (\left (-d^2+\frac {5 a e^2}{7 b+2 b p}\right ) \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int x^4 \left (1+\frac {b x^2}{a}\right )^p \, dx \\ & = \frac {e^2 x^5 \left (a+b x^2\right )^{1+p}}{b (7+2 p)}+\frac {1}{5} \left (d^2-\frac {5 a e^2}{7 b+2 b p}\right ) x^5 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {5}{2},-p;\frac {7}{2};-\frac {b x^2}{a}\right )+(d e) \text {Subst}\left (\int \left (\frac {a^2 (a+b x)^p}{b^2}-\frac {2 a (a+b x)^{1+p}}{b^2}+\frac {(a+b x)^{2+p}}{b^2}\right ) \, dx,x,x^2\right ) \\ & = \frac {a^2 d e \left (a+b x^2\right )^{1+p}}{b^3 (1+p)}+\frac {e^2 x^5 \left (a+b x^2\right )^{1+p}}{b (7+2 p)}-\frac {2 a d e \left (a+b x^2\right )^{2+p}}{b^3 (2+p)}+\frac {d e \left (a+b x^2\right )^{3+p}}{b^3 (3+p)}+\frac {1}{5} \left (d^2-\frac {5 a e^2}{7 b+2 b p}\right ) x^5 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {5}{2},-p;\frac {7}{2};-\frac {b x^2}{a}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.88 \[ \int x^4 (d+e x)^2 \left (a+b x^2\right )^p \, dx=\frac {1}{35} \left (a+b x^2\right )^p \left (\frac {35 d e \left (a+b x^2\right ) \left (2 a^2-2 a b (1+p) x^2+b^2 \left (2+3 p+p^2\right ) x^4\right )}{b^3 (1+p) (2+p) (3+p)}+7 d^2 x^5 \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-p,\frac {7}{2},-\frac {b x^2}{a}\right )+5 e^2 x^7 \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-p,\frac {9}{2},-\frac {b x^2}{a}\right )\right ) \]

[In]

Integrate[x^4*(d + e*x)^2*(a + b*x^2)^p,x]

[Out]

((a + b*x^2)^p*((35*d*e*(a + b*x^2)*(2*a^2 - 2*a*b*(1 + p)*x^2 + b^2*(2 + 3*p + p^2)*x^4))/(b^3*(1 + p)*(2 + p
)*(3 + p)) + (7*d^2*x^5*Hypergeometric2F1[5/2, -p, 7/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p + (5*e^2*x^7*Hypergeo
metric2F1[7/2, -p, 9/2, -((b*x^2)/a)])/(1 + (b*x^2)/a)^p))/35

Maple [F]

\[\int x^{4} \left (e x +d \right )^{2} \left (b \,x^{2}+a \right )^{p}d x\]

[In]

int(x^4*(e*x+d)^2*(b*x^2+a)^p,x)

[Out]

int(x^4*(e*x+d)^2*(b*x^2+a)^p,x)

Fricas [F]

\[ \int x^4 (d+e x)^2 \left (a+b x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (b x^{2} + a\right )}^{p} x^{4} \,d x } \]

[In]

integrate(x^4*(e*x+d)^2*(b*x^2+a)^p,x, algorithm="fricas")

[Out]

integral((e^2*x^6 + 2*d*e*x^5 + d^2*x^4)*(b*x^2 + a)^p, x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 925 vs. \(2 (151) = 302\).

Time = 16.94 (sec) , antiderivative size = 986, normalized size of antiderivative = 5.57 \[ \int x^4 (d+e x)^2 \left (a+b x^2\right )^p \, dx=\frac {a^{p} d^{2} x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{2}, - p \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{5} + \frac {a^{p} e^{2} x^{7} {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{2}, - p \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{7} + 2 d e \left (\begin {cases} \frac {a^{p} x^{6}}{6} & \text {for}\: b = 0 \\\frac {2 a^{2} \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{4 a^{2} b^{3} + 8 a b^{4} x^{2} + 4 b^{5} x^{4}} + \frac {2 a^{2} \log {\left (x + \sqrt {- \frac {a}{b}} \right )}}{4 a^{2} b^{3} + 8 a b^{4} x^{2} + 4 b^{5} x^{4}} + \frac {3 a^{2}}{4 a^{2} b^{3} + 8 a b^{4} x^{2} + 4 b^{5} x^{4}} + \frac {4 a b x^{2} \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{4 a^{2} b^{3} + 8 a b^{4} x^{2} + 4 b^{5} x^{4}} + \frac {4 a b x^{2} \log {\left (x + \sqrt {- \frac {a}{b}} \right )}}{4 a^{2} b^{3} + 8 a b^{4} x^{2} + 4 b^{5} x^{4}} + \frac {4 a b x^{2}}{4 a^{2} b^{3} + 8 a b^{4} x^{2} + 4 b^{5} x^{4}} + \frac {2 b^{2} x^{4} \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{4 a^{2} b^{3} + 8 a b^{4} x^{2} + 4 b^{5} x^{4}} + \frac {2 b^{2} x^{4} \log {\left (x + \sqrt {- \frac {a}{b}} \right )}}{4 a^{2} b^{3} + 8 a b^{4} x^{2} + 4 b^{5} x^{4}} & \text {for}\: p = -3 \\- \frac {2 a^{2} \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{2 a b^{3} + 2 b^{4} x^{2}} - \frac {2 a^{2} \log {\left (x + \sqrt {- \frac {a}{b}} \right )}}{2 a b^{3} + 2 b^{4} x^{2}} - \frac {2 a^{2}}{2 a b^{3} + 2 b^{4} x^{2}} - \frac {2 a b x^{2} \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{2 a b^{3} + 2 b^{4} x^{2}} - \frac {2 a b x^{2} \log {\left (x + \sqrt {- \frac {a}{b}} \right )}}{2 a b^{3} + 2 b^{4} x^{2}} + \frac {b^{2} x^{4}}{2 a b^{3} + 2 b^{4} x^{2}} & \text {for}\: p = -2 \\\frac {a^{2} \log {\left (x - \sqrt {- \frac {a}{b}} \right )}}{2 b^{3}} + \frac {a^{2} \log {\left (x + \sqrt {- \frac {a}{b}} \right )}}{2 b^{3}} - \frac {a x^{2}}{2 b^{2}} + \frac {x^{4}}{4 b} & \text {for}\: p = -1 \\\frac {2 a^{3} \left (a + b x^{2}\right )^{p}}{2 b^{3} p^{3} + 12 b^{3} p^{2} + 22 b^{3} p + 12 b^{3}} - \frac {2 a^{2} b p x^{2} \left (a + b x^{2}\right )^{p}}{2 b^{3} p^{3} + 12 b^{3} p^{2} + 22 b^{3} p + 12 b^{3}} + \frac {a b^{2} p^{2} x^{4} \left (a + b x^{2}\right )^{p}}{2 b^{3} p^{3} + 12 b^{3} p^{2} + 22 b^{3} p + 12 b^{3}} + \frac {a b^{2} p x^{4} \left (a + b x^{2}\right )^{p}}{2 b^{3} p^{3} + 12 b^{3} p^{2} + 22 b^{3} p + 12 b^{3}} + \frac {b^{3} p^{2} x^{6} \left (a + b x^{2}\right )^{p}}{2 b^{3} p^{3} + 12 b^{3} p^{2} + 22 b^{3} p + 12 b^{3}} + \frac {3 b^{3} p x^{6} \left (a + b x^{2}\right )^{p}}{2 b^{3} p^{3} + 12 b^{3} p^{2} + 22 b^{3} p + 12 b^{3}} + \frac {2 b^{3} x^{6} \left (a + b x^{2}\right )^{p}}{2 b^{3} p^{3} + 12 b^{3} p^{2} + 22 b^{3} p + 12 b^{3}} & \text {otherwise} \end {cases}\right ) \]

[In]

integrate(x**4*(e*x+d)**2*(b*x**2+a)**p,x)

[Out]

a**p*d**2*x**5*hyper((5/2, -p), (7/2,), b*x**2*exp_polar(I*pi)/a)/5 + a**p*e**2*x**7*hyper((7/2, -p), (9/2,),
b*x**2*exp_polar(I*pi)/a)/7 + 2*d*e*Piecewise((a**p*x**6/6, Eq(b, 0)), (2*a**2*log(x - sqrt(-a/b))/(4*a**2*b**
3 + 8*a*b**4*x**2 + 4*b**5*x**4) + 2*a**2*log(x + sqrt(-a/b))/(4*a**2*b**3 + 8*a*b**4*x**2 + 4*b**5*x**4) + 3*
a**2/(4*a**2*b**3 + 8*a*b**4*x**2 + 4*b**5*x**4) + 4*a*b*x**2*log(x - sqrt(-a/b))/(4*a**2*b**3 + 8*a*b**4*x**2
+ 4*b**5*x**4) + 4*a*b*x**2*log(x + sqrt(-a/b))/(4*a**2*b**3 + 8*a*b**4*x**2 + 4*b**5*x**4) + 4*a*b*x**2/(4*a
**2*b**3 + 8*a*b**4*x**2 + 4*b**5*x**4) + 2*b**2*x**4*log(x - sqrt(-a/b))/(4*a**2*b**3 + 8*a*b**4*x**2 + 4*b**
5*x**4) + 2*b**2*x**4*log(x + sqrt(-a/b))/(4*a**2*b**3 + 8*a*b**4*x**2 + 4*b**5*x**4), Eq(p, -3)), (-2*a**2*lo
g(x - sqrt(-a/b))/(2*a*b**3 + 2*b**4*x**2) - 2*a**2*log(x + sqrt(-a/b))/(2*a*b**3 + 2*b**4*x**2) - 2*a**2/(2*a
*b**3 + 2*b**4*x**2) - 2*a*b*x**2*log(x - sqrt(-a/b))/(2*a*b**3 + 2*b**4*x**2) - 2*a*b*x**2*log(x + sqrt(-a/b)
)/(2*a*b**3 + 2*b**4*x**2) + b**2*x**4/(2*a*b**3 + 2*b**4*x**2), Eq(p, -2)), (a**2*log(x - sqrt(-a/b))/(2*b**3
) + a**2*log(x + sqrt(-a/b))/(2*b**3) - a*x**2/(2*b**2) + x**4/(4*b), Eq(p, -1)), (2*a**3*(a + b*x**2)**p/(2*b
**3*p**3 + 12*b**3*p**2 + 22*b**3*p + 12*b**3) - 2*a**2*b*p*x**2*(a + b*x**2)**p/(2*b**3*p**3 + 12*b**3*p**2 +
22*b**3*p + 12*b**3) + a*b**2*p**2*x**4*(a + b*x**2)**p/(2*b**3*p**3 + 12*b**3*p**2 + 22*b**3*p + 12*b**3) +
a*b**2*p*x**4*(a + b*x**2)**p/(2*b**3*p**3 + 12*b**3*p**2 + 22*b**3*p + 12*b**3) + b**3*p**2*x**6*(a + b*x**2)
**p/(2*b**3*p**3 + 12*b**3*p**2 + 22*b**3*p + 12*b**3) + 3*b**3*p*x**6*(a + b*x**2)**p/(2*b**3*p**3 + 12*b**3*
p**2 + 22*b**3*p + 12*b**3) + 2*b**3*x**6*(a + b*x**2)**p/(2*b**3*p**3 + 12*b**3*p**2 + 22*b**3*p + 12*b**3),
True))

Maxima [F]

\[ \int x^4 (d+e x)^2 \left (a+b x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (b x^{2} + a\right )}^{p} x^{4} \,d x } \]

[In]

integrate(x^4*(e*x+d)^2*(b*x^2+a)^p,x, algorithm="maxima")

[Out]

integrate((e*x + d)^2*(b*x^2 + a)^p*x^4, x)

Giac [F]

\[ \int x^4 (d+e x)^2 \left (a+b x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{2} {\left (b x^{2} + a\right )}^{p} x^{4} \,d x } \]

[In]

integrate(x^4*(e*x+d)^2*(b*x^2+a)^p,x, algorithm="giac")

[Out]

integrate((e*x + d)^2*(b*x^2 + a)^p*x^4, x)

Mupad [F(-1)]

Timed out. \[ \int x^4 (d+e x)^2 \left (a+b x^2\right )^p \, dx=\int x^4\,{\left (b\,x^2+a\right )}^p\,{\left (d+e\,x\right )}^2 \,d x \]

[In]

int(x^4*(a + b*x^2)^p*(d + e*x)^2,x)

[Out]

int(x^4*(a + b*x^2)^p*(d + e*x)^2, x)