\(\int (g x)^m (d+e x)^2 (a+c x^2)^p \, dx\) [431]

Optimal result

Integrand size = 22, antiderivative size = 205 \[ \int (g x)^m (d+e x)^2 \left (a+c x^2\right )^p \, dx=\frac {e^2 (g x)^{1+m} \left (a+c x^2\right )^{1+p}}{c g (3+m+2 p)}-\frac {\left (a e^2 (1+m)-c d^2 (3+m+2 p)\right ) (g x)^{1+m} \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},-\frac {c x^2}{a}\right )}{c g (1+m) (3+m+2 p)}+\frac {2 d e (g x)^{2+m} \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},-p,\frac {4+m}{2},-\frac {c x^2}{a}\right )}{g^2 (2+m)} \]

[Out]

e^2*(g*x)^(1+m)*(c*x^2+a)^(p+1)/c/g/(3+m+2*p)-(a*e^2*(1+m)-c*d^2*(3+m+2*p))*(g*x)^(1+m)*(c*x^2+a)^p*hypergeom(
[-p, 1/2+1/2*m],[3/2+1/2*m],-c*x^2/a)/c/g/(1+m)/(3+m+2*p)/((1+c*x^2/a)^p)+2*d*e*(g*x)^(2+m)*(c*x^2+a)^p*hyperg
eom([-p, 1+1/2*m],[2+1/2*m],-c*x^2/a)/g^2/(2+m)/((1+c*x^2/a)^p)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1823, 822, 372, 371} \[ \int (g x)^m (d+e x)^2 \left (a+c x^2\right )^p \, dx=\frac {(g x)^{m+1} \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \left (\frac {d^2}{m+1}-\frac {a e^2}{c (m+2 p+3)}\right ) \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},-p,\frac {m+3}{2},-\frac {c x^2}{a}\right )}{g}+\frac {2 d e (g x)^{m+2} \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+2}{2},-p,\frac {m+4}{2},-\frac {c x^2}{a}\right )}{g^2 (m+2)}+\frac {e^2 (g x)^{m+1} \left (a+c x^2\right )^{p+1}}{c g (m+2 p+3)} \]

[In]

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

[Out]

(e^2*(g*x)^(1 + m)*(a + c*x^2)^(1 + p))/(c*g*(3 + m + 2*p)) + ((d^2/(1 + m) - (a*e^2)/(c*(3 + m + 2*p)))*(g*x)
^(1 + m)*(a + c*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((c*x^2)/a)])/(g*(1 + (c*x^2)/a)^p) + (2*d
*e*(g*x)^(2 + m)*(a + c*x^2)^p*Hypergeometric2F1[(2 + m)/2, -p, (4 + m)/2, -((c*x^2)/a)])/(g^2*(2 + m)*(1 + (c
*x^2)/a)^p)

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 822

Int[((e_.)*(x_))^(m_)*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[f, Int[(e*x)^m*(a + c*
x^2)^p, x], x] + Dist[g/e, Int[(e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, p}, x] &&  !Ration
alQ[m] &&  !IGtQ[p, 0]

Rule 1823

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rubi steps \begin{align*} \text {integral}& = \frac {e^2 (g x)^{1+m} \left (a+c x^2\right )^{1+p}}{c g (3+m+2 p)}+\frac {\int (g x)^m \left (-a e^2 (1+m)+c d^2 (3+m+2 p)+2 c d e (3+m+2 p) x\right ) \left (a+c x^2\right )^p \, dx}{c (3+m+2 p)} \\ & = \frac {e^2 (g x)^{1+m} \left (a+c x^2\right )^{1+p}}{c g (3+m+2 p)}+\frac {(2 d e) \int (g x)^{1+m} \left (a+c x^2\right )^p \, dx}{g}+\left (d^2-\frac {a e^2 (1+m)}{c (3+m+2 p)}\right ) \int (g x)^m \left (a+c x^2\right )^p \, dx \\ & = \frac {e^2 (g x)^{1+m} \left (a+c x^2\right )^{1+p}}{c g (3+m+2 p)}+\frac {\left (2 d e \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int (g x)^{1+m} \left (1+\frac {c x^2}{a}\right )^p \, dx}{g}+\left (\left (d^2-\frac {a e^2 (1+m)}{c (3+m+2 p)}\right ) \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int (g x)^m \left (1+\frac {c x^2}{a}\right )^p \, dx \\ & = \frac {e^2 (g x)^{1+m} \left (a+c x^2\right )^{1+p}}{c g (3+m+2 p)}+\frac {\left (d^2-\frac {a e^2 (1+m)}{c (3+m+2 p)}\right ) (g x)^{1+m} \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1+m}{2},-p;\frac {3+m}{2};-\frac {c x^2}{a}\right )}{g (1+m)}+\frac {2 d e (g x)^{2+m} \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \, _2F_1\left (\frac {2+m}{2},-p;\frac {4+m}{2};-\frac {c x^2}{a}\right )}{g^2 (2+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.77 \[ \int (g x)^m (d+e x)^2 \left (a+c x^2\right )^p \, dx=\frac {x (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \left (d^2 \left (6+5 m+m^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},-\frac {c x^2}{a}\right )+e (1+m) x \left (2 d (3+m) \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},-p,\frac {4+m}{2},-\frac {c x^2}{a}\right )+e (2+m) x \operatorname {Hypergeometric2F1}\left (\frac {3+m}{2},-p,\frac {5+m}{2},-\frac {c x^2}{a}\right )\right )\right )}{(1+m) (2+m) (3+m)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

int((g*x)^m*(e*x+d)^2*(c*x^2+a)^p,x)

[Out]

int((g*x)^m*(e*x+d)^2*(c*x^2+a)^p,x)

Fricas [F]

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

[In]

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

[Out]

integral((e^2*x^2 + 2*d*e*x + d^2)*(c*x^2 + a)^p*(g*x)^m, x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 70.15 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.82 \[ \int (g x)^m (d+e x)^2 \left (a+c x^2\right )^p \, dx=\frac {a^{p} d^{2} g^{m} x^{m + 1} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {a^{p} d e g^{m} x^{m + 2} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (\frac {m}{2} + 2\right )} + \frac {a^{p} e^{2} g^{m} x^{m + 3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + \frac {3}{2} \\ \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} \]

[In]

integrate((g*x)**m*(e*x+d)**2*(c*x**2+a)**p,x)

[Out]

a**p*d**2*g**m*x**(m + 1)*gamma(m/2 + 1/2)*hyper((-p, m/2 + 1/2), (m/2 + 3/2,), c*x**2*exp_polar(I*pi)/a)/(2*g
amma(m/2 + 3/2)) + a**p*d*e*g**m*x**(m + 2)*gamma(m/2 + 1)*hyper((-p, m/2 + 1), (m/2 + 2,), c*x**2*exp_polar(I
*pi)/a)/gamma(m/2 + 2) + a**p*e**2*g**m*x**(m + 3)*gamma(m/2 + 3/2)*hyper((-p, m/2 + 3/2), (m/2 + 5/2,), c*x**
2*exp_polar(I*pi)/a)/(2*gamma(m/2 + 5/2))

Maxima [F]

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

[In]

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

[Out]

integrate((e*x + d)^2*(c*x^2 + a)^p*(g*x)^m, x)

Giac [F]

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

[In]

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

[Out]

integrate((e*x + d)^2*(c*x^2 + a)^p*(g*x)^m, x)

Mupad [F(-1)]

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

[In]

int((g*x)^m*(a + c*x^2)^p*(d + e*x)^2,x)

[Out]

int((g*x)^m*(a + c*x^2)^p*(d + e*x)^2, x)