\(\int (g x)^m (d+e x)^3 (d^2-e^2 x^2)^p \, dx\) [306]

Optimal result

Integrand size = 27, antiderivative size = 264 \[ \int (g x)^m (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=-\frac {3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{1+p}}{g (3+m+2 p)}-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{1+p}}{g^2 (4+m+2 p)}+\frac {2 d^3 (3+2 m+p) (g x)^{1+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )}{g (1+m) (3+m+2 p)}+\frac {2 d^2 e (7+2 m+3 p) (g x)^{2+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},-p,\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )}{g^2 (2+m) (4+m+2 p)} \]

[Out]

-3*d*(g*x)^(1+m)*(-e^2*x^2+d^2)^(p+1)/g/(3+m+2*p)-e*(g*x)^(2+m)*(-e^2*x^2+d^2)^(p+1)/g^2/(4+m+2*p)+2*d^3*(3+2*
m+p)*(g*x)^(1+m)*(-e^2*x^2+d^2)^p*hypergeom([-p, 1/2+1/2*m],[3/2+1/2*m],e^2*x^2/d^2)/g/(1+m)/(3+m+2*p)/((1-e^2
*x^2/d^2)^p)+2*d^2*e*(7+2*m+3*p)*(g*x)^(2+m)*(-e^2*x^2+d^2)^p*hypergeom([-p, 1+1/2*m],[2+1/2*m],e^2*x^2/d^2)/g
^2/(2+m)/(4+m+2*p)/((1-e^2*x^2/d^2)^p)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1823, 822, 372, 371} \[ \int (g x)^m (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=\frac {2 d^2 e (2 m+3 p+7) (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+2}{2},-p,\frac {m+4}{2},\frac {e^2 x^2}{d^2}\right )}{g^2 (m+2) (m+2 p+4)}-\frac {e (g x)^{m+2} \left (d^2-e^2 x^2\right )^{p+1}}{g^2 (m+2 p+4)}-\frac {3 d (g x)^{m+1} \left (d^2-e^2 x^2\right )^{p+1}}{g (m+2 p+3)}+\frac {2 d^3 (2 m+p+3) (g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},-p,\frac {m+3}{2},\frac {e^2 x^2}{d^2}\right )}{g (m+1) (m+2 p+3)} \]

[In]

Int[(g*x)^m*(d + e*x)^3*(d^2 - e^2*x^2)^p,x]

[Out]

(-3*d*(g*x)^(1 + m)*(d^2 - e^2*x^2)^(1 + p))/(g*(3 + m + 2*p)) - (e*(g*x)^(2 + m)*(d^2 - e^2*x^2)^(1 + p))/(g^
2*(4 + m + 2*p)) + (2*d^3*(3 + 2*m + p)*(g*x)^(1 + m)*(d^2 - e^2*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 +
m)/2, (e^2*x^2)/d^2])/(g*(1 + m)*(3 + m + 2*p)*(1 - (e^2*x^2)/d^2)^p) + (2*d^2*e*(7 + 2*m + 3*p)*(g*x)^(2 + m)
*(d^2 - e^2*x^2)^p*Hypergeometric2F1[(2 + m)/2, -p, (4 + m)/2, (e^2*x^2)/d^2])/(g^2*(2 + m)*(4 + m + 2*p)*(1 -
(e^2*x^2)/d^2)^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 (g x)^{2+m} \left (d^2-e^2 x^2\right )^{1+p}}{g^2 (4+m+2 p)}-\frac {\int (g x)^m \left (d^2-e^2 x^2\right )^p \left (-d^3 e^2 (4+m+2 p)-2 d^2 e^3 (7+2 m+3 p) x-3 d e^4 (4+m+2 p) x^2\right ) \, dx}{e^2 (4+m+2 p)} \\ & = -\frac {3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{1+p}}{g (3+m+2 p)}-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{1+p}}{g^2 (4+m+2 p)}+\frac {\int (g x)^m \left (2 d^3 e^4 (3+2 m+p) (4+m+2 p)+2 d^2 e^5 (3+m+2 p) (7+2 m+3 p) x\right ) \left (d^2-e^2 x^2\right )^p \, dx}{e^4 (3+m+2 p) (4+m+2 p)} \\ & = -\frac {3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{1+p}}{g (3+m+2 p)}-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{1+p}}{g^2 (4+m+2 p)}+\frac {\left (2 d^3 (3+2 m+p)\right ) \int (g x)^m \left (d^2-e^2 x^2\right )^p \, dx}{3+m+2 p}+\frac {\left (2 d^2 e (7+2 m+3 p)\right ) \int (g x)^{1+m} \left (d^2-e^2 x^2\right )^p \, dx}{g (4+m+2 p)} \\ & = -\frac {3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{1+p}}{g (3+m+2 p)}-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{1+p}}{g^2 (4+m+2 p)}+\frac {\left (2 d^3 (3+2 m+p) \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int (g x)^m \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx}{3+m+2 p}+\frac {\left (2 d^2 e (7+2 m+3 p) \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int (g x)^{1+m} \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx}{g (4+m+2 p)} \\ & = -\frac {3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{1+p}}{g (3+m+2 p)}-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{1+p}}{g^2 (4+m+2 p)}+\frac {2 d^3 (3+2 m+p) (g x)^{1+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {1+m}{2},-p;\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{g (1+m) (3+m+2 p)}+\frac {2 d^2 e (7+2 m+3 p) (g x)^{2+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {2+m}{2},-p;\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )}{g^2 (2+m) (4+m+2 p)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.73 \[ \int (g x)^m (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=x (g x)^m \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (\frac {d^3 \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )}{1+m}+e x \left (\frac {3 d^2 \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},-p,\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )}{2+m}+e x \left (\frac {3 d \operatorname {Hypergeometric2F1}\left (\frac {3+m}{2},-p,\frac {5+m}{2},\frac {e^2 x^2}{d^2}\right )}{3+m}+\frac {e x \operatorname {Hypergeometric2F1}\left (\frac {4+m}{2},-p,\frac {6+m}{2},\frac {e^2 x^2}{d^2}\right )}{4+m}\right )\right )\right ) \]

[In]

Integrate[(g*x)^m*(d + e*x)^3*(d^2 - e^2*x^2)^p,x]

[Out]

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

Maple [F]

\[\int \left (g x \right )^{m} \left (e x +d \right )^{3} \left (-e^{2} x^{2}+d^{2}\right )^{p}d x\]

[In]

int((g*x)^m*(e*x+d)^3*(-e^2*x^2+d^2)^p,x)

[Out]

int((g*x)^m*(e*x+d)^3*(-e^2*x^2+d^2)^p,x)

Fricas [F]

\[ \int (g x)^m (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{3} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} \left (g x\right )^{m} \,d x } \]

[In]

integrate((g*x)^m*(e*x+d)^3*(-e^2*x^2+d^2)^p,x, algorithm="fricas")

[Out]

integral((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*(-e^2*x^2 + d^2)^p*(g*x)^m, x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 9.15 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.97 \[ \int (g x)^m (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=\frac {d^{3} d^{2 p} g^{m} x^{m + 1} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {3 d^{2} d^{2 p} e g^{m} x^{m + 2} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + 2\right )} + \frac {3 d d^{2 p} e^{2} g^{m} x^{m + 3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + \frac {3}{2} \\ \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {d^{2 p} e^{3} g^{m} x^{m + 4} \Gamma \left (\frac {m}{2} + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + 2 \\ \frac {m}{2} + 3 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + 3\right )} \]

[In]

integrate((g*x)**m*(e*x+d)**3*(-e**2*x**2+d**2)**p,x)

[Out]

d**3*d**(2*p)*g**m*x**(m + 1)*gamma(m/2 + 1/2)*hyper((-p, m/2 + 1/2), (m/2 + 3/2,), e**2*x**2*exp_polar(2*I*pi
)/d**2)/(2*gamma(m/2 + 3/2)) + 3*d**2*d**(2*p)*e*g**m*x**(m + 2)*gamma(m/2 + 1)*hyper((-p, m/2 + 1), (m/2 + 2,
), e**2*x**2*exp_polar(2*I*pi)/d**2)/(2*gamma(m/2 + 2)) + 3*d*d**(2*p)*e**2*g**m*x**(m + 3)*gamma(m/2 + 3/2)*h
yper((-p, m/2 + 3/2), (m/2 + 5/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/(2*gamma(m/2 + 5/2)) + d**(2*p)*e**3*g**
m*x**(m + 4)*gamma(m/2 + 2)*hyper((-p, m/2 + 2), (m/2 + 3,), e**2*x**2*exp_polar(2*I*pi)/d**2)/(2*gamma(m/2 +
3))

Maxima [F]

\[ \int (g x)^m (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{3} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} \left (g x\right )^{m} \,d x } \]

[In]

integrate((g*x)^m*(e*x+d)^3*(-e^2*x^2+d^2)^p,x, algorithm="maxima")

[Out]

integrate((e*x + d)^3*(-e^2*x^2 + d^2)^p*(g*x)^m, x)

Giac [F]

\[ \int (g x)^m (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{3} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} \left (g x\right )^{m} \,d x } \]

[In]

integrate((g*x)^m*(e*x+d)^3*(-e^2*x^2+d^2)^p,x, algorithm="giac")

[Out]

integrate((e*x + d)^3*(-e^2*x^2 + d^2)^p*(g*x)^m, x)

Mupad [F(-1)]

Timed out. \[ \int (g x)^m (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=\int {\left (d^2-e^2\,x^2\right )}^p\,{\left (g\,x\right )}^m\,{\left (d+e\,x\right )}^3 \,d x \]

[In]

int((d^2 - e^2*x^2)^p*(g*x)^m*(d + e*x)^3,x)

[Out]

int((d^2 - e^2*x^2)^p*(g*x)^m*(d + e*x)^3, x)