Integrand size = 22, antiderivative size = 399 \[ \int (g x)^m (d+e x)^n \left (a+c x^2\right )^2 \, dx=-\frac {c d (2+m) \left (c d^2 \left (12+7 m+m^2\right )+2 a e^2 \left (20+m^2+9 n+n^2+m (9+2 n)\right )\right ) (g x)^{1+m} (d+e x)^{1+n}}{e^4 g (2+m+n) (3+m+n) (4+m+n) (5+m+n)}+\frac {c \left (c d^2 \left (12+7 m+m^2\right )+2 a e^2 \left (20+m^2+9 n+n^2+m (9+2 n)\right )\right ) (g x)^{2+m} (d+e x)^{1+n}}{e^3 g^2 (3+m+n) (4+m+n) (5+m+n)}-\frac {c^2 d (4+m) (g x)^{3+m} (d+e x)^{1+n}}{e^2 g^3 (4+m+n) (5+m+n)}+\frac {c^2 (g x)^{4+m} (d+e x)^{1+n}}{e g^4 (5+m+n)}+\frac {\left (a^2 e^4 (2+m+n) (3+m+n) (4+m+n) (5+m+n)+c d^2 (1+m) (2+m) \left (c d^2 \left (12+7 m+m^2\right )+2 a e^2 \left (20+m^2+9 n+n^2+m (9+2 n)\right )\right )\right ) (g x)^{1+m} (d+e x)^n \left (1+\frac {e x}{d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {e x}{d}\right )}{e^4 g (1+m) (2+m+n) (3+m+n) (4+m+n) (5+m+n)} \]
-c*d*(2+m)*(c*d^2*(m^2+7*m+12)+2*a*e^2*(20+m^2+9*n+n^2+m*(9+2*n)))*(g*x)^( 1+m)*(e*x+d)^(1+n)/e^4/g/(2+m+n)/(3+m+n)/(4+m+n)/(5+m+n)+c*(c*d^2*(m^2+7*m +12)+2*a*e^2*(20+m^2+9*n+n^2+m*(9+2*n)))*(g*x)^(2+m)*(e*x+d)^(1+n)/e^3/g^2 /(3+m+n)/(4+m+n)/(5+m+n)-c^2*d*(4+m)*(g*x)^(3+m)*(e*x+d)^(1+n)/e^2/g^3/(4+ m+n)/(5+m+n)+c^2*(g*x)^(4+m)*(e*x+d)^(1+n)/e/g^4/(5+m+n)+(a^2*e^4*(2+m+n)* (3+m+n)*(4+m+n)*(5+m+n)+c*d^2*(1+m)*(2+m)*(c*d^2*(m^2+7*m+12)+2*a*e^2*(20+ m^2+9*n+n^2+m*(9+2*n))))*(g*x)^(1+m)*(e*x+d)^n*hypergeom([-n, 1+m],[2+m],- e*x/d)/e^4/g/(1+m)/(2+m+n)/(3+m+n)/(4+m+n)/(5+m+n)/((1+e*x/d)^n)
Time = 0.13 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.69 \[ \int (g x)^m (d+e x)^n \left (a+c x^2\right )^2 \, dx=\frac {x (g x)^m (d+e x)^n \left (1+\frac {e x}{d}\right )^{-n} \left (c^2 d^4 \operatorname {Hypergeometric2F1}\left (1+m,-4-n,2+m,-\frac {e x}{d}\right )-4 c^2 d^4 \operatorname {Hypergeometric2F1}\left (1+m,-3-n,2+m,-\frac {e x}{d}\right )+6 c^2 d^4 \operatorname {Hypergeometric2F1}\left (1+m,-2-n,2+m,-\frac {e x}{d}\right )+2 a c d^2 e^2 \operatorname {Hypergeometric2F1}\left (1+m,-2-n,2+m,-\frac {e x}{d}\right )-4 c^2 d^4 \operatorname {Hypergeometric2F1}\left (1+m,-1-n,2+m,-\frac {e x}{d}\right )-4 a c d^2 e^2 \operatorname {Hypergeometric2F1}\left (1+m,-1-n,2+m,-\frac {e x}{d}\right )+c^2 d^4 \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {e x}{d}\right )+2 a c d^2 e^2 \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {e x}{d}\right )+a^2 e^4 \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {e x}{d}\right )\right )}{e^4 (1+m)} \]
(x*(g*x)^m*(d + e*x)^n*(c^2*d^4*Hypergeometric2F1[1 + m, -4 - n, 2 + m, -( (e*x)/d)] - 4*c^2*d^4*Hypergeometric2F1[1 + m, -3 - n, 2 + m, -((e*x)/d)] + 6*c^2*d^4*Hypergeometric2F1[1 + m, -2 - n, 2 + m, -((e*x)/d)] + 2*a*c*d^ 2*e^2*Hypergeometric2F1[1 + m, -2 - n, 2 + m, -((e*x)/d)] - 4*c^2*d^4*Hype rgeometric2F1[1 + m, -1 - n, 2 + m, -((e*x)/d)] - 4*a*c*d^2*e^2*Hypergeome tric2F1[1 + m, -1 - n, 2 + m, -((e*x)/d)] + c^2*d^4*Hypergeometric2F1[1 + m, -n, 2 + m, -((e*x)/d)] + 2*a*c*d^2*e^2*Hypergeometric2F1[1 + m, -n, 2 + m, -((e*x)/d)] + a^2*e^4*Hypergeometric2F1[1 + m, -n, 2 + m, -((e*x)/d)]) )/(e^4*(1 + m)*(1 + (e*x)/d)^n)
Time = 0.73 (sec) , antiderivative size = 378, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {521, 2125, 27, 521, 27, 90, 76, 74}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \left (a+c x^2\right )^2 (g x)^m (d+e x)^n \, dx\) |
| \(\Big \downarrow \) 521 |
| \(\displaystyle \frac {\int (g x)^m (d+e x)^n \left (-c^2 d (m+4) x^3 g^4+2 a c e (m+n+5) x^2 g^4+a^2 e (m+n+5) g^4\right )dx}{e g^4 (m+n+5)}+\frac {c^2 (g x)^{m+4} (d+e x)^{n+1}}{e g^4 (m+n+5)}\) |
| \(\Big \downarrow \) 2125 |
| \(\displaystyle \frac {\frac {\int g^7 (g x)^m (d+e x)^n \left (a^2 (m+n+4) (m+n+5) e^2+c \left (c \left (m^2+7 m+12\right ) d^2+2 a e^2 \left (m^2+(2 n+9) m+n^2+9 n+20\right )\right ) x^2\right )dx}{e g^3 (m+n+4)}-\frac {c^2 d g (m+4) (g x)^{m+3} (d+e x)^{n+1}}{e (m+n+4)}}{e g^4 (m+n+5)}+\frac {c^2 (g x)^{m+4} (d+e x)^{n+1}}{e g^4 (m+n+5)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {g^4 \int (g x)^m (d+e x)^n \left (a^2 (m+n+4) (m+n+5) e^2+c \left (c \left (m^2+7 m+12\right ) d^2+2 a e^2 \left (m^2+(2 n+9) m+n^2+9 n+20\right )\right ) x^2\right )dx}{e (m+n+4)}-\frac {c^2 d g (m+4) (g x)^{m+3} (d+e x)^{n+1}}{e (m+n+4)}}{e g^4 (m+n+5)}+\frac {c^2 (g x)^{m+4} (d+e x)^{n+1}}{e g^4 (m+n+5)}\) |
| \(\Big \downarrow \) 521 |
| \(\displaystyle \frac {\frac {g^4 \left (\frac {\int g^2 (g x)^m (d+e x)^n \left (a^2 e^3 (m+n+3) (m+n+4) (m+n+5)-c d (m+2) \left (c \left (m^2+7 m+12\right ) d^2+2 a e^2 \left (m^2+(2 n+9) m+n^2+9 n+20\right )\right ) x\right )dx}{e g^2 (m+n+3)}+\frac {c (g x)^{m+2} (d+e x)^{n+1} \left (2 a e^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+c d^2 \left (m^2+7 m+12\right )\right )}{e g^2 (m+n+3)}\right )}{e (m+n+4)}-\frac {c^2 d g (m+4) (g x)^{m+3} (d+e x)^{n+1}}{e (m+n+4)}}{e g^4 (m+n+5)}+\frac {c^2 (g x)^{m+4} (d+e x)^{n+1}}{e g^4 (m+n+5)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {g^4 \left (\frac {\int (g x)^m (d+e x)^n \left (a^2 e^3 (m+n+3) (m+n+4) (m+n+5)-c d (m+2) \left (c \left (m^2+7 m+12\right ) d^2+2 a e^2 \left (m^2+(2 n+9) m+n^2+9 n+20\right )\right ) x\right )dx}{e (m+n+3)}+\frac {c (g x)^{m+2} (d+e x)^{n+1} \left (2 a e^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+c d^2 \left (m^2+7 m+12\right )\right )}{e g^2 (m+n+3)}\right )}{e (m+n+4)}-\frac {c^2 d g (m+4) (g x)^{m+3} (d+e x)^{n+1}}{e (m+n+4)}}{e g^4 (m+n+5)}+\frac {c^2 (g x)^{m+4} (d+e x)^{n+1}}{e g^4 (m+n+5)}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {\frac {g^4 \left (\frac {\frac {\left (a^2 e^4 (m+n+3) (m+n+4) (m+n+5)+\frac {c d^2 (m+1) (m+2) \left (2 a e^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+c d^2 \left (m^2+7 m+12\right )\right )}{m+n+2}\right ) \int (g x)^m (d+e x)^ndx}{e}-\frac {c d (m+2) (g x)^{m+1} (d+e x)^{n+1} \left (2 a e^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+c d^2 \left (m^2+7 m+12\right )\right )}{e g (m+n+2)}}{e (m+n+3)}+\frac {c (g x)^{m+2} (d+e x)^{n+1} \left (2 a e^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+c d^2 \left (m^2+7 m+12\right )\right )}{e g^2 (m+n+3)}\right )}{e (m+n+4)}-\frac {c^2 d g (m+4) (g x)^{m+3} (d+e x)^{n+1}}{e (m+n+4)}}{e g^4 (m+n+5)}+\frac {c^2 (g x)^{m+4} (d+e x)^{n+1}}{e g^4 (m+n+5)}\) |
| \(\Big \downarrow \) 76 |
| \(\displaystyle \frac {\frac {g^4 \left (\frac {\frac {(d+e x)^n \left (\frac {e x}{d}+1\right )^{-n} \left (a^2 e^4 (m+n+3) (m+n+4) (m+n+5)+\frac {c d^2 (m+1) (m+2) \left (2 a e^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+c d^2 \left (m^2+7 m+12\right )\right )}{m+n+2}\right ) \int (g x)^m \left (\frac {e x}{d}+1\right )^ndx}{e}-\frac {c d (m+2) (g x)^{m+1} (d+e x)^{n+1} \left (2 a e^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+c d^2 \left (m^2+7 m+12\right )\right )}{e g (m+n+2)}}{e (m+n+3)}+\frac {c (g x)^{m+2} (d+e x)^{n+1} \left (2 a e^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+c d^2 \left (m^2+7 m+12\right )\right )}{e g^2 (m+n+3)}\right )}{e (m+n+4)}-\frac {c^2 d g (m+4) (g x)^{m+3} (d+e x)^{n+1}}{e (m+n+4)}}{e g^4 (m+n+5)}+\frac {c^2 (g x)^{m+4} (d+e x)^{n+1}}{e g^4 (m+n+5)}\) |
| \(\Big \downarrow \) 74 |
| \(\displaystyle \frac {\frac {g^4 \left (\frac {\frac {(g x)^{m+1} (d+e x)^n \left (\frac {e x}{d}+1\right )^{-n} \left (a^2 e^4 (m+n+3) (m+n+4) (m+n+5)+\frac {c d^2 (m+1) (m+2) \left (2 a e^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+c d^2 \left (m^2+7 m+12\right )\right )}{m+n+2}\right ) \operatorname {Hypergeometric2F1}\left (m+1,-n,m+2,-\frac {e x}{d}\right )}{e g (m+1)}-\frac {c d (m+2) (g x)^{m+1} (d+e x)^{n+1} \left (2 a e^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+c d^2 \left (m^2+7 m+12\right )\right )}{e g (m+n+2)}}{e (m+n+3)}+\frac {c (g x)^{m+2} (d+e x)^{n+1} \left (2 a e^2 \left (m^2+m (2 n+9)+n^2+9 n+20\right )+c d^2 \left (m^2+7 m+12\right )\right )}{e g^2 (m+n+3)}\right )}{e (m+n+4)}-\frac {c^2 d g (m+4) (g x)^{m+3} (d+e x)^{n+1}}{e (m+n+4)}}{e g^4 (m+n+5)}+\frac {c^2 (g x)^{m+4} (d+e x)^{n+1}}{e g^4 (m+n+5)}\) |
(c^2*(g*x)^(4 + m)*(d + e*x)^(1 + n))/(e*g^4*(5 + m + n)) + (-((c^2*d*g*(4 + m)*(g*x)^(3 + m)*(d + e*x)^(1 + n))/(e*(4 + m + n))) + (g^4*((c*(c*d^2* (12 + 7*m + m^2) + 2*a*e^2*(20 + m^2 + 9*n + n^2 + m*(9 + 2*n)))*(g*x)^(2 + m)*(d + e*x)^(1 + n))/(e*g^2*(3 + m + n)) + (-((c*d*(2 + m)*(c*d^2*(12 + 7*m + m^2) + 2*a*e^2*(20 + m^2 + 9*n + n^2 + m*(9 + 2*n)))*(g*x)^(1 + m)* (d + e*x)^(1 + n))/(e*g*(2 + m + n))) + ((a^2*e^4*(3 + m + n)*(4 + m + n)* (5 + m + n) + (c*d^2*(1 + m)*(2 + m)*(c*d^2*(12 + 7*m + m^2) + 2*a*e^2*(20 + m^2 + 9*n + n^2 + m*(9 + 2*n))))/(2 + m + n))*(g*x)^(1 + m)*(d + e*x)^n *Hypergeometric2F1[1 + m, -n, 2 + m, -((e*x)/d)])/(e*g*(1 + m)*(1 + (e*x)/ d)^n))/(e*(3 + m + n))))/(e*(4 + m + n)))/(e*g^4*(5 + m + n))
3.4.77.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x )^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] && !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^IntPart [n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n]) Int[(b*x)^m*(1 + d* (x/c))^n, x], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && !Integer Q[n] && !GtQ[c, 0] && !GtQ[-d/(b*c), 0] && ((RationalQ[m] && !(EqQ[n, -2 ^(-1)] && EqQ[c^2 - d^2, 0])) || !RationalQ[n])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[b^p*(e*x)^(m + 2*p)*((c + d*x)^(n + 1)/(d*e^(2*p)*(m + n + 2*p + 1))), x] + Simp[1/(d*e^(2*p)*(m + n + 2*p + 1)) Int[(e*x)^m*(c + d*x)^n*ExpandToSum[d*(m + n + 2*p + 1)*(e^(2*p)*(a + b*x^2)^p - b^p*(e*x)^ (2*p)) - b^p*(e*c)*(m + 2*p)*(e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && NeQ[m + n + 2*p + 1, 0] && !IntegerQ[m] && !I ntegerQ[n]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> With[{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, Simp[k*(a + b*x )^(m + q)*((c + d*x)^(n + 1)/(d*b^q*(m + n + q + 1))), x] + Simp[1/(d*b^q*( m + n + q + 1)) Int[(a + b*x)^m*(c + d*x)^n*ExpandToSum[d*b^q*(m + n + q + 1)*Px - d*k*(m + n + q + 1)*(a + b*x)^q - k*(b*c - a*d)*(m + q)*(a + b*x) ^(q - 1), x], x], x] /; NeQ[m + n + q + 1, 0]] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x]
\[\int \left (g x \right )^{m} \left (e x +d \right )^{n} \left (c \,x^{2}+a \right )^{2}d x\]
\[ \int (g x)^m (d+e x)^n \left (a+c x^2\right )^2 \, dx=\int { {\left (c x^{2} + a\right )}^{2} {\left (e x + d\right )}^{n} \left (g x\right )^{m} \,d x } \]
Result contains complex when optimal does not.
Time = 16.57 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.32 \[ \int (g x)^m (d+e x)^n \left (a+c x^2\right )^2 \, dx=\frac {a^{2} d^{n} g^{m} x^{m + 1} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {e x e^{i \pi }}{d}} \right )}}{\Gamma \left (m + 2\right )} + \frac {2 a c d^{n} g^{m} x^{m + 3} \Gamma \left (m + 3\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 3 \\ m + 4 \end {matrix}\middle | {\frac {e x e^{i \pi }}{d}} \right )}}{\Gamma \left (m + 4\right )} + \frac {c^{2} d^{n} g^{m} x^{m + 5} \Gamma \left (m + 5\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 5 \\ m + 6 \end {matrix}\middle | {\frac {e x e^{i \pi }}{d}} \right )}}{\Gamma \left (m + 6\right )} \]
a**2*d**n*g**m*x**(m + 1)*gamma(m + 1)*hyper((-n, m + 1), (m + 2,), e*x*ex p_polar(I*pi)/d)/gamma(m + 2) + 2*a*c*d**n*g**m*x**(m + 3)*gamma(m + 3)*hy per((-n, m + 3), (m + 4,), e*x*exp_polar(I*pi)/d)/gamma(m + 4) + c**2*d**n *g**m*x**(m + 5)*gamma(m + 5)*hyper((-n, m + 5), (m + 6,), e*x*exp_polar(I *pi)/d)/gamma(m + 6)
\[ \int (g x)^m (d+e x)^n \left (a+c x^2\right )^2 \, dx=\int { {\left (c x^{2} + a\right )}^{2} {\left (e x + d\right )}^{n} \left (g x\right )^{m} \,d x } \]
\[ \int (g x)^m (d+e x)^n \left (a+c x^2\right )^2 \, dx=\int { {\left (c x^{2} + a\right )}^{2} {\left (e x + d\right )}^{n} \left (g x\right )^{m} \,d x } \]
Timed out. \[ \int (g x)^m (d+e x)^n \left (a+c x^2\right )^2 \, dx=\int {\left (g\,x\right )}^m\,{\left (c\,x^2+a\right )}^2\,{\left (d+e\,x\right )}^n \,d x \]