Integrand size = 35, antiderivative size = 515 \[ \int \frac {\left (7+3 x-6 x^2\right )^q \left (3+2 x+4 x^2\right )}{\left (1+5 x-2 x^2\right )^3} \, dx=-\frac {(23+14 x) \left (7+3 x-6 x^2\right )^{1+q}}{528 \left (1+5 x-2 x^2\right )^2}-\frac {(51+61 q+142 (1-q) x) \left (7+3 x-6 x^2\right )^{1+q}}{15488 \left (1+5 x-2 x^2\right )}+\frac {3^{-\frac {1}{2}+2 q} \left (6144-13505 q+3 \sqrt {33} (741-517 q) q+9825 q^2\right ) \left (\frac {3-\sqrt {177}-12 x}{5-\sqrt {33}-4 x}\right )^{-q} \left (\frac {3+\sqrt {177}-12 x}{5-\sqrt {33}-4 x}\right )^{-q} \left (7+3 x-6 x^2\right )^q \operatorname {AppellF1}\left (-2 q,-q,-q,1-2 q,\frac {12-3 \sqrt {33}-\sqrt {177}}{3 \left (5-\sqrt {33}-4 x\right )},\frac {12-3 \sqrt {33}+\sqrt {177}}{3 \left (5-\sqrt {33}-4 x\right )}\right )}{61952 \sqrt {11} q}-\frac {3^{-\frac {1}{2}+2 q} \left (6144-13505 q-3 \sqrt {33} (741-517 q) q+9825 q^2\right ) \left (\frac {3-\sqrt {177}-12 x}{5+\sqrt {33}-4 x}\right )^{-q} \left (\frac {3+\sqrt {177}-12 x}{5+\sqrt {33}-4 x}\right )^{-q} \left (7+3 x-6 x^2\right )^q \operatorname {AppellF1}\left (-2 q,-q,-q,1-2 q,\frac {12+3 \sqrt {33}-\sqrt {177}}{3 \left (5+\sqrt {33}-4 x\right )},\frac {12+3 \sqrt {33}+\sqrt {177}}{3 \left (5+\sqrt {33}-4 x\right )}\right )}{61952 \sqrt {11} q}-\frac {213}{121} 2^{-8-3 q} 59^q (1-q) (1+2 q) (1-4 x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-q,\frac {3}{2},\frac {3}{59} (1-4 x)^2\right ) \] Output:
-1/528*(23+14*x)*(-6*x^2+3*x+7)^(1+q)/(-2*x^2+5*x+1)^2-(51+61*q+142*(1-q)* x)*(-6*x^2+3*x+7)^(1+q)/(-30976*x^2+77440*x+15488)+1/681472*3^(-1/2+2*q)*( 6144-13505*q+3*33^(1/2)*(741-517*q)*q+9825*q^2)*(-6*x^2+3*x+7)^q*AppellF1( -2*q,-q,-q,1-2*q,(12-3*33^(1/2)-177^(1/2))/(15-3*33^(1/2)-12*x),(12-3*33^( 1/2)+177^(1/2))/(15-3*33^(1/2)-12*x))*11^(1/2)/q/(((3-177^(1/2)-12*x)/(5-3 3^(1/2)-4*x))^q)/(((3+177^(1/2)-12*x)/(5-33^(1/2)-4*x))^q)-1/681472*3^(-1/ 2+2*q)*(6144-13505*q-3*33^(1/2)*(741-517*q)*q+9825*q^2)*(-6*x^2+3*x+7)^q*A ppellF1(-2*q,-q,-q,1-2*q,(12+3*33^(1/2)-177^(1/2))/(15+3*33^(1/2)-12*x),(1 2+3*33^(1/2)+177^(1/2))/(15+3*33^(1/2)-12*x))*11^(1/2)/q/(((3-177^(1/2)-12 *x)/(5+33^(1/2)-4*x))^q)/(((3+177^(1/2)-12*x)/(5+33^(1/2)-4*x))^q)-213/121 *2^(-8-3*q)*59^q*(1-q)*(1+2*q)*(1-4*x)*hypergeom([1/2, -q],[3/2],3/59*(1-4 *x)^2)
\[ \int \frac {\left (7+3 x-6 x^2\right )^q \left (3+2 x+4 x^2\right )}{\left (1+5 x-2 x^2\right )^3} \, dx=\int \frac {\left (7+3 x-6 x^2\right )^q \left (3+2 x+4 x^2\right )}{\left (1+5 x-2 x^2\right )^3} \, dx \] Input:
Integrate[((7 + 3*x - 6*x^2)^q*(3 + 2*x + 4*x^2))/(1 + 5*x - 2*x^2)^3,x]
Output:
Integrate[((7 + 3*x - 6*x^2)^q*(3 + 2*x + 4*x^2))/(1 + 5*x - 2*x^2)^3, x]
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.
| \(\displaystyle \int \frac {\left (4 x^2+2 x+3\right ) \left (-6 x^2+3 x+7\right )^q}{\left (-2 x^2+5 x+1\right )^3} \, dx\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {\int \frac {264 \left (-6 x^2+3 x+7\right )^q \left (28 (1-2 q) x^2+6 (34-13 q) x+23 q+47\right )}{\left (-2 x^2+5 x+1\right )^2}dx}{46464}-\frac {(14 x+23) \left (-6 x^2+3 x+7\right )^{q+1}}{528 \left (-2 x^2+5 x+1\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{176} \int \frac {\left (-6 x^2+3 x+7\right )^q \left (28 (1-2 q) x^2+6 (34-13 q) x+23 q+47\right )}{\left (-2 x^2+5 x+1\right )^2}dx-\frac {(14 x+23) \left (-6 x^2+3 x+7\right )^{q+1}}{528 \left (-2 x^2+5 x+1\right )^2}\) |
| \(\Big \downarrow \) 2135 |
| \(\displaystyle \frac {1}{176} \left (\frac {\int \frac {264 \left (-6 x^2+3 x+7\right )^q \left (183 q^2-769 q-852 (1-q) (2 q+1) x^2+6 \left (-193 q^2-386 q+355\right ) x+3498\right )}{-2 x^2+5 x+1}dx}{23232}-\frac {(142 (1-q) x+61 q+51) \left (-6 x^2+3 x+7\right )^{q+1}}{88 \left (-2 x^2+5 x+1\right )}\right )-\frac {(14 x+23) \left (-6 x^2+3 x+7\right )^{q+1}}{528 \left (-2 x^2+5 x+1\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{176} \left (\frac {1}{88} \int \frac {\left (-6 x^2+3 x+7\right )^q \left (183 q^2-769 q-852 (1-q) (2 q+1) x^2+6 \left (-193 q^2-386 q+355\right ) x+3498\right )}{-2 x^2+5 x+1}dx-\frac {(142 (1-q) x+61 q+51) \left (-6 x^2+3 x+7\right )^{q+1}}{88 \left (-2 x^2+5 x+1\right )}\right )-\frac {(14 x+23) \left (-6 x^2+3 x+7\right )^{q+1}}{528 \left (-2 x^2+5 x+1\right )^2}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \frac {1}{176} \left (\frac {1}{88} \int \left (426 (1-q) (2 q+1) \left (-6 x^2+3 x+7\right )^q+\frac {\left (1035 q^2-6 (741-517 q) x q-1195 q+3072\right ) \left (-6 x^2+3 x+7\right )^q}{-2 x^2+5 x+1}\right )dx-\frac {(142 (1-q) x+61 q+51) \left (-6 x^2+3 x+7\right )^{q+1}}{88 \left (-2 x^2+5 x+1\right )}\right )-\frac {(14 x+23) \left (-6 x^2+3 x+7\right )^{q+1}}{528 \left (-2 x^2+5 x+1\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{176} \left (\frac {1}{88} \left (\int \frac {\left (1035 q^2-6 (741-517 q) x q-1195 q+3072\right ) \left (-6 x^2+3 x+7\right )^q}{-2 x^2+5 x+1}dx-213\ 2^{-3 q-1} 59^q (1-q) (2 q+1) (1-4 x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-q,\frac {3}{2},\frac {3}{59} (1-4 x)^2\right )\right )-\frac {(142 (1-q) x+61 q+51) \left (-6 x^2+3 x+7\right )^{q+1}}{88 \left (-2 x^2+5 x+1\right )}\right )-\frac {(14 x+23) \left (-6 x^2+3 x+7\right )^{q+1}}{528 \left (-2 x^2+5 x+1\right )^2}\) |
Input:
Int[((7 + 3*x - 6*x^2)^q*(3 + 2*x + 4*x^2))/(1 + 5*x - 2*x^2)^3,x]
Output:
$Aborted
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Px_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_. )*(x_)^2)^(q_), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1] , C = Coeff[Px, x, 2]}, Simp[(a + b*x + c*x^2)^(p + 1)*((d + e*x + f*x^2)^( q + 1)/((b^2 - 4*a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)))*( (A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b *e + 2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)) - B*(b*c*d - 2*a*c* e + a*b*f) + C*(b^2*d - a*b*e - 2*a*(c*d - a*f)))*x), x] + Simp[1/((b^2 - 4 *a*c)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1)) Int[(a + b*x + c *x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(b*B - 2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f))*(p + 1) + (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e - b*(c*d + a*f)) + (A*b - a*B )*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a *c*e - b*(c*d + a*f)) + (A*b - a*B)*(2*c^2*d + b^2*f - c*(b*e + 2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*( c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(b*f*(p + 1) - c*e*(2*p + q + 4))) *x - c*f*(b^2*(C*d + A*f) - b*(B*c*d + A*c*e + a*C*e + a*B*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x]] /; F reeQ[{a, b, c, d, e, f, q}, x] && PolyQ[Px, x, 2] && LtQ[p, -1] && NeQ[(c*d - a*f)^2 - (b*d - a*e)*(c*e - b*f), 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q, 0]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
\[\int \frac {\left (-6 x^{2}+3 x +7\right )^{q} \left (4 x^{2}+2 x +3\right )}{\left (-2 x^{2}+5 x +1\right )^{3}}d x\]
Input:
int((-6*x^2+3*x+7)^q*(4*x^2+2*x+3)/(-2*x^2+5*x+1)^3,x)
Output:
int((-6*x^2+3*x+7)^q*(4*x^2+2*x+3)/(-2*x^2+5*x+1)^3,x)
\[ \int \frac {\left (7+3 x-6 x^2\right )^q \left (3+2 x+4 x^2\right )}{\left (1+5 x-2 x^2\right )^3} \, dx=\int { -\frac {{\left (4 \, x^{2} + 2 \, x + 3\right )} {\left (-6 \, x^{2} + 3 \, x + 7\right )}^{q}}{{\left (2 \, x^{2} - 5 \, x - 1\right )}^{3}} \,d x } \] Input:
integrate((-6*x^2+3*x+7)^q*(4*x^2+2*x+3)/(-2*x^2+5*x+1)^3,x, algorithm="fr icas")
Output:
integral(-(4*x^2 + 2*x + 3)*(-6*x^2 + 3*x + 7)^q/(8*x^6 - 60*x^5 + 138*x^4 - 65*x^3 - 69*x^2 - 15*x - 1), x)
Timed out. \[ \int \frac {\left (7+3 x-6 x^2\right )^q \left (3+2 x+4 x^2\right )}{\left (1+5 x-2 x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((-6*x**2+3*x+7)**q*(4*x**2+2*x+3)/(-2*x**2+5*x+1)**3,x)
Output:
Timed out
\[ \int \frac {\left (7+3 x-6 x^2\right )^q \left (3+2 x+4 x^2\right )}{\left (1+5 x-2 x^2\right )^3} \, dx=\int { -\frac {{\left (4 \, x^{2} + 2 \, x + 3\right )} {\left (-6 \, x^{2} + 3 \, x + 7\right )}^{q}}{{\left (2 \, x^{2} - 5 \, x - 1\right )}^{3}} \,d x } \] Input:
integrate((-6*x^2+3*x+7)^q*(4*x^2+2*x+3)/(-2*x^2+5*x+1)^3,x, algorithm="ma xima")
Output:
-integrate((4*x^2 + 2*x + 3)*(-6*x^2 + 3*x + 7)^q/(2*x^2 - 5*x - 1)^3, x)
\[ \int \frac {\left (7+3 x-6 x^2\right )^q \left (3+2 x+4 x^2\right )}{\left (1+5 x-2 x^2\right )^3} \, dx=\int { -\frac {{\left (4 \, x^{2} + 2 \, x + 3\right )} {\left (-6 \, x^{2} + 3 \, x + 7\right )}^{q}}{{\left (2 \, x^{2} - 5 \, x - 1\right )}^{3}} \,d x } \] Input:
integrate((-6*x^2+3*x+7)^q*(4*x^2+2*x+3)/(-2*x^2+5*x+1)^3,x, algorithm="gi ac")
Output:
integrate(-(4*x^2 + 2*x + 3)*(-6*x^2 + 3*x + 7)^q/(2*x^2 - 5*x - 1)^3, x)
Timed out. \[ \int \frac {\left (7+3 x-6 x^2\right )^q \left (3+2 x+4 x^2\right )}{\left (1+5 x-2 x^2\right )^3} \, dx=\int \frac {\left (4\,x^2+2\,x+3\right )\,{\left (-6\,x^2+3\,x+7\right )}^q}{{\left (-2\,x^2+5\,x+1\right )}^3} \,d x \] Input:
int(((2*x + 4*x^2 + 3)*(3*x - 6*x^2 + 7)^q)/(5*x - 2*x^2 + 1)^3,x)
Output:
int(((2*x + 4*x^2 + 3)*(3*x - 6*x^2 + 7)^q)/(5*x - 2*x^2 + 1)^3, x)
\[ \int \frac {\left (7+3 x-6 x^2\right )^q \left (3+2 x+4 x^2\right )}{\left (1+5 x-2 x^2\right )^3} \, dx=\text {too large to display} \] Input:
int((-6*x^2+3*x+7)^q*(4*x^2+2*x+3)/(-2*x^2+5*x+1)^3,x)
Output:
( - 66*( - 6*x**2 + 3*x + 7)**q*q*x - 19*( - 6*x**2 + 3*x + 7)**q*q + 84*( - 6*x**2 + 3*x + 7)**q*x + 3*( - 6*x**2 + 3*x + 7)**q + 126984*int(( - 6* x**2 + 3*x + 7)**q/(1056*q**2*x**8 - 8448*q**2*x**7 + 20944*q**2*x**6 - 84 48*q**2*x**5 - 26070*q**2*x**4 + 12584*q**2*x**3 + 11484*q**2*x**2 + 2376* q**2*x + 154*q**2 - 2928*q*x**8 + 23424*q*x**7 - 58072*q*x**6 + 23424*q*x* *5 + 72285*q*x**4 - 34892*q*x**3 - 31842*q*x**2 - 6588*q*x - 427*q + 2016* x**8 - 16128*x**7 + 39984*x**6 - 16128*x**5 - 49770*x**4 + 24024*x**3 + 21 924*x**2 + 4536*x + 294),x)*q**4*x**4 - 634920*int(( - 6*x**2 + 3*x + 7)** q/(1056*q**2*x**8 - 8448*q**2*x**7 + 20944*q**2*x**6 - 8448*q**2*x**5 - 26 070*q**2*x**4 + 12584*q**2*x**3 + 11484*q**2*x**2 + 2376*q**2*x + 154*q**2 - 2928*q*x**8 + 23424*q*x**7 - 58072*q*x**6 + 23424*q*x**5 + 72285*q*x**4 - 34892*q*x**3 - 31842*q*x**2 - 6588*q*x - 427*q + 2016*x**8 - 16128*x**7 + 39984*x**6 - 16128*x**5 - 49770*x**4 + 24024*x**3 + 21924*x**2 + 4536*x + 294),x)*q**4*x**3 + 666666*int(( - 6*x**2 + 3*x + 7)**q/(1056*q**2*x**8 - 8448*q**2*x**7 + 20944*q**2*x**6 - 8448*q**2*x**5 - 26070*q**2*x**4 + 1 2584*q**2*x**3 + 11484*q**2*x**2 + 2376*q**2*x + 154*q**2 - 2928*q*x**8 + 23424*q*x**7 - 58072*q*x**6 + 23424*q*x**5 + 72285*q*x**4 - 34892*q*x**3 - 31842*q*x**2 - 6588*q*x - 427*q + 2016*x**8 - 16128*x**7 + 39984*x**6 - 1 6128*x**5 - 49770*x**4 + 24024*x**3 + 21924*x**2 + 4536*x + 294),x)*q**4*x **2 + 317460*int(( - 6*x**2 + 3*x + 7)**q/(1056*q**2*x**8 - 8448*q**2*x...