Integrand size = 26, antiderivative size = 238 \[ \int (5-4 x)^5 (1+2 x)^{-5-m} (2+3 x)^m \, dx=-\frac {16807 (1+2 x)^{-4-m} (2+3 x)^{1+m}}{4+m}+\frac {2401 (103+10 m) (1+2 x)^{-3-m} (2+3 x)^{1+m}}{(3+m) (4+m)}-\frac {686 \left (2403+350 m+20 m^2\right ) (1+2 x)^{-2-m} (2+3 x)^{1+m}}{(2+m) (3+m) (4+m)}+\frac {98 \left (51423+8390 m+780 m^2+40 m^3\right ) (1+2 x)^{-1-m} (2+3 x)^{1+m}}{(1+m) (2+m) (3+m) (4+m)}-\frac {32}{3} (1+2 x)^{-m} (2+3 x)^{1+m}-\frac {2^{3-m} (105-2 m) (1+2 x)^{-m} \operatorname {Hypergeometric2F1}(-m,-m,1-m,-3 (1+2 x))}{3 m} \] Output:
-16807*(1+2*x)^(-4-m)*(2+3*x)^(1+m)/(4+m)+2401*(103+10*m)*(1+2*x)^(-3-m)*( 2+3*x)^(1+m)/(3+m)/(4+m)-686*(20*m^2+350*m+2403)*(1+2*x)^(-2-m)*(2+3*x)^(1 +m)/(2+m)/(3+m)/(4+m)+98*(40*m^3+780*m^2+8390*m+51423)*(1+2*x)^(-1-m)*(2+3 *x)^(1+m)/(1+m)/(2+m)/(3+m)/(4+m)-32/3*(2+3*x)^(1+m)/((1+2*x)^m)-1/3*2^(3- m)*(105-2*m)*hypergeom([-m, -m],[1-m],-3-6*x)/m/((1+2*x)^m)
Time = 11.82 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.15 \[ \int (5-4 x)^5 (1+2 x)^{-5-m} (2+3 x)^m \, dx=\frac {3920 (1+2 x)^{-1-m} (2+3 x)^{1+m}}{1+m}+\frac {2^{4-m} (1+2 x)^{1-m} \operatorname {Hypergeometric2F1}(1-m,-m,2-m,-3-6 x)}{-1+m}-\frac {560 (-3-6 x)^m (1+2 x)^{-m} (2+3 x)^{1+m} \operatorname {Hypergeometric2F1}(1+m,1+m,2+m,4+6 x)}{1+m}-\frac {13720\ 3^{2+m} (-1-2 x)^m (1+2 x)^{-m} (2+3 x)^{1+m} \operatorname {Hypergeometric2F1}(1+m,3+m,2+m,4+6 x)}{1+m}-\frac {12005\ 3^{3+m} (-1-2 x)^m (2+4 x)^{-m} (4+6 x)^{1+m} \operatorname {Hypergeometric2F1}(1+m,4+m,2+m,4+6 x)}{1+m}-\frac {16807\ 3^{4+m} (-1-2 x)^m (1+2 x)^{-m} (2+3 x)^{1+m} \operatorname {Hypergeometric2F1}(1+m,5+m,2+m,4+6 x)}{1+m} \] Input:
Integrate[(5 - 4*x)^5*(1 + 2*x)^(-5 - m)*(2 + 3*x)^m,x]
Output:
(3920*(1 + 2*x)^(-1 - m)*(2 + 3*x)^(1 + m))/(1 + m) + (2^(4 - m)*(1 + 2*x) ^(1 - m)*Hypergeometric2F1[1 - m, -m, 2 - m, -3 - 6*x])/(-1 + m) - (560*(- 3 - 6*x)^m*(2 + 3*x)^(1 + m)*Hypergeometric2F1[1 + m, 1 + m, 2 + m, 4 + 6* x])/((1 + m)*(1 + 2*x)^m) - (13720*3^(2 + m)*(-1 - 2*x)^m*(2 + 3*x)^(1 + m )*Hypergeometric2F1[1 + m, 3 + m, 2 + m, 4 + 6*x])/((1 + m)*(1 + 2*x)^m) - (12005*3^(3 + m)*(-1 - 2*x)^m*(4 + 6*x)^(1 + m)*Hypergeometric2F1[1 + m, 4 + m, 2 + m, 4 + 6*x])/((1 + m)*(2 + 4*x)^m) - (16807*3^(4 + m)*(-1 - 2*x )^m*(2 + 3*x)^(1 + m)*Hypergeometric2F1[1 + m, 5 + m, 2 + m, 4 + 6*x])/((1 + m)*(1 + 2*x)^m)
Leaf count is larger than twice the leaf count of optimal. \(482\) vs. \(2(238)=476\).
Time = 0.61 (sec) , antiderivative size = 482, normalized size of antiderivative = 2.03, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {111, 27, 177, 105, 101, 27, 88, 55, 48, 137, 2009}
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.
| \(\displaystyle \int (5-4 x)^5 (2 x+1)^{-m-5} (3 x+2)^m \, dx\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {1}{6} \int -2 (5-4 x)^3 (2 x+1)^{-m-5} (3 x+2)^m (10 m+4 (105-2 m) x+119)dx-\frac {2}{3} (5-4 x)^4 (2 x+1)^{-m-4} (3 x+2)^{m+1}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{3} \int (5-4 x)^3 (2 x+1)^{-m-5} (3 x+2)^m (10 m+4 (105-2 m) x+119)dx-\frac {2}{3} (5-4 x)^4 (3 x+2)^{m+1} (2 x+1)^{-m-4}\) |
| \(\Big \downarrow \) 177 |
| \(\displaystyle \frac {1}{3} \left (7 (13-2 m) \int (5-4 x)^3 (2 x+1)^{-m-5} (3 x+2)^mdx-2 (105-2 m) \int (5-4 x)^3 (2 x+1)^{-m-4} (3 x+2)^mdx\right )-\frac {2}{3} (5-4 x)^4 (2 x+1)^{-m-4} (3 x+2)^{m+1}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {1}{3} \left (7 (13-2 m) \left (-\frac {69 \int (5-4 x)^2 (2 x+1)^{-m-4} (3 x+2)^mdx}{m+4}-\frac {(5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-4}}{m+4}\right )-2 (105-2 m) \int (5-4 x)^3 (2 x+1)^{-m-4} (3 x+2)^mdx\right )-\frac {2}{3} (5-4 x)^4 (2 x+1)^{-m-4} (3 x+2)^{m+1}\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle \frac {1}{3} \left (7 (13-2 m) \left (-\frac {69 \left (\frac {2}{3} (5-4 x) (2 x+1)^{-m-3} (3 x+2)^{m+1}-\frac {1}{6} \int -2 (2 x+1)^{-m-4} (3 x+2)^m (10 m-8 (m+2) x+181)dx\right )}{m+4}-\frac {(5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-4}}{m+4}\right )-2 (105-2 m) \int (5-4 x)^3 (2 x+1)^{-m-4} (3 x+2)^mdx\right )-\frac {2}{3} (5-4 x)^4 (2 x+1)^{-m-4} (3 x+2)^{m+1}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (7 (13-2 m) \left (-\frac {69 \left (\frac {1}{3} \int (2 x+1)^{-m-4} (3 x+2)^m (10 m-8 (m+2) x+181)dx+\frac {2}{3} (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m-3}\right )}{m+4}-\frac {(5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-4}}{m+4}\right )-2 (105-2 m) \int (5-4 x)^3 (2 x+1)^{-m-4} (3 x+2)^mdx\right )-\frac {2}{3} (5-4 x)^4 (2 x+1)^{-m-4} (3 x+2)^{m+1}\) |
| \(\Big \downarrow \) 88 |
| \(\displaystyle \frac {1}{3} \left (7 (13-2 m) \left (-\frac {69 \left (\frac {1}{3} \left (-\frac {2 \left (2 m^2+52 m+579\right ) \int (2 x+1)^{-m-3} (3 x+2)^mdx}{m+3}-\frac {7 (2 m+27) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{m+3}\right )+\frac {2}{3} (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m-3}\right )}{m+4}-\frac {(5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-4}}{m+4}\right )-2 (105-2 m) \int (5-4 x)^3 (2 x+1)^{-m-4} (3 x+2)^mdx\right )-\frac {2}{3} (5-4 x)^4 (2 x+1)^{-m-4} (3 x+2)^{m+1}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {1}{3} \left (7 (13-2 m) \left (-\frac {69 \left (\frac {1}{3} \left (-\frac {2 \left (2 m^2+52 m+579\right ) \left (-\frac {3 \int (2 x+1)^{-m-2} (3 x+2)^mdx}{m+2}-\frac {(3 x+2)^{m+1} (2 x+1)^{-m-2}}{m+2}\right )}{m+3}-\frac {7 (2 m+27) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{m+3}\right )+\frac {2}{3} (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m-3}\right )}{m+4}-\frac {(5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-4}}{m+4}\right )-2 (105-2 m) \int (5-4 x)^3 (2 x+1)^{-m-4} (3 x+2)^mdx\right )-\frac {2}{3} (5-4 x)^4 (2 x+1)^{-m-4} (3 x+2)^{m+1}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {1}{3} \left (7 (13-2 m) \left (-\frac {69 \left (\frac {1}{3} \left (-\frac {2 \left (2 m^2+52 m+579\right ) \left (\frac {3 (2 x+1)^{-m-1} (3 x+2)^{m+1}}{(m+1) (m+2)}-\frac {(2 x+1)^{-m-2} (3 x+2)^{m+1}}{m+2}\right )}{m+3}-\frac {7 (2 m+27) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{m+3}\right )+\frac {2}{3} (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m-3}\right )}{m+4}-\frac {(5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-4}}{m+4}\right )-2 (105-2 m) \int (5-4 x)^3 (2 x+1)^{-m-4} (3 x+2)^mdx\right )-\frac {2}{3} (5-4 x)^4 (2 x+1)^{-m-4} (3 x+2)^{m+1}\) |
| \(\Big \downarrow \) 137 |
| \(\displaystyle \frac {1}{3} \left (7 (13-2 m) \left (-\frac {69 \left (\frac {1}{3} \left (-\frac {2 \left (2 m^2+52 m+579\right ) \left (\frac {3 (2 x+1)^{-m-1} (3 x+2)^{m+1}}{(m+1) (m+2)}-\frac {(2 x+1)^{-m-2} (3 x+2)^{m+1}}{m+2}\right )}{m+3}-\frac {7 (2 m+27) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{m+3}\right )+\frac {2}{3} (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m-3}\right )}{m+4}-\frac {(5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-4}}{m+4}\right )-2 (105-2 m) \int \left (\frac {12167}{27} (3 x+2)^m (2 x+1)^{-m-4}-\frac {2116}{9} (3 x+2)^{m+1} (2 x+1)^{-m-4}+\frac {368}{9} (3 x+2)^{m+2} (2 x+1)^{-m-4}-\frac {64}{27} (3 x+2)^{m+3} (2 x+1)^{-m-4}\right )dx\right )-\frac {2}{3} (5-4 x)^4 (2 x+1)^{-m-4} (3 x+2)^{m+1}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \left (7 (13-2 m) \left (-\frac {69 \left (\frac {1}{3} \left (-\frac {2 \left (2 m^2+52 m+579\right ) \left (\frac {3 (2 x+1)^{-m-1} (3 x+2)^{m+1}}{(m+1) (m+2)}-\frac {(2 x+1)^{-m-2} (3 x+2)^{m+1}}{m+2}\right )}{m+3}-\frac {7 (2 m+27) (3 x+2)^{m+1} (2 x+1)^{-m-3}}{m+3}\right )+\frac {2}{3} (5-4 x) (3 x+2)^{m+1} (2 x+1)^{-m-3}\right )}{m+4}-\frac {(5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{-m-4}}{m+4}\right )-2 (105-2 m) \left (\frac {2^{2-m} (2 x+1)^{-m-3} \operatorname {Hypergeometric2F1}(-m-3,-m-3,-m-2,-3 (2 x+1))}{27 (m+3)}+\frac {24334 (3 x+2)^{m+1} (2 x+1)^{-m-2}}{9 \left (m^2+5 m+6\right )}-\frac {2116 (3 x+2)^{m+2} (2 x+1)^{-m-2}}{3 \left (m^2+5 m+6\right )}-\frac {24334 (3 x+2)^{m+1} (2 x+1)^{-m-1}}{3 \left (m^3+6 m^2+11 m+6\right )}-\frac {12167 (3 x+2)^{m+1} (2 x+1)^{-m-3}}{27 (m+3)}+\frac {2116 (3 x+2)^{m+2} (2 x+1)^{-m-3}}{9 (m+3)}-\frac {368 (3 x+2)^{m+3} (2 x+1)^{-m-3}}{9 (m+3)}\right )\right )-\frac {2}{3} (5-4 x)^4 (2 x+1)^{-m-4} (3 x+2)^{m+1}\) |
Input:
Int[(5 - 4*x)^5*(1 + 2*x)^(-5 - m)*(2 + 3*x)^m,x]
Output:
(-2*(5 - 4*x)^4*(1 + 2*x)^(-4 - m)*(2 + 3*x)^(1 + m))/3 + (7*(13 - 2*m)*(- (((5 - 4*x)^3*(1 + 2*x)^(-4 - m)*(2 + 3*x)^(1 + m))/(4 + m)) - (69*((2*(5 - 4*x)*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/3 + ((-7*(27 + 2*m)*(1 + 2*x) ^(-3 - m)*(2 + 3*x)^(1 + m))/(3 + m) - (2*(579 + 52*m + 2*m^2)*(-(((1 + 2* x)^(-2 - m)*(2 + 3*x)^(1 + m))/(2 + m)) + (3*(1 + 2*x)^(-1 - m)*(2 + 3*x)^ (1 + m))/((1 + m)*(2 + m))))/(3 + m))/3))/(4 + m)) - 2*(105 - 2*m)*((-1216 7*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(1 + m))/(27*(3 + m)) + (24334*(1 + 2*x)^(- 2 - m)*(2 + 3*x)^(1 + m))/(9*(6 + 5*m + m^2)) - (24334*(1 + 2*x)^(-1 - m)* (2 + 3*x)^(1 + m))/(3*(6 + 11*m + 6*m^2 + m^3)) + (2116*(1 + 2*x)^(-3 - m) *(2 + 3*x)^(2 + m))/(9*(3 + m)) - (2116*(1 + 2*x)^(-2 - m)*(2 + 3*x)^(2 + m))/(3*(6 + 5*m + m^2)) - (368*(1 + 2*x)^(-3 - m)*(2 + 3*x)^(3 + m))/(9*(3 + m)) + (2^(2 - m)*(1 + 2*x)^(-3 - m)*Hypergeometric2F1[-3 - m, -3 - m, - 2 - m, -3*(1 + 2*x)])/(27*(3 + m))))/3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && !RationalQ[p] && SumSimpl erQ[p, 1]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (IGtQ[m, 0] || (ILtQ[m, 0] && ILtQ[n, 0]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h/b Int[(a + b*x)^(m + 1)*(c + d *x)^n*(e + f*x)^p, x], x] + Simp[(b*g - a*h)/b Int[(a + b*x)^m*(c + d*x)^ n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && (Su mSimplerQ[m, 1] || ( !SumSimplerQ[n, 1] && !SumSimplerQ[p, 1]))
\[\int \left (5-4 x \right )^{5} \left (1+2 x \right )^{-5-m} \left (2+3 x \right )^{m}d x\]
Input:
int((5-4*x)^5*(1+2*x)^(-5-m)*(2+3*x)^m,x)
Output:
int((5-4*x)^5*(1+2*x)^(-5-m)*(2+3*x)^m,x)
\[ \int (5-4 x)^5 (1+2 x)^{-5-m} (2+3 x)^m \, dx=\int { -{\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 5} {\left (4 \, x - 5\right )}^{5} \,d x } \] Input:
integrate((5-4*x)^5*(1+2*x)^(-5-m)*(2+3*x)^m,x, algorithm="fricas")
Output:
integral(-(1024*x^5 - 6400*x^4 + 16000*x^3 - 20000*x^2 + 12500*x - 3125)*( 3*x + 2)^m*(2*x + 1)^(-m - 5), x)
\[ \int (5-4 x)^5 (1+2 x)^{-5-m} (2+3 x)^m \, dx=- \int \left (- 3125 \left (2 x + 1\right )^{- m - 5} \left (3 x + 2\right )^{m}\right )\, dx - \int 12500 x \left (2 x + 1\right )^{- m - 5} \left (3 x + 2\right )^{m}\, dx - \int \left (- 20000 x^{2} \left (2 x + 1\right )^{- m - 5} \left (3 x + 2\right )^{m}\right )\, dx - \int 16000 x^{3} \left (2 x + 1\right )^{- m - 5} \left (3 x + 2\right )^{m}\, dx - \int \left (- 6400 x^{4} \left (2 x + 1\right )^{- m - 5} \left (3 x + 2\right )^{m}\right )\, dx - \int 1024 x^{5} \left (2 x + 1\right )^{- m - 5} \left (3 x + 2\right )^{m}\, dx \] Input:
integrate((5-4*x)**5*(1+2*x)**(-5-m)*(2+3*x)**m,x)
Output:
-Integral(-3125*(2*x + 1)**(-m - 5)*(3*x + 2)**m, x) - Integral(12500*x*(2 *x + 1)**(-m - 5)*(3*x + 2)**m, x) - Integral(-20000*x**2*(2*x + 1)**(-m - 5)*(3*x + 2)**m, x) - Integral(16000*x**3*(2*x + 1)**(-m - 5)*(3*x + 2)** m, x) - Integral(-6400*x**4*(2*x + 1)**(-m - 5)*(3*x + 2)**m, x) - Integra l(1024*x**5*(2*x + 1)**(-m - 5)*(3*x + 2)**m, x)
\[ \int (5-4 x)^5 (1+2 x)^{-5-m} (2+3 x)^m \, dx=\int { -{\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 5} {\left (4 \, x - 5\right )}^{5} \,d x } \] Input:
integrate((5-4*x)^5*(1+2*x)^(-5-m)*(2+3*x)^m,x, algorithm="maxima")
Output:
-integrate((3*x + 2)^m*(2*x + 1)^(-m - 5)*(4*x - 5)^5, x)
\[ \int (5-4 x)^5 (1+2 x)^{-5-m} (2+3 x)^m \, dx=\int { -{\left (3 \, x + 2\right )}^{m} {\left (2 \, x + 1\right )}^{-m - 5} {\left (4 \, x - 5\right )}^{5} \,d x } \] Input:
integrate((5-4*x)^5*(1+2*x)^(-5-m)*(2+3*x)^m,x, algorithm="giac")
Output:
integrate(-(3*x + 2)^m*(2*x + 1)^(-m - 5)*(4*x - 5)^5, x)
Timed out. \[ \int (5-4 x)^5 (1+2 x)^{-5-m} (2+3 x)^m \, dx=-\int \frac {{\left (3\,x+2\right )}^m\,{\left (4\,x-5\right )}^5}{{\left (2\,x+1\right )}^{m+5}} \,d x \] Input:
int(-((3*x + 2)^m*(4*x - 5)^5)/(2*x + 1)^(m + 5),x)
Output:
-int(((3*x + 2)^m*(4*x - 5)^5)/(2*x + 1)^(m + 5), x)
\[ \int (5-4 x)^5 (1+2 x)^{-5-m} (2+3 x)^m \, dx=3125 \left (\int \frac {\left (3 x +2\right )^{m}}{32 \left (2 x +1\right )^{m} x^{5}+80 \left (2 x +1\right )^{m} x^{4}+80 \left (2 x +1\right )^{m} x^{3}+40 \left (2 x +1\right )^{m} x^{2}+10 \left (2 x +1\right )^{m} x +\left (2 x +1\right )^{m}}d x \right )-1024 \left (\int \frac {\left (3 x +2\right )^{m} x^{5}}{32 \left (2 x +1\right )^{m} x^{5}+80 \left (2 x +1\right )^{m} x^{4}+80 \left (2 x +1\right )^{m} x^{3}+40 \left (2 x +1\right )^{m} x^{2}+10 \left (2 x +1\right )^{m} x +\left (2 x +1\right )^{m}}d x \right )+6400 \left (\int \frac {\left (3 x +2\right )^{m} x^{4}}{32 \left (2 x +1\right )^{m} x^{5}+80 \left (2 x +1\right )^{m} x^{4}+80 \left (2 x +1\right )^{m} x^{3}+40 \left (2 x +1\right )^{m} x^{2}+10 \left (2 x +1\right )^{m} x +\left (2 x +1\right )^{m}}d x \right )-16000 \left (\int \frac {\left (3 x +2\right )^{m} x^{3}}{32 \left (2 x +1\right )^{m} x^{5}+80 \left (2 x +1\right )^{m} x^{4}+80 \left (2 x +1\right )^{m} x^{3}+40 \left (2 x +1\right )^{m} x^{2}+10 \left (2 x +1\right )^{m} x +\left (2 x +1\right )^{m}}d x \right )+20000 \left (\int \frac {\left (3 x +2\right )^{m} x^{2}}{32 \left (2 x +1\right )^{m} x^{5}+80 \left (2 x +1\right )^{m} x^{4}+80 \left (2 x +1\right )^{m} x^{3}+40 \left (2 x +1\right )^{m} x^{2}+10 \left (2 x +1\right )^{m} x +\left (2 x +1\right )^{m}}d x \right )-12500 \left (\int \frac {\left (3 x +2\right )^{m} x}{32 \left (2 x +1\right )^{m} x^{5}+80 \left (2 x +1\right )^{m} x^{4}+80 \left (2 x +1\right )^{m} x^{3}+40 \left (2 x +1\right )^{m} x^{2}+10 \left (2 x +1\right )^{m} x +\left (2 x +1\right )^{m}}d x \right ) \] Input:
int((5-4*x)^5*(1+2*x)^(-5-m)*(2+3*x)^m,x)
Output:
3125*int((3*x + 2)**m/(32*(2*x + 1)**m*x**5 + 80*(2*x + 1)**m*x**4 + 80*(2 *x + 1)**m*x**3 + 40*(2*x + 1)**m*x**2 + 10*(2*x + 1)**m*x + (2*x + 1)**m) ,x) - 1024*int(((3*x + 2)**m*x**5)/(32*(2*x + 1)**m*x**5 + 80*(2*x + 1)**m *x**4 + 80*(2*x + 1)**m*x**3 + 40*(2*x + 1)**m*x**2 + 10*(2*x + 1)**m*x + (2*x + 1)**m),x) + 6400*int(((3*x + 2)**m*x**4)/(32*(2*x + 1)**m*x**5 + 80 *(2*x + 1)**m*x**4 + 80*(2*x + 1)**m*x**3 + 40*(2*x + 1)**m*x**2 + 10*(2*x + 1)**m*x + (2*x + 1)**m),x) - 16000*int(((3*x + 2)**m*x**3)/(32*(2*x + 1 )**m*x**5 + 80*(2*x + 1)**m*x**4 + 80*(2*x + 1)**m*x**3 + 40*(2*x + 1)**m* x**2 + 10*(2*x + 1)**m*x + (2*x + 1)**m),x) + 20000*int(((3*x + 2)**m*x**2 )/(32*(2*x + 1)**m*x**5 + 80*(2*x + 1)**m*x**4 + 80*(2*x + 1)**m*x**3 + 40 *(2*x + 1)**m*x**2 + 10*(2*x + 1)**m*x + (2*x + 1)**m),x) - 12500*int(((3* x + 2)**m*x)/(32*(2*x + 1)**m*x**5 + 80*(2*x + 1)**m*x**4 + 80*(2*x + 1)** m*x**3 + 40*(2*x + 1)**m*x**2 + 10*(2*x + 1)**m*x + (2*x + 1)**m),x)