\(\int \frac {(2+3 x)^m}{(1-2 x) (3+5 x)^3} \, dx\) [3198]
Optimal result
Integrand size = 22, antiderivative size = 124 \[
\int \frac {(2+3 x)^m}{(1-2 x) (3+5 x)^3} \, dx=-\frac {5 (2+3 x)^{1+m}}{22 (3+5 x)^2}+\frac {5 (29-33 m) (2+3 x)^{1+m}}{242 (3+5 x)}+\frac {8 (2+3 x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {2}{7} (2+3 x)\right )}{9317 (1+m)}-\frac {5 \left (8-957 m+1089 m^2\right ) (2+3 x)^{1+m} \operatorname {Hypergeometric2F1}(1,1+m,2+m,5 (2+3 x))}{2662 (1+m)}
\]
[Out]
-5/22*(2+3*x)^(1+m)/(3+5*x)^2+5/242*(29-33*m)*(2+3*x)^(1+m)/(3+5*x)+8/9317*(2+3*x)^(1+m)*hypergeom([1, 1+m],[2
+m],4/7+6/7*x)/(1+m)-5/2662*(1089*m^2-957*m+8)*(2+3*x)^(1+m)*hypergeom([1, 1+m],[2+m],10+15*x)/(1+m)
Rubi [A] (verified)
Time = 0.07 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of
steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {105, 156, 162, 70}
\[
\int \frac {(2+3 x)^m}{(1-2 x) (3+5 x)^3} \, dx=-\frac {5 \left (1089 m^2-957 m+8\right ) (3 x+2)^{m+1} \operatorname {Hypergeometric2F1}(1,m+1,m+2,5 (3 x+2))}{2662 (m+1)}+\frac {8 (3 x+2)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {2}{7} (3 x+2)\right )}{9317 (m+1)}+\frac {5 (29-33 m) (3 x+2)^{m+1}}{242 (5 x+3)}-\frac {5 (3 x+2)^{m+1}}{22 (5 x+3)^2}
\]
[In]
Int[(2 + 3*x)^m/((1 - 2*x)*(3 + 5*x)^3),x]
[Out]
(-5*(2 + 3*x)^(1 + m))/(22*(3 + 5*x)^2) + (5*(29 - 33*m)*(2 + 3*x)^(1 + m))/(242*(3 + 5*x)) + (8*(2 + 3*x)^(1
+ m)*Hypergeometric2F1[1, 1 + m, 2 + m, (2*(2 + 3*x))/7])/(9317*(1 + m)) - (5*(8 - 957*m + 1089*m^2)*(2 + 3*x)
^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, 5*(2 + 3*x)])/(2662*(1 + m))
Rule 70
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/(b^(
n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] && !IntegerQ[m] && IntegerQ[n]
Rule 105
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Rule 156
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f
))), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
Rule 162
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Rubi steps \begin{align*}
\text {integral}& = -\frac {5 (2+3 x)^{1+m}}{22 (3+5 x)^2}-\frac {1}{22} \int \frac {(2+3 x)^m (11-15 m-30 (1-m) x)}{(1-2 x) (3+5 x)^2} \, dx \\ & = -\frac {5 (2+3 x)^{1+m}}{22 (3+5 x)^2}+\frac {5 (29-33 m) (2+3 x)^{1+m}}{242 (3+5 x)}+\frac {1}{242} \int \frac {(2+3 x)^m \left (8-435 m+495 m^2+30 (29-33 m) m x\right )}{(1-2 x) (3+5 x)} \, dx \\ & = -\frac {5 (2+3 x)^{1+m}}{22 (3+5 x)^2}+\frac {5 (29-33 m) (2+3 x)^{1+m}}{242 (3+5 x)}+\frac {8 \int \frac {(2+3 x)^m}{1-2 x} \, dx}{1331}+\frac {\left (5 \left (8-957 m+1089 m^2\right )\right ) \int \frac {(2+3 x)^m}{3+5 x} \, dx}{2662} \\ & = -\frac {5 (2+3 x)^{1+m}}{22 (3+5 x)^2}+\frac {5 (29-33 m) (2+3 x)^{1+m}}{242 (3+5 x)}+\frac {8 (2+3 x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {2}{7} (2+3 x)\right )}{9317 (1+m)}-\frac {5 \left (8-957 m+1089 m^2\right ) (2+3 x)^{1+m} \, _2F_1(1,1+m;2+m;5 (2+3 x))}{2662 (1+m)} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.13 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.84
\[
\int \frac {(2+3 x)^m}{(1-2 x) (3+5 x)^3} \, dx=\frac {(2+3 x)^{1+m} \left (-385 (1+m) (-76-145 x+33 m (3+5 x))+16 (3+5 x)^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {2}{7} (2+3 x)\right )-35 \left (8-957 m+1089 m^2\right ) (3+5 x)^2 \operatorname {Hypergeometric2F1}(1,1+m,2+m,5 (2+3 x))\right )}{18634 (1+m) (3+5 x)^2}
\]
[In]
Integrate[(2 + 3*x)^m/((1 - 2*x)*(3 + 5*x)^3),x]
[Out]
((2 + 3*x)^(1 + m)*(-385*(1 + m)*(-76 - 145*x + 33*m*(3 + 5*x)) + 16*(3 + 5*x)^2*Hypergeometric2F1[1, 1 + m, 2
+ m, (2*(2 + 3*x))/7] - 35*(8 - 957*m + 1089*m^2)*(3 + 5*x)^2*Hypergeometric2F1[1, 1 + m, 2 + m, 5*(2 + 3*x)]
))/(18634*(1 + m)*(3 + 5*x)^2)
Maple [F]
\[\int \frac {\left (2+3 x \right )^{m}}{\left (1-2 x \right ) \left (3+5 x \right )^{3}}d x\]
[In]
int((2+3*x)^m/(1-2*x)/(3+5*x)^3,x)
[Out]
int((2+3*x)^m/(1-2*x)/(3+5*x)^3,x)
Fricas [F]
\[
\int \frac {(2+3 x)^m}{(1-2 x) (3+5 x)^3} \, dx=\int { -\frac {{\left (3 \, x + 2\right )}^{m}}{{\left (5 \, x + 3\right )}^{3} {\left (2 \, x - 1\right )}} \,d x }
\]
[In]
integrate((2+3*x)^m/(1-2*x)/(3+5*x)^3,x, algorithm="fricas")
[Out]
integral(-(3*x + 2)^m/(250*x^4 + 325*x^3 + 45*x^2 - 81*x - 27), x)
Sympy [C] (verification not implemented)
Result contains complex when optimal does not.
Time = 2.21 (sec) , antiderivative size = 1062, normalized size of antiderivative = 8.56
\[
\int \frac {(2+3 x)^m}{(1-2 x) (3+5 x)^3} \, dx=\text {Too large to display}
\]
[In]
integrate((2+3*x)**m/(1-2*x)/(3+5*x)**3,x)
[Out]
245025*3**m*m**3*(x + 2/3)**2*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(598950*
(x + 2/3)**2*gamma(1 - m) - 79860*(x + 2/3)*gamma(1 - m) + 2662*gamma(1 - m)) - 32670*3**m*m**3*(x + 2/3)*(x +
2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(598950*(x + 2/3)**2*gamma(1 - m) - 79860*
(x + 2/3)*gamma(1 - m) + 2662*gamma(1 - m)) + 1089*3**m*m**3*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_
polar(I*pi))*gamma(-m)/(598950*(x + 2/3)**2*gamma(1 - m) - 79860*(x + 2/3)*gamma(1 - m) + 2662*gamma(1 - m)) -
215325*3**m*m**2*(x + 2/3)**2*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(598950
*(x + 2/3)**2*gamma(1 - m) - 79860*(x + 2/3)*gamma(1 - m) + 2662*gamma(1 - m)) + 245025*3**m*m**2*(x + 2/3)**2
*(x + 2/3)**m*gamma(-m)/(598950*(x + 2/3)**2*gamma(1 - m) - 79860*(x + 2/3)*gamma(1 - m) + 2662*gamma(1 - m))
+ 28710*3**m*m**2*(x + 2/3)*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(598950*(x
+ 2/3)**2*gamma(1 - m) - 79860*(x + 2/3)*gamma(1 - m) + 2662*gamma(1 - m)) - 16335*3**m*m**2*(x + 2/3)*(x + 2
/3)**m*gamma(-m)/(598950*(x + 2/3)**2*gamma(1 - m) - 79860*(x + 2/3)*gamma(1 - m) + 2662*gamma(1 - m)) - 957*3
**m*m**2*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(598950*(x + 2/3)**2*gamma(1
- m) - 79860*(x + 2/3)*gamma(1 - m) + 2662*gamma(1 - m)) + 1800*3**m*m*(x + 2/3)**2*(x + 2/3)**m*lerchphi(1/(1
5*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(598950*(x + 2/3)**2*gamma(1 - m) - 79860*(x + 2/3)*gamma(1 - m)
+ 2662*gamma(1 - m)) - 1800*3**m*m*(x + 2/3)**2*(x + 2/3)**m*lerchphi(7/(6*(x + 2/3)), 1, m*exp_polar(I*pi))*
gamma(-m)/(598950*(x + 2/3)**2*gamma(1 - m) - 79860*(x + 2/3)*gamma(1 - m) + 2662*gamma(1 - m)) - 215325*3**m*
m*(x + 2/3)**2*(x + 2/3)**m*gamma(-m)/(598950*(x + 2/3)**2*gamma(1 - m) - 79860*(x + 2/3)*gamma(1 - m) + 2662*
gamma(1 - m)) - 240*3**m*m*(x + 2/3)*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(
598950*(x + 2/3)**2*gamma(1 - m) - 79860*(x + 2/3)*gamma(1 - m) + 2662*gamma(1 - m)) + 240*3**m*m*(x + 2/3)*(x
+ 2/3)**m*lerchphi(7/(6*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(598950*(x + 2/3)**2*gamma(1 - m) - 79860
*(x + 2/3)*gamma(1 - m) + 2662*gamma(1 - m)) + 30690*3**m*m*(x + 2/3)*(x + 2/3)**m*gamma(-m)/(598950*(x + 2/3)
**2*gamma(1 - m) - 79860*(x + 2/3)*gamma(1 - m) + 2662*gamma(1 - m)) + 8*3**m*m*(x + 2/3)**m*lerchphi(1/(15*(x
+ 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(598950*(x + 2/3)**2*gamma(1 - m) - 79860*(x + 2/3)*gamma(1 - m) + 2
662*gamma(1 - m)) - 8*3**m*m*(x + 2/3)**m*lerchphi(7/(6*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(598950*(x
+ 2/3)**2*gamma(1 - m) - 79860*(x + 2/3)*gamma(1 - m) + 2662*gamma(1 - m))
Maxima [F]
\[
\int \frac {(2+3 x)^m}{(1-2 x) (3+5 x)^3} \, dx=\int { -\frac {{\left (3 \, x + 2\right )}^{m}}{{\left (5 \, x + 3\right )}^{3} {\left (2 \, x - 1\right )}} \,d x }
\]
[In]
integrate((2+3*x)^m/(1-2*x)/(3+5*x)^3,x, algorithm="maxima")
[Out]
-integrate((3*x + 2)^m/((5*x + 3)^3*(2*x - 1)), x)
Giac [F]
\[
\int \frac {(2+3 x)^m}{(1-2 x) (3+5 x)^3} \, dx=\int { -\frac {{\left (3 \, x + 2\right )}^{m}}{{\left (5 \, x + 3\right )}^{3} {\left (2 \, x - 1\right )}} \,d x }
\]
[In]
integrate((2+3*x)^m/(1-2*x)/(3+5*x)^3,x, algorithm="giac")
[Out]
integrate(-(3*x + 2)^m/((5*x + 3)^3*(2*x - 1)), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {(2+3 x)^m}{(1-2 x) (3+5 x)^3} \, dx=\int -\frac {{\left (3\,x+2\right )}^m}{\left (2\,x-1\right )\,{\left (5\,x+3\right )}^3} \,d x
\]
[In]
int(-(3*x + 2)^m/((2*x - 1)*(5*x + 3)^3),x)
[Out]
int(-(3*x + 2)^m/((2*x - 1)*(5*x + 3)^3), x)