3.3198 \(\int \frac{(2+3 x)^m}{(1-2 x) (3+5 x)^3} \, dx\)
Optimal. Leaf size=124 \[ -\frac{5 \left (1089 m^2-957 m+8\right ) (3 x+2)^{m+1} \, _2F_1(1,m+1;m+2;5 (3 x+2))}{2662 (m+1)}+\frac{8 (3 x+2)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{2}{7} (3 x+2)\right )}{9317 (m+1)}+\frac{5 (29-33 m) (3 x+2)^{m+1}}{242 (5 x+3)}-\frac{5 (3 x+2)^{m+1}}{22 (5 x+3)^2} \]
[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))
________________________________________________________________________________________
Rubi [A] time = 0.094268, antiderivative size = 124, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used =
{103, 151, 156, 68} \[ -\frac{5 \left (1089 m^2-957 m+8\right ) (3 x+2)^{m+1} \, _2F_1(1,m+1;m+2;5 (3 x+2))}{2662 (m+1)}+\frac{8 (3 x+2)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{2}{7} (3 x+2)\right )}{9317 (m+1)}+\frac{5 (29-33 m) (3 x+2)^{m+1}}{242 (5 x+3)}-\frac{5 (3 x+2)^{m+1}}{22 (5 x+3)^2} \]
Antiderivative was successfully verified.
[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 103
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] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])
Rule 151
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] && LtQ[m, -1] && IntegerQ[m]
Rule 156
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]
Rule 68
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] && !IntegerQ[m] && IntegerQ[n]
Rubi steps
\begin{align*} \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{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] time = 0.0627021, size = 104, normalized size = 0.84 \[ \frac{(3 x+2)^{m+1} \left (-35 \left (1089 m^2-957 m+8\right ) (5 x+3)^2 \, _2F_1(1,m+1;m+2;5 (3 x+2))+16 (5 x+3)^2 \, _2F_1\left (1,m+1;m+2;\frac{2}{7} (3 x+2)\right )-385 (m+1) (33 m (5 x+3)-145 x-76)\right )}{18634 (m+1) (5 x+3)^2} \]
Antiderivative was successfully verified.
[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] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( 2+3\,x \right ) ^{m}}{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (3 \, x + 2\right )}^{m}}{{\left (5 \, x + 3\right )}^{3}{\left (2 \, x - 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (3 \, x + 2\right )}^{m}}{250 \, x^{4} + 325 \, x^{3} + 45 \, x^{2} - 81 \, x - 27}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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] time = 3.48532, size = 1590, normalized size = 12.82 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2+3*x)**m/(1-2*x)/(3+5*x)**3,x)
[Out]
462150*15**(2*m)*3**m*m*(x + 2/3)**2*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(
598950*15**(2*m)*(x + 2/3)**2*gamma(1 - m) - 79860*15**(2*m)*(x + 2/3)*gamma(1 - m) + 2662*15**(2*m)*gamma(1 -
m)) - 1800*15**(2*m)*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*15**(2*m)*(x + 2/3)**2*gamma(1 - m) - 79860*15**(2*m)*(x + 2/3)*gamma(1 - m) + 2662*15**(2*m)*gamma
(1 - m)) - 61620*15**(2*m)*3**m*m*(x + 2/3)*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))*gamm
a(-m)/(598950*15**(2*m)*(x + 2/3)**2*gamma(1 - m) - 79860*15**(2*m)*(x + 2/3)*gamma(1 - m) + 2662*15**(2*m)*ga
mma(1 - m)) + 240*15**(2*m)*3**m*m*(x + 2/3)*(x + 2/3)**m*lerchphi(7/(6*(x + 2/3)), 1, m*exp_polar(I*pi))*gamm
a(-m)/(598950*15**(2*m)*(x + 2/3)**2*gamma(1 - m) - 79860*15**(2*m)*(x + 2/3)*gamma(1 - m) + 2662*15**(2*m)*ga
mma(1 - m)) + 2054*15**(2*m)*3**m*m*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(5
98950*15**(2*m)*(x + 2/3)**2*gamma(1 - m) - 79860*15**(2*m)*(x + 2/3)*gamma(1 - m) + 2662*15**(2*m)*gamma(1 -
m)) - 8*15**(2*m)*3**m*m*(x + 2/3)**m*lerchphi(7/(6*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(598950*15**(2
*m)*(x + 2/3)**2*gamma(1 - m) - 79860*15**(2*m)*(x + 2/3)*gamma(1 - m) + 2662*15**(2*m)*gamma(1 - m)) + 245025
*675**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*15**
(2*m)*(x + 2/3)**2*gamma(1 - m) - 79860*15**(2*m)*(x + 2/3)*gamma(1 - m) + 2662*15**(2*m)*gamma(1 - m)) - 3267
0*675**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*15**(2
*m)*(x + 2/3)**2*gamma(1 - m) - 79860*15**(2*m)*(x + 2/3)*gamma(1 - m) + 2662*15**(2*m)*gamma(1 - m)) + 1089*6
75**m*m**3*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(598950*15**(2*m)*(x + 2/3)
**2*gamma(1 - m) - 79860*15**(2*m)*(x + 2/3)*gamma(1 - m) + 2662*15**(2*m)*gamma(1 - m)) - 215325*675**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*15**(2*m)*(x + 2/
3)**2*gamma(1 - m) - 79860*15**(2*m)*(x + 2/3)*gamma(1 - m) + 2662*15**(2*m)*gamma(1 - m)) + 245025*675**m*m**
2*(x + 2/3)**2*(x + 2/3)**m*gamma(-m)/(598950*15**(2*m)*(x + 2/3)**2*gamma(1 - m) - 79860*15**(2*m)*(x + 2/3)*
gamma(1 - m) + 2662*15**(2*m)*gamma(1 - m)) + 28710*675**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*15**(2*m)*(x + 2/3)**2*gamma(1 - m) - 79860*15**(2*m)*(x + 2/3)*ga
mma(1 - m) + 2662*15**(2*m)*gamma(1 - m)) - 16335*675**m*m**2*(x + 2/3)*(x + 2/3)**m*gamma(-m)/(598950*15**(2*
m)*(x + 2/3)**2*gamma(1 - m) - 79860*15**(2*m)*(x + 2/3)*gamma(1 - m) + 2662*15**(2*m)*gamma(1 - m)) - 957*675
**m*m**2*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(598950*15**(2*m)*(x + 2/3)**
2*gamma(1 - m) - 79860*15**(2*m)*(x + 2/3)*gamma(1 - m) + 2662*15**(2*m)*gamma(1 - m)) - 460350*675**m*m*(x +
2/3)**2*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(598950*15**(2*m)*(x + 2/3)**2
*gamma(1 - m) - 79860*15**(2*m)*(x + 2/3)*gamma(1 - m) + 2662*15**(2*m)*gamma(1 - m)) - 215325*675**m*m*(x + 2
/3)**2*(x + 2/3)**m*gamma(-m)/(598950*15**(2*m)*(x + 2/3)**2*gamma(1 - m) - 79860*15**(2*m)*(x + 2/3)*gamma(1
- m) + 2662*15**(2*m)*gamma(1 - m)) + 61380*675**m*m*(x + 2/3)*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*ex
p_polar(I*pi))*gamma(-m)/(598950*15**(2*m)*(x + 2/3)**2*gamma(1 - m) - 79860*15**(2*m)*(x + 2/3)*gamma(1 - m)
+ 2662*15**(2*m)*gamma(1 - m)) + 30690*675**m*m*(x + 2/3)*(x + 2/3)**m*gamma(-m)/(598950*15**(2*m)*(x + 2/3)**
2*gamma(1 - m) - 79860*15**(2*m)*(x + 2/3)*gamma(1 - m) + 2662*15**(2*m)*gamma(1 - m)) - 2046*675**m*m*(x + 2/
3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(598950*15**(2*m)*(x + 2/3)**2*gamma(1 - m) -
79860*15**(2*m)*(x + 2/3)*gamma(1 - m) + 2662*15**(2*m)*gamma(1 - m))
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (3 \, x + 2\right )}^{m}}{{\left (5 \, x + 3\right )}^{3}{\left (2 \, x - 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)