Integrand size = 22, antiderivative size = 94 \[ \int \frac {(2+3 x)^m}{(1-2 x) (3+5 x)^2} \, dx=-\frac {5 (2+3 x)^{1+m}}{11 (3+5 x)}+\frac {4 (2+3 x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {2}{7} (2+3 x)\right )}{847 (1+m)}-\frac {5 (2+33 m) (2+3 x)^{1+m} \operatorname {Hypergeometric2F1}(1,1+m,2+m,5 (2+3 x))}{121 (1+m)} \] Output:
-5*(2+3*x)^(1+m)/(33+55*x)+4*(2+3*x)^(1+m)*hypergeom([1, 1+m],[2+m],4/7+6/ 7*x)/(847+847*m)-5*(2+33*m)*(2+3*x)^(1+m)*hypergeom([1, 1+m],[2+m],10+15*x )/(121+121*m)
Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.87 \[ \int \frac {(2+3 x)^m}{(1-2 x) (3+5 x)^2} \, dx=\frac {(2+3 x)^{1+m} \left (-385 (1+m)+4 (3+5 x) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {2}{7} (2+3 x)\right )-35 (2+33 m) (3+5 x) \operatorname {Hypergeometric2F1}(1,1+m,2+m,5 (2+3 x))\right )}{847 (1+m) (3+5 x)} \] Input:
Integrate[(2 + 3*x)^m/((1 - 2*x)*(3 + 5*x)^2),x]
Output:
((2 + 3*x)^(1 + m)*(-385*(1 + m) + 4*(3 + 5*x)*Hypergeometric2F1[1, 1 + m, 2 + m, (2*(2 + 3*x))/7] - 35*(2 + 33*m)*(3 + 5*x)*Hypergeometric2F1[1, 1 + m, 2 + m, 5*(2 + 3*x)]))/(847*(1 + m)*(3 + 5*x))
Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {114, 25, 174, 78}
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.
| \(\displaystyle \int \frac {(3 x+2)^m}{(1-2 x) (5 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {1}{11} \int -\frac {(3 x+2)^m (-30 x m+15 m+2)}{(1-2 x) (5 x+3)}dx-\frac {5 (3 x+2)^{m+1}}{11 (5 x+3)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{11} \int \frac {(3 x+2)^m (-30 x m+15 m+2)}{(1-2 x) (5 x+3)}dx-\frac {5 (3 x+2)^{m+1}}{11 (5 x+3)}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {1}{11} \left (\frac {4}{11} \int \frac {(3 x+2)^m}{1-2 x}dx+\frac {5}{11} (33 m+2) \int \frac {(3 x+2)^m}{5 x+3}dx\right )-\frac {5 (3 x+2)^{m+1}}{11 (5 x+3)}\) |
| \(\Big \downarrow \) 78 |
| \(\displaystyle \frac {1}{11} \left (\frac {4 (3 x+2)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {2}{7} (3 x+2)\right )}{77 (m+1)}-\frac {5 (33 m+2) (3 x+2)^{m+1} \operatorname {Hypergeometric2F1}(1,m+1,m+2,5 (3 x+2))}{11 (m+1)}\right )-\frac {5 (3 x+2)^{m+1}}{11 (5 x+3)}\) |
Input:
Int[(2 + 3*x)^m/((1 - 2*x)*(3 + 5*x)^2),x]
Output:
(-5*(2 + 3*x)^(1 + m))/(11*(3 + 5*x)) + ((4*(2 + 3*x)^(1 + m)*Hypergeometr ic2F1[1, 1 + m, 2 + m, (2*(2 + 3*x))/7])/(77*(1 + m)) - (5*(2 + 33*m)*(2 + 3*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, 5*(2 + 3*x)])/(11*(1 + m) ))/11
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] && !IntegerQ[m] && IntegerQ[n]
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 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[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])
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(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]
\[\int \frac {\left (2+3 x \right )^{m}}{\left (1-2 x \right ) \left (3+5 x \right )^{2}}d x\]
Input:
int((2+3*x)^m/(1-2*x)/(3+5*x)^2,x)
Output:
int((2+3*x)^m/(1-2*x)/(3+5*x)^2,x)
\[ \int \frac {(2+3 x)^m}{(1-2 x) (3+5 x)^2} \, dx=\int { -\frac {{\left (3 \, x + 2\right )}^{m}}{{\left (5 \, x + 3\right )}^{2} {\left (2 \, x - 1\right )}} \,d x } \] Input:
integrate((2+3*x)^m/(1-2*x)/(3+5*x)^2,x, algorithm="fricas")
Output:
integral(-(3*x + 2)^m/(50*x^3 + 35*x^2 - 12*x - 9), x)
Result contains complex when optimal does not.
Time = 1.41 (sec) , antiderivative size = 366, normalized size of antiderivative = 3.89 \[ \int \frac {(2+3 x)^m}{(1-2 x) (3+5 x)^2} \, dx=\frac {495 \cdot 3^{m} m^{2} \left (x + \frac {2}{3}\right ) \left (x + \frac {2}{3}\right )^{m} \Phi \left (\frac {1}{15 \left (x + \frac {2}{3}\right )}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{1815 \left (x + \frac {2}{3}\right ) \Gamma \left (1 - m\right ) - 121 \Gamma \left (1 - m\right )} - \frac {33 \cdot 3^{m} m^{2} \left (x + \frac {2}{3}\right )^{m} \Phi \left (\frac {1}{15 \left (x + \frac {2}{3}\right )}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{1815 \left (x + \frac {2}{3}\right ) \Gamma \left (1 - m\right ) - 121 \Gamma \left (1 - m\right )} + \frac {30 \cdot 3^{m} m \left (x + \frac {2}{3}\right ) \left (x + \frac {2}{3}\right )^{m} \Phi \left (\frac {1}{15 \left (x + \frac {2}{3}\right )}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{1815 \left (x + \frac {2}{3}\right ) \Gamma \left (1 - m\right ) - 121 \Gamma \left (1 - m\right )} - \frac {30 \cdot 3^{m} m \left (x + \frac {2}{3}\right ) \left (x + \frac {2}{3}\right )^{m} \Phi \left (\frac {7}{6 \left (x + \frac {2}{3}\right )}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{1815 \left (x + \frac {2}{3}\right ) \Gamma \left (1 - m\right ) - 121 \Gamma \left (1 - m\right )} + \frac {495 \cdot 3^{m} m \left (x + \frac {2}{3}\right ) \left (x + \frac {2}{3}\right )^{m} \Gamma \left (- m\right )}{1815 \left (x + \frac {2}{3}\right ) \Gamma \left (1 - m\right ) - 121 \Gamma \left (1 - m\right )} - \frac {2 \cdot 3^{m} m \left (x + \frac {2}{3}\right )^{m} \Phi \left (\frac {1}{15 \left (x + \frac {2}{3}\right )}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{1815 \left (x + \frac {2}{3}\right ) \Gamma \left (1 - m\right ) - 121 \Gamma \left (1 - m\right )} + \frac {2 \cdot 3^{m} m \left (x + \frac {2}{3}\right )^{m} \Phi \left (\frac {7}{6 \left (x + \frac {2}{3}\right )}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{1815 \left (x + \frac {2}{3}\right ) \Gamma \left (1 - m\right ) - 121 \Gamma \left (1 - m\right )} \] Input:
integrate((2+3*x)**m/(1-2*x)/(3+5*x)**2,x)
Output:
495*3**m*m**2*(x + 2/3)*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_p olar(I*pi))*gamma(-m)/(1815*(x + 2/3)*gamma(1 - m) - 121*gamma(1 - m)) - 3 3*3**m*m**2*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))* gamma(-m)/(1815*(x + 2/3)*gamma(1 - m) - 121*gamma(1 - m)) + 30*3**m*m*(x + 2/3)*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma (-m)/(1815*(x + 2/3)*gamma(1 - m) - 121*gamma(1 - m)) - 30*3**m*m*(x + 2/3 )*(x + 2/3)**m*lerchphi(7/(6*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/( 1815*(x + 2/3)*gamma(1 - m) - 121*gamma(1 - m)) + 495*3**m*m*(x + 2/3)*(x + 2/3)**m*gamma(-m)/(1815*(x + 2/3)*gamma(1 - m) - 121*gamma(1 - m)) - 2*3 **m*m*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma( -m)/(1815*(x + 2/3)*gamma(1 - m) - 121*gamma(1 - m)) + 2*3**m*m*(x + 2/3)* *m*lerchphi(7/(6*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(1815*(x + 2/ 3)*gamma(1 - m) - 121*gamma(1 - m))
\[ \int \frac {(2+3 x)^m}{(1-2 x) (3+5 x)^2} \, dx=\int { -\frac {{\left (3 \, x + 2\right )}^{m}}{{\left (5 \, x + 3\right )}^{2} {\left (2 \, x - 1\right )}} \,d x } \] Input:
integrate((2+3*x)^m/(1-2*x)/(3+5*x)^2,x, algorithm="maxima")
Output:
-integrate((3*x + 2)^m/((5*x + 3)^2*(2*x - 1)), x)
\[ \int \frac {(2+3 x)^m}{(1-2 x) (3+5 x)^2} \, dx=\int { -\frac {{\left (3 \, x + 2\right )}^{m}}{{\left (5 \, x + 3\right )}^{2} {\left (2 \, x - 1\right )}} \,d x } \] Input:
integrate((2+3*x)^m/(1-2*x)/(3+5*x)^2,x, algorithm="giac")
Output:
integrate(-(3*x + 2)^m/((5*x + 3)^2*(2*x - 1)), x)
Timed out. \[ \int \frac {(2+3 x)^m}{(1-2 x) (3+5 x)^2} \, dx=-\int \frac {{\left (3\,x+2\right )}^m}{\left (2\,x-1\right )\,{\left (5\,x+3\right )}^2} \,d x \] Input:
int(-(3*x + 2)^m/((2*x - 1)*(5*x + 3)^2),x)
Output:
-int((3*x + 2)^m/((2*x - 1)*(5*x + 3)^2), x)
\[ \int \frac {(2+3 x)^m}{(1-2 x) (3+5 x)^2} \, dx=-\left (\int \frac {\left (3 x +2\right )^{m}}{50 x^{3}+35 x^{2}-12 x -9}d x \right ) \] Input:
int((2+3*x)^m/(1-2*x)/(3+5*x)^2,x)
Output:
- int((3*x + 2)**m/(50*x**3 + 35*x**2 - 12*x - 9),x)