Integrand size = 21, antiderivative size = 105 \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{x^3} \, dx=-\frac {a (d+e x)^{1+m}}{2 d x^2}+\frac {c (d+e x)^{1+m}}{e m x}+\frac {\left (2 c d^2+e (2 b d-a e (1-m)) m\right ) (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,1+\frac {e x}{d}\right )}{2 d^3 m (1+m)} \] Output:
-1/2*a*(e*x+d)^(1+m)/d/x^2+c*(e*x+d)^(1+m)/e/m/x+1/2*(2*c*d^2+e*(2*b*d-a*e *(1-m))*m)*(e*x+d)^(1+m)*hypergeom([2, 1+m],[2+m],1+e*x/d)/d^3/m/(1+m)
Time = 0.14 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.79 \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{x^3} \, dx=-\frac {(d+e x)^{1+m} \left (c d^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,1+\frac {e x}{d}\right )+e \left (-b d \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,1+\frac {e x}{d}\right )+a e \operatorname {Hypergeometric2F1}\left (3,1+m,2+m,1+\frac {e x}{d}\right )\right )\right )}{d^3 (1+m)} \] Input:
Integrate[((d + e*x)^m*(a + b*x + c*x^2))/x^3,x]
Output:
-(((d + e*x)^(1 + m)*(c*d^2*Hypergeometric2F1[1, 1 + m, 2 + m, 1 + (e*x)/d ] + e*(-(b*d*Hypergeometric2F1[2, 1 + m, 2 + m, 1 + (e*x)/d]) + a*e*Hyperg eometric2F1[3, 1 + m, 2 + m, 1 + (e*x)/d])))/(d^3*(1 + m)))
Time = 0.43 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1193, 25, 87, 75}
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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right ) (d+e x)^m}{x^3} \, dx\) |
| \(\Big \downarrow \) 1193 |
| \(\displaystyle -\frac {\int -\frac {(2 b d+2 c x d-a e (1-m)) (d+e x)^m}{x^2}dx}{2 d}-\frac {a (d+e x)^{m+1}}{2 d x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {(2 b d+2 c x d-a e (1-m)) (d+e x)^m}{x^2}dx}{2 d}-\frac {a (d+e x)^{m+1}}{2 d x^2}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {\frac {\left (e m (2 b d-a e (1-m))+2 c d^2\right ) \int \frac {(d+e x)^m}{x}dx}{d}-\frac {(d+e x)^{m+1} (2 b d-a e (1-m))}{d x}}{2 d}-\frac {a (d+e x)^{m+1}}{2 d x^2}\) |
| \(\Big \downarrow \) 75 |
| \(\displaystyle \frac {-\frac {(d+e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {e x}{d}+1\right ) \left (e m (2 b d-a e (1-m))+2 c d^2\right )}{d^2 (m+1)}-\frac {(d+e x)^{m+1} (2 b d-a e (1-m))}{d x}}{2 d}-\frac {a (d+e x)^{m+1}}{2 d x^2}\) |
Input:
Int[((d + e*x)^m*(a + b*x + c*x^2))/x^3,x]
Output:
-1/2*(a*(d + e*x)^(1 + m))/(d*x^2) + (-(((2*b*d - a*e*(1 - m))*(d + e*x)^( 1 + m))/(d*x)) - ((2*c*d^2 + e*(2*b*d - a*e*(1 - m))*m)*(d + e*x)^(1 + m)* Hypergeometric2F1[1, 1 + m, 2 + m, 1 + (e*x)/d])/(d^2*(1 + m)))/(2*d)
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x )^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[n] && (IntegerQ[m] || GtQ[-d/(b*c), 0])
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)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p, d + e*x, x]}, Simp[R*(d + e*x)^(m + 1)*((f + g*x)^(n + 1)/((m + 1)*(e*f - d*g)) ), x] + Simp[1/((m + 1)*(e*f - d*g)) Int[(d + e*x)^(m + 1)*(f + g*x)^n*Ex pandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /; FreeQ[{a , b, c, d, e, f, g, n}, x] && IGtQ[p, 0] && ILtQ[2*m, -2] && !IntegerQ[n] && !(EqQ[m, -2] && EqQ[p, 1] && EqQ[2*c*d - b*e, 0])
\[\int \frac {\left (e x +d \right )^{m} \left (c \,x^{2}+b x +a \right )}{x^{3}}d x\]
Input:
int((e*x+d)^m*(c*x^2+b*x+a)/x^3,x)
Output:
int((e*x+d)^m*(c*x^2+b*x+a)/x^3,x)
\[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{x^3} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m}}{x^{3}} \,d x } \] Input:
integrate((e*x+d)^m*(c*x^2+b*x+a)/x^3,x, algorithm="fricas")
Output:
integral((c*x^2 + b*x + a)*(e*x + d)^m/x^3, x)
Leaf count of result is larger than twice the leaf count of optimal. 1006 vs. \(2 (82) = 164\).
Time = 5.07 (sec) , antiderivative size = 1006, normalized size of antiderivative = 9.58 \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{x^3} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)**m*(c*x**2+b*x+a)/x**3,x)
Output:
a*d**2*e**(m + 3)*m**3*(d/e + x)**(m + 1)*lerchphi(1 + e*x/d, 1, m + 1)*ga mma(m + 1)/(-2*d**5*gamma(m + 2) - 4*d**4*e*x*gamma(m + 2) + 2*d**3*e**2*( d/e + x)**2*gamma(m + 2)) - a*d**2*e**(m + 3)*m*(d/e + x)**(m + 1)*lerchph i(1 + e*x/d, 1, m + 1)*gamma(m + 1)/(-2*d**5*gamma(m + 2) - 4*d**4*e*x*gam ma(m + 2) + 2*d**3*e**2*(d/e + x)**2*gamma(m + 2)) - a*d**2*e**(m + 3)*m*( d/e + x)**(m + 1)*gamma(m + 1)/(-2*d**5*gamma(m + 2) - 4*d**4*e*x*gamma(m + 2) + 2*d**3*e**2*(d/e + x)**2*gamma(m + 2)) - a*d**2*e**(m + 3)*(d/e + x )**(m + 1)*gamma(m + 1)/(-2*d**5*gamma(m + 2) - 4*d**4*e*x*gamma(m + 2) + 2*d**3*e**2*(d/e + x)**2*gamma(m + 2)) + 2*a*d*e*e**(m + 3)*m**3*x*(d/e + x)**(m + 1)*lerchphi(1 + e*x/d, 1, m + 1)*gamma(m + 1)/(-2*d**5*gamma(m + 2) - 4*d**4*e*x*gamma(m + 2) + 2*d**3*e**2*(d/e + x)**2*gamma(m + 2)) - a* d*e*e**(m + 3)*m**2*x*(d/e + x)**(m + 1)*gamma(m + 1)/(-2*d**5*gamma(m + 2 ) - 4*d**4*e*x*gamma(m + 2) + 2*d**3*e**2*(d/e + x)**2*gamma(m + 2)) - 2*a *d*e*e**(m + 3)*m*x*(d/e + x)**(m + 1)*lerchphi(1 + e*x/d, 1, m + 1)*gamma (m + 1)/(-2*d**5*gamma(m + 2) - 4*d**4*e*x*gamma(m + 2) + 2*d**3*e**2*(d/e + x)**2*gamma(m + 2)) + a*d*e*e**(m + 3)*x*(d/e + x)**(m + 1)*gamma(m + 1 )/(-2*d**5*gamma(m + 2) - 4*d**4*e*x*gamma(m + 2) + 2*d**3*e**2*(d/e + x)* *2*gamma(m + 2)) - a*e**2*e**(m + 3)*m**3*(d/e + x)**2*(d/e + x)**(m + 1)* lerchphi(1 + e*x/d, 1, m + 1)*gamma(m + 1)/(-2*d**5*gamma(m + 2) - 4*d**4* e*x*gamma(m + 2) + 2*d**3*e**2*(d/e + x)**2*gamma(m + 2)) + a*e**2*e**(...
\[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{x^3} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m}}{x^{3}} \,d x } \] Input:
integrate((e*x+d)^m*(c*x^2+b*x+a)/x^3,x, algorithm="maxima")
Output:
integrate((c*x^2 + b*x + a)*(e*x + d)^m/x^3, x)
\[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{x^3} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )} {\left (e x + d\right )}^{m}}{x^{3}} \,d x } \] Input:
integrate((e*x+d)^m*(c*x^2+b*x+a)/x^3,x, algorithm="giac")
Output:
integrate((c*x^2 + b*x + a)*(e*x + d)^m/x^3, x)
Timed out. \[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{x^3} \, dx=\int \frac {{\left (d+e\,x\right )}^m\,\left (c\,x^2+b\,x+a\right )}{x^3} \,d x \] Input:
int(((d + e*x)^m*(a + b*x + c*x^2))/x^3,x)
Output:
int(((d + e*x)^m*(a + b*x + c*x^2))/x^3, x)
\[ \int \frac {(d+e x)^m \left (a+b x+c x^2\right )}{x^3} \, dx=\frac {-\left (e x +d \right )^{m} a d m -\left (e x +d \right )^{m} a e \,m^{2} x -2 \left (e x +d \right )^{m} b d m x +2 \left (e x +d \right )^{m} c d \,x^{2}+\left (\int \frac {\left (e x +d \right )^{m}}{e \,x^{2}+d x}d x \right ) a \,e^{2} m^{3} x^{2}-\left (\int \frac {\left (e x +d \right )^{m}}{e \,x^{2}+d x}d x \right ) a \,e^{2} m^{2} x^{2}+2 \left (\int \frac {\left (e x +d \right )^{m}}{e \,x^{2}+d x}d x \right ) b d e \,m^{2} x^{2}+2 \left (\int \frac {\left (e x +d \right )^{m}}{e \,x^{2}+d x}d x \right ) c \,d^{2} m \,x^{2}}{2 d m \,x^{2}} \] Input:
int((e*x+d)^m*(c*x^2+b*x+a)/x^3,x)
Output:
( - (d + e*x)**m*a*d*m - (d + e*x)**m*a*e*m**2*x - 2*(d + e*x)**m*b*d*m*x + 2*(d + e*x)**m*c*d*x**2 + int((d + e*x)**m/(d*x + e*x**2),x)*a*e**2*m**3 *x**2 - int((d + e*x)**m/(d*x + e*x**2),x)*a*e**2*m**2*x**2 + 2*int((d + e *x)**m/(d*x + e*x**2),x)*b*d*e*m**2*x**2 + 2*int((d + e*x)**m/(d*x + e*x** 2),x)*c*d**2*m*x**2)/(2*d*m*x**2)