Integrand size = 18, antiderivative size = 115 \[ \int \frac {(a+b x)^n (c+d x)^2}{x^3} \, dx=-\frac {c^2 (a+b x)^{1+n}}{2 a x^2}+\frac {d^2 (a+b x)^{1+n}}{b n x}+\frac {\left (2 a^2 d^2+4 a b c d n-b^2 c^2 (1-n) n\right ) (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {b x}{a}\right )}{2 a^3 n (1+n)} \] Output:
-1/2*c^2*(b*x+a)^(1+n)/a/x^2+d^2*(b*x+a)^(1+n)/b/n/x+1/2*(2*a^2*d^2+4*a*b* c*d*n-b^2*c^2*(1-n)*n)*(b*x+a)^(1+n)*hypergeom([2, 1+n],[2+n],1+b*x/a)/a^3 /n/(1+n)
Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^n (c+d x)^2}{x^3} \, dx=-\frac {(a+b x)^{1+n} \left (a c (1+n) (b c (-1+n) x+a (c+4 d x))+\left (2 a^2 d^2+4 a b c d n+b^2 c^2 (-1+n) n\right ) x^2 \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )\right )}{2 a^3 (1+n) x^2} \] Input:
Integrate[((a + b*x)^n*(c + d*x)^2)/x^3,x]
Output:
-1/2*((a + b*x)^(1 + n)*(a*c*(1 + n)*(b*c*(-1 + n)*x + a*(c + 4*d*x)) + (2 *a^2*d^2 + 4*a*b*c*d*n + b^2*c^2*(-1 + n)*n)*x^2*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a]))/(a^3*(1 + n)*x^2)
Time = 0.24 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {100, 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.
| \(\displaystyle \int \frac {(c+d x)^2 (a+b x)^n}{x^3} \, dx\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle \frac {\int \frac {(a+b x)^n \left (2 a x d^2+c (4 a d-b c (1-n))\right )}{x^2}dx}{2 a}-\frac {c^2 (a+b x)^{n+1}}{2 a x^2}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {\frac {\left (2 a^2 d^2+4 a b c d n-b^2 c^2 (1-n) n\right ) \int \frac {(a+b x)^n}{x}dx}{a}-\frac {c (a+b x)^{n+1} (4 a d-b c (1-n))}{a x}}{2 a}-\frac {c^2 (a+b x)^{n+1}}{2 a x^2}\) |
| \(\Big \downarrow \) 75 |
| \(\displaystyle \frac {-\frac {(a+b x)^{n+1} \left (2 a^2 d^2+4 a b c d n-b^2 c^2 (1-n) n\right ) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b x}{a}+1\right )}{a^2 (n+1)}-\frac {c (a+b x)^{n+1} (4 a d-b c (1-n))}{a x}}{2 a}-\frac {c^2 (a+b x)^{n+1}}{2 a x^2}\) |
Input:
Int[((a + b*x)^n*(c + d*x)^2)/x^3,x]
Output:
-1/2*(c^2*(a + b*x)^(1 + n))/(a*x^2) + (-((c*(4*a*d - b*c*(1 - n))*(a + b* x)^(1 + n))/(a*x)) - ((2*a^2*d^2 + 4*a*b*c*d*n - b^2*c^2*(1 - n)*n)*(a + b *x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/(a^2*(1 + n)) )/(2*a)
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[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
\[\int \frac {\left (b x +a \right )^{n} \left (x d +c \right )^{2}}{x^{3}}d x\]
Input:
int((b*x+a)^n*(d*x+c)^2/x^3,x)
Output:
int((b*x+a)^n*(d*x+c)^2/x^3,x)
\[ \int \frac {(a+b x)^n (c+d x)^2}{x^3} \, dx=\int { \frac {{\left (d x + c\right )}^{2} {\left (b x + a\right )}^{n}}{x^{3}} \,d x } \] Input:
integrate((b*x+a)^n*(d*x+c)^2/x^3,x, algorithm="fricas")
Output:
integral((d^2*x^2 + 2*c*d*x + c^2)*(b*x + a)^n/x^3, x)
Leaf count of result is larger than twice the leaf count of optimal. 1040 vs. \(2 (94) = 188\).
Time = 4.16 (sec) , antiderivative size = 1040, normalized size of antiderivative = 9.04 \[ \int \frac {(a+b x)^n (c+d x)^2}{x^3} \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)**n*(d*x+c)**2/x**3,x)
Output:
a**2*b**(n + 3)*c**2*n**3*(a/b + x)**(n + 1)*lerchphi(1 + b*x/a, 1, n + 1) *gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b** 2*(a/b + x)**2*gamma(n + 2)) - a**2*b**(n + 3)*c**2*n*(a/b + x)**(n + 1)*l erchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b *x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) - a**2*b**(n + 3) *c**2*n*(a/b + x)**(n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x *gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) - a**2*b**(n + 3)*c **2*(a/b + x)**(n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gam ma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) + 2*a*b*b**(n + 3)*c**2 *n**3*x*(a/b + x)**(n + 1)*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(-2* a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gam ma(n + 2)) - a*b*b**(n + 3)*c**2*n**2*x*(a/b + x)**(n + 1)*gamma(n + 1)/(- 2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*g amma(n + 2)) - 2*a*b*b**(n + 3)*c**2*n*x*(a/b + x)**(n + 1)*lerchphi(1 + b *x/a, 1, n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) + a*b*b**(n + 3)*c**2*x*(a/b + x)**(n + 1)*gamma(n + 1)/(-2*a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) - b**2*b**(n + 3)*c**2*n**3*(a/b + x)**2*(a/b + x)**(n + 1)*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(-2* a**5*gamma(n + 2) - 4*a**4*b*x*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*...
\[ \int \frac {(a+b x)^n (c+d x)^2}{x^3} \, dx=\int { \frac {{\left (d x + c\right )}^{2} {\left (b x + a\right )}^{n}}{x^{3}} \,d x } \] Input:
integrate((b*x+a)^n*(d*x+c)^2/x^3,x, algorithm="maxima")
Output:
integrate((d*x + c)^2*(b*x + a)^n/x^3, x)
\[ \int \frac {(a+b x)^n (c+d x)^2}{x^3} \, dx=\int { \frac {{\left (d x + c\right )}^{2} {\left (b x + a\right )}^{n}}{x^{3}} \,d x } \] Input:
integrate((b*x+a)^n*(d*x+c)^2/x^3,x, algorithm="giac")
Output:
integrate((d*x + c)^2*(b*x + a)^n/x^3, x)
Timed out. \[ \int \frac {(a+b x)^n (c+d x)^2}{x^3} \, dx=\int \frac {{\left (a+b\,x\right )}^n\,{\left (c+d\,x\right )}^2}{x^3} \,d x \] Input:
int(((a + b*x)^n*(c + d*x)^2)/x^3,x)
Output:
int(((a + b*x)^n*(c + d*x)^2)/x^3, x)
\[ \int \frac {(a+b x)^n (c+d x)^2}{x^3} \, dx=\frac {-\left (b x +a \right )^{n} a \,c^{2} n -4 \left (b x +a \right )^{n} a c d n x +2 \left (b x +a \right )^{n} a \,d^{2} x^{2}-\left (b x +a \right )^{n} b \,c^{2} n^{2} x +2 \left (\int \frac {\left (b x +a \right )^{n}}{b \,x^{2}+a x}d x \right ) a^{2} d^{2} n \,x^{2}+4 \left (\int \frac {\left (b x +a \right )^{n}}{b \,x^{2}+a x}d x \right ) a b c d \,n^{2} x^{2}+\left (\int \frac {\left (b x +a \right )^{n}}{b \,x^{2}+a x}d x \right ) b^{2} c^{2} n^{3} x^{2}-\left (\int \frac {\left (b x +a \right )^{n}}{b \,x^{2}+a x}d x \right ) b^{2} c^{2} n^{2} x^{2}}{2 a n \,x^{2}} \] Input:
int((b*x+a)^n*(d*x+c)^2/x^3,x)
Output:
( - (a + b*x)**n*a*c**2*n - 4*(a + b*x)**n*a*c*d*n*x + 2*(a + b*x)**n*a*d* *2*x**2 - (a + b*x)**n*b*c**2*n**2*x + 2*int((a + b*x)**n/(a*x + b*x**2),x )*a**2*d**2*n*x**2 + 4*int((a + b*x)**n/(a*x + b*x**2),x)*a*b*c*d*n**2*x** 2 + int((a + b*x)**n/(a*x + b*x**2),x)*b**2*c**2*n**3*x**2 - int((a + b*x) **n/(a*x + b*x**2),x)*b**2*c**2*n**2*x**2)/(2*a*n*x**2)