Integrand size = 18, antiderivative size = 143 \[ \int \frac {(a+b x)^n (c+d x)^3}{x^2} \, dx=\frac {d (a+b x)^{1+n} (c+d x)^2}{b (2+n) x}-\frac {(a+b x)^{1+n} \left (b c^2 (1+n) (a d+b c (2+n))+a d^2 (a d-b c (4+n)) x\right )}{a b^2 (1+n) (2+n) x}-\frac {c^2 (3 a d+b c n) (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )}{a^2 (1+n)} \]
d*(b*x+a)^(1+n)*(d*x+c)^2/b/(2+n)/x-(b*x+a)^(1+n)*(b*c^2*(1+n)*(a*d+b*c*(2 +n))+a*d^2*(a*d-b*c*(4+n))*x)/a/b^2/(1+n)/(2+n)/x-c^2*(b*c*n+3*a*d)*(b*x+a )^(1+n)*hypergeom([1, 1+n],[2+n],1+b*x/a)/a^2/(1+n)
Time = 0.09 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b x)^n (c+d x)^3}{x^2} \, dx=-\frac {(a+b x)^{1+n} \left (a \left (b^2 c^3 \left (2+3 n+n^2\right )+a^2 d^3 x-a b d^2 x (3 c (2+n)+d (1+n) x)\right )+b^2 c^2 (2+n) (3 a d+b c n) x \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )\right )}{a^2 b^2 (1+n) (2+n) x} \]
-(((a + b*x)^(1 + n)*(a*(b^2*c^3*(2 + 3*n + n^2) + a^2*d^3*x - a*b*d^2*x*( 3*c*(2 + n) + d*(1 + n)*x)) + b^2*c^2*(2 + n)*(3*a*d + b*c*n)*x*Hypergeome tric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a]))/(a^2*b^2*(1 + n)*(2 + n)*x))
Time = 0.30 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {111, 163, 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.
| \(\displaystyle \int \frac {(c+d x)^3 (a+b x)^n}{x^2} \, dx\) |
| \(\Big \downarrow \) 111 |
| \(\displaystyle \frac {\int \frac {(a+b x)^n (c+d x) (c (a d+b c (n+2))-d (a d-b c (n+4)) x)}{x^2}dx}{b (n+2)}+\frac {d (c+d x)^2 (a+b x)^{n+1}}{b (n+2) x}\) |
| \(\Big \downarrow \) 163 |
| \(\displaystyle \frac {\frac {b c^2 (n+2) (3 a d+b c n) \int \frac {(a+b x)^n}{x}dx}{a}-\frac {(a+b x)^{n+1} \left (b c^2 (n+1) (a d+b c (n+2))+a d^2 x (a d-b c (n+4))\right )}{a b (n+1) x}}{b (n+2)}+\frac {d (c+d x)^2 (a+b x)^{n+1}}{b (n+2) x}\) |
| \(\Big \downarrow \) 75 |
| \(\displaystyle \frac {-\frac {b c^2 (n+2) (a+b x)^{n+1} (3 a d+b c n) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b x}{a}+1\right )}{a^2 (n+1)}-\frac {(a+b x)^{n+1} \left (b c^2 (n+1) (a d+b c (n+2))+a d^2 x (a d-b c (n+4))\right )}{a b (n+1) x}}{b (n+2)}+\frac {d (c+d x)^2 (a+b x)^{n+1}}{b (n+2) x}\) |
(d*(a + b*x)^(1 + n)*(c + d*x)^2)/(b*(2 + n)*x) + (-(((a + b*x)^(1 + n)*(b *c^2*(1 + n)*(a*d + b*c*(2 + n)) + a*d^2*(a*d - b*c*(4 + n))*x))/(a*b*(1 + n)*x)) - (b*c^2*(2 + n)*(3*a*d + b*c*n)*(a + b*x)^(1 + n)*Hypergeometric2 F1[1, 1 + n, 2 + n, 1 + (b*x)/a])/(a^2*(1 + n)))/(b*(2 + n))
3.10.34.3.1 Defintions of rubi rules used
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_))^(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 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) )*((g_.) + (h_.)*(x_)), x_] :> Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n + 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(b*c - a*d)* (m + 1)*x)/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] - Simp[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f *h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c* d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2* d*(b*c - a*d)*(m + 1)*(m + n + 3)) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((GeQ[m, -2] && LtQ[m, - 1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 0]
\[\int \frac {\left (b x +a \right )^{n} \left (d x +c \right )^{3}}{x^{2}}d x\]
\[ \int \frac {(a+b x)^n (c+d x)^3}{x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{3} {\left (b x + a\right )}^{n}}{x^{2}} \,d x } \]
Time = 2.94 (sec) , antiderivative size = 478, normalized size of antiderivative = 3.34 \[ \int \frac {(a+b x)^n (c+d x)^3}{x^2} \, dx=3 c d^{2} \left (\begin {cases} a^{n} x & \text {for}\: b = 0 \\\frac {\begin {cases} \frac {\left (a + b x\right )^{n + 1}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (a + b x \right )} & \text {otherwise} \end {cases}}{b} & \text {otherwise} \end {cases}\right ) + d^{3} \left (\begin {cases} \frac {a^{n} x^{2}}{2} & \text {for}\: b = 0 \\\frac {a \log {\left (\frac {a}{b} + x \right )}}{a b^{2} + b^{3} x} + \frac {a}{a b^{2} + b^{3} x} + \frac {b x \log {\left (\frac {a}{b} + x \right )}}{a b^{2} + b^{3} x} & \text {for}\: n = -2 \\- \frac {a \log {\left (\frac {a}{b} + x \right )}}{b^{2}} + \frac {x}{b} & \text {for}\: n = -1 \\- \frac {a^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac {a b n x \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac {b^{2} n x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} + \frac {b^{2} x^{2} \left (a + b x\right )^{n}}{b^{2} n^{2} + 3 b^{2} n + 2 b^{2}} & \text {otherwise} \end {cases}\right ) - \frac {3 b^{n + 1} c^{2} d n \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} - \frac {3 b^{n + 1} c^{2} d \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} - \frac {b^{n + 2} c^{3} n \left (\frac {a}{b} + x\right )^{n + 1} \Gamma \left (n + 1\right )}{a b x \Gamma \left (n + 2\right )} - \frac {b^{n + 2} c^{3} \left (\frac {a}{b} + x\right )^{n + 1} \Gamma \left (n + 1\right )}{a b x \Gamma \left (n + 2\right )} - \frac {b^{n + 2} c^{3} n^{2} \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a^{2} \Gamma \left (n + 2\right )} - \frac {b^{n + 2} c^{3} n \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a^{2} \Gamma \left (n + 2\right )} \]
3*c*d**2*Piecewise((a**n*x, Eq(b, 0)), (Piecewise(((a + b*x)**(n + 1)/(n + 1), Ne(n, -1)), (log(a + b*x), True))/b, True)) + d**3*Piecewise((a**n*x* *2/2, Eq(b, 0)), (a*log(a/b + x)/(a*b**2 + b**3*x) + a/(a*b**2 + b**3*x) + b*x*log(a/b + x)/(a*b**2 + b**3*x), Eq(n, -2)), (-a*log(a/b + x)/b**2 + x /b, Eq(n, -1)), (-a**2*(a + b*x)**n/(b**2*n**2 + 3*b**2*n + 2*b**2) + a*b* n*x*(a + b*x)**n/(b**2*n**2 + 3*b**2*n + 2*b**2) + b**2*n*x**2*(a + b*x)** n/(b**2*n**2 + 3*b**2*n + 2*b**2) + b**2*x**2*(a + b*x)**n/(b**2*n**2 + 3* b**2*n + 2*b**2), True)) - 3*b**(n + 1)*c**2*d*n*(a/b + x)**(n + 1)*lerchp hi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2)) - 3*b**(n + 1)*c**2* d*(a/b + x)**(n + 1)*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2)) - b**(n + 2)*c**3*n*(a/b + x)**(n + 1)*gamma(n + 1)/(a*b*x*gamma(n + 2)) - b**(n + 2)*c**3*(a/b + x)**(n + 1)*gamma(n + 1)/(a*b*x*gamma(n + 2 )) - b**(n + 2)*c**3*n**2*(a/b + x)**(n + 1)*lerchphi(1 + b*x/a, 1, n + 1) *gamma(n + 1)/(a**2*gamma(n + 2)) - b**(n + 2)*c**3*n*(a/b + x)**(n + 1)*l erchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a**2*gamma(n + 2))
\[ \int \frac {(a+b x)^n (c+d x)^3}{x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{3} {\left (b x + a\right )}^{n}}{x^{2}} \,d x } \]
3*(b*x + a)^(n + 1)*c*d^2/(b*(n + 1)) + integrate((d^3*x^3 + 3*c^2*d*x + c ^3)*(b*x + a)^n/x^2, x)
\[ \int \frac {(a+b x)^n (c+d x)^3}{x^2} \, dx=\int { \frac {{\left (d x + c\right )}^{3} {\left (b x + a\right )}^{n}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {(a+b x)^n (c+d x)^3}{x^2} \, dx=\int \frac {{\left (a+b\,x\right )}^n\,{\left (c+d\,x\right )}^3}{x^2} \,d x \]