\(\int \frac {(a+b x)^n (c+d x)}{x^2} \, dx\) [922]

Optimal result

Integrand size = 16, antiderivative size = 62 \[ \int \frac {(a+b x)^n (c+d x)}{x^2} \, dx=-\frac {c (a+b x)^{1+n}}{a x}-\frac {(a d+b c n) (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )}{a^2 (1+n)} \]

[Out]

-c*(b*x+a)^(1+n)/a/x-(b*c*n+a*d)*(b*x+a)^(1+n)*hypergeom([1, 1+n],[2+n],1+b*x/a)/a^2/(1+n)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 67} \[ \int \frac {(a+b x)^n (c+d x)}{x^2} \, dx=-\frac {(a+b x)^{n+1} (a d+b c n) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b x}{a}+1\right )}{a^2 (n+1)}-\frac {c (a+b x)^{n+1}}{a x} \]

[In]

Int[((a + b*x)^n*(c + d*x))/x^2,x]

[Out]

-((c*(a + b*x)^(1 + n))/(a*x)) - ((a*d + b*c*n)*(a + b*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)
/a])/(a^2*(1 + n))

Rule 67

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] && (Intege
rQ[m] || GtQ[-d/(b*c), 0])

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(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] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rubi steps \begin{align*} \text {integral}& = -\frac {c (a+b x)^{1+n}}{a x}+\frac {(a d+b c n) \int \frac {(a+b x)^n}{x} \, dx}{a} \\ & = -\frac {c (a+b x)^{1+n}}{a x}-\frac {(a d+b c n) (a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {b x}{a}\right )}{a^2 (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^n (c+d x)}{x^2} \, dx=-\frac {(a+b x)^{1+n} \left (a c (1+n)+(a d+b c n) x \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )\right )}{a^2 (1+n) x} \]

[In]

Integrate[((a + b*x)^n*(c + d*x))/x^2,x]

[Out]

-(((a + b*x)^(1 + n)*(a*c*(1 + n) + (a*d + b*c*n)*x*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (b*x)/a]))/(a^2*(1
+ n)*x))

Maple [F]

\[\int \frac {\left (b x +a \right )^{n} \left (d x +c \right )}{x^{2}}d x\]

[In]

int((b*x+a)^n*(d*x+c)/x^2,x)

[Out]

int((b*x+a)^n*(d*x+c)/x^2,x)

Fricas [F]

\[ \int \frac {(a+b x)^n (c+d x)}{x^2} \, dx=\int { \frac {{\left (d x + c\right )} {\left (b x + a\right )}^{n}}{x^{2}} \,d x } \]

[In]

integrate((b*x+a)^n*(d*x+c)/x^2,x, algorithm="fricas")

[Out]

integral((d*x + c)*(b*x + a)^n/x^2, x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (49) = 98\).

Time = 2.87 (sec) , antiderivative size = 223, normalized size of antiderivative = 3.60 \[ \int \frac {(a+b x)^n (c+d x)}{x^2} \, dx=- \frac {b^{n + 1} d n \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} - \frac {b^{n + 1} d \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} - \frac {b^{n + 2} c n \left (\frac {a}{b} + x\right )^{n + 1} \Gamma \left (n + 1\right )}{a b x \Gamma \left (n + 2\right )} - \frac {b^{n + 2} c \left (\frac {a}{b} + x\right )^{n + 1} \Gamma \left (n + 1\right )}{a b x \Gamma \left (n + 2\right )} - \frac {b^{n + 2} c n^{2} \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a^{2} \Gamma \left (n + 2\right )} - \frac {b^{n + 2} c n \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a^{2} \Gamma \left (n + 2\right )} \]

[In]

integrate((b*x+a)**n*(d*x+c)/x**2,x)

[Out]

-b**(n + 1)*d*n*(a/b + x)**(n + 1)*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2)) - b**(n + 1)*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*n*(a/b + x)**(n
+ 1)*gamma(n + 1)/(a*b*x*gamma(n + 2)) - b**(n + 2)*c*(a/b + x)**(n + 1)*gamma(n + 1)/(a*b*x*gamma(n + 2)) - b
**(n + 2)*c*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*n*(a/b + x)**(n + 1)*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a**2*gamma(n + 2))

Maxima [F]

\[ \int \frac {(a+b x)^n (c+d x)}{x^2} \, dx=\int { \frac {{\left (d x + c\right )} {\left (b x + a\right )}^{n}}{x^{2}} \,d x } \]

[In]

integrate((b*x+a)^n*(d*x+c)/x^2,x, algorithm="maxima")

[Out]

integrate((d*x + c)*(b*x + a)^n/x^2, x)

Giac [F]

\[ \int \frac {(a+b x)^n (c+d x)}{x^2} \, dx=\int { \frac {{\left (d x + c\right )} {\left (b x + a\right )}^{n}}{x^{2}} \,d x } \]

[In]

integrate((b*x+a)^n*(d*x+c)/x^2,x, algorithm="giac")

[Out]

integrate((d*x + c)*(b*x + a)^n/x^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^n (c+d x)}{x^2} \, dx=\int \frac {{\left (a+b\,x\right )}^n\,\left (c+d\,x\right )}{x^2} \,d x \]

[In]

int(((a + b*x)^n*(c + d*x))/x^2,x)

[Out]

int(((a + b*x)^n*(c + d*x))/x^2, x)