\(\int \frac {(a+b x)^n}{x^3} \, dx\) [738]
Optimal result
Integrand size = 11, antiderivative size = 38 \[
\int \frac {(a+b x)^n}{x^3} \, dx=-\frac {b^2 (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (3,1+n,2+n,1+\frac {b x}{a}\right )}{a^3 (1+n)}
\]
[Out]
-b^2*(b*x+a)^(1+n)*hypergeom([3, 1+n],[2+n],1+b*x/a)/a^3/(1+n)
Rubi [A] (verified)
Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00,
number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {67}
\[
\int \frac {(a+b x)^n}{x^3} \, dx=-\frac {b^2 (a+b x)^{n+1} \operatorname {Hypergeometric2F1}\left (3,n+1,n+2,\frac {b x}{a}+1\right )}{a^3 (n+1)}
\]
[In]
Int[(a + b*x)^n/x^3,x]
[Out]
-((b^2*(a + b*x)^(1 + n)*Hypergeometric2F1[3, 1 + n, 2 + n, 1 + (b*x)/a])/(a^3*(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])
Rubi steps \begin{align*}
\text {integral}& = -\frac {b^2 (a+b x)^{1+n} \, _2F_1\left (3,1+n;2+n;1+\frac {b x}{a}\right )}{a^3 (1+n)} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00
\[
\int \frac {(a+b x)^n}{x^3} \, dx=-\frac {b^2 (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (3,1+n,2+n,1+\frac {b x}{a}\right )}{a^3 (1+n)}
\]
[In]
Integrate[(a + b*x)^n/x^3,x]
[Out]
-((b^2*(a + b*x)^(1 + n)*Hypergeometric2F1[3, 1 + n, 2 + n, 1 + (b*x)/a])/(a^3*(1 + n)))
Maple [F]
\[\int \frac {\left (b x +a \right )^{n}}{x^{3}}d x\]
[In]
int((b*x+a)^n/x^3,x)
[Out]
int((b*x+a)^n/x^3,x)
Fricas [F]
\[
\int \frac {(a+b x)^n}{x^3} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{x^{3}} \,d x }
\]
[In]
integrate((b*x+a)^n/x^3,x, algorithm="fricas")
[Out]
integral((b*x + a)^n/x^3, x)
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 899 vs. \(2 (31) = 62\).
Time = 3.72 (sec) , antiderivative size = 899, normalized size of antiderivative = 23.66
\[
\int \frac {(a+b x)^n}{x^3} \, dx=\text {Too large to display}
\]
[In]
integrate((b*x+a)**n/x**3,x)
[Out]
-a**2*b**(n + 3)*n**3*(a/b + x)**(n + 1)*lerchphi(b*(a/b + x)/a, 1, n + 1)*gamma(n + 1)/(2*a**5*gamma(n + 2) -
4*a**4*b*(a/b + x)*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) + a**2*b**(n + 3)*n**2*(a/b + x)**(n
+ 1)*gamma(n + 1)/(2*a**5*gamma(n + 2) - 4*a**4*b*(a/b + x)*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n +
2)) + a**2*b**(n + 3)*n*(a/b + x)**(n + 1)*lerchphi(b*(a/b + x)/a, 1, n + 1)*gamma(n + 1)/(2*a**5*gamma(n + 2
) - 4*a**4*b*(a/b + x)*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) - a**2*b**(n + 3)*n*(a/b + x)**(n
+ 1)*gamma(n + 1)/(2*a**5*gamma(n + 2) - 4*a**4*b*(a/b + x)*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n +
2)) - 2*a**2*b**(n + 3)*(a/b + x)**(n + 1)*gamma(n + 1)/(2*a**5*gamma(n + 2) - 4*a**4*b*(a/b + x)*gamma(n + 2
) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) + 2*a*b*b**(n + 3)*n**3*(a/b + x)*(a/b + x)**(n + 1)*lerchphi(b*(a/
b + x)/a, 1, n + 1)*gamma(n + 1)/(2*a**5*gamma(n + 2) - 4*a**4*b*(a/b + x)*gamma(n + 2) + 2*a**3*b**2*(a/b + x
)**2*gamma(n + 2)) - a*b*b**(n + 3)*n**2*(a/b + x)*(a/b + x)**(n + 1)*gamma(n + 1)/(2*a**5*gamma(n + 2) - 4*a*
*4*b*(a/b + x)*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) - 2*a*b*b**(n + 3)*n*(a/b + x)*(a/b + x)*
*(n + 1)*lerchphi(b*(a/b + x)/a, 1, n + 1)*gamma(n + 1)/(2*a**5*gamma(n + 2) - 4*a**4*b*(a/b + x)*gamma(n + 2)
+ 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) + a*b*b**(n + 3)*(a/b + x)*(a/b + x)**(n + 1)*gamma(n + 1)/(2*a**5*g
amma(n + 2) - 4*a**4*b*(a/b + x)*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) - b**2*b**(n + 3)*n**3*
(a/b + x)**2*(a/b + x)**(n + 1)*lerchphi(b*(a/b + x)/a, 1, n + 1)*gamma(n + 1)/(2*a**5*gamma(n + 2) - 4*a**4*b
*(a/b + x)*gamma(n + 2) + 2*a**3*b**2*(a/b + x)**2*gamma(n + 2)) + b**2*b**(n + 3)*n*(a/b + x)**2*(a/b + x)**(
n + 1)*lerchphi(b*(a/b + x)/a, 1, n + 1)*gamma(n + 1)/(2*a**5*gamma(n + 2) - 4*a**4*b*(a/b + x)*gamma(n + 2) +
2*a**3*b**2*(a/b + x)**2*gamma(n + 2))
Maxima [F]
\[
\int \frac {(a+b x)^n}{x^3} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{x^{3}} \,d x }
\]
[In]
integrate((b*x+a)^n/x^3,x, algorithm="maxima")
[Out]
integrate((b*x + a)^n/x^3, x)
Giac [F]
\[
\int \frac {(a+b x)^n}{x^3} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{x^{3}} \,d x }
\]
[In]
integrate((b*x+a)^n/x^3,x, algorithm="giac")
[Out]
integrate((b*x + a)^n/x^3, x)
Mupad [F(-1)]
Timed out. \[
\int \frac {(a+b x)^n}{x^3} \, dx=\int \frac {{\left (a+b\,x\right )}^n}{x^3} \,d x
\]
[In]
int((a + b*x)^n/x^3,x)
[Out]
int((a + b*x)^n/x^3, x)