3.738 \(\int \frac {(a+b x)^n}{x^3} \, dx\)

Optimal. Leaf size=38 \[ -\frac {b^2 (a+b x)^{n+1} \, _2F_1\left (3,n+1;n+2;\frac {b x}{a}+1\right )}{a^3 (n+1)} \]

[Out]

-b^2*(b*x+a)^(1+n)*hypergeom([3, 1+n],[2+n],1+b*x/a)/a^3/(1+n)

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {65} \[ -\frac {b^2 (a+b x)^{n+1} \, _2F_1\left (3,n+1;n+2;\frac {b x}{a}+1\right )}{a^3 (n+1)} \]

Antiderivative was successfully verified.

[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 65

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)*Hypergeometric2F1[-m, n +
1, n + 2, 1 + (d*x)/c])/(d*(n + 1)*(-(d/(b*c)))^m), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Inte
gerQ[m] || GtQ[-(d/(b*c)), 0])

Rubi steps

\begin {align*} \int \frac {(a+b x)^n}{x^3} \, dx &=-\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]  time = 0.01, size = 38, normalized size = 1.00 \[ -\frac {b^2 (a+b x)^{n+1} \, _2F_1\left (3,n+1;n+2;\frac {b x}{a}+1\right )}{a^3 (n+1)} \]

Antiderivative was successfully verified.

[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)))

________________________________________________________________________________________

fricas [F]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x + a\right )}^{n}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^n/x^3,x, algorithm="fricas")

[Out]

integral((b*x + a)^n/x^3, x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{n}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^n/x^3,x, algorithm="giac")

[Out]

integrate((b*x + a)^n/x^3, x)

________________________________________________________________________________________

maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{n}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^n/x^3,x)

[Out]

int((b*x+a)^n/x^3,x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x + a\right )}^{n}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^n/x^3,x, algorithm="maxima")

[Out]

integrate((b*x + a)^n/x^3, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (a+b\,x\right )}^n}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^n/x^3,x)

[Out]

int((a + b*x)^n/x^3, x)

________________________________________________________________________________________

sympy [B]  time = 2.84, size = 918, normalized size = 24.16 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**n/x**3,x)

[Out]

-a**2*b**3*b**n*n**3*(a/b + x)*(a/b + x)**n*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**3*b**n*n**2*(a/b + x)*(
a/b + x)**n*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*gam
ma(n + 2)) + a**2*b**3*b**n*n*(a/b + x)*(a/b + x)**n*lerchphi(b*(a/b + x)/a, 1, n + 1)*gamma(n + 1)/(2*a**5*ga
mma(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**3*b**n*n*(a/b
+ x)*(a/b + x)**n*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**3*b**n*(a/b + x)*(a/b + x)**n*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**4*b**n*n**3*(a/b + x)**2*(a/b + x)**n*lerch
phi(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**4*b**n*n**2*(a/b + x)**2*(a/b + x)**n*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**4*b**n*n*(a/b + x)**2*(a/b +
x)**n*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**4*b**n*(a/b + x)**2*(a/b + x)**n*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**5*b**n*n**3*(a/b + x)**3*(a
/b + x)**n*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**5*b**n*n*(a/b + x)**3*(a/b + x)**n*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))

________________________________________________________________________________________