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*(a + b*x)^(1 + n)*Hypergeometric2F1[3, 1 + n, 2 + n, 1 + (b*x)/a])/(a^3*(1 + n)))
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Rubi [A] time = 0.0076062, antiderivative size = 38, normalized size of antiderivative = 1.,
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 = 0.0059817, size = 38, normalized size = 1. \[ -\frac{b^2 (a+b x)^{n+1} \, _2F_1\left (3,n+1;n+2;\frac{b x}{a}+1\right )}{a^3 (n+1)} \]
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)))
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n}}{{x}^{3}}}\, dx \end{align*}
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)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n}}{x^{3}}\,{d x} \end{align*}
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)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x + a\right )}^{n}}{x^{3}}, x\right ) \end{align*}
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)
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Sympy [B] time = 2.50652, size = 918, normalized size = 24.16 \begin{align*} \text{result too large to display} \end{align*}
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))
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n}}{x^{3}}\,{d x} \end{align*}
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)