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 = 0.0246275, 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 \[ -\frac{b^2 (a+b x)^{n+1} \, _2F_1\left (3,n+1;n+2;\frac{b x}{a}+1\right )}{a^3 (n+1)} \]

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)))

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Rubi in Sympy [A]  time = 3.86528, size = 31, normalized size = 0.82 \[ - \frac{b^{2} \left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 3, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a^{3} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**n/x**3,x)

[Out]

-b**2*(a + b*x)**(n + 1)*hyper((3, n + 1), (n + 2,), 1 + b*x/a)/(a**3*(n + 1))

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Mathematica [A]  time = 0.0243385, size = 53, normalized size = 1.39 \[ \frac{\left (\frac{a}{b x}+1\right )^{-n} (a+b x)^n \, _2F_1\left (2-n,-n;3-n;-\frac{a}{b x}\right )}{(n-2) x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^n/x^3,x]

[Out]

((a + b*x)^n*Hypergeometric2F1[2 - n, -n, 3 - n, -(a/(b*x))])/((-2 + n)*(1 + a/(
b*x))^n*x^2)

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n}}{{x}^{3}}}\, dx \]

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)

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

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)

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

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)

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Sympy [A]  time = 8.77164, 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)*ga
mma(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*gamma(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*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*(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*g
amma(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)*ga
mma(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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n}}{x^{3}}\,{d x} \]

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)