3.8.38 \(\int \frac {(a+b x)^n}{x^3} \, dx\) [738]

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

[Out]

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

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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 = {67} \begin {gather*} -\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)} \end {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.02, size = 38, normalized size = 1.00 \begin {gather*} -\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 {gather*}

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 = 0.03, 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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 918 vs. \(2 (31) = 62\).
time = 2.02, size = 918, normalized size = 24.16 \begin {gather*} - \frac {a^{2} b^{3} b^{n} n^{3} \left (\frac {a}{b} + x\right ) \left (\frac {a}{b} + x\right )^{n} \Phi \left (\frac {b \left (\frac {a}{b} + x\right )}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{2 a^{5} \Gamma \left (n + 2\right ) - 4 a^{4} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right ) + 2 a^{3} b^{2} \left (\frac {a}{b} + x\right )^{2} \Gamma \left (n + 2\right )} + \frac {a^{2} b^{3} b^{n} n^{2} \left (\frac {a}{b} + x\right ) \left (\frac {a}{b} + x\right )^{n} \Gamma \left (n + 1\right )}{2 a^{5} \Gamma \left (n + 2\right ) - 4 a^{4} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right ) + 2 a^{3} b^{2} \left (\frac {a}{b} + x\right )^{2} \Gamma \left (n + 2\right )} + \frac {a^{2} b^{3} b^{n} n \left (\frac {a}{b} + x\right ) \left (\frac {a}{b} + x\right )^{n} \Phi \left (\frac {b \left (\frac {a}{b} + x\right )}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{2 a^{5} \Gamma \left (n + 2\right ) - 4 a^{4} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right ) + 2 a^{3} b^{2} \left (\frac {a}{b} + x\right )^{2} \Gamma \left (n + 2\right )} - \frac {a^{2} b^{3} b^{n} n \left (\frac {a}{b} + x\right ) \left (\frac {a}{b} + x\right )^{n} \Gamma \left (n + 1\right )}{2 a^{5} \Gamma \left (n + 2\right ) - 4 a^{4} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right ) + 2 a^{3} b^{2} \left (\frac {a}{b} + x\right )^{2} \Gamma \left (n + 2\right )} - \frac {2 a^{2} b^{3} b^{n} \left (\frac {a}{b} + x\right ) \left (\frac {a}{b} + x\right )^{n} \Gamma \left (n + 1\right )}{2 a^{5} \Gamma \left (n + 2\right ) - 4 a^{4} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right ) + 2 a^{3} b^{2} \left (\frac {a}{b} + x\right )^{2} \Gamma \left (n + 2\right )} + \frac {2 a b^{4} b^{n} n^{3} \left (\frac {a}{b} + x\right )^{2} \left (\frac {a}{b} + x\right )^{n} \Phi \left (\frac {b \left (\frac {a}{b} + x\right )}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{2 a^{5} \Gamma \left (n + 2\right ) - 4 a^{4} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right ) + 2 a^{3} b^{2} \left (\frac {a}{b} + x\right )^{2} \Gamma \left (n + 2\right )} - \frac {a b^{4} b^{n} n^{2} \left (\frac {a}{b} + x\right )^{2} \left (\frac {a}{b} + x\right )^{n} \Gamma \left (n + 1\right )}{2 a^{5} \Gamma \left (n + 2\right ) - 4 a^{4} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right ) + 2 a^{3} b^{2} \left (\frac {a}{b} + x\right )^{2} \Gamma \left (n + 2\right )} - \frac {2 a b^{4} b^{n} n \left (\frac {a}{b} + x\right )^{2} \left (\frac {a}{b} + x\right )^{n} \Phi \left (\frac {b \left (\frac {a}{b} + x\right )}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{2 a^{5} \Gamma \left (n + 2\right ) - 4 a^{4} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right ) + 2 a^{3} b^{2} \left (\frac {a}{b} + x\right )^{2} \Gamma \left (n + 2\right )} + \frac {a b^{4} b^{n} \left (\frac {a}{b} + x\right )^{2} \left (\frac {a}{b} + x\right )^{n} \Gamma \left (n + 1\right )}{2 a^{5} \Gamma \left (n + 2\right ) - 4 a^{4} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right ) + 2 a^{3} b^{2} \left (\frac {a}{b} + x\right )^{2} \Gamma \left (n + 2\right )} - \frac {b^{5} b^{n} n^{3} \left (\frac {a}{b} + x\right )^{3} \left (\frac {a}{b} + x\right )^{n} \Phi \left (\frac {b \left (\frac {a}{b} + x\right )}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{2 a^{5} \Gamma \left (n + 2\right ) - 4 a^{4} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right ) + 2 a^{3} b^{2} \left (\frac {a}{b} + x\right )^{2} \Gamma \left (n + 2\right )} + \frac {b^{5} b^{n} n \left (\frac {a}{b} + x\right )^{3} \left (\frac {a}{b} + x\right )^{n} \Phi \left (\frac {b \left (\frac {a}{b} + x\right )}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{2 a^{5} \Gamma \left (n + 2\right ) - 4 a^{4} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right ) + 2 a^{3} b^{2} \left (\frac {a}{b} + x\right )^{2} \Gamma \left (n + 2\right )} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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

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)

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