3.8.37 \(\int \frac {(a+b x)^n}{x^2} \, dx\) [737]

Optimal. Leaf size=35 \[ \frac {b (a+b x)^{1+n} \, _2F_1\left (2,1+n;2+n;1+\frac {b x}{a}\right )}{a^2 (1+n)} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 35, 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 (a+b x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac {b x}{a}+1\right )}{a^2 (n+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^n/x^2,x]

[Out]

(b*(a + b*x)^(1 + n)*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (b*x)/a])/(a^2*(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^2} \, dx &=\frac {b (a+b x)^{1+n} \, _2F_1\left (2,1+n;2+n;1+\frac {b x}{a}\right )}{a^2 (1+n)}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 35, normalized size = 1.00 \begin {gather*} \frac {b (a+b x)^{1+n} \, _2F_1\left (2,1+n;2+n;1+\frac {b x}{a}\right )}{a^2 (1+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^n/x^2,x]

[Out]

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

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
time = 5.61, size = 75, normalized size = 2.14 \begin {gather*} \frac {\left (-a^2-a b x-a b n x \text {LerchPhi}\left [\frac {a+b x}{a},1,1+n\right ]-b^2 n x^2 \text {LerchPhi}\left [\frac {a+b x}{a},1,1+n\right ]\right ) \left (\frac {a+b x}{b}\right )^n b^n}{a^2 x} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(a + b*x)^n/x^2,x]')

[Out]

(-a ^ 2 - a b x - a b n x LerchPhi[(a + b x) / a, 1, 1 + n] - b ^ 2 n x ^ 2 LerchPhi[(a + b x) / a, 1, 1 + n])
((a + b x) / b) ^ n b ^ n / (a ^ 2 x)

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (b x +a \right )^{n}}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^n/x^2,x)

[Out]

int((b*x+a)^n/x^2,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^2,x, algorithm="maxima")

[Out]

integrate((b*x + a)^n/x^2, x)

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Fricas [F]
time = 0.31, 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^2,x, algorithm="fricas")

[Out]

integral((b*x + a)^n/x^2, x)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (27) = 54\).
time = 0.96, size = 354, normalized size = 10.11 \begin {gather*} \frac {a b^{2} b^{n} n^{2} \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 )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right )} + \frac {a b^{2} 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 )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right )} - \frac {a b^{2} b^{n} n \left (\frac {a}{b} + x\right ) \left (\frac {a}{b} + x\right )^{n} \Gamma \left (n + 1\right )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right )} - \frac {a b^{2} b^{n} \left (\frac {a}{b} + x\right ) \left (\frac {a}{b} + x\right )^{n} \Gamma \left (n + 1\right )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right )} - \frac {b^{3} b^{n} n^{2} \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 )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right )} - \frac {b^{3} 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 )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac {a}{b} + x\right ) \Gamma \left (n + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**n/x**2,x)

[Out]

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

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Giac [F] N/A
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^2,x)

[Out]

Could not integrate

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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^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^n/x^2,x)

[Out]

int((a + b*x)^n/x^2, x)

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