3.8.53 \(\int x^{-1+n} (a+b x)^{-1-n} \, dx\) [753]

Optimal. Leaf size=19 \[ \frac {x^n (a+b x)^{-n}}{a n} \]

[Out]

x^n/a/n/((b*x+a)^n)

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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {37} \begin {gather*} \frac {x^n (a+b x)^{-n}}{a n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n)*(a + b*x)^(-1 - n),x]

[Out]

x^n/(a*n*(a + b*x)^n)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin {align*} \int x^{-1+n} (a+b x)^{-1-n} \, dx &=\frac {x^n (a+b x)^{-n}}{a n}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 19, normalized size = 1.00 \begin {gather*} \frac {x^n (a+b x)^{-n}}{a n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + n)*(a + b*x)^(-1 - n),x]

[Out]

x^n/(a*n*(a + b*x)^n)

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 202.58, size = 236, normalized size = 12.42 \begin {gather*} \text {Piecewise}\left [\left \{\left \{-\frac {x^{-1+n} \left (b x\right )^{-n}}{b},a\text {==}0\right \},\left \{\frac {x^n \text {ComplexInfinity}^{1+n}}{n},a\text {==}-b x\right \},\left \{\frac {x^n {\left (0^{\frac {1}{n}}\right )}^{-1-n}}{n},a\text {==}-b x+0^{\frac {1}{n}}\right \},\left \{\frac {\text {Log}\left [x\right ]-\text {Log}\left [\frac {a}{b}+x\right ]}{a},n\text {==}0\right \}\right \},\frac {a^2 x^n}{a^3 n \left (a+b x\right )^n+2 a^2 b n x \left (a+b x\right )^n+a b^2 n x^2 \left (a+b x\right )^n}+\frac {b x x^n}{a^2 n \left (a+b x\right )^n+a b n x \left (a+b x\right )^n}+\frac {a b x x^n}{a^3 n \left (a+b x\right )^n+2 a^2 b n x \left (a+b x\right )^n+a b^2 n x^2 \left (a+b x\right )^n}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x^(-1 + n)*(a + b*x)^(-1 - n),x]')

[Out]

Piecewise[{{-x ^ (-1 + n) (b x) ^ (-n) / b, a == 0}, {x ^ n ComplexInfinity ^ (1 + n) / n, a == -b x}, {x ^ n
(0 ^ (1 / n)) ^ (-1 - n) / n, a == -b x + 0 ^ (1 / n)}, {(Log[x] - Log[a / b + x]) / a, n == 0}}, a ^ 2 x ^ n
/ (a ^ 3 n (a + b x) ^ n + 2 a ^ 2 b n x (a + b x) ^ n + a b ^ 2 n x ^ 2 (a + b x) ^ n) + b x x ^ n / (a ^ 2 n
(a + b x) ^ n + a b n x (a + b x) ^ n) + a b x x ^ n / (a ^ 3 n (a + b x) ^ n + 2 a ^ 2 b n x (a + b x) ^ n +
a b ^ 2 n x ^ 2 (a + b x) ^ n)]

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Maple [A]
time = 0.11, size = 20, normalized size = 1.05

method	result	size
gosper	\(\frac {x^{n} \left (b x +a \right )^{-n}}{a n}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+n)*(b*x+a)^(-1-n),x,method=_RETURNVERBOSE)

[Out]

x^n*(b*x+a)^(-n)/a/n

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Maxima [A]
time = 0.33, size = 22, normalized size = 1.16 \begin {gather*} \frac {e^{\left (-n \log \left (b x + a\right ) + n \log \left (x\right )\right )}}{a n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*(b*x+a)^(-1-n),x, algorithm="maxima")

[Out]

e^(-n*log(b*x + a) + n*log(x))/(a*n)

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Fricas [A]
time = 0.32, size = 32, normalized size = 1.68 \begin {gather*} \frac {{\left (b x^{2} + a x\right )} {\left (b x + a\right )}^{-n - 1} x^{n - 1}}{a n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*(b*x+a)^(-1-n),x, algorithm="fricas")

[Out]

(b*x^2 + a*x)*(b*x + a)^(-n - 1)*x^(n - 1)/(a*n)

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Sympy [A]
time = 128.51, size = 197, normalized size = 10.37 \begin {gather*} \begin {cases} - \frac {x^{n} \left (b x\right )^{- n}}{b x} & \text {for}\: a = 0 \\\frac {0^{- n - 1} x^{n}}{n} & \text {for}\: a = - b x \\\frac {x^{n} \left (0^{\frac {1}{n}}\right )^{- n - 1}}{n} & \text {for}\: a = 0^{\frac {1}{n}} - b x \\\frac {\log {\left (x \right )}}{a} - \frac {\log {\left (\frac {a}{b} + x \right )}}{a} & \text {for}\: n = 0 \\\frac {a^{2} x^{n}}{a^{3} n \left (a + b x\right )^{n} + 2 a^{2} b n x \left (a + b x\right )^{n} + a b^{2} n x^{2} \left (a + b x\right )^{n}} + \frac {a b x x^{n}}{a^{3} n \left (a + b x\right )^{n} + 2 a^{2} b n x \left (a + b x\right )^{n} + a b^{2} n x^{2} \left (a + b x\right )^{n}} + \frac {b x x^{n}}{a^{2} n \left (a + b x\right )^{n} + a b n x \left (a + b x\right )^{n}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+n)*(b*x+a)**(-1-n),x)

[Out]

Piecewise((-x**n/(b*x*(b*x)**n), Eq(a, 0)), (0**(-n - 1)*x**n/n, Eq(a, -b*x)), (x**n*(0**(1/n))**(-n - 1)/n, E
q(a, 0**(1/n) - b*x)), (log(x)/a - log(a/b + x)/a, Eq(n, 0)), (a**2*x**n/(a**3*n*(a + b*x)**n + 2*a**2*b*n*x*(
a + b*x)**n + a*b**2*n*x**2*(a + b*x)**n) + a*b*x*x**n/(a**3*n*(a + b*x)**n + 2*a**2*b*n*x*(a + b*x)**n + a*b*
*2*n*x**2*(a + b*x)**n) + b*x*x**n/(a**2*n*(a + b*x)**n + a*b*n*x*(a + b*x)**n), True))

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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(x^(-1+n)*(b*x+a)^(-1-n),x)

[Out]

Could not integrate

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Mupad [B]
time = 0.50, size = 19, normalized size = 1.00 \begin {gather*} \frac {x^n}{a\,n\,{\left (a+b\,x\right )}^n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(n - 1)/(a + b*x)^(n + 1),x)

[Out]

x^n/(a*n*(a + b*x)^n)

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