3.8.55 \(\int x^{2 n-3 (1+n)} (a+b x)^n \, dx\) [755]

Optimal. Leaf size=58 \[ -\frac {x^{-2-n} (a+b x)^{1+n}}{a (2+n)}+\frac {b x^{-1-n} (a+b x)^{1+n}}{a^2 (1+n) (2+n)} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37} \begin {gather*} \frac {b x^{-n-1} (a+b x)^{n+1}}{a^2 (n+1) (n+2)}-\frac {x^{-n-2} (a+b x)^{n+1}}{a (n+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 47

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] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps

\begin {align*} \int x^{2 n-3 (1+n)} (a+b x)^n \, dx &=-\frac {x^{-2-n} (a+b x)^{1+n}}{a (2+n)}-\frac {b \int x^{-2-n} (a+b x)^n \, dx}{a (2+n)}\\ &=-\frac {x^{-2-n} (a+b x)^{1+n}}{a (2+n)}+\frac {b x^{-1-n} (a+b x)^{1+n}}{a^2 (1+n) (2+n)}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 40, normalized size = 0.69 \begin {gather*} -\frac {x^{-2-n} (a+a n-b x) (a+b x)^{1+n}}{a^2 (1+n) (2+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.12, size = 41, normalized size = 0.71

method	result	size
gosper	\(-\frac {\left (b x +a \right )^{1+n} x^{-2-n} \left (a n -b x +a \right )}{\left (2+n \right ) \left (1+n \right ) a^{2}}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-3-n)*(b*x+a)^n,x,method=_RETURNVERBOSE)

[Out]

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

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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(x^(-3-n)*(b*x+a)^n,x, algorithm="maxima")

[Out]

integrate((b*x + a)^n*x^(-n - 3), x)

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Fricas [A]
time = 1.02, size = 64, normalized size = 1.10 \begin {gather*} -\frac {{\left (a b n x^{2} - b^{2} x^{3} + {\left (a^{2} n + a^{2}\right )} x\right )} {\left (b x + a\right )}^{n} x^{-n - 3}}{a^{2} n^{2} + 3 \, a^{2} n + 2 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-3-n)*(b*x+a)^n,x, algorithm="fricas")

[Out]

-(a*b*n*x^2 - b^2*x^3 + (a^2*n + a^2)*x)*(b*x + a)^n*x^(-n - 3)/(a^2*n^2 + 3*a^2*n + 2*a^2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (48) = 96\).
time = 16.75, size = 328, normalized size = 5.66 \begin {gather*} \begin {cases} - \frac {x^{- n} \left (b x\right )^{n}}{2 x^{2}} & \text {for}\: a = 0 \\\frac {a \log {\left (x \right )}}{a^{3} + a^{2} b x} - \frac {a \log {\left (\frac {a}{b} + x \right )}}{a^{3} + a^{2} b x} + \frac {a}{a^{3} + a^{2} b x} + \frac {b x \log {\left (x \right )}}{a^{3} + a^{2} b x} - \frac {b x \log {\left (\frac {a}{b} + x \right )}}{a^{3} + a^{2} b x} & \text {for}\: n = -2 \\- \frac {1}{a x} - \frac {b \log {\left (x \right )}}{a^{2}} + \frac {b \log {\left (\frac {a}{b} + x \right )}}{a^{2}} & \text {for}\: n = -1 \\- \frac {a^{2} n \left (a + b x\right )^{n}}{a^{2} n^{2} x^{2} x^{n} + 3 a^{2} n x^{2} x^{n} + 2 a^{2} x^{2} x^{n}} - \frac {a^{2} \left (a + b x\right )^{n}}{a^{2} n^{2} x^{2} x^{n} + 3 a^{2} n x^{2} x^{n} + 2 a^{2} x^{2} x^{n}} - \frac {a b n x \left (a + b x\right )^{n}}{a^{2} n^{2} x^{2} x^{n} + 3 a^{2} n x^{2} x^{n} + 2 a^{2} x^{2} x^{n}} + \frac {b^{2} x^{2} \left (a + b x\right )^{n}}{a^{2} n^{2} x^{2} x^{n} + 3 a^{2} n x^{2} x^{n} + 2 a^{2} x^{2} x^{n}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-3-n)*(b*x+a)**n,x)

[Out]

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

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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(x^(-3-n)*(b*x+a)^n,x, algorithm="giac")

[Out]

integrate((b*x + a)^n*x^(-n - 3), x)

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Mupad [B]
time = 0.00, size = 86, normalized size = 1.48 \begin {gather*} -{\left (a+b\,x\right )}^n\,\left (\frac {x\,\left (n+1\right )}{x^{n+3}\,\left (n^2+3\,n+2\right )}-\frac {b^2\,x^3}{a^2\,x^{n+3}\,\left (n^2+3\,n+2\right )}+\frac {b\,n\,x^2}{a\,x^{n+3}\,\left (n^2+3\,n+2\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^n/x^(n + 3),x)

[Out]

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

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