3.755 \(\int x^{2 n-3 (1+n)} (a+b x)^n \, dx\)

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

[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 = {45, 37} \[ \frac {b x^{-n-1} (a+b x)^{n+1}}{a^2 (n+1) (n+2)}-\frac {x^{-n-2} (a+b x)^{n+1}}{a (n+2)} \]

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 45

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 \[ -\frac {x^{-n-2} (a n+a-b x) (a+b x)^{n+1}}{a^2 (n+1) (n+2)} \]

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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fricas [A]  time = 0.47, size = 64, normalized size = 1.10 \[ -\frac {{\left (a b n x^{2} - b^{2} x^{3} + {\left (a^{2} n + a^{2}\right )} x\right )} {\left (b x + a\right )}^{n} x^{-n - 3}}{a^{2} n^{2} + 3 \, a^{2} n + 2 \, a^{2}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{n} x^{-n - 3}\,{d x} \]

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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maple [A]  time = 0.00, size = 41, normalized size = 0.71 \[ -\frac {\left (a n -b x +a \right ) x^{-n -2} \left (b x +a \right )^{n +1}}{\left (n +2\right ) \left (n +1\right ) a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{n} x^{-n - 3}\,{d x} \]

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

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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sympy [A]  time = 94.49, size = 323, normalized size = 5.57 \[ \begin {cases} - \frac {b^{n}}{2 x^{2}} & \text {for}\: a = 0 \\\frac {a \log {\relax (x )}}{a^{3} + a^{2} b x} - \frac {a \log {\left (\frac {a}{b} + x \right )}}{a^{3} + a^{2} b x} + \frac {a}{a^{3} + a^{2} b x} + \frac {b x \log {\relax (x )}}{a^{3} + a^{2} b x} - \frac {b x \log {\left (\frac {a}{b} + x \right )}}{a^{3} + a^{2} b x} & \text {for}\: n = -2 \\- \frac {1}{a x} - \frac {b \log {\relax (x )}}{a^{2}} + \frac {b \log {\left (\frac {a}{b} + x \right )}}{a^{2}} & \text {for}\: n = -1 \\- \frac {a^{2} n \left (a + b x\right )^{n}}{a^{2} n^{2} x^{2} x^{n} + 3 a^{2} n x^{2} x^{n} + 2 a^{2} x^{2} x^{n}} - \frac {a^{2} \left (a + b x\right )^{n}}{a^{2} n^{2} x^{2} x^{n} + 3 a^{2} n x^{2} x^{n} + 2 a^{2} x^{2} x^{n}} - \frac {a b n x \left (a + b x\right )^{n}}{a^{2} n^{2} x^{2} x^{n} + 3 a^{2} n x^{2} x^{n} + 2 a^{2} x^{2} x^{n}} + \frac {b^{2} x^{2} \left (a + b x\right )^{n}}{a^{2} n^{2} x^{2} x^{n} + 3 a^{2} n x^{2} x^{n} + 2 a^{2} x^{2} x^{n}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-b**n/(2*x**2), 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*lo
g(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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