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

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Rubi [A]  time = 0.0111185, antiderivative size = 58, normalized size of antiderivative = 1., 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 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])

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^{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*}

Mathematica [A]  time = 0.0021056, 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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Maple [A]  time = 0., size = 41, normalized size = 0.7 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{1+n}{x}^{-2-n} \left ( an-bx+a \right ) }{ \left ( 2+n \right ) \left ( 1+n \right ){a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{n} x^{-n - 3}\,{d x} \end{align*}

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 = 1.56186, size = 126, normalized size = 2.17 \begin{align*} -\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{align*}

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 [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{n} x^{-n - 3}\,{d x} \end{align*}

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)