3.579 \(\int (c+d x^{-1+n}) (a+b x^n) \, dx\)

Optimal. Leaf size=41 \[ a c x+\frac{a d x^n}{n}+\frac{b c x^{n+1}}{n+1}+\frac{b d x^{2 n}}{2 n} \]

[Out]

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

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Rubi [A]  time = 0.0227337, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1891, 14} \[ a c x+\frac{a d x^n}{n}+\frac{b c x^{n+1}}{n+1}+\frac{b d x^{2 n}}{2 n} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^(-1 + n))*(a + b*x^n),x]

[Out]

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

Rule 1891

Int[((A_) + (B_.)*(x_)^(m_.))*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[A, Int[(a + b*x^n)^p, x], x] +
Dist[B, Int[x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, A, B, m, n, p}, x] && EqQ[m - n + 1, 0]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0962162, size = 42, normalized size = 1.02 \[ \frac{2 a \left (c n x+d x^n\right )+b x^n \left (\frac{2 c n x}{n+1}+d x^n\right )}{2 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.011, size = 45, normalized size = 1.1 \begin{align*} acx+{\frac{ad{{\rm e}^{n\ln \left ( x \right ) }}}{n}}+{\frac{bcx{{\rm e}^{n\ln \left ( x \right ) }}}{1+n}}+{\frac{bd \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c*x+a*d/n*exp(n*ln(x))+b*c/(1+n)*x*exp(n*ln(x))+1/2*b*d/n*exp(n*ln(x))^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.48533, size = 128, normalized size = 3.12 \begin{align*} \frac{2 \,{\left (a c n^{2} + a c n\right )} x +{\left (b d n + b d\right )} x^{2 \, n} + 2 \,{\left (b c n x + a d n + a d\right )} x^{n}}{2 \,{\left (n^{2} + n\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.50908, size = 163, normalized size = 3.98 \begin{align*} \begin{cases} a c x - \frac{a d}{x} + b c \log{\left (x \right )} - \frac{b d}{2 x^{2}} & \text{for}\: n = -1 \\\left (a + b\right ) \left (c x + d \log{\left (x \right )}\right ) & \text{for}\: n = 0 \\\frac{2 a c n^{2} x}{2 n^{2} + 2 n} + \frac{2 a c n x}{2 n^{2} + 2 n} + \frac{2 a d n x^{n}}{2 n^{2} + 2 n} + \frac{2 a d x^{n}}{2 n^{2} + 2 n} + \frac{2 b c n x x^{n}}{2 n^{2} + 2 n} + \frac{b d n x^{2 n}}{2 n^{2} + 2 n} + \frac{b d x^{2 n}}{2 n^{2} + 2 n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.05632, size = 88, normalized size = 2.15 \begin{align*} \frac{2 \, a c n^{2} x + 2 \, b c n x x^{n} + 2 \, a c n x + b d n x^{2 \, n} + 2 \, a d n x^{n} + b d x^{2 \, n} + 2 \, a d x^{n}}{2 \,{\left (n^{2} + n\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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