3.578 \(\int (c+d x^{-1+n}) (a+b x^n)^2 \, dx\)

Optimal. Leaf size=61 \[ a^2 c x+\frac{2 a b c x^{n+1}}{n+1}+\frac{d \left (a+b x^n\right )^3}{3 b n}+\frac{b^2 c x^{2 n+1}}{2 n+1} \]

[Out]

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

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Rubi [A]  time = 0.0402319, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1891, 244, 261} \[ a^2 c x+\frac{2 a b c x^{n+1}}{n+1}+\frac{d \left (a+b x^n\right )^3}{3 b n}+\frac{b^2 c x^{2 n+1}}{2 n+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*c*x + (2*a*b*c*x^(1 + n))/(1 + n) + (b^2*c*x^(1 + 2*n))/(1 + 2*n) + (d*(a + b*x^n)^3)/(3*b*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 244

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n},
x] && IGtQ[p, 0]

Rule 261

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

Rubi steps

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

Mathematica [A]  time = 0.112459, size = 120, normalized size = 1.97 \[ \frac{3 a^2 b \left (2 n^2+3 n+1\right ) \left (c n x+d x^n\right )+a^3 d \left (2 n^2+3 n+1\right )+3 a b^2 (2 n+1) x^n \left (2 c n x+d (n+1) x^n\right )+b^3 (n+1) x^{2 n} \left (3 c n x+d (2 n+1) x^n\right )}{3 b n (n+1) (2 n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.012, size = 87, normalized size = 1.4 \begin{align*}{a}^{2}cx+{\frac{{a}^{2}d{{\rm e}^{n\ln \left ( x \right ) }}}{n}}+{\frac{bda \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{n}}+{\frac{{b}^{2}cx \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{1+2\,n}}+{\frac{{b}^{2}d \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{3\,n}}+2\,{\frac{abcx{{\rm e}^{n\ln \left ( x \right ) }}}{1+n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*x^(-1+n))*(a+b*x^n)^2,x)

[Out]

a^2*c*x+a^2*d/n*exp(n*ln(x))+b*d*a/n*exp(n*ln(x))^2+b^2*c/(1+2*n)*x*exp(n*ln(x))^2+1/3*b^2*d/n*exp(n*ln(x))^3+
2*a*b*c/(1+n)*x*exp(n*ln(x))

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.53706, size = 347, normalized size = 5.69 \begin{align*} \frac{3 \,{\left (2 \, a^{2} c n^{3} + 3 \, a^{2} c n^{2} + a^{2} c n\right )} x +{\left (2 \, b^{2} d n^{2} + 3 \, b^{2} d n + b^{2} d\right )} x^{3 \, n} + 3 \,{\left (2 \, a b d n^{2} + 3 \, a b d n + a b d +{\left (b^{2} c n^{2} + b^{2} c n\right )} x\right )} x^{2 \, n} + 3 \,{\left (2 \, a^{2} d n^{2} + 3 \, a^{2} d n + a^{2} d + 2 \,{\left (2 \, a b c n^{2} + a b c n\right )} x\right )} x^{n}}{3 \,{\left (2 \, n^{3} + 3 \, 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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.25191, size = 552, normalized size = 9.05 \begin{align*} \begin{cases} a^{2} c x - \frac{a^{2} d}{x} + 2 a b c \log{\left (x \right )} - \frac{a b d}{x^{2}} - \frac{b^{2} c}{x} - \frac{b^{2} d}{3 x^{3}} & \text{for}\: n = -1 \\a^{2} c x - \frac{2 a^{2} d}{\sqrt{x}} + 4 a b c \sqrt{x} - \frac{2 a b d}{x} + b^{2} c \log{\left (x \right )} - \frac{2 b^{2} d}{3 x^{\frac{3}{2}}} & \text{for}\: n = - \frac{1}{2} \\\left (a + b\right )^{2} \left (c x + d \log{\left (x \right )}\right ) & \text{for}\: n = 0 \\\frac{6 a^{2} c n^{3} x}{6 n^{3} + 9 n^{2} + 3 n} + \frac{9 a^{2} c n^{2} x}{6 n^{3} + 9 n^{2} + 3 n} + \frac{3 a^{2} c n x}{6 n^{3} + 9 n^{2} + 3 n} + \frac{6 a^{2} d n^{2} x^{n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{9 a^{2} d n x^{n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{3 a^{2} d x^{n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{12 a b c n^{2} x x^{n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{6 a b c n x x^{n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{6 a b d n^{2} x^{2 n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{9 a b d n x^{2 n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{3 a b d x^{2 n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{3 b^{2} c n^{2} x x^{2 n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{3 b^{2} c n x x^{2 n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{2 b^{2} d n^{2} x^{3 n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{3 b^{2} d n x^{3 n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{b^{2} d x^{3 n}}{6 n^{3} + 9 n^{2} + 3 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)**2,x)

[Out]

Piecewise((a**2*c*x - a**2*d/x + 2*a*b*c*log(x) - a*b*d/x**2 - b**2*c/x - b**2*d/(3*x**3), Eq(n, -1)), (a**2*c
*x - 2*a**2*d/sqrt(x) + 4*a*b*c*sqrt(x) - 2*a*b*d/x + b**2*c*log(x) - 2*b**2*d/(3*x**(3/2)), Eq(n, -1/2)), ((a
+ b)**2*(c*x + d*log(x)), Eq(n, 0)), (6*a**2*c*n**3*x/(6*n**3 + 9*n**2 + 3*n) + 9*a**2*c*n**2*x/(6*n**3 + 9*n
**2 + 3*n) + 3*a**2*c*n*x/(6*n**3 + 9*n**2 + 3*n) + 6*a**2*d*n**2*x**n/(6*n**3 + 9*n**2 + 3*n) + 9*a**2*d*n*x*
*n/(6*n**3 + 9*n**2 + 3*n) + 3*a**2*d*x**n/(6*n**3 + 9*n**2 + 3*n) + 12*a*b*c*n**2*x*x**n/(6*n**3 + 9*n**2 + 3
*n) + 6*a*b*c*n*x*x**n/(6*n**3 + 9*n**2 + 3*n) + 6*a*b*d*n**2*x**(2*n)/(6*n**3 + 9*n**2 + 3*n) + 9*a*b*d*n*x**
(2*n)/(6*n**3 + 9*n**2 + 3*n) + 3*a*b*d*x**(2*n)/(6*n**3 + 9*n**2 + 3*n) + 3*b**2*c*n**2*x*x**(2*n)/(6*n**3 +
9*n**2 + 3*n) + 3*b**2*c*n*x*x**(2*n)/(6*n**3 + 9*n**2 + 3*n) + 2*b**2*d*n**2*x**(3*n)/(6*n**3 + 9*n**2 + 3*n)
+ 3*b**2*d*n*x**(3*n)/(6*n**3 + 9*n**2 + 3*n) + b**2*d*x**(3*n)/(6*n**3 + 9*n**2 + 3*n), True))

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Giac [B]  time = 1.07394, size = 265, normalized size = 4.34 \begin{align*} \frac{6 \, a^{2} c n^{3} x + 3 \, b^{2} c n^{2} x x^{2 \, n} + 12 \, a b c n^{2} x x^{n} + 9 \, a^{2} c n^{2} x + 2 \, b^{2} d n^{2} x^{3 \, n} + 6 \, a b d n^{2} x^{2 \, n} + 3 \, b^{2} c n x x^{2 \, n} + 6 \, a^{2} d n^{2} x^{n} + 6 \, a b c n x x^{n} + 3 \, a^{2} c n x + 3 \, b^{2} d n x^{3 \, n} + 9 \, a b d n x^{2 \, n} + 9 \, a^{2} d n x^{n} + b^{2} d x^{3 \, n} + 3 \, a b d x^{2 \, n} + 3 \, a^{2} d x^{n}}{3 \,{\left (2 \, n^{3} + 3 \, 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)^2,x, algorithm="giac")

[Out]

1/3*(6*a^2*c*n^3*x + 3*b^2*c*n^2*x*x^(2*n) + 12*a*b*c*n^2*x*x^n + 9*a^2*c*n^2*x + 2*b^2*d*n^2*x^(3*n) + 6*a*b*
d*n^2*x^(2*n) + 3*b^2*c*n*x*x^(2*n) + 6*a^2*d*n^2*x^n + 6*a*b*c*n*x*x^n + 3*a^2*c*n*x + 3*b^2*d*n*x^(3*n) + 9*
a*b*d*n*x^(2*n) + 9*a^2*d*n*x^n + b^2*d*x^(3*n) + 3*a*b*d*x^(2*n) + 3*a^2*d*x^n)/(2*n^3 + 3*n^2 + n)