3.577 \(\int (c+d x^{-1+n}) (a+b x^n)^3 \, dx\)

Optimal. Leaf size=84 \[ \frac{3 a^2 b c x^{n+1}}{n+1}+a^3 c x+\frac{3 a b^2 c x^{2 n+1}}{2 n+1}+\frac{d \left (a+b x^n\right )^4}{4 b n}+\frac{b^3 c x^{3 n+1}}{3 n+1} \]

[Out]

a^3*c*x + (3*a^2*b*c*x^(1 + n))/(1 + n) + (3*a*b^2*c*x^(1 + 2*n))/(1 + 2*n) + (b^3*c*x^(1 + 3*n))/(1 + 3*n) +
(d*(a + b*x^n)^4)/(4*b*n)

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Rubi [A]  time = 0.0557268, antiderivative size = 84, 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} \[ \frac{3 a^2 b c x^{n+1}}{n+1}+a^3 c x+\frac{3 a b^2 c x^{2 n+1}}{2 n+1}+\frac{d \left (a+b x^n\right )^4}{4 b n}+\frac{b^3 c x^{3 n+1}}{3 n+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^3*c*x + (3*a^2*b*c*x^(1 + n))/(1 + n) + (3*a*b^2*c*x^(1 + 2*n))/(1 + 2*n) + (b^3*c*x^(1 + 3*n))/(1 + 3*n) +
(d*(a + b*x^n)^4)/(4*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 )^3 \, dx &=c \int \left (a+b x^n\right )^3 \, dx+d \int x^{-1+n} \left (a+b x^n\right )^3 \, dx\\ &=\frac{d \left (a+b x^n\right )^4}{4 b n}+c \int \left (a^3+3 a^2 b x^n+3 a b^2 x^{2 n}+b^3 x^{3 n}\right ) \, dx\\ &=a^3 c x+\frac{3 a^2 b c x^{1+n}}{1+n}+\frac{3 a b^2 c x^{1+2 n}}{1+2 n}+\frac{b^3 c x^{1+3 n}}{1+3 n}+\frac{d \left (a+b x^n\right )^4}{4 b n}\\ \end{align*}

Mathematica [A]  time = 0.163315, size = 108, normalized size = 1.29 \[ \frac{x \left (c+d x^{n-1}\right ) \left (\frac{12 a^2 b c x^{n+1}}{n+1}+4 a^3 c x+\frac{12 a b^2 c x^{2 n+1}}{2 n+1}+\frac{d \left (a+b x^n\right )^4}{b n}+\frac{4 b^3 c x^{3 n+1}}{3 n+1}\right )}{4 \left (c x+d x^n\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(c + d*x^(-1 + n))*(4*a^3*c*x + (12*a^2*b*c*x^(1 + n))/(1 + n) + (12*a*b^2*c*x^(1 + 2*n))/(1 + 2*n) + (4*b^
3*c*x^(1 + 3*n))/(1 + 3*n) + (d*(a + b*x^n)^4)/(b*n)))/(4*(c*x + d*x^n))

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Maple [A]  time = 0.015, size = 130, normalized size = 1.6 \begin{align*}{a}^{3}cx+{\frac{{a}^{3}d{{\rm e}^{n\ln \left ( x \right ) }}}{n}}+{\frac{a{b}^{2}d \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{n}}+{\frac{{b}^{3}cx \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{1+3\,n}}+{\frac{{b}^{3}d \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{4\,n}}+{\frac{3\,{a}^{2}bd \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{2\,n}}+3\,{\frac{ac{b}^{2}x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{1+2\,n}}+3\,{\frac{{a}^{2}bcx{{\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)^3,x)

[Out]

a^3*c*x+a^3*d/n*exp(n*ln(x))+a*b^2*d/n*exp(n*ln(x))^3+b^3*c/(1+3*n)*x*exp(n*ln(x))^3+1/4*b^3*d/n*exp(n*ln(x))^
4+3/2*a^2*b*d/n*exp(n*ln(x))^2+3*a*c*b^2/(1+2*n)*x*exp(n*ln(x))^2+3*b*a^2*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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.54533, size = 660, normalized size = 7.86 \begin{align*} \frac{4 \,{\left (6 \, a^{3} c n^{4} + 11 \, a^{3} c n^{3} + 6 \, a^{3} c n^{2} + a^{3} c n\right )} x +{\left (6 \, b^{3} d n^{3} + 11 \, b^{3} d n^{2} + 6 \, b^{3} d n + b^{3} d\right )} x^{4 \, n} + 4 \,{\left (6 \, a b^{2} d n^{3} + 11 \, a b^{2} d n^{2} + 6 \, a b^{2} d n + a b^{2} d +{\left (2 \, b^{3} c n^{3} + 3 \, b^{3} c n^{2} + b^{3} c n\right )} x\right )} x^{3 \, n} + 6 \,{\left (6 \, a^{2} b d n^{3} + 11 \, a^{2} b d n^{2} + 6 \, a^{2} b d n + a^{2} b d + 2 \,{\left (3 \, a b^{2} c n^{3} + 4 \, a b^{2} c n^{2} + a b^{2} c n\right )} x\right )} x^{2 \, n} + 4 \,{\left (6 \, a^{3} d n^{3} + 11 \, a^{3} d n^{2} + 6 \, a^{3} d n + a^{3} d + 3 \,{\left (6 \, a^{2} b c n^{3} + 5 \, a^{2} b c n^{2} + a^{2} b c n\right )} x\right )} x^{n}}{4 \,{\left (6 \, n^{4} + 11 \, n^{3} + 6 \, 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)^3,x, algorithm="fricas")

[Out]

1/4*(4*(6*a^3*c*n^4 + 11*a^3*c*n^3 + 6*a^3*c*n^2 + a^3*c*n)*x + (6*b^3*d*n^3 + 11*b^3*d*n^2 + 6*b^3*d*n + b^3*
d)*x^(4*n) + 4*(6*a*b^2*d*n^3 + 11*a*b^2*d*n^2 + 6*a*b^2*d*n + a*b^2*d + (2*b^3*c*n^3 + 3*b^3*c*n^2 + b^3*c*n)
*x)*x^(3*n) + 6*(6*a^2*b*d*n^3 + 11*a^2*b*d*n^2 + 6*a^2*b*d*n + a^2*b*d + 2*(3*a*b^2*c*n^3 + 4*a*b^2*c*n^2 + a
*b^2*c*n)*x)*x^(2*n) + 4*(6*a^3*d*n^3 + 11*a^3*d*n^2 + 6*a^3*d*n + a^3*d + 3*(6*a^2*b*c*n^3 + 5*a^2*b*c*n^2 +
a^2*b*c*n)*x)*x^n)/(6*n^4 + 11*n^3 + 6*n^2 + n)

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Sympy [A]  time = 6.76931, size = 1251, normalized size = 14.89 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*x**(-1+n))*(a+b*x**n)**3,x)

[Out]

Piecewise((a**3*c*x - a**3*d/x + 3*a**2*b*c*log(x) - 3*a**2*b*d/(2*x**2) - 3*a*b**2*c/x - a*b**2*d/x**3 - b**3
*c/(2*x**2) - b**3*d/(4*x**4), Eq(n, -1)), (a**3*c*x - 2*a**3*d/sqrt(x) + 6*a**2*b*c*sqrt(x) - 3*a**2*b*d/x +
3*a*b**2*c*log(x) - 2*a*b**2*d/x**(3/2) - 2*b**3*c/sqrt(x) - b**3*d/(2*x**2), Eq(n, -1/2)), (a**3*c*x - 3*a**3
*d/x**(1/3) + 9*a**2*b*c*x**(2/3)/2 - 9*a**2*b*d/(2*x**(2/3)) + 9*a*b**2*c*x**(1/3) - 3*a*b**2*d/x + b**3*c*lo
g(x) - 3*b**3*d/(4*x**(4/3)), Eq(n, -1/3)), ((a + b)**3*(c*x + d*log(x)), Eq(n, 0)), (24*a**3*c*n**4*x/(24*n**
4 + 44*n**3 + 24*n**2 + 4*n) + 44*a**3*c*n**3*x/(24*n**4 + 44*n**3 + 24*n**2 + 4*n) + 24*a**3*c*n**2*x/(24*n**
4 + 44*n**3 + 24*n**2 + 4*n) + 4*a**3*c*n*x/(24*n**4 + 44*n**3 + 24*n**2 + 4*n) + 24*a**3*d*n**3*x**n/(24*n**4
+ 44*n**3 + 24*n**2 + 4*n) + 44*a**3*d*n**2*x**n/(24*n**4 + 44*n**3 + 24*n**2 + 4*n) + 24*a**3*d*n*x**n/(24*n
**4 + 44*n**3 + 24*n**2 + 4*n) + 4*a**3*d*x**n/(24*n**4 + 44*n**3 + 24*n**2 + 4*n) + 72*a**2*b*c*n**3*x*x**n/(
24*n**4 + 44*n**3 + 24*n**2 + 4*n) + 60*a**2*b*c*n**2*x*x**n/(24*n**4 + 44*n**3 + 24*n**2 + 4*n) + 12*a**2*b*c
*n*x*x**n/(24*n**4 + 44*n**3 + 24*n**2 + 4*n) + 36*a**2*b*d*n**3*x**(2*n)/(24*n**4 + 44*n**3 + 24*n**2 + 4*n)
+ 66*a**2*b*d*n**2*x**(2*n)/(24*n**4 + 44*n**3 + 24*n**2 + 4*n) + 36*a**2*b*d*n*x**(2*n)/(24*n**4 + 44*n**3 +
24*n**2 + 4*n) + 6*a**2*b*d*x**(2*n)/(24*n**4 + 44*n**3 + 24*n**2 + 4*n) + 36*a*b**2*c*n**3*x*x**(2*n)/(24*n**
4 + 44*n**3 + 24*n**2 + 4*n) + 48*a*b**2*c*n**2*x*x**(2*n)/(24*n**4 + 44*n**3 + 24*n**2 + 4*n) + 12*a*b**2*c*n
*x*x**(2*n)/(24*n**4 + 44*n**3 + 24*n**2 + 4*n) + 24*a*b**2*d*n**3*x**(3*n)/(24*n**4 + 44*n**3 + 24*n**2 + 4*n
) + 44*a*b**2*d*n**2*x**(3*n)/(24*n**4 + 44*n**3 + 24*n**2 + 4*n) + 24*a*b**2*d*n*x**(3*n)/(24*n**4 + 44*n**3
+ 24*n**2 + 4*n) + 4*a*b**2*d*x**(3*n)/(24*n**4 + 44*n**3 + 24*n**2 + 4*n) + 8*b**3*c*n**3*x*x**(3*n)/(24*n**4
+ 44*n**3 + 24*n**2 + 4*n) + 12*b**3*c*n**2*x*x**(3*n)/(24*n**4 + 44*n**3 + 24*n**2 + 4*n) + 4*b**3*c*n*x*x**
(3*n)/(24*n**4 + 44*n**3 + 24*n**2 + 4*n) + 6*b**3*d*n**3*x**(4*n)/(24*n**4 + 44*n**3 + 24*n**2 + 4*n) + 11*b*
*3*d*n**2*x**(4*n)/(24*n**4 + 44*n**3 + 24*n**2 + 4*n) + 6*b**3*d*n*x**(4*n)/(24*n**4 + 44*n**3 + 24*n**2 + 4*
n) + b**3*d*x**(4*n)/(24*n**4 + 44*n**3 + 24*n**2 + 4*n), True))

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Giac [B]  time = 1.06869, size = 529, normalized size = 6.3 \begin{align*} \frac{24 \, a^{3} c n^{4} x + 8 \, b^{3} c n^{3} x x^{3 \, n} + 36 \, a b^{2} c n^{3} x x^{2 \, n} + 72 \, a^{2} b c n^{3} x x^{n} + 44 \, a^{3} c n^{3} x + 6 \, b^{3} d n^{3} x^{4 \, n} + 24 \, a b^{2} d n^{3} x^{3 \, n} + 12 \, b^{3} c n^{2} x x^{3 \, n} + 36 \, a^{2} b d n^{3} x^{2 \, n} + 48 \, a b^{2} c n^{2} x x^{2 \, n} + 24 \, a^{3} d n^{3} x^{n} + 60 \, a^{2} b c n^{2} x x^{n} + 24 \, a^{3} c n^{2} x + 11 \, b^{3} d n^{2} x^{4 \, n} + 44 \, a b^{2} d n^{2} x^{3 \, n} + 4 \, b^{3} c n x x^{3 \, n} + 66 \, a^{2} b d n^{2} x^{2 \, n} + 12 \, a b^{2} c n x x^{2 \, n} + 44 \, a^{3} d n^{2} x^{n} + 12 \, a^{2} b c n x x^{n} + 4 \, a^{3} c n x + 6 \, b^{3} d n x^{4 \, n} + 24 \, a b^{2} d n x^{3 \, n} + 36 \, a^{2} b d n x^{2 \, n} + 24 \, a^{3} d n x^{n} + b^{3} d x^{4 \, n} + 4 \, a b^{2} d x^{3 \, n} + 6 \, a^{2} b d x^{2 \, n} + 4 \, a^{3} d x^{n}}{4 \,{\left (6 \, n^{4} + 11 \, n^{3} + 6 \, 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)^3,x, algorithm="giac")

[Out]

1/4*(24*a^3*c*n^4*x + 8*b^3*c*n^3*x*x^(3*n) + 36*a*b^2*c*n^3*x*x^(2*n) + 72*a^2*b*c*n^3*x*x^n + 44*a^3*c*n^3*x
+ 6*b^3*d*n^3*x^(4*n) + 24*a*b^2*d*n^3*x^(3*n) + 12*b^3*c*n^2*x*x^(3*n) + 36*a^2*b*d*n^3*x^(2*n) + 48*a*b^2*c
*n^2*x*x^(2*n) + 24*a^3*d*n^3*x^n + 60*a^2*b*c*n^2*x*x^n + 24*a^3*c*n^2*x + 11*b^3*d*n^2*x^(4*n) + 44*a*b^2*d*
n^2*x^(3*n) + 4*b^3*c*n*x*x^(3*n) + 66*a^2*b*d*n^2*x^(2*n) + 12*a*b^2*c*n*x*x^(2*n) + 44*a^3*d*n^2*x^n + 12*a^
2*b*c*n*x*x^n + 4*a^3*c*n*x + 6*b^3*d*n*x^(4*n) + 24*a*b^2*d*n*x^(3*n) + 36*a^2*b*d*n*x^(2*n) + 24*a^3*d*n*x^n
+ b^3*d*x^(4*n) + 4*a*b^2*d*x^(3*n) + 6*a^2*b*d*x^(2*n) + 4*a^3*d*x^n)/(6*n^4 + 11*n^3 + 6*n^2 + n)