3.6.78 \(\int (c+d x^{-1+n}) (a+b x^n)^2 \, dx\) [578]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, 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 = {1905, 250, 267} \begin {gather*} 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} \end {gather*}

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 250

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 267

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]

Rule 1905

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]

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*}

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Mathematica [A]
time = 0.11, size = 99, normalized size = 1.62 \begin {gather*} \frac {3 a^2 \left (1+3 n+2 n^2\right ) \left (c n x+d x^n\right )+3 a b (1+2 n) x^n \left (2 c n x+d (1+n) x^n\right )+b^2 (1+n) x^{2 n} \left (3 c n x+d (1+2 n) x^n\right )}{3 n (1+n) (1+2 n)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.34, size = 80, normalized size = 1.31

method	result	size
risch	\(a^{2} c x +\frac {b^{2} d \,x^{3 n}}{3 n}+\frac {b \left (n b c x +2 a d n +a d \right ) x^{2 n}}{n \left (1+2 n \right )}+\frac {a \left (2 n b c x +a d n +a d \right ) x^{n}}{n \left (1+n \right )}\)	\(80\)
norman	\(a^{2} c x +\frac {a^{2} d \,{\mathrm e}^{n \ln \left (x \right )}}{n}+\frac {a b d \,{\mathrm e}^{2 n \ln \left (x \right )}}{n}+\frac {b^{2} c x \,{\mathrm e}^{2 n \ln \left (x \right )}}{1+2 n}+\frac {b^{2} d \,{\mathrm e}^{3 n \ln \left (x \right )}}{3 n}+\frac {2 a b c x \,{\mathrm e}^{n \ln \left (x \right )}}{1+n}\)	\(87\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 78, normalized size = 1.28 \begin {gather*} a^{2} c x + \frac {b^{2} d x^{3 \, n}}{3 \, n} + \frac {a b d x^{2 \, n}}{n} + \frac {b^{2} c x^{2 \, n + 1}}{2 \, n + 1} + \frac {2 \, a b c x^{n + 1}}{n + 1} + \frac {a^{2} d x^{n}}{n} \end {gather*}

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]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (59) = 118\).
time = 0.39, size = 160, normalized size = 2.62 \begin {gather*} \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 552 vs. \(2 (53) = 106\).
time = 0.58, size = 552, normalized size = 9.05 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (59) = 118\).
time = 0.91, size = 196, normalized size = 3.21 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 5.06, size = 76, normalized size = 1.25 \begin {gather*} a^2\,c\,x+\frac {a^2\,d\,x^n}{n}+\frac {b^2\,d\,x^{3\,n}}{3\,n}+\frac {b^2\,c\,x\,x^{2\,n}}{2\,n+1}+\frac {a\,b\,d\,x^{2\,n}}{n}+\frac {2\,a\,b\,c\,x\,x^n}{n+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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