3.294 \(\int (a+b x^n)^2 (c+d x^n) \, dx\)

Optimal. Leaf size=70 \[ a^2 c x+\frac{a x^{n+1} (a d+2 b c)}{n+1}+\frac{b x^{2 n+1} (2 a d+b c)}{2 n+1}+\frac{b^2 d x^{3 n+1}}{3 n+1} \]

[Out]

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

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Rubi [A]  time = 0.0471064, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {373} \[ a^2 c x+\frac{a x^{n+1} (a d+2 b c)}{n+1}+\frac{b x^{2 n+1} (2 a d+b c)}{2 n+1}+\frac{b^2 d x^{3 n+1}}{3 n+1} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^2*(c + d*x^n),x]

[Out]

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

Rule 373

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

Rubi steps

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

Mathematica [A]  time = 0.0860143, size = 70, normalized size = 1. \[ \frac{d x \left (a+b x^n\right )^3-x \left (a^2+\frac{2 a b x^n}{n+1}+\frac{b^2 x^{2 n}}{2 n+1}\right ) (a d-b (3 c n+c))}{3 b n+b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.008, size = 74, normalized size = 1.1 \begin{align*}{a}^{2}cx+{\frac{a \left ( ad+2\,bc \right ) x{{\rm e}^{n\ln \left ( x \right ) }}}{1+n}}+{\frac{b \left ( 2\,ad+bc \right ) x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{1+2\,n}}+{\frac{{b}^{2}dx \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{1+3\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((a+b*x^n)^2*(c+d*x^n),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.63465, size = 382, normalized size = 5.46 \begin{align*} \frac{{\left (2 \, b^{2} d n^{2} + 3 \, b^{2} d n + b^{2} d\right )} x x^{3 \, n} +{\left (b^{2} c + 2 \, a b d + 3 \,{\left (b^{2} c + 2 \, a b d\right )} n^{2} + 4 \,{\left (b^{2} c + 2 \, a b d\right )} n\right )} x x^{2 \, n} +{\left (2 \, a b c + a^{2} d + 6 \,{\left (2 \, a b c + a^{2} d\right )} n^{2} + 5 \,{\left (2 \, a b c + a^{2} d\right )} n\right )} x x^{n} +{\left (6 \, a^{2} c n^{3} + 11 \, a^{2} c n^{2} + 6 \, a^{2} c n + a^{2} c\right )} x}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^2*(c+d*x^n),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.970842, size = 726, normalized size = 10.37 \begin{align*} \begin{cases} a^{2} c x + a^{2} d \log{\left (x \right )} + 2 a b c \log{\left (x \right )} - \frac{2 a b d}{x} - \frac{b^{2} c}{x} - \frac{b^{2} d}{2 x^{2}} & \text{for}\: n = -1 \\a^{2} c x + 2 a^{2} d \sqrt{x} + 4 a b c \sqrt{x} + 2 a b d \log{\left (x \right )} + b^{2} c \log{\left (x \right )} - \frac{2 b^{2} d}{\sqrt{x}} & \text{for}\: n = - \frac{1}{2} \\a^{2} c x + \frac{3 a^{2} d x^{\frac{2}{3}}}{2} + 3 a b c x^{\frac{2}{3}} + 6 a b d \sqrt [3]{x} + 3 b^{2} c \sqrt [3]{x} + b^{2} d \log{\left (x \right )} & \text{for}\: n = - \frac{1}{3} \\\frac{6 a^{2} c n^{3} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{11 a^{2} c n^{2} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{6 a^{2} c n x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{a^{2} c x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{6 a^{2} d n^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{5 a^{2} d n x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{a^{2} d x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{12 a b c n^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{10 a b c n x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{2 a b c x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{6 a b d n^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{8 a b d n x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{2 a b d x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{3 b^{2} c n^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{4 b^{2} c n x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{b^{2} c x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{2 b^{2} d n^{2} x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{3 b^{2} d n x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{b^{2} d x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**n)**2*(c+d*x**n),x)

[Out]

Piecewise((a**2*c*x + a**2*d*log(x) + 2*a*b*c*log(x) - 2*a*b*d/x - b**2*c/x - b**2*d/(2*x**2), Eq(n, -1)), (a*
*2*c*x + 2*a**2*d*sqrt(x) + 4*a*b*c*sqrt(x) + 2*a*b*d*log(x) + b**2*c*log(x) - 2*b**2*d/sqrt(x), Eq(n, -1/2)),
(a**2*c*x + 3*a**2*d*x**(2/3)/2 + 3*a*b*c*x**(2/3) + 6*a*b*d*x**(1/3) + 3*b**2*c*x**(1/3) + b**2*d*log(x), Eq
(n, -1/3)), (6*a**2*c*n**3*x/(6*n**3 + 11*n**2 + 6*n + 1) + 11*a**2*c*n**2*x/(6*n**3 + 11*n**2 + 6*n + 1) + 6*
a**2*c*n*x/(6*n**3 + 11*n**2 + 6*n + 1) + a**2*c*x/(6*n**3 + 11*n**2 + 6*n + 1) + 6*a**2*d*n**2*x*x**n/(6*n**3
+ 11*n**2 + 6*n + 1) + 5*a**2*d*n*x*x**n/(6*n**3 + 11*n**2 + 6*n + 1) + a**2*d*x*x**n/(6*n**3 + 11*n**2 + 6*n
+ 1) + 12*a*b*c*n**2*x*x**n/(6*n**3 + 11*n**2 + 6*n + 1) + 10*a*b*c*n*x*x**n/(6*n**3 + 11*n**2 + 6*n + 1) + 2
*a*b*c*x*x**n/(6*n**3 + 11*n**2 + 6*n + 1) + 6*a*b*d*n**2*x*x**(2*n)/(6*n**3 + 11*n**2 + 6*n + 1) + 8*a*b*d*n*
x*x**(2*n)/(6*n**3 + 11*n**2 + 6*n + 1) + 2*a*b*d*x*x**(2*n)/(6*n**3 + 11*n**2 + 6*n + 1) + 3*b**2*c*n**2*x*x*
*(2*n)/(6*n**3 + 11*n**2 + 6*n + 1) + 4*b**2*c*n*x*x**(2*n)/(6*n**3 + 11*n**2 + 6*n + 1) + b**2*c*x*x**(2*n)/(
6*n**3 + 11*n**2 + 6*n + 1) + 2*b**2*d*n**2*x*x**(3*n)/(6*n**3 + 11*n**2 + 6*n + 1) + 3*b**2*d*n*x*x**(3*n)/(6
*n**3 + 11*n**2 + 6*n + 1) + b**2*d*x*x**(3*n)/(6*n**3 + 11*n**2 + 6*n + 1), True))

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Giac [B]  time = 1.11375, size = 313, normalized size = 4.47 \begin{align*} \frac{6 \, a^{2} c n^{3} x + 2 \, b^{2} d n^{2} x x^{3 \, n} + 3 \, b^{2} c n^{2} x x^{2 \, n} + 6 \, a b d n^{2} x x^{2 \, n} + 12 \, a b c n^{2} x x^{n} + 6 \, a^{2} d n^{2} x x^{n} + 11 \, a^{2} c n^{2} x + 3 \, b^{2} d n x x^{3 \, n} + 4 \, b^{2} c n x x^{2 \, n} + 8 \, a b d n x x^{2 \, n} + 10 \, a b c n x x^{n} + 5 \, a^{2} d n x x^{n} + 6 \, a^{2} c n x + b^{2} d x x^{3 \, n} + b^{2} c x x^{2 \, n} + 2 \, a b d x x^{2 \, n} + 2 \, a b c x x^{n} + a^{2} d x x^{n} + a^{2} c x}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^2*(c+d*x^n),x, algorithm="giac")

[Out]

(6*a^2*c*n^3*x + 2*b^2*d*n^2*x*x^(3*n) + 3*b^2*c*n^2*x*x^(2*n) + 6*a*b*d*n^2*x*x^(2*n) + 12*a*b*c*n^2*x*x^n +
6*a^2*d*n^2*x*x^n + 11*a^2*c*n^2*x + 3*b^2*d*n*x*x^(3*n) + 4*b^2*c*n*x*x^(2*n) + 8*a*b*d*n*x*x^(2*n) + 10*a*b*
c*n*x*x^n + 5*a^2*d*n*x*x^n + 6*a^2*c*n*x + b^2*d*x*x^(3*n) + b^2*c*x*x^(2*n) + 2*a*b*d*x*x^(2*n) + 2*a*b*c*x*
x^n + a^2*d*x*x^n + a^2*c*x)/(6*n^3 + 11*n^2 + 6*n + 1)