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

Optimal. Leaf size=40 \[ \frac{x^{n+1} (a d+b c)}{n+1}+a c x+\frac{b d x^{2 n+1}}{2 n+1} \]

[Out]

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

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Rubi [A]  time = 0.0209865, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {373} \[ \frac{x^{n+1} (a d+b c)}{n+1}+a c x+\frac{b d x^{2 n+1}}{2 n+1} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)*(c + d*x^n),x]

[Out]

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

Mathematica [A]  time = 0.0618715, size = 37, normalized size = 0.92 \[ x \left (\frac{x^n (a d+b c)}{n+1}+a c+\frac{b d x^{2 n}}{2 n+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^n)*(c + d*x^n),x]

[Out]

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

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Maple [A]  time = 0.006, size = 43, normalized size = 1.1 \begin{align*} acx+{\frac{ \left ( ad+bc \right ) x{{\rm e}^{n\ln \left ( x \right ) }}}{1+n}}+{\frac{bdx \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{1+2\,n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.60407, size = 155, normalized size = 3.88 \begin{align*} \frac{{\left (b d n + b d\right )} x x^{2 \, n} +{\left (b c + a d + 2 \,{\left (b c + a d\right )} n\right )} x x^{n} +{\left (2 \, a c n^{2} + 3 \, a c n + a c\right )} x}{2 \, n^{2} + 3 \, n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.439729, size = 236, normalized size = 5.9 \begin{align*} \begin{cases} a c x + a d \log{\left (x \right )} + b c \log{\left (x \right )} - \frac{b d}{x} & \text{for}\: n = -1 \\a c x + 2 a d \sqrt{x} + 2 b c \sqrt{x} + b d \log{\left (x \right )} & \text{for}\: n = - \frac{1}{2} \\\frac{2 a c n^{2} x}{2 n^{2} + 3 n + 1} + \frac{3 a c n x}{2 n^{2} + 3 n + 1} + \frac{a c x}{2 n^{2} + 3 n + 1} + \frac{2 a d n x x^{n}}{2 n^{2} + 3 n + 1} + \frac{a d x x^{n}}{2 n^{2} + 3 n + 1} + \frac{2 b c n x x^{n}}{2 n^{2} + 3 n + 1} + \frac{b c x x^{n}}{2 n^{2} + 3 n + 1} + \frac{b d n x x^{2 n}}{2 n^{2} + 3 n + 1} + \frac{b d x x^{2 n}}{2 n^{2} + 3 n + 1} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*c*x + a*d*log(x) + b*c*log(x) - b*d/x, Eq(n, -1)), (a*c*x + 2*a*d*sqrt(x) + 2*b*c*sqrt(x) + b*d*l
og(x), Eq(n, -1/2)), (2*a*c*n**2*x/(2*n**2 + 3*n + 1) + 3*a*c*n*x/(2*n**2 + 3*n + 1) + a*c*x/(2*n**2 + 3*n + 1
) + 2*a*d*n*x*x**n/(2*n**2 + 3*n + 1) + a*d*x*x**n/(2*n**2 + 3*n + 1) + 2*b*c*n*x*x**n/(2*n**2 + 3*n + 1) + b*
c*x*x**n/(2*n**2 + 3*n + 1) + b*d*n*x*x**(2*n)/(2*n**2 + 3*n + 1) + b*d*x*x**(2*n)/(2*n**2 + 3*n + 1), True))

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Giac [B]  time = 1.09448, size = 112, normalized size = 2.8 \begin{align*} \frac{2 \, a c n^{2} x + b d n x x^{2 \, n} + 2 \, b c n x x^{n} + 2 \, a d n x x^{n} + 3 \, a c n x + b d x x^{2 \, n} + b c x x^{n} + a d x x^{n} + a c x}{2 \, n^{2} + 3 \, n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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