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

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

[Out]

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

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Rubi [A]  time = 0.0451552, 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} \[ \frac{c x^{n+1} (2 a d+b c)}{n+1}+\frac{d x^{2 n+1} (a d+2 b c)}{2 n+1}+a c^2 x+\frac{b d^2 x^{3 n+1}}{3 n+1} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)*(c + d*x^n)^2,x]

[Out]

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

Mathematica [A]  time = 0.0812045, size = 70, normalized size = 1. \[ \frac{b x \left (c+d x^n\right )^3-x \left (c^2+\frac{2 c d x^n}{n+1}+\frac{d^2 x^{2 n}}{2 n+1}\right ) (b c-a d (3 n+1))}{3 d n+d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.009, size = 74, normalized size = 1.1 \begin{align*} a{c}^{2}x+{\frac{b{d}^{2}x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{1+3\,n}}+{\frac{c \left ( 2\,ad+bc \right ) x{{\rm e}^{n\ln \left ( x \right ) }}}{1+n}}+{\frac{d \left ( ad+2\,bc \right ) x \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)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.65614, size = 382, normalized size = 5.46 \begin{align*} \frac{{\left (2 \, b d^{2} n^{2} + 3 \, b d^{2} n + b d^{2}\right )} x x^{3 \, n} +{\left (2 \, b c d + a d^{2} + 3 \,{\left (2 \, b c d + a d^{2}\right )} n^{2} + 4 \,{\left (2 \, b c d + a d^{2}\right )} n\right )} x x^{2 \, n} +{\left (b c^{2} + 2 \, a c d + 6 \,{\left (b c^{2} + 2 \, a c d\right )} n^{2} + 5 \,{\left (b c^{2} + 2 \, a c d\right )} n\right )} x x^{n} +{\left (6 \, a c^{2} n^{3} + 11 \, a c^{2} n^{2} + 6 \, a c^{2} n + a c^{2}\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)*(c+d*x^n)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.927817, size = 726, normalized size = 10.37 \begin{align*} \begin{cases} a c^{2} x + 2 a c d \log{\left (x \right )} - \frac{a d^{2}}{x} + b c^{2} \log{\left (x \right )} - \frac{2 b c d}{x} - \frac{b d^{2}}{2 x^{2}} & \text{for}\: n = -1 \\a c^{2} x + 4 a c d \sqrt{x} + a d^{2} \log{\left (x \right )} + 2 b c^{2} \sqrt{x} + 2 b c d \log{\left (x \right )} - \frac{2 b d^{2}}{\sqrt{x}} & \text{for}\: n = - \frac{1}{2} \\a c^{2} x + 3 a c d x^{\frac{2}{3}} + 3 a d^{2} \sqrt [3]{x} + \frac{3 b c^{2} x^{\frac{2}{3}}}{2} + 6 b c d \sqrt [3]{x} + b d^{2} \log{\left (x \right )} & \text{for}\: n = - \frac{1}{3} \\\frac{6 a c^{2} n^{3} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{11 a c^{2} n^{2} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{6 a c^{2} n x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{a c^{2} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{12 a c d n^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{10 a c d n x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{2 a c d x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{3 a d^{2} n^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{4 a d^{2} n x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{a d^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{6 b c^{2} n^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{5 b c^{2} n x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{b c^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{6 b c d n^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{8 b c d n x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{2 b c d x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{2 b d^{2} n^{2} x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{3 b d^{2} n x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{b d^{2} 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)*(c+d*x**n)**2,x)

[Out]

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

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Giac [B]  time = 1.12922, size = 313, normalized size = 4.47 \begin{align*} \frac{6 \, a c^{2} n^{3} x + 2 \, b d^{2} n^{2} x x^{3 \, n} + 6 \, b c d n^{2} x x^{2 \, n} + 3 \, a d^{2} n^{2} x x^{2 \, n} + 6 \, b c^{2} n^{2} x x^{n} + 12 \, a c d n^{2} x x^{n} + 11 \, a c^{2} n^{2} x + 3 \, b d^{2} n x x^{3 \, n} + 8 \, b c d n x x^{2 \, n} + 4 \, a d^{2} n x x^{2 \, n} + 5 \, b c^{2} n x x^{n} + 10 \, a c d n x x^{n} + 6 \, a c^{2} n x + b d^{2} x x^{3 \, n} + 2 \, b c d x x^{2 \, n} + a d^{2} x x^{2 \, n} + b c^{2} x x^{n} + 2 \, a c d x x^{n} + a c^{2} 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)*(c+d*x^n)^2,x, algorithm="giac")

[Out]

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