3.293 \(\int (a+b x^n)^2 (d+e x^n)^2 \, dx\)

Optimal. Leaf size=112 \[ \frac{x^{2 n+1} \left (a^2 e^2+4 a b d e+b^2 d^2\right )}{2 n+1}+a^2 d^2 x+\frac{2 a d x^{n+1} (a e+b d)}{n+1}+\frac{2 b e x^{3 n+1} (a e+b d)}{3 n+1}+\frac{b^2 e^2 x^{4 n+1}}{4 n+1} \]

[Out]

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

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Rubi [A]  time = 0.0816809, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {373} \[ \frac{x^{2 n+1} \left (a^2 e^2+4 a b d e+b^2 d^2\right )}{2 n+1}+a^2 d^2 x+\frac{2 a d x^{n+1} (a e+b d)}{n+1}+\frac{2 b e x^{3 n+1} (a e+b d)}{3 n+1}+\frac{b^2 e^2 x^{4 n+1}}{4 n+1} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^n)^2*(d + e*x^n)^2,x]

[Out]

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

Mathematica [A]  time = 0.157799, size = 105, normalized size = 0.94 \[ x \left (\frac{x^{2 n} \left (a^2 e^2+4 a b d e+b^2 d^2\right )}{2 n+1}+a^2 d^2+\frac{2 a d x^n (a e+b d)}{n+1}+\frac{2 b e x^{3 n} (a e+b d)}{3 n+1}+\frac{b^2 e^2 x^{4 n}}{4 n+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.01, size = 117, normalized size = 1. \begin{align*}{a}^{2}{d}^{2}x+{\frac{ \left ({a}^{2}{e}^{2}+4\,abde+{b}^{2}{d}^{2} \right ) x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{1+2\,n}}+{\frac{{b}^{2}{e}^{2}x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{1+4\,n}}+2\,{\frac{ad \left ( ae+bd \right ) x{{\rm e}^{n\ln \left ( x \right ) }}}{1+n}}+2\,{\frac{be \left ( ae+bd \right ) x \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*(d+e*x^n)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.64533, size = 787, normalized size = 7.03 \begin{align*} \frac{{\left (6 \, b^{2} e^{2} n^{3} + 11 \, b^{2} e^{2} n^{2} + 6 \, b^{2} e^{2} n + b^{2} e^{2}\right )} x x^{4 \, n} + 2 \,{\left (b^{2} d e + a b e^{2} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} n^{3} + 14 \,{\left (b^{2} d e + a b e^{2}\right )} n^{2} + 7 \,{\left (b^{2} d e + a b e^{2}\right )} n\right )} x x^{3 \, n} +{\left (b^{2} d^{2} + 4 \, a b d e + a^{2} e^{2} + 12 \,{\left (b^{2} d^{2} + 4 \, a b d e + a^{2} e^{2}\right )} n^{3} + 19 \,{\left (b^{2} d^{2} + 4 \, a b d e + a^{2} e^{2}\right )} n^{2} + 8 \,{\left (b^{2} d^{2} + 4 \, a b d e + a^{2} e^{2}\right )} n\right )} x x^{2 \, n} + 2 \,{\left (a b d^{2} + a^{2} d e + 24 \,{\left (a b d^{2} + a^{2} d e\right )} n^{3} + 26 \,{\left (a b d^{2} + a^{2} d e\right )} n^{2} + 9 \,{\left (a b d^{2} + a^{2} d e\right )} n\right )} x x^{n} +{\left (24 \, a^{2} d^{2} n^{4} + 50 \, a^{2} d^{2} n^{3} + 35 \, a^{2} d^{2} n^{2} + 10 \, a^{2} d^{2} n + a^{2} d^{2}\right )} x}{24 \, n^{4} + 50 \, n^{3} + 35 \, n^{2} + 10 \, n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((6*b^2*e^2*n^3 + 11*b^2*e^2*n^2 + 6*b^2*e^2*n + b^2*e^2)*x*x^(4*n) + 2*(b^2*d*e + a*b*e^2 + 8*(b^2*d*e + a*b*
e^2)*n^3 + 14*(b^2*d*e + a*b*e^2)*n^2 + 7*(b^2*d*e + a*b*e^2)*n)*x*x^(3*n) + (b^2*d^2 + 4*a*b*d*e + a^2*e^2 +
12*(b^2*d^2 + 4*a*b*d*e + a^2*e^2)*n^3 + 19*(b^2*d^2 + 4*a*b*d*e + a^2*e^2)*n^2 + 8*(b^2*d^2 + 4*a*b*d*e + a^2
*e^2)*n)*x*x^(2*n) + 2*(a*b*d^2 + a^2*d*e + 24*(a*b*d^2 + a^2*d*e)*n^3 + 26*(a*b*d^2 + a^2*d*e)*n^2 + 9*(a*b*d
^2 + a^2*d*e)*n)*x*x^n + (24*a^2*d^2*n^4 + 50*a^2*d^2*n^3 + 35*a^2*d^2*n^2 + 10*a^2*d^2*n + a^2*d^2)*x)/(24*n^
4 + 50*n^3 + 35*n^2 + 10*n + 1)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [B]  time = 1.13428, size = 728, normalized size = 6.5 \begin{align*} \frac{24 \, a^{2} d^{2} n^{4} x + 12 \, b^{2} d^{2} n^{3} x x^{2 \, n} + 48 \, a b d^{2} n^{3} x x^{n} + 16 \, b^{2} d n^{3} x x^{3 \, n} e + 48 \, a b d n^{3} x x^{2 \, n} e + 48 \, a^{2} d n^{3} x x^{n} e + 50 \, a^{2} d^{2} n^{3} x + 19 \, b^{2} d^{2} n^{2} x x^{2 \, n} + 52 \, a b d^{2} n^{2} x x^{n} + 6 \, b^{2} n^{3} x x^{4 \, n} e^{2} + 16 \, a b n^{3} x x^{3 \, n} e^{2} + 12 \, a^{2} n^{3} x x^{2 \, n} e^{2} + 28 \, b^{2} d n^{2} x x^{3 \, n} e + 76 \, a b d n^{2} x x^{2 \, n} e + 52 \, a^{2} d n^{2} x x^{n} e + 35 \, a^{2} d^{2} n^{2} x + 8 \, b^{2} d^{2} n x x^{2 \, n} + 18 \, a b d^{2} n x x^{n} + 11 \, b^{2} n^{2} x x^{4 \, n} e^{2} + 28 \, a b n^{2} x x^{3 \, n} e^{2} + 19 \, a^{2} n^{2} x x^{2 \, n} e^{2} + 14 \, b^{2} d n x x^{3 \, n} e + 32 \, a b d n x x^{2 \, n} e + 18 \, a^{2} d n x x^{n} e + 10 \, a^{2} d^{2} n x + b^{2} d^{2} x x^{2 \, n} + 2 \, a b d^{2} x x^{n} + 6 \, b^{2} n x x^{4 \, n} e^{2} + 14 \, a b n x x^{3 \, n} e^{2} + 8 \, a^{2} n x x^{2 \, n} e^{2} + 2 \, b^{2} d x x^{3 \, n} e + 4 \, a b d x x^{2 \, n} e + 2 \, a^{2} d x x^{n} e + a^{2} d^{2} x + b^{2} x x^{4 \, n} e^{2} + 2 \, a b x x^{3 \, n} e^{2} + a^{2} x x^{2 \, n} e^{2}}{24 \, n^{4} + 50 \, n^{3} + 35 \, n^{2} + 10 \, n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(24*a^2*d^2*n^4*x + 12*b^2*d^2*n^3*x*x^(2*n) + 48*a*b*d^2*n^3*x*x^n + 16*b^2*d*n^3*x*x^(3*n)*e + 48*a*b*d*n^3*
x*x^(2*n)*e + 48*a^2*d*n^3*x*x^n*e + 50*a^2*d^2*n^3*x + 19*b^2*d^2*n^2*x*x^(2*n) + 52*a*b*d^2*n^2*x*x^n + 6*b^
2*n^3*x*x^(4*n)*e^2 + 16*a*b*n^3*x*x^(3*n)*e^2 + 12*a^2*n^3*x*x^(2*n)*e^2 + 28*b^2*d*n^2*x*x^(3*n)*e + 76*a*b*
d*n^2*x*x^(2*n)*e + 52*a^2*d*n^2*x*x^n*e + 35*a^2*d^2*n^2*x + 8*b^2*d^2*n*x*x^(2*n) + 18*a*b*d^2*n*x*x^n + 11*
b^2*n^2*x*x^(4*n)*e^2 + 28*a*b*n^2*x*x^(3*n)*e^2 + 19*a^2*n^2*x*x^(2*n)*e^2 + 14*b^2*d*n*x*x^(3*n)*e + 32*a*b*
d*n*x*x^(2*n)*e + 18*a^2*d*n*x*x^n*e + 10*a^2*d^2*n*x + b^2*d^2*x*x^(2*n) + 2*a*b*d^2*x*x^n + 6*b^2*n*x*x^(4*n
)*e^2 + 14*a*b*n*x*x^(3*n)*e^2 + 8*a^2*n*x*x^(2*n)*e^2 + 2*b^2*d*x*x^(3*n)*e + 4*a*b*d*x*x^(2*n)*e + 2*a^2*d*x
*x^n*e + a^2*d^2*x + b^2*x*x^(4*n)*e^2 + 2*a*b*x*x^(3*n)*e^2 + a^2*x*x^(2*n)*e^2)/(24*n^4 + 50*n^3 + 35*n^2 +
10*n + 1)