3.195 \(\int \left (a+b x^n\right )^2 \left (c+d x^n\right ) \, dx\)

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 = 0.114034, 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 \[ a^2 c x+\frac{a x^{n+1} (a d+2 b c)}{n+1}+\frac{b x^{2 n+1} (2 a d+b c)}{2 n+1}+\frac{b^2 d x^{3 n+1}}{3 n+1} \]

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)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{2} \int c\, dx + \frac{a x^{n + 1} \left (a d + 2 b c\right )}{n + 1} + \frac{b^{2} d x^{3 n + 1}}{3 n + 1} + \frac{b x^{2 n + 1} \left (2 a d + b c\right )}{2 n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**n)**2*(c+d*x**n),x)

[Out]

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

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Mathematica [A]  time = 0.110819, size = 65, normalized size = 0.93 \[ x \left (a^2 c+\frac{b x^{2 n} (2 a d+b c)}{2 n+1}+\frac{a x^n (a d+2 b c)}{n+1}+\frac{b^2 d x^{3 n}}{3 n+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 74, normalized size = 1.1 \[{a}^{2}cx+{\frac{a \left ( ad+2\,bc \right ) x{{\rm e}^{n\ln \left ( x \right ) }}}{1+n}}+{\frac{b \left ( 2\,ad+bc \right ) x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{1+2\,n}}+{\frac{{b}^{2}dx \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{1+3\,n}} \]

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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.254831, size = 236, normalized size = 3.37 \[ \frac{{\left (2 \, b^{2} d n^{2} + 3 \, b^{2} d n + b^{2} d\right )} x x^{3 \, n} +{\left (b^{2} c + 2 \, a b d + 3 \,{\left (b^{2} c + 2 \, a b d\right )} n^{2} + 4 \,{\left (b^{2} c + 2 \, a b d\right )} n\right )} x x^{2 \, n} +{\left (2 \, a b c + a^{2} d + 6 \,{\left (2 \, a b c + a^{2} d\right )} n^{2} + 5 \,{\left (2 \, a b c + a^{2} d\right )} n\right )} x x^{n} +{\left (6 \, a^{2} c n^{3} + 11 \, a^{2} c n^{2} + 6 \, a^{2} c n + a^{2} c\right )} x}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^2*(d*x^n + c),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 = 2.77067, size = 726, normalized size = 10.37 \[ \begin{cases} a^{2} c x + a^{2} d \log{\left (x \right )} + 2 a b c \log{\left (x \right )} - \frac{2 a b d}{x} - \frac{b^{2} c}{x} - \frac{b^{2} d}{2 x^{2}} & \text{for}\: n = -1 \\a^{2} c x + 2 a^{2} d \sqrt{x} + 4 a b c \sqrt{x} + 2 a b d \log{\left (x \right )} + b^{2} c \log{\left (x \right )} - \frac{2 b^{2} d}{\sqrt{x}} & \text{for}\: n = - \frac{1}{2} \\a^{2} c x + \frac{3 a^{2} d x^{\frac{2}{3}}}{2} + 3 a b c x^{\frac{2}{3}} + 6 a b d \sqrt [3]{x} + 3 b^{2} c \sqrt [3]{x} + b^{2} d \log{\left (x \right )} & \text{for}\: n = - \frac{1}{3} \\\frac{6 a^{2} c n^{3} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{11 a^{2} c n^{2} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{6 a^{2} c n x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{a^{2} c x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{6 a^{2} d n^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{5 a^{2} d n x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{a^{2} d x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{12 a b c n^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{10 a b c n x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{2 a b c x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{6 a b d n^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{8 a b d n x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{2 a b d x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{3 b^{2} c n^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{4 b^{2} c n x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{b^{2} c x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{2 b^{2} d n^{2} x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{3 b^{2} d n x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{b^{2} d x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} & \text{otherwise} \end{cases} \]

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/XCAS [A]  time = 0.219823, size = 342, normalized size = 4.89 \[ \frac{6 \, a^{2} c n^{3} x + 2 \, b^{2} d n^{2} x e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 3 \, b^{2} c n^{2} x e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 6 \, a b d n^{2} x e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 12 \, a b c n^{2} x e^{\left (n{\rm ln}\left (x\right )\right )} + 6 \, a^{2} d n^{2} x e^{\left (n{\rm ln}\left (x\right )\right )} + 11 \, a^{2} c n^{2} x + 3 \, b^{2} d n x e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 4 \, b^{2} c n x e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 8 \, a b d n x e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 10 \, a b c n x e^{\left (n{\rm ln}\left (x\right )\right )} + 5 \, a^{2} d n x e^{\left (n{\rm ln}\left (x\right )\right )} + 6 \, a^{2} c n x + b^{2} d x e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + b^{2} c x e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 2 \, a b d x e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 2 \, a b c x e^{\left (n{\rm ln}\left (x\right )\right )} + a^{2} d x e^{\left (n{\rm ln}\left (x\right )\right )} + a^{2} c x}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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