3.566 \(\int \left (c+d x^{-1+n}\right ) \left (a+b x^n\right )^2 \, dx\)

Optimal. Leaf size=61 \[ a^2 c x+\frac{2 a b c x^{n+1}}{n+1}+\frac{d \left (a+b x^n\right )^3}{3 b n}+\frac{b^2 c x^{2 n+1}}{2 n+1} \]

[Out]

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

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Rubi [A]  time = 0.0824043, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ a^2 c x+\frac{2 a b c x^{n+1}}{n+1}+\frac{d \left (a+b x^n\right )^3}{3 b n}+\frac{b^2 c x^{2 n+1}}{2 n+1} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^(-1 + n))*(a + b*x^n)^2,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c+d*x**(-1+n))*(a+b*x**n)**2,x)

[Out]

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

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Mathematica [A]  time = 0.109347, size = 99, normalized size = 1.62 \[ \frac{3 a^2 \left (2 n^2+3 n+1\right ) \left (c n x+d x^n\right )+3 a b (2 n+1) x^n \left (2 c n x+d (n+1) x^n\right )+b^2 (n+1) x^{2 n} \left (3 c n x+d (2 n+1) x^n\right )}{3 n (n+1) (2 n+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^(-1 + n))*(a + b*x^n)^2,x]

[Out]

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

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Maple [A]  time = 0.024, size = 87, normalized size = 1.4 \[{a}^{2}cx+{\frac{{a}^{2}d{{\rm e}^{n\ln \left ( x \right ) }}}{n}}+{\frac{bda \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{n}}+{\frac{{b}^{2}cx \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{1+2\,n}}+{\frac{{b}^{2}d \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{3\,n}}+2\,{\frac{abcx{{\rm e}^{n\ln \left ( x \right ) }}}{1+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d*x^(-1+n))*(a+b*x^n)^2,x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227066, size = 216, normalized size = 3.54 \[ \frac{3 \,{\left (2 \, a^{2} c n^{3} + 3 \, a^{2} c n^{2} + a^{2} c n\right )} x +{\left (2 \, b^{2} d n^{2} + 3 \, b^{2} d n + b^{2} d\right )} x^{3 \, n} + 3 \,{\left (2 \, a b d n^{2} + 3 \, a b d n + a b d +{\left (b^{2} c n^{2} + b^{2} c n\right )} x\right )} x^{2 \, n} + 3 \,{\left (2 \, a^{2} d n^{2} + 3 \, a^{2} d n + a^{2} d + 2 \,{\left (2 \, a b c n^{2} + a b c n\right )} x\right )} x^{n}}{3 \,{\left (2 \, n^{3} + 3 \, n^{2} + n\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^(n - 1) + c)*(b*x^n + a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.457, size = 552, normalized size = 9.05 \[ \begin{cases} a^{2} c x - \frac{a^{2} d}{x} + 2 a b c \log{\left (x \right )} - \frac{a b d}{x^{2}} - \frac{b^{2} c}{x} - \frac{b^{2} d}{3 x^{3}} & \text{for}\: n = -1 \\a^{2} c x - \frac{2 a^{2} d}{\sqrt{x}} + 4 a b c \sqrt{x} - \frac{2 a b d}{x} + b^{2} c \log{\left (x \right )} - \frac{2 b^{2} d}{3 x^{\frac{3}{2}}} & \text{for}\: n = - \frac{1}{2} \\\left (a + b\right )^{2} \left (c x + d \log{\left (x \right )}\right ) & \text{for}\: n = 0 \\\frac{6 a^{2} c n^{3} x}{6 n^{3} + 9 n^{2} + 3 n} + \frac{9 a^{2} c n^{2} x}{6 n^{3} + 9 n^{2} + 3 n} + \frac{3 a^{2} c n x}{6 n^{3} + 9 n^{2} + 3 n} + \frac{6 a^{2} d n^{2} x^{n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{9 a^{2} d n x^{n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{3 a^{2} d x^{n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{12 a b c n^{2} x x^{n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{6 a b c n x x^{n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{6 a b d n^{2} x^{2 n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{9 a b d n x^{2 n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{3 a b d x^{2 n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{3 b^{2} c n^{2} x x^{2 n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{3 b^{2} c n x x^{2 n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{2 b^{2} d n^{2} x^{3 n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{3 b^{2} d n x^{3 n}}{6 n^{3} + 9 n^{2} + 3 n} + \frac{b^{2} d x^{3 n}}{6 n^{3} + 9 n^{2} + 3 n} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c+d*x**(-1+n))*(a+b*x**n)**2,x)

[Out]

Piecewise((a**2*c*x - a**2*d/x + 2*a*b*c*log(x) - a*b*d/x**2 - b**2*c/x - b**2*d
/(3*x**3), Eq(n, -1)), (a**2*c*x - 2*a**2*d/sqrt(x) + 4*a*b*c*sqrt(x) - 2*a*b*d/
x + b**2*c*log(x) - 2*b**2*d/(3*x**(3/2)), Eq(n, -1/2)), ((a + b)**2*(c*x + d*lo
g(x)), Eq(n, 0)), (6*a**2*c*n**3*x/(6*n**3 + 9*n**2 + 3*n) + 9*a**2*c*n**2*x/(6*
n**3 + 9*n**2 + 3*n) + 3*a**2*c*n*x/(6*n**3 + 9*n**2 + 3*n) + 6*a**2*d*n**2*x**n
/(6*n**3 + 9*n**2 + 3*n) + 9*a**2*d*n*x**n/(6*n**3 + 9*n**2 + 3*n) + 3*a**2*d*x*
*n/(6*n**3 + 9*n**2 + 3*n) + 12*a*b*c*n**2*x*x**n/(6*n**3 + 9*n**2 + 3*n) + 6*a*
b*c*n*x*x**n/(6*n**3 + 9*n**2 + 3*n) + 6*a*b*d*n**2*x**(2*n)/(6*n**3 + 9*n**2 +
3*n) + 9*a*b*d*n*x**(2*n)/(6*n**3 + 9*n**2 + 3*n) + 3*a*b*d*x**(2*n)/(6*n**3 + 9
*n**2 + 3*n) + 3*b**2*c*n**2*x*x**(2*n)/(6*n**3 + 9*n**2 + 3*n) + 3*b**2*c*n*x*x
**(2*n)/(6*n**3 + 9*n**2 + 3*n) + 2*b**2*d*n**2*x**(3*n)/(6*n**3 + 9*n**2 + 3*n)
+ 3*b**2*d*n*x**(3*n)/(6*n**3 + 9*n**2 + 3*n) + b**2*d*x**(3*n)/(6*n**3 + 9*n**
2 + 3*n), True))

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GIAC/XCAS [A]  time = 0.224274, size = 289, normalized size = 4.74 \[ \frac{6 \, a^{2} c n^{3} x + 3 \, b^{2} c n^{2} x e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 12 \, a b c n^{2} x e^{\left (n{\rm ln}\left (x\right )\right )} + 9 \, a^{2} c n^{2} x + 2 \, b^{2} d n^{2} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 6 \, a b d n^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 3 \, b^{2} c n x e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 6 \, a^{2} d n^{2} e^{\left (n{\rm ln}\left (x\right )\right )} + 6 \, a b c n x e^{\left (n{\rm ln}\left (x\right )\right )} + 3 \, a^{2} c n x + 3 \, b^{2} d n e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 9 \, a b d n e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 9 \, a^{2} d n e^{\left (n{\rm ln}\left (x\right )\right )} + b^{2} d e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 3 \, a b d e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 3 \, a^{2} d e^{\left (n{\rm ln}\left (x\right )\right )}}{3 \,{\left (2 \, n^{3} + 3 \, n^{2} + n\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^(n - 1) + c)*(b*x^n + a)^2,x, algorithm="giac")

[Out]

1/3*(6*a^2*c*n^3*x + 3*b^2*c*n^2*x*e^(2*n*ln(x)) + 12*a*b*c*n^2*x*e^(n*ln(x)) +
9*a^2*c*n^2*x + 2*b^2*d*n^2*e^(3*n*ln(x)) + 6*a*b*d*n^2*e^(2*n*ln(x)) + 3*b^2*c*
n*x*e^(2*n*ln(x)) + 6*a^2*d*n^2*e^(n*ln(x)) + 6*a*b*c*n*x*e^(n*ln(x)) + 3*a^2*c*
n*x + 3*b^2*d*n*e^(3*n*ln(x)) + 9*a*b*d*n*e^(2*n*ln(x)) + 9*a^2*d*n*e^(n*ln(x))
+ b^2*d*e^(3*n*ln(x)) + 3*a*b*d*e^(2*n*ln(x)) + 3*a^2*d*e^(n*ln(x)))/(2*n^3 + 3*
n^2 + n)