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

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)

_______________________________________________________________________________________

Rubi [A]  time = 0.114146, 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 \[ \frac{c x^{n+1} (2 a d+b c)}{n+1}+\frac{d x^{2 n+1} (a d+2 b c)}{2 n+1}+a c^2 x+\frac{b d^2 x^{3 n+1}}{3 n+1} \]

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)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.109143, size = 65, normalized size = 0.93 \[ x \left (\frac{d x^{2 n} (a d+2 b c)}{2 n+1}+\frac{c x^n (2 a d+b c)}{n+1}+a c^2+\frac{b d^2 x^{3 n}}{3 n+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

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

_______________________________________________________________________________________

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)*(d*x^n + c)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.257042, size = 236, normalized size = 3.37 \[ \frac{{\left (2 \, b d^{2} n^{2} + 3 \, b d^{2} n + b d^{2}\right )} x x^{3 \, n} +{\left (2 \, b c d + a d^{2} + 3 \,{\left (2 \, b c d + a d^{2}\right )} n^{2} + 4 \,{\left (2 \, b c d + a d^{2}\right )} n\right )} x x^{2 \, n} +{\left (b c^{2} + 2 \, a c d + 6 \,{\left (b c^{2} + 2 \, a c d\right )} n^{2} + 5 \,{\left (b c^{2} + 2 \, a c d\right )} n\right )} x x^{n} +{\left (6 \, a c^{2} n^{3} + 11 \, a c^{2} n^{2} + 6 \, a c^{2} n + a c^{2}\right )} x}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)*(d*x^n + c)^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)

_______________________________________________________________________________________

Sympy [A]  time = 2.79227, size = 726, normalized size = 10.37 \[ \begin{cases} a c^{2} x + 2 a c d \log{\left (x \right )} - \frac{a d^{2}}{x} + b c^{2} \log{\left (x \right )} - \frac{2 b c d}{x} - \frac{b d^{2}}{2 x^{2}} & \text{for}\: n = -1 \\a c^{2} x + 4 a c d \sqrt{x} + a d^{2} \log{\left (x \right )} + 2 b c^{2} \sqrt{x} + 2 b c d \log{\left (x \right )} - \frac{2 b d^{2}}{\sqrt{x}} & \text{for}\: n = - \frac{1}{2} \\a c^{2} x + 3 a c d x^{\frac{2}{3}} + 3 a d^{2} \sqrt [3]{x} + \frac{3 b c^{2} x^{\frac{2}{3}}}{2} + 6 b c d \sqrt [3]{x} + b d^{2} \log{\left (x \right )} & \text{for}\: n = - \frac{1}{3} \\\frac{6 a c^{2} n^{3} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{11 a c^{2} n^{2} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{6 a c^{2} n x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{a c^{2} x}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{12 a c d n^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{10 a c d n x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{2 a c d x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{3 a d^{2} n^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{4 a d^{2} n x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{a d^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{6 b c^{2} n^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{5 b c^{2} n x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{b c^{2} x x^{n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{6 b c d n^{2} x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{8 b c d n x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{2 b c d x x^{2 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{2 b d^{2} n^{2} x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{3 b d^{2} n x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} + \frac{b d^{2} x x^{3 n}}{6 n^{3} + 11 n^{2} + 6 n + 1} & \text{otherwise} \end{cases} \]

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))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.221124, size = 342, normalized size = 4.89 \[ \frac{6 \, a c^{2} n^{3} x + 2 \, b d^{2} n^{2} x e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 6 \, b c d n^{2} x e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 3 \, a d^{2} n^{2} x e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 6 \, b c^{2} n^{2} x e^{\left (n{\rm ln}\left (x\right )\right )} + 12 \, a c d n^{2} x e^{\left (n{\rm ln}\left (x\right )\right )} + 11 \, a c^{2} n^{2} x + 3 \, b d^{2} n x e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 8 \, b c d n x e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 4 \, a d^{2} n x e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + 5 \, b c^{2} n x e^{\left (n{\rm ln}\left (x\right )\right )} + 10 \, a c d n x e^{\left (n{\rm ln}\left (x\right )\right )} + 6 \, a c^{2} n x + b d^{2} x e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 2 \, b c d x e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + a d^{2} x e^{\left (2 \, n{\rm ln}\left (x\right )\right )} + b c^{2} x e^{\left (n{\rm ln}\left (x\right )\right )} + 2 \, a c d x e^{\left (n{\rm ln}\left (x\right )\right )} + a c^{2} x}{6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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