3.186 \(\int \left (a+b x^n\right ) \left (c+d x^n\right )^3 \, dx\)

Optimal. Leaf size=99 \[ \frac{c^2 x^{n+1} (3 a d+b c)}{n+1}+\frac{d^2 x^{3 n+1} (a d+3 b c)}{3 n+1}+\frac{3 c d x^{2 n+1} (a d+b c)}{2 n+1}+a c^3 x+\frac{b d^3 x^{4 n+1}}{4 n+1} \]

[Out]

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

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Rubi [A]  time = 0.167138, antiderivative size = 99, 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 \[ \frac{c^2 x^{n+1} (3 a d+b c)}{n+1}+\frac{d^2 x^{3 n+1} (a d+3 b c)}{3 n+1}+\frac{3 c d x^{2 n+1} (a d+b c)}{2 n+1}+a c^3 x+\frac{b d^3 x^{4 n+1}}{4 n+1} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^n)*(c + d*x^n)^3,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b d^{3} x^{4 n + 1}}{4 n + 1} + c^{3} \int a\, dx + \frac{c^{2} x^{n + 1} \left (3 a d + b c\right )}{n + 1} + \frac{3 c d x^{2 n + 1} \left (a d + b c\right )}{2 n + 1} + \frac{d^{2} x^{3 n + 1} \left (a d + 3 b c\right )}{3 n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**n)*(c+d*x**n)**3,x)

[Out]

b*d**3*x**(4*n + 1)/(4*n + 1) + c**3*Integral(a, x) + c**2*x**(n + 1)*(3*a*d + b
*c)/(n + 1) + 3*c*d*x**(2*n + 1)*(a*d + b*c)/(2*n + 1) + d**2*x**(3*n + 1)*(a*d
+ 3*b*c)/(3*n + 1)

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Mathematica [A]  time = 0.144884, size = 92, normalized size = 0.93 \[ x \left (\frac{c^2 x^n (3 a d+b c)}{n+1}+\frac{d^2 x^{3 n} (a d+3 b c)}{3 n+1}+\frac{3 c d x^{2 n} (a d+b c)}{2 n+1}+a c^3+\frac{b d^3 x^{4 n}}{4 n+1}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^n)*(c + d*x^n)^3,x]

[Out]

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

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Maple [A]  time = 0.016, size = 104, normalized size = 1.1 \[ a{c}^{3}x+{\frac{b{d}^{3}x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{1+4\,n}}+{\frac{{c}^{2} \left ( 3\,ad+bc \right ) x{{\rm e}^{n\ln \left ( x \right ) }}}{1+n}}+{\frac{{d}^{2} \left ( ad+3\,bc \right ) x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{1+3\,n}}+3\,{\frac{cd \left ( ad+bc \right ) x \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{1+2\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)*(c+d*x^n)^3,x)

[Out]

a*c^3*x+b*d^3/(1+4*n)*x*exp(n*ln(x))^4+c^2*(3*a*d+b*c)/(1+n)*x*exp(n*ln(x))+d^2*
(a*d+3*b*c)/(1+3*n)*x*exp(n*ln(x))^3+3*c*d*(a*d+b*c)/(1+2*n)*x*exp(n*ln(x))^2

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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)*(d*x^n + c)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24791, size = 431, normalized size = 4.35 \[ \frac{{\left (6 \, b d^{3} n^{3} + 11 \, b d^{3} n^{2} + 6 \, b d^{3} n + b d^{3}\right )} x x^{4 \, n} +{\left (3 \, b c d^{2} + a d^{3} + 8 \,{\left (3 \, b c d^{2} + a d^{3}\right )} n^{3} + 14 \,{\left (3 \, b c d^{2} + a d^{3}\right )} n^{2} + 7 \,{\left (3 \, b c d^{2} + a d^{3}\right )} n\right )} x x^{3 \, n} + 3 \,{\left (b c^{2} d + a c d^{2} + 12 \,{\left (b c^{2} d + a c d^{2}\right )} n^{3} + 19 \,{\left (b c^{2} d + a c d^{2}\right )} n^{2} + 8 \,{\left (b c^{2} d + a c d^{2}\right )} n\right )} x x^{2 \, n} +{\left (b c^{3} + 3 \, a c^{2} d + 24 \,{\left (b c^{3} + 3 \, a c^{2} d\right )} n^{3} + 26 \,{\left (b c^{3} + 3 \, a c^{2} d\right )} n^{2} + 9 \,{\left (b c^{3} + 3 \, a c^{2} d\right )} n\right )} x x^{n} +{\left (24 \, a c^{3} n^{4} + 50 \, a c^{3} n^{3} + 35 \, a c^{3} n^{2} + 10 \, a c^{3} n + a c^{3}\right )} x}{24 \, n^{4} + 50 \, n^{3} + 35 \, n^{2} + 10 \, n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)*(d*x^n + c)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.2599, size = 1540, normalized size = 15.56 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**n)*(c+d*x**n)**3,x)

[Out]

Piecewise((a*c**3*x + 3*a*c**2*d*log(x) - 3*a*c*d**2/x - a*d**3/(2*x**2) + b*c**
3*log(x) - 3*b*c**2*d/x - 3*b*c*d**2/(2*x**2) - b*d**3/(3*x**3), Eq(n, -1)), (a*
c**3*x + 6*a*c**2*d*sqrt(x) + 3*a*c*d**2*log(x) - 2*a*d**3/sqrt(x) + 2*b*c**3*sq
rt(x) + 3*b*c**2*d*log(x) - 6*b*c*d**2/sqrt(x) - b*d**3/x, Eq(n, -1/2)), (a*c**3
*x + 9*a*c**2*d*x**(2/3)/2 + 9*a*c*d**2*x**(1/3) + a*d**3*log(x) + 3*b*c**3*x**(
2/3)/2 + 9*b*c**2*d*x**(1/3) + 3*b*c*d**2*log(x) - 3*b*d**3/x**(1/3), Eq(n, -1/3
)), (a*c**3*x + 4*a*c**2*d*x**(3/4) + 6*a*c*d**2*sqrt(x) + 4*a*d**3*x**(1/4) + 4
*b*c**3*x**(3/4)/3 + 6*b*c**2*d*sqrt(x) + 12*b*c*d**2*x**(1/4) + b*d**3*log(x),
Eq(n, -1/4)), (24*a*c**3*n**4*x/(24*n**4 + 50*n**3 + 35*n**2 + 10*n + 1) + 50*a*
c**3*n**3*x/(24*n**4 + 50*n**3 + 35*n**2 + 10*n + 1) + 35*a*c**3*n**2*x/(24*n**4
+ 50*n**3 + 35*n**2 + 10*n + 1) + 10*a*c**3*n*x/(24*n**4 + 50*n**3 + 35*n**2 +
10*n + 1) + a*c**3*x/(24*n**4 + 50*n**3 + 35*n**2 + 10*n + 1) + 72*a*c**2*d*n**3
*x*x**n/(24*n**4 + 50*n**3 + 35*n**2 + 10*n + 1) + 78*a*c**2*d*n**2*x*x**n/(24*n
**4 + 50*n**3 + 35*n**2 + 10*n + 1) + 27*a*c**2*d*n*x*x**n/(24*n**4 + 50*n**3 +
35*n**2 + 10*n + 1) + 3*a*c**2*d*x*x**n/(24*n**4 + 50*n**3 + 35*n**2 + 10*n + 1)
+ 36*a*c*d**2*n**3*x*x**(2*n)/(24*n**4 + 50*n**3 + 35*n**2 + 10*n + 1) + 57*a*c
*d**2*n**2*x*x**(2*n)/(24*n**4 + 50*n**3 + 35*n**2 + 10*n + 1) + 24*a*c*d**2*n*x
*x**(2*n)/(24*n**4 + 50*n**3 + 35*n**2 + 10*n + 1) + 3*a*c*d**2*x*x**(2*n)/(24*n
**4 + 50*n**3 + 35*n**2 + 10*n + 1) + 8*a*d**3*n**3*x*x**(3*n)/(24*n**4 + 50*n**
3 + 35*n**2 + 10*n + 1) + 14*a*d**3*n**2*x*x**(3*n)/(24*n**4 + 50*n**3 + 35*n**2
+ 10*n + 1) + 7*a*d**3*n*x*x**(3*n)/(24*n**4 + 50*n**3 + 35*n**2 + 10*n + 1) +
a*d**3*x*x**(3*n)/(24*n**4 + 50*n**3 + 35*n**2 + 10*n + 1) + 24*b*c**3*n**3*x*x*
*n/(24*n**4 + 50*n**3 + 35*n**2 + 10*n + 1) + 26*b*c**3*n**2*x*x**n/(24*n**4 + 5
0*n**3 + 35*n**2 + 10*n + 1) + 9*b*c**3*n*x*x**n/(24*n**4 + 50*n**3 + 35*n**2 +
10*n + 1) + b*c**3*x*x**n/(24*n**4 + 50*n**3 + 35*n**2 + 10*n + 1) + 36*b*c**2*d
*n**3*x*x**(2*n)/(24*n**4 + 50*n**3 + 35*n**2 + 10*n + 1) + 57*b*c**2*d*n**2*x*x
**(2*n)/(24*n**4 + 50*n**3 + 35*n**2 + 10*n + 1) + 24*b*c**2*d*n*x*x**(2*n)/(24*
n**4 + 50*n**3 + 35*n**2 + 10*n + 1) + 3*b*c**2*d*x*x**(2*n)/(24*n**4 + 50*n**3
+ 35*n**2 + 10*n + 1) + 24*b*c*d**2*n**3*x*x**(3*n)/(24*n**4 + 50*n**3 + 35*n**2
+ 10*n + 1) + 42*b*c*d**2*n**2*x*x**(3*n)/(24*n**4 + 50*n**3 + 35*n**2 + 10*n +
1) + 21*b*c*d**2*n*x*x**(3*n)/(24*n**4 + 50*n**3 + 35*n**2 + 10*n + 1) + 3*b*c*
d**2*x*x**(3*n)/(24*n**4 + 50*n**3 + 35*n**2 + 10*n + 1) + 6*b*d**3*n**3*x*x**(4
*n)/(24*n**4 + 50*n**3 + 35*n**2 + 10*n + 1) + 11*b*d**3*n**2*x*x**(4*n)/(24*n**
4 + 50*n**3 + 35*n**2 + 10*n + 1) + 6*b*d**3*n*x*x**(4*n)/(24*n**4 + 50*n**3 + 3
5*n**2 + 10*n + 1) + b*d**3*x*x**(4*n)/(24*n**4 + 50*n**3 + 35*n**2 + 10*n + 1),
True))

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GIAC/XCAS [A]  time = 0.224958, size = 656, normalized size = 6.63 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)*(d*x^n + c)^3,x, algorithm="giac")

[Out]

(24*a*c^3*n^4*x + 6*b*d^3*n^3*x*e^(4*n*ln(x)) + 24*b*c*d^2*n^3*x*e^(3*n*ln(x)) +
8*a*d^3*n^3*x*e^(3*n*ln(x)) + 36*b*c^2*d*n^3*x*e^(2*n*ln(x)) + 36*a*c*d^2*n^3*x
*e^(2*n*ln(x)) + 24*b*c^3*n^3*x*e^(n*ln(x)) + 72*a*c^2*d*n^3*x*e^(n*ln(x)) + 50*
a*c^3*n^3*x + 11*b*d^3*n^2*x*e^(4*n*ln(x)) + 42*b*c*d^2*n^2*x*e^(3*n*ln(x)) + 14
*a*d^3*n^2*x*e^(3*n*ln(x)) + 57*b*c^2*d*n^2*x*e^(2*n*ln(x)) + 57*a*c*d^2*n^2*x*e
^(2*n*ln(x)) + 26*b*c^3*n^2*x*e^(n*ln(x)) + 78*a*c^2*d*n^2*x*e^(n*ln(x)) + 35*a*
c^3*n^2*x + 6*b*d^3*n*x*e^(4*n*ln(x)) + 21*b*c*d^2*n*x*e^(3*n*ln(x)) + 7*a*d^3*n
*x*e^(3*n*ln(x)) + 24*b*c^2*d*n*x*e^(2*n*ln(x)) + 24*a*c*d^2*n*x*e^(2*n*ln(x)) +
9*b*c^3*n*x*e^(n*ln(x)) + 27*a*c^2*d*n*x*e^(n*ln(x)) + 10*a*c^3*n*x + b*d^3*x*e
^(4*n*ln(x)) + 3*b*c*d^2*x*e^(3*n*ln(x)) + a*d^3*x*e^(3*n*ln(x)) + 3*b*c^2*d*x*e
^(2*n*ln(x)) + 3*a*c*d^2*x*e^(2*n*ln(x)) + b*c^3*x*e^(n*ln(x)) + 3*a*c^2*d*x*e^(
n*ln(x)) + a*c^3*x)/(24*n^4 + 50*n^3 + 35*n^2 + 10*n + 1)