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

Optimal. Leaf size=94 \[ \frac{\left (b^3 c-a^3 d\right ) (a+b x)^{n+1}}{b^4 (n+1)}+\frac{3 a^2 d (a+b x)^{n+2}}{b^4 (n+2)}-\frac{3 a d (a+b x)^{n+3}}{b^4 (n+3)}+\frac{d (a+b x)^{n+4}}{b^4 (n+4)} \]

[Out]

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

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Rubi [A]  time = 0.0960538, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{\left (b^3 c-a^3 d\right ) (a+b x)^{n+1}}{b^4 (n+1)}+\frac{3 a^2 d (a+b x)^{n+2}}{b^4 (n+2)}-\frac{3 a d (a+b x)^{n+3}}{b^4 (n+3)}+\frac{d (a+b x)^{n+4}}{b^4 (n+4)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^n*(c + d*x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 24.1069, size = 83, normalized size = 0.88 \[ \frac{3 a^{2} d \left (a + b x\right )^{n + 2}}{b^{4} \left (n + 2\right )} - \frac{3 a d \left (a + b x\right )^{n + 3}}{b^{4} \left (n + 3\right )} + \frac{d \left (a + b x\right )^{n + 4}}{b^{4} \left (n + 4\right )} - \frac{\left (a + b x\right )^{n + 1} \left (a^{3} d - b^{3} c\right )}{b^{4} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.108475, size = 95, normalized size = 1.01 \[ \frac{(a+b x)^{n+1} \left (-6 a^3 d+6 a^2 b d (n+1) x-3 a b^2 d \left (n^2+3 n+2\right ) x^2+b^3 \left (n^2+5 n+6\right ) \left (c (n+4)+d (n+1) x^3\right )\right )}{b^4 (n+1) (n+2) (n+3) (n+4)} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^(1 + n)*(-6*a^3*d + 6*a^2*b*d*(1 + n)*x - 3*a*b^2*d*(2 + 3*n + n^2)*x
^2 + b^3*(6 + 5*n + n^2)*(c*(4 + n) + d*(1 + n)*x^3)))/(b^4*(1 + n)*(2 + n)*(3 +
n)*(4 + n))

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Maple [A]  time = 0.007, size = 167, normalized size = 1.8 \[ -{\frac{ \left ( bx+a \right ) ^{1+n} \left ( -{b}^{3}d{n}^{3}{x}^{3}-6\,{b}^{3}d{n}^{2}{x}^{3}+3\,a{b}^{2}d{n}^{2}{x}^{2}-11\,{b}^{3}dn{x}^{3}+9\,a{b}^{2}dn{x}^{2}-{b}^{3}c{n}^{3}-6\,d{x}^{3}{b}^{3}-6\,{a}^{2}bdnx+6\,ad{x}^{2}{b}^{2}-9\,{b}^{3}c{n}^{2}-6\,d{a}^{2}xb-26\,{b}^{3}cn+6\,{a}^{3}d-24\,{b}^{3}c \right ) }{{b}^{4} \left ({n}^{4}+10\,{n}^{3}+35\,{n}^{2}+50\,n+24 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n*(d*x^3+c),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.285765, size = 300, normalized size = 3.19 \[ \frac{{\left (a b^{3} c n^{3} + 9 \, a b^{3} c n^{2} + 26 \, a b^{3} c n + 24 \, a b^{3} c - 6 \, a^{4} d +{\left (b^{4} d n^{3} + 6 \, b^{4} d n^{2} + 11 \, b^{4} d n + 6 \, b^{4} d\right )} x^{4} +{\left (a b^{3} d n^{3} + 3 \, a b^{3} d n^{2} + 2 \, a b^{3} d n\right )} x^{3} - 3 \,{\left (a^{2} b^{2} d n^{2} + a^{2} b^{2} d n\right )} x^{2} +{\left (b^{4} c n^{3} + 9 \, b^{4} c n^{2} + 24 \, b^{4} c + 2 \,{\left (13 \, b^{4} c + 3 \, a^{3} b d\right )} n\right )} x\right )}{\left (b x + a\right )}^{n}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a*b^3*c*n^3 + 9*a*b^3*c*n^2 + 26*a*b^3*c*n + 24*a*b^3*c - 6*a^4*d + (b^4*d*n^3
+ 6*b^4*d*n^2 + 11*b^4*d*n + 6*b^4*d)*x^4 + (a*b^3*d*n^3 + 3*a*b^3*d*n^2 + 2*a*b
^3*d*n)*x^3 - 3*(a^2*b^2*d*n^2 + a^2*b^2*d*n)*x^2 + (b^4*c*n^3 + 9*b^4*c*n^2 + 2
4*b^4*c + 2*(13*b^4*c + 3*a^3*b*d)*n)*x)*(b*x + a)^n/(b^4*n^4 + 10*b^4*n^3 + 35*
b^4*n^2 + 50*b^4*n + 24*b^4)

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Sympy [A]  time = 9.16395, size = 1904, normalized size = 20.26 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((a**n*(c*x + d*x**4/4), Eq(b, 0)), (6*a**3*d*log(a/b + x)/(6*a**3*b**4
+ 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 5*a**3*d/(6*a**3*b**4 + 18*a
**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 18*a**2*b*d*x*log(a/b + x)/(6*a**3*
b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 9*a**2*b*d*x/(6*a**3*b**
4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 18*a*b**2*d*x**2*log(a/b +
x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 2*b**3*c/(6*a
**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) + 6*b**3*d*x**3*log(a/
b + x)/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3) - 6*b**3*d*
x**3/(6*a**3*b**4 + 18*a**2*b**5*x + 18*a*b**6*x**2 + 6*b**7*x**3), Eq(n, -4)),
(-6*a**3*d*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 9*a**3*d/(2*a
**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - 12*a**2*b*d*x*log(a/b + x)/(2*a**2*b**4 +
4*a*b**5*x + 2*b**6*x**2) - 12*a**2*b*d*x/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x*
*2) - 6*a*b**2*d*x**2*log(a/b + x)/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) - b*
*3*c/(2*a**2*b**4 + 4*a*b**5*x + 2*b**6*x**2) + 2*b**3*d*x**3/(2*a**2*b**4 + 4*a
*b**5*x + 2*b**6*x**2), Eq(n, -3)), (6*a**3*d*log(a/b + x)/(2*a*b**4 + 2*b**5*x)
+ 6*a**3*d/(2*a*b**4 + 2*b**5*x) + 6*a**2*b*d*x*log(a/b + x)/(2*a*b**4 + 2*b**5
*x) - 3*a*b**2*d*x**2/(2*a*b**4 + 2*b**5*x) - 2*b**3*c/(2*a*b**4 + 2*b**5*x) + b
**3*d*x**3/(2*a*b**4 + 2*b**5*x), Eq(n, -2)), (-a**3*d*log(a/b + x)/b**4 + a**2*
d*x/b**3 - a*d*x**2/(2*b**2) + c*log(a/b + x)/b + d*x**3/(3*b), Eq(n, -1)), (-6*
a**4*d*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b*
*4) + 6*a**3*b*d*n*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*
b**4*n + 24*b**4) - 3*a**2*b**2*d*n**2*x**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n*
*3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) - 3*a**2*b**2*d*n*x**2*(a + b*x)**n/(b*
*4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + a*b**3*c*n**3*(a
+ b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 9*a*
b**3*c*n**2*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n +
24*b**4) + 26*a*b**3*c*n*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 +
50*b**4*n + 24*b**4) + 24*a*b**3*c*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*
b**4*n**2 + 50*b**4*n + 24*b**4) + a*b**3*d*n**3*x**3*(a + b*x)**n/(b**4*n**4 +
10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 3*a*b**3*d*n**2*x**3*(a + b
*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 2*a*b**
3*d*n*x**3*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 2
4*b**4) + b**4*c*n**3*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 +
50*b**4*n + 24*b**4) + 9*b**4*c*n**2*x*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 +
35*b**4*n**2 + 50*b**4*n + 24*b**4) + 26*b**4*c*n*x*(a + b*x)**n/(b**4*n**4 + 10
*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 24*b**4*c*x*(a + b*x)**n/(b**
4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + b**4*d*n**3*x**4*(
a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4) + 6*
b**4*d*n**2*x**4*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**4*n**2 + 50*b**4
*n + 24*b**4) + 11*b**4*d*n*x**4*(a + b*x)**n/(b**4*n**4 + 10*b**4*n**3 + 35*b**
4*n**2 + 50*b**4*n + 24*b**4) + 6*b**4*d*x**4*(a + b*x)**n/(b**4*n**4 + 10*b**4*
n**3 + 35*b**4*n**2 + 50*b**4*n + 24*b**4), True))

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GIAC/XCAS [A]  time = 0.268955, size = 539, normalized size = 5.73 \[ \frac{b^{4} d n^{3} x^{4} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + a b^{3} d n^{3} x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 6 \, b^{4} d n^{2} x^{4} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 3 \, a b^{3} d n^{2} x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 11 \, b^{4} d n x^{4} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + b^{4} c n^{3} x e^{\left (n{\rm ln}\left (b x + a\right )\right )} - 3 \, a^{2} b^{2} d n^{2} x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 2 \, a b^{3} d n x^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 6 \, b^{4} d x^{4} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + a b^{3} c n^{3} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 9 \, b^{4} c n^{2} x e^{\left (n{\rm ln}\left (b x + a\right )\right )} - 3 \, a^{2} b^{2} d n x^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 9 \, a b^{3} c n^{2} e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 26 \, b^{4} c n x e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 6 \, a^{3} b d n x e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 26 \, a b^{3} c n e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 24 \, b^{4} c x e^{\left (n{\rm ln}\left (b x + a\right )\right )} + 24 \, a b^{3} c e^{\left (n{\rm ln}\left (b x + a\right )\right )} - 6 \, a^{4} d e^{\left (n{\rm ln}\left (b x + a\right )\right )}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b^4*d*n^3*x^4*e^(n*ln(b*x + a)) + a*b^3*d*n^3*x^3*e^(n*ln(b*x + a)) + 6*b^4*d*n
^2*x^4*e^(n*ln(b*x + a)) + 3*a*b^3*d*n^2*x^3*e^(n*ln(b*x + a)) + 11*b^4*d*n*x^4*
e^(n*ln(b*x + a)) + b^4*c*n^3*x*e^(n*ln(b*x + a)) - 3*a^2*b^2*d*n^2*x^2*e^(n*ln(
b*x + a)) + 2*a*b^3*d*n*x^3*e^(n*ln(b*x + a)) + 6*b^4*d*x^4*e^(n*ln(b*x + a)) +
a*b^3*c*n^3*e^(n*ln(b*x + a)) + 9*b^4*c*n^2*x*e^(n*ln(b*x + a)) - 3*a^2*b^2*d*n*
x^2*e^(n*ln(b*x + a)) + 9*a*b^3*c*n^2*e^(n*ln(b*x + a)) + 26*b^4*c*n*x*e^(n*ln(b
*x + a)) + 6*a^3*b*d*n*x*e^(n*ln(b*x + a)) + 26*a*b^3*c*n*e^(n*ln(b*x + a)) + 24
*b^4*c*x*e^(n*ln(b*x + a)) + 24*a*b^3*c*e^(n*ln(b*x + a)) - 6*a^4*d*e^(n*ln(b*x
+ a)))/(b^4*n^4 + 10*b^4*n^3 + 35*b^4*n^2 + 50*b^4*n + 24*b^4)