3.316 \(\int \left (c+d x+e x^2\right ) \left (a+b x^3\right )^2 \, dx\)

Optimal. Leaf size=77 \[ a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{2} a b c x^4+\frac{2}{5} a b d x^5+\frac{e \left (a+b x^3\right )^3}{9 b}+\frac{1}{7} b^2 c x^7+\frac{1}{8} b^2 d x^8 \]

[Out]

a^2*c*x + (a^2*d*x^2)/2 + (a*b*c*x^4)/2 + (2*a*b*d*x^5)/5 + (b^2*c*x^7)/7 + (b^2
*d*x^8)/8 + (e*(a + b*x^3)^3)/(9*b)

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Rubi [A]  time = 0.114055, antiderivative size = 92, normalized size of antiderivative = 1.19, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{3} a^2 e x^3+\frac{1}{2} a b c x^4+\frac{2}{5} a b d x^5+\frac{1}{3} a b e x^6+\frac{1}{7} b^2 c x^7+\frac{1}{8} b^2 d x^8+\frac{1}{9} b^2 e x^9 \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x + e*x^2)*(a + b*x^3)^2,x]

[Out]

a^2*c*x + (a^2*d*x^2)/2 + (a^2*e*x^3)/3 + (a*b*c*x^4)/2 + (2*a*b*d*x^5)/5 + (a*b
*e*x^6)/3 + (b^2*c*x^7)/7 + (b^2*d*x^8)/8 + (b^2*e*x^9)/9

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{2} d \int x\, dx + a^{2} \int c\, dx + \frac{a b c x^{4}}{2} + \frac{2 a b d x^{5}}{5} + \frac{b^{2} c x^{7}}{7} + \frac{b^{2} d x^{8}}{8} + \frac{e \left (a + b x^{3}\right )^{3}}{9 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x**2+d*x+c)*(b*x**3+a)**2,x)

[Out]

a**2*d*Integral(x, x) + a**2*Integral(c, x) + a*b*c*x**4/2 + 2*a*b*d*x**5/5 + b*
*2*c*x**7/7 + b**2*d*x**8/8 + e*(a + b*x**3)**3/(9*b)

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Mathematica [A]  time = 0.00542147, size = 92, normalized size = 1.19 \[ a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{3} a^2 e x^3+\frac{1}{2} a b c x^4+\frac{2}{5} a b d x^5+\frac{1}{3} a b e x^6+\frac{1}{7} b^2 c x^7+\frac{1}{8} b^2 d x^8+\frac{1}{9} b^2 e x^9 \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x + e*x^2)*(a + b*x^3)^2,x]

[Out]

a^2*c*x + (a^2*d*x^2)/2 + (a^2*e*x^3)/3 + (a*b*c*x^4)/2 + (2*a*b*d*x^5)/5 + (a*b
*e*x^6)/3 + (b^2*c*x^7)/7 + (b^2*d*x^8)/8 + (b^2*e*x^9)/9

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Maple [A]  time = 0.002, size = 77, normalized size = 1. \[{a}^{2}cx+{\frac{{a}^{2}d{x}^{2}}{2}}+{\frac{{a}^{2}e{x}^{3}}{3}}+{\frac{abc{x}^{4}}{2}}+{\frac{2\,abd{x}^{5}}{5}}+{\frac{abe{x}^{6}}{3}}+{\frac{{b}^{2}c{x}^{7}}{7}}+{\frac{{b}^{2}d{x}^{8}}{8}}+{\frac{{b}^{2}e{x}^{9}}{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+d*x+c)*(b*x^3+a)^2,x)

[Out]

a^2*c*x+1/2*a^2*d*x^2+1/3*a^2*e*x^3+1/2*a*b*c*x^4+2/5*a*b*d*x^5+1/3*a*b*e*x^6+1/
7*b^2*c*x^7+1/8*b^2*d*x^8+1/9*b^2*e*x^9

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Maxima [A]  time = 1.41973, size = 103, normalized size = 1.34 \[ \frac{1}{9} \, b^{2} e x^{9} + \frac{1}{8} \, b^{2} d x^{8} + \frac{1}{7} \, b^{2} c x^{7} + \frac{1}{3} \, a b e x^{6} + \frac{2}{5} \, a b d x^{5} + \frac{1}{2} \, a b c x^{4} + \frac{1}{3} \, a^{2} e x^{3} + \frac{1}{2} \, a^{2} d x^{2} + a^{2} c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2*(e*x^2 + d*x + c),x, algorithm="maxima")

[Out]

1/9*b^2*e*x^9 + 1/8*b^2*d*x^8 + 1/7*b^2*c*x^7 + 1/3*a*b*e*x^6 + 2/5*a*b*d*x^5 +
1/2*a*b*c*x^4 + 1/3*a^2*e*x^3 + 1/2*a^2*d*x^2 + a^2*c*x

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Fricas [A]  time = 0.191922, size = 1, normalized size = 0.01 \[ \frac{1}{9} x^{9} e b^{2} + \frac{1}{8} x^{8} d b^{2} + \frac{1}{7} x^{7} c b^{2} + \frac{1}{3} x^{6} e b a + \frac{2}{5} x^{5} d b a + \frac{1}{2} x^{4} c b a + \frac{1}{3} x^{3} e a^{2} + \frac{1}{2} x^{2} d a^{2} + x c a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2*(e*x^2 + d*x + c),x, algorithm="fricas")

[Out]

1/9*x^9*e*b^2 + 1/8*x^8*d*b^2 + 1/7*x^7*c*b^2 + 1/3*x^6*e*b*a + 2/5*x^5*d*b*a +
1/2*x^4*c*b*a + 1/3*x^3*e*a^2 + 1/2*x^2*d*a^2 + x*c*a^2

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Sympy [A]  time = 0.065512, size = 88, normalized size = 1.14 \[ a^{2} c x + \frac{a^{2} d x^{2}}{2} + \frac{a^{2} e x^{3}}{3} + \frac{a b c x^{4}}{2} + \frac{2 a b d x^{5}}{5} + \frac{a b e x^{6}}{3} + \frac{b^{2} c x^{7}}{7} + \frac{b^{2} d x^{8}}{8} + \frac{b^{2} e x^{9}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x**2+d*x+c)*(b*x**3+a)**2,x)

[Out]

a**2*c*x + a**2*d*x**2/2 + a**2*e*x**3/3 + a*b*c*x**4/2 + 2*a*b*d*x**5/5 + a*b*e
*x**6/3 + b**2*c*x**7/7 + b**2*d*x**8/8 + b**2*e*x**9/9

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GIAC/XCAS [A]  time = 0.207988, size = 107, normalized size = 1.39 \[ \frac{1}{9} \, b^{2} x^{9} e + \frac{1}{8} \, b^{2} d x^{8} + \frac{1}{7} \, b^{2} c x^{7} + \frac{1}{3} \, a b x^{6} e + \frac{2}{5} \, a b d x^{5} + \frac{1}{2} \, a b c x^{4} + \frac{1}{3} \, a^{2} x^{3} e + \frac{1}{2} \, a^{2} d x^{2} + a^{2} c x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2*(e*x^2 + d*x + c),x, algorithm="giac")

[Out]

1/9*b^2*x^9*e + 1/8*b^2*d*x^8 + 1/7*b^2*c*x^7 + 1/3*a*b*x^6*e + 2/5*a*b*d*x^5 +
1/2*a*b*c*x^4 + 1/3*a^2*x^3*e + 1/2*a^2*d*x^2 + a^2*c*x