3.210 \(\int \left (a+c x^2\right ) \left (1+\left (a x+\frac{c x^3}{3}\right )^5\right ) \, dx\)

Optimal. Leaf size=30 \[ \frac{1}{6} \left (a x+\frac{c x^3}{3}\right )^6+a x+\frac{c x^3}{3} \]

[Out]

a*x + (c*x^3)/3 + (a*x + (c*x^3)/3)^6/6

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Rubi [A]  time = 0.0295571, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{\left (3 a x+c x^3\right )^6}{4374}+a x+\frac{c x^3}{3} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)*(1 + (a*x + (c*x^3)/3)^5),x]

[Out]

a*x + (c*x^3)/3 + (3*a*x + c*x^3)^6/4374

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Rubi in Sympy [A]  time = 4.07227, size = 22, normalized size = 0.73 \[ a x + \frac{c x^{3}}{3} + \frac{\left (a x + \frac{c x^{3}}{3}\right )^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)*(1+(a*x+1/3*c*x**3)**5),x)

[Out]

a*x + c*x**3/3 + (a*x + c*x**3/3)**6/6

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Mathematica [B]  time = 0.00919983, size = 93, normalized size = 3.1 \[ \frac{a^6 x^6}{6}+\frac{1}{3} a^5 c x^8+\frac{5}{18} a^4 c^2 x^{10}+\frac{10}{81} a^3 c^3 x^{12}+\frac{5}{162} a^2 c^4 x^{14}+\frac{1}{243} a c^5 x^{16}+a x+\frac{c^6 x^{18}}{4374}+\frac{c x^3}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)*(1 + (a*x + (c*x^3)/3)^5),x]

[Out]

a*x + (c*x^3)/3 + (a^6*x^6)/6 + (a^5*c*x^8)/3 + (5*a^4*c^2*x^10)/18 + (10*a^3*c^
3*x^12)/81 + (5*a^2*c^4*x^14)/162 + (a*c^5*x^16)/243 + (c^6*x^18)/4374

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Maple [B]  time = 0.005, size = 78, normalized size = 2.6 \[{\frac{{c}^{6}{x}^{18}}{4374}}+{\frac{a{c}^{5}{x}^{16}}{243}}+{\frac{5\,{a}^{2}{c}^{4}{x}^{14}}{162}}+{\frac{10\,{a}^{3}{c}^{3}{x}^{12}}{81}}+{\frac{5\,{a}^{4}{c}^{2}{x}^{10}}{18}}+{\frac{{a}^{5}c{x}^{8}}{3}}+{\frac{{a}^{6}{x}^{6}}{6}}+{\frac{c{x}^{3}}{3}}+ax \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)*(1+(a*x+1/3*c*x^3)^5),x)

[Out]

1/4374*c^6*x^18+1/243*a*c^5*x^16+5/162*a^2*c^4*x^14+10/81*a^3*c^3*x^12+5/18*a^4*
c^2*x^10+1/3*a^5*c*x^8+1/6*a^6*x^6+1/3*c*x^3+a*x

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Maxima [A]  time = 0.80384, size = 104, normalized size = 3.47 \[ \frac{1}{4374} \, c^{6} x^{18} + \frac{1}{243} \, a c^{5} x^{16} + \frac{5}{162} \, a^{2} c^{4} x^{14} + \frac{10}{81} \, a^{3} c^{3} x^{12} + \frac{5}{18} \, a^{4} c^{2} x^{10} + \frac{1}{3} \, a^{5} c x^{8} + \frac{1}{6} \, a^{6} x^{6} + \frac{1}{3} \, c x^{3} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/243*((c*x^3 + 3*a*x)^5 + 243)*(c*x^2 + a),x, algorithm="maxima")

[Out]

1/4374*c^6*x^18 + 1/243*a*c^5*x^16 + 5/162*a^2*c^4*x^14 + 10/81*a^3*c^3*x^12 + 5
/18*a^4*c^2*x^10 + 1/3*a^5*c*x^8 + 1/6*a^6*x^6 + 1/3*c*x^3 + a*x

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Fricas [A]  time = 0.256895, size = 104, normalized size = 3.47 \[ \frac{1}{4374} \, c^{6} x^{18} + \frac{1}{243} \, a c^{5} x^{16} + \frac{5}{162} \, a^{2} c^{4} x^{14} + \frac{10}{81} \, a^{3} c^{3} x^{12} + \frac{5}{18} \, a^{4} c^{2} x^{10} + \frac{1}{3} \, a^{5} c x^{8} + \frac{1}{6} \, a^{6} x^{6} + \frac{1}{3} \, c x^{3} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/243*((c*x^3 + 3*a*x)^5 + 243)*(c*x^2 + a),x, algorithm="fricas")

[Out]

1/4374*c^6*x^18 + 1/243*a*c^5*x^16 + 5/162*a^2*c^4*x^14 + 10/81*a^3*c^3*x^12 + 5
/18*a^4*c^2*x^10 + 1/3*a^5*c*x^8 + 1/6*a^6*x^6 + 1/3*c*x^3 + a*x

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Sympy [A]  time = 0.174432, size = 87, normalized size = 2.9 \[ \frac{a^{6} x^{6}}{6} + \frac{a^{5} c x^{8}}{3} + \frac{5 a^{4} c^{2} x^{10}}{18} + \frac{10 a^{3} c^{3} x^{12}}{81} + \frac{5 a^{2} c^{4} x^{14}}{162} + \frac{a c^{5} x^{16}}{243} + a x + \frac{c^{6} x^{18}}{4374} + \frac{c x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)*(1+(a*x+1/3*c*x**3)**5),x)

[Out]

a**6*x**6/6 + a**5*c*x**8/3 + 5*a**4*c**2*x**10/18 + 10*a**3*c**3*x**12/81 + 5*a
**2*c**4*x**14/162 + a*c**5*x**16/243 + a*x + c**6*x**18/4374 + c*x**3/3

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GIAC/XCAS [A]  time = 0.258867, size = 104, normalized size = 3.47 \[ \frac{1}{4374} \, c^{6} x^{18} + \frac{1}{243} \, a c^{5} x^{16} + \frac{5}{162} \, a^{2} c^{4} x^{14} + \frac{10}{81} \, a^{3} c^{3} x^{12} + \frac{5}{18} \, a^{4} c^{2} x^{10} + \frac{1}{3} \, a^{5} c x^{8} + \frac{1}{6} \, a^{6} x^{6} + \frac{1}{3} \, c x^{3} + a x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/243*((c*x^3 + 3*a*x)^5 + 243)*(c*x^2 + a),x, algorithm="giac")

[Out]

1/4374*c^6*x^18 + 1/243*a*c^5*x^16 + 5/162*a^2*c^4*x^14 + 10/81*a^3*c^3*x^12 + 5
/18*a^4*c^2*x^10 + 1/3*a^5*c*x^8 + 1/6*a^6*x^6 + 1/3*c*x^3 + a*x