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

Optimal. Leaf size=56 \[ a^3 d x+a^2 c d x^3+\frac{3}{5} a c^2 d x^5+\frac{e \left (a+c x^2\right )^4}{8 c}+\frac{1}{7} c^3 d x^7 \]

[Out]

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

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Rubi [A]  time = 0.0674953, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ a^3 d x+a^2 c d x^3+\frac{3}{5} a c^2 d x^5+\frac{e \left (a+c x^2\right )^4}{8 c}+\frac{1}{7} c^3 d x^7 \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)*(a + c*x^2)^3,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00476199, size = 85, normalized size = 1.52 \[ a^3 d x+\frac{1}{2} a^3 e x^2+a^2 c d x^3+\frac{3}{4} a^2 c e x^4+\frac{3}{5} a c^2 d x^5+\frac{1}{2} a c^2 e x^6+\frac{1}{7} c^3 d x^7+\frac{1}{8} c^3 e x^8 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 74, normalized size = 1.3 \[{\frac{{c}^{3}e{x}^{8}}{8}}+{\frac{{c}^{3}d{x}^{7}}{7}}+{\frac{ea{c}^{2}{x}^{6}}{2}}+{\frac{3\,a{c}^{2}d{x}^{5}}{5}}+{\frac{3\,e{a}^{2}c{x}^{4}}{4}}+{a}^{2}cd{x}^{3}+{\frac{{a}^{3}e{x}^{2}}{2}}+{a}^{3}dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(c*x^2+a)^3,x)

[Out]

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

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Maxima [A]  time = 0.698356, size = 99, normalized size = 1.77 \[ \frac{1}{8} \, c^{3} e x^{8} + \frac{1}{7} \, c^{3} d x^{7} + \frac{1}{2} \, a c^{2} e x^{6} + \frac{3}{5} \, a c^{2} d x^{5} + \frac{3}{4} \, a^{2} c e x^{4} + a^{2} c d x^{3} + \frac{1}{2} \, a^{3} e x^{2} + a^{3} d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.181758, size = 1, normalized size = 0.02 \[ \frac{1}{8} x^{8} e c^{3} + \frac{1}{7} x^{7} d c^{3} + \frac{1}{2} x^{6} e c^{2} a + \frac{3}{5} x^{5} d c^{2} a + \frac{3}{4} x^{4} e c a^{2} + x^{3} d c a^{2} + \frac{1}{2} x^{2} e a^{3} + x d a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.122274, size = 85, normalized size = 1.52 \[ a^{3} d x + \frac{a^{3} e x^{2}}{2} + a^{2} c d x^{3} + \frac{3 a^{2} c e x^{4}}{4} + \frac{3 a c^{2} d x^{5}}{5} + \frac{a c^{2} e x^{6}}{2} + \frac{c^{3} d x^{7}}{7} + \frac{c^{3} e x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.210784, size = 104, normalized size = 1.86 \[ \frac{1}{8} \, c^{3} x^{8} e + \frac{1}{7} \, c^{3} d x^{7} + \frac{1}{2} \, a c^{2} x^{6} e + \frac{3}{5} \, a c^{2} d x^{5} + \frac{3}{4} \, a^{2} c x^{4} e + a^{2} c d x^{3} + \frac{1}{2} \, a^{3} x^{2} e + a^{3} d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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