3.1318 \(\int (A+B x) \left (a+c x^2\right )^3 \, dx\)

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

[Out]

a^3*A*x + a^2*A*c*x^3 + (3*a*A*c^2*x^5)/5 + (A*c^3*x^7)/7 + (B*(a + c*x^2)^4)/(8
*c)

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Rubi [A]  time = 0.0579803, 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 \[ a^3 A x+a^2 A c x^3+\frac{3}{5} a A c^2 x^5+\frac{B \left (a+c x^2\right )^4}{8 c}+\frac{1}{7} A c^3 x^7 \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)*(a + c*x^2)^3,x]

[Out]

a^3*A*x + a^2*A*c*x^3 + (3*a*A*c^2*x^5)/5 + (A*c^3*x^7)/7 + (B*(a + c*x^2)^4)/(8
*c)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**3,x)

[Out]

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

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Mathematica [A]  time = 0.00449928, size = 85, normalized size = 1.52 \[ a^3 A x+\frac{1}{2} a^3 B x^2+a^2 A c x^3+\frac{3}{4} a^2 B c x^4+\frac{3}{5} a A c^2 x^5+\frac{1}{2} a B c^2 x^6+\frac{1}{7} A c^3 x^7+\frac{1}{8} B c^3 x^8 \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)*(a + c*x^2)^3,x]

[Out]

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

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Maple [A]  time = 0., size = 74, normalized size = 1.3 \[{\frac{B{c}^{3}{x}^{8}}{8}}+{\frac{A{c}^{3}{x}^{7}}{7}}+{\frac{aB{c}^{2}{x}^{6}}{2}}+{\frac{3\,aA{c}^{2}{x}^{5}}{5}}+{\frac{3\,{a}^{2}Bc{x}^{4}}{4}}+{a}^{2}Ac{x}^{3}+{\frac{{a}^{3}B{x}^{2}}{2}}+{a}^{3}Ax \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^3,x)

[Out]

1/8*B*c^3*x^8+1/7*A*c^3*x^7+1/2*a*B*c^2*x^6+3/5*a*A*c^2*x^5+3/4*a^2*B*c*x^4+a^2*
A*c*x^3+1/2*a^3*B*x^2+a^3*A*x

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Maxima [A]  time = 0.705357, size = 99, normalized size = 1.77 \[ \frac{1}{8} \, B c^{3} x^{8} + \frac{1}{7} \, A c^{3} x^{7} + \frac{1}{2} \, B a c^{2} x^{6} + \frac{3}{5} \, A a c^{2} x^{5} + \frac{3}{4} \, B a^{2} c x^{4} + A a^{2} c x^{3} + \frac{1}{2} \, B a^{3} x^{2} + A a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*B*c^3*x^8 + 1/7*A*c^3*x^7 + 1/2*B*a*c^2*x^6 + 3/5*A*a*c^2*x^5 + 3/4*B*a^2*c*
x^4 + A*a^2*c*x^3 + 1/2*B*a^3*x^2 + A*a^3*x

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Fricas [A]  time = 0.244376, size = 1, normalized size = 0.02 \[ \frac{1}{8} x^{8} c^{3} B + \frac{1}{7} x^{7} c^{3} A + \frac{1}{2} x^{6} c^{2} a B + \frac{3}{5} x^{5} c^{2} a A + \frac{3}{4} x^{4} c a^{2} B + x^{3} c a^{2} A + \frac{1}{2} x^{2} a^{3} B + x a^{3} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*x^8*c^3*B + 1/7*x^7*c^3*A + 1/2*x^6*c^2*a*B + 3/5*x^5*c^2*a*A + 3/4*x^4*c*a^
2*B + x^3*c*a^2*A + 1/2*x^2*a^3*B + x*a^3*A

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Sympy [A]  time = 0.132696, size = 85, normalized size = 1.52 \[ A a^{3} x + A a^{2} c x^{3} + \frac{3 A a c^{2} x^{5}}{5} + \frac{A c^{3} x^{7}}{7} + \frac{B a^{3} x^{2}}{2} + \frac{3 B a^{2} c x^{4}}{4} + \frac{B a c^{2} x^{6}}{2} + \frac{B c^{3} x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+a)**3,x)

[Out]

A*a**3*x + A*a**2*c*x**3 + 3*A*a*c**2*x**5/5 + A*c**3*x**7/7 + B*a**3*x**2/2 + 3
*B*a**2*c*x**4/4 + B*a*c**2*x**6/2 + B*c**3*x**8/8

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GIAC/XCAS [A]  time = 0.282112, size = 99, normalized size = 1.77 \[ \frac{1}{8} \, B c^{3} x^{8} + \frac{1}{7} \, A c^{3} x^{7} + \frac{1}{2} \, B a c^{2} x^{6} + \frac{3}{5} \, A a c^{2} x^{5} + \frac{3}{4} \, B a^{2} c x^{4} + A a^{2} c x^{3} + \frac{1}{2} \, B a^{3} x^{2} + A a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*B*c^3*x^8 + 1/7*A*c^3*x^7 + 1/2*B*a*c^2*x^6 + 3/5*A*a*c^2*x^5 + 3/4*B*a^2*c*
x^4 + A*a^2*c*x^3 + 1/2*B*a^3*x^2 + A*a^3*x