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

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

[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 = 0.0221575, 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, Rules used = {641, 194} \[ a^2 A c x^3+a^3 A x+\frac{3}{5} a A c^2 x^5+\frac{B \left (a+c x^2\right )^4}{8 c}+\frac{1}{7} A c^3 x^7 \]

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)

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int (A+B x) \left (a+c x^2\right )^3 \, dx &=\frac{B \left (a+c x^2\right )^4}{8 c}+A \int \left (a+c x^2\right )^3 \, dx\\ &=\frac{B \left (a+c x^2\right )^4}{8 c}+A \int \left (a^3+3 a^2 c x^2+3 a c^2 x^4+c^3 x^6\right ) \, dx\\ &=a^3 A x+a^2 A c x^3+\frac{3}{5} a A c^2 x^5+\frac{1}{7} A c^3 x^7+\frac{B \left (a+c x^2\right )^4}{8 c}\\ \end{align*}

Mathematica [A]  time = 0.0024509, size = 85, normalized size = 1.52 \[ a^2 A c x^3+a^3 A x+\frac{3}{4} a^2 B c x^4+\frac{1}{2} a^3 B x^2+\frac{3}{5} a A c^2 x^5+\frac{1}{2} a B c^2 x^6+\frac{1}{7} A c^3 x^7+\frac{1}{8} B c^3 x^8 \]

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 = 0.002, size = 74, normalized size = 1.3 \begin{align*}{\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 \end{align*}

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 = 0.998353, size = 99, normalized size = 1.77 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3,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 = 1.58575, size = 169, normalized size = 3.02 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3,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 = 0.097281, size = 85, normalized size = 1.52 \begin{align*} 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} \end{align*}

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 [A]  time = 1.16705, size = 99, normalized size = 1.77 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3,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