3.54 \(\int (a+b x^3)^2 (a c+a d x+b c x^3+b d x^4) \, dx\)

Optimal. Leaf size=88 \[ \frac{3}{4} a^2 b c x^4+\frac{3}{5} a^2 b d x^5+a^3 c x+\frac{1}{2} a^3 d x^2+\frac{3}{7} a b^2 c x^7+\frac{3}{8} a b^2 d x^8+\frac{1}{10} b^3 c x^{10}+\frac{1}{11} b^3 d x^{11} \]

[Out]

a^3*c*x + (a^3*d*x^2)/2 + (3*a^2*b*c*x^4)/4 + (3*a^2*b*d*x^5)/5 + (3*a*b^2*c*x^7)/7 + (3*a*b^2*d*x^8)/8 + (b^3
*c*x^10)/10 + (b^3*d*x^11)/11

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Rubi [A]  time = 0.0602554, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used = {1850} \[ \frac{3}{4} a^2 b c x^4+\frac{3}{5} a^2 b d x^5+a^3 c x+\frac{1}{2} a^3 d x^2+\frac{3}{7} a b^2 c x^7+\frac{3}{8} a b^2 d x^8+\frac{1}{10} b^3 c x^{10}+\frac{1}{11} b^3 d x^{11} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2*(a*c + a*d*x + b*c*x^3 + b*d*x^4),x]

[Out]

a^3*c*x + (a^3*d*x^2)/2 + (3*a^2*b*c*x^4)/4 + (3*a^2*b*d*x^5)/5 + (3*a*b^2*c*x^7)/7 + (3*a*b^2*d*x^8)/8 + (b^3
*c*x^10)/10 + (b^3*d*x^11)/11

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin{align*} \int \left (a+b x^3\right )^2 \left (a c+a d x+b c x^3+b d x^4\right ) \, dx &=\int \left (a^3 c+a^3 d x+3 a^2 b c x^3+3 a^2 b d x^4+3 a b^2 c x^6+3 a b^2 d x^7+b^3 c x^9+b^3 d x^{10}\right ) \, dx\\ &=a^3 c x+\frac{1}{2} a^3 d x^2+\frac{3}{4} a^2 b c x^4+\frac{3}{5} a^2 b d x^5+\frac{3}{7} a b^2 c x^7+\frac{3}{8} a b^2 d x^8+\frac{1}{10} b^3 c x^{10}+\frac{1}{11} b^3 d x^{11}\\ \end{align*}

Mathematica [A]  time = 0.0026354, size = 88, normalized size = 1. \[ \frac{3}{4} a^2 b c x^4+\frac{3}{5} a^2 b d x^5+a^3 c x+\frac{1}{2} a^3 d x^2+\frac{3}{7} a b^2 c x^7+\frac{3}{8} a b^2 d x^8+\frac{1}{10} b^3 c x^{10}+\frac{1}{11} b^3 d x^{11} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2*(a*c + a*d*x + b*c*x^3 + b*d*x^4),x]

[Out]

a^3*c*x + (a^3*d*x^2)/2 + (3*a^2*b*c*x^4)/4 + (3*a^2*b*d*x^5)/5 + (3*a*b^2*c*x^7)/7 + (3*a*b^2*d*x^8)/8 + (b^3
*c*x^10)/10 + (b^3*d*x^11)/11

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Maple [A]  time = 0.001, size = 75, normalized size = 0.9 \begin{align*}{a}^{3}cx+{\frac{{a}^{3}d{x}^{2}}{2}}+{\frac{3\,{a}^{2}bc{x}^{4}}{4}}+{\frac{3\,{a}^{2}bd{x}^{5}}{5}}+{\frac{3\,a{b}^{2}c{x}^{7}}{7}}+{\frac{3\,a{b}^{2}d{x}^{8}}{8}}+{\frac{{b}^{3}c{x}^{10}}{10}}+{\frac{{b}^{3}d{x}^{11}}{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2*(b*d*x^4+b*c*x^3+a*d*x+a*c),x)

[Out]

a^3*c*x+1/2*a^3*d*x^2+3/4*a^2*b*c*x^4+3/5*a^2*b*d*x^5+3/7*a*b^2*c*x^7+3/8*a*b^2*d*x^8+1/10*b^3*c*x^10+1/11*b^3
*d*x^11

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Maxima [A]  time = 0.960131, size = 100, normalized size = 1.14 \begin{align*} \frac{1}{11} \, b^{3} d x^{11} + \frac{1}{10} \, b^{3} c x^{10} + \frac{3}{8} \, a b^{2} d x^{8} + \frac{3}{7} \, a b^{2} c x^{7} + \frac{3}{5} \, a^{2} b d x^{5} + \frac{3}{4} \, a^{2} b c x^{4} + \frac{1}{2} \, a^{3} d x^{2} + a^{3} c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(b*d*x^4+b*c*x^3+a*d*x+a*c),x, algorithm="maxima")

[Out]

1/11*b^3*d*x^11 + 1/10*b^3*c*x^10 + 3/8*a*b^2*d*x^8 + 3/7*a*b^2*c*x^7 + 3/5*a^2*b*d*x^5 + 3/4*a^2*b*c*x^4 + 1/
2*a^3*d*x^2 + a^3*c*x

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Fricas [A]  time = 1.10836, size = 180, normalized size = 2.05 \begin{align*} \frac{1}{11} x^{11} d b^{3} + \frac{1}{10} x^{10} c b^{3} + \frac{3}{8} x^{8} d b^{2} a + \frac{3}{7} x^{7} c b^{2} a + \frac{3}{5} x^{5} d b a^{2} + \frac{3}{4} x^{4} c b a^{2} + \frac{1}{2} x^{2} d a^{3} + x c a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(b*d*x^4+b*c*x^3+a*d*x+a*c),x, algorithm="fricas")

[Out]

1/11*x^11*d*b^3 + 1/10*x^10*c*b^3 + 3/8*x^8*d*b^2*a + 3/7*x^7*c*b^2*a + 3/5*x^5*d*b*a^2 + 3/4*x^4*c*b*a^2 + 1/
2*x^2*d*a^3 + x*c*a^3

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Sympy [A]  time = 0.078548, size = 90, normalized size = 1.02 \begin{align*} a^{3} c x + \frac{a^{3} d x^{2}}{2} + \frac{3 a^{2} b c x^{4}}{4} + \frac{3 a^{2} b d x^{5}}{5} + \frac{3 a b^{2} c x^{7}}{7} + \frac{3 a b^{2} d x^{8}}{8} + \frac{b^{3} c x^{10}}{10} + \frac{b^{3} d x^{11}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2*(b*d*x**4+b*c*x**3+a*d*x+a*c),x)

[Out]

a**3*c*x + a**3*d*x**2/2 + 3*a**2*b*c*x**4/4 + 3*a**2*b*d*x**5/5 + 3*a*b**2*c*x**7/7 + 3*a*b**2*d*x**8/8 + b**
3*c*x**10/10 + b**3*d*x**11/11

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Giac [A]  time = 1.08721, size = 100, normalized size = 1.14 \begin{align*} \frac{1}{11} \, b^{3} d x^{11} + \frac{1}{10} \, b^{3} c x^{10} + \frac{3}{8} \, a b^{2} d x^{8} + \frac{3}{7} \, a b^{2} c x^{7} + \frac{3}{5} \, a^{2} b d x^{5} + \frac{3}{4} \, a^{2} b c x^{4} + \frac{1}{2} \, a^{3} d x^{2} + a^{3} c x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2*(b*d*x^4+b*c*x^3+a*d*x+a*c),x, algorithm="giac")

[Out]

1/11*b^3*d*x^11 + 1/10*b^3*c*x^10 + 3/8*a*b^2*d*x^8 + 3/7*a*b^2*c*x^7 + 3/5*a^2*b*d*x^5 + 3/4*a^2*b*c*x^4 + 1/
2*a^3*d*x^2 + a^3*c*x