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

Optimal. Leaf size=122 \[ \frac{1}{10} d x^{10} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{7} c x^7 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 x+\frac{1}{4} a c^2 x^4 (3 a d+2 b c)+\frac{1}{13} b d^2 x^{13} (2 a d+3 b c)+\frac{1}{16} b^2 d^3 x^{16} \]

[Out]

a^2*c^3*x + (a*c^2*(2*b*c + 3*a*d)*x^4)/4 + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^7)/7 + (d*(3*b^2*c^2 + 6*a*
b*c*d + a^2*d^2)*x^10)/10 + (b*d^2*(3*b*c + 2*a*d)*x^13)/13 + (b^2*d^3*x^16)/16

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Rubi [A]  time = 0.0710383, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {373} \[ \frac{1}{10} d x^{10} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{7} c x^7 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 x+\frac{1}{4} a c^2 x^4 (3 a d+2 b c)+\frac{1}{13} b d^2 x^{13} (2 a d+3 b c)+\frac{1}{16} b^2 d^3 x^{16} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2*(c + d*x^3)^3,x]

[Out]

a^2*c^3*x + (a*c^2*(2*b*c + 3*a*d)*x^4)/4 + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^7)/7 + (d*(3*b^2*c^2 + 6*a*
b*c*d + a^2*d^2)*x^10)/10 + (b*d^2*(3*b*c + 2*a*d)*x^13)/13 + (b^2*d^3*x^16)/16

Rule 373

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

Rubi steps

\begin{align*} \int \left (a+b x^3\right )^2 \left (c+d x^3\right )^3 \, dx &=\int \left (a^2 c^3+a c^2 (2 b c+3 a d) x^3+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^6+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^9+b d^2 (3 b c+2 a d) x^{12}+b^2 d^3 x^{15}\right ) \, dx\\ &=a^2 c^3 x+\frac{1}{4} a c^2 (2 b c+3 a d) x^4+\frac{1}{7} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^7+\frac{1}{10} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{10}+\frac{1}{13} b d^2 (3 b c+2 a d) x^{13}+\frac{1}{16} b^2 d^3 x^{16}\\ \end{align*}

Mathematica [A]  time = 0.0161661, size = 122, normalized size = 1. \[ \frac{1}{10} d x^{10} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{7} c x^7 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 x+\frac{1}{4} a c^2 x^4 (3 a d+2 b c)+\frac{1}{13} b d^2 x^{13} (2 a d+3 b c)+\frac{1}{16} b^2 d^3 x^{16} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*c^3*x + (a*c^2*(2*b*c + 3*a*d)*x^4)/4 + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^7)/7 + (d*(3*b^2*c^2 + 6*a*
b*c*d + a^2*d^2)*x^10)/10 + (b*d^2*(3*b*c + 2*a*d)*x^13)/13 + (b^2*d^3*x^16)/16

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Maple [A]  time = 0., size = 125, normalized size = 1. \begin{align*}{\frac{{b}^{2}{d}^{3}{x}^{16}}{16}}+{\frac{ \left ( 2\,ab{d}^{3}+3\,{b}^{2}c{d}^{2} \right ){x}^{13}}{13}}+{\frac{ \left ({a}^{2}{d}^{3}+6\,abc{d}^{2}+3\,{b}^{2}{c}^{2}d \right ){x}^{10}}{10}}+{\frac{ \left ( 3\,{a}^{2}c{d}^{2}+6\,ab{c}^{2}d+{b}^{2}{c}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,{a}^{2}{c}^{2}d+2\,ab{c}^{3} \right ){x}^{4}}{4}}+{a}^{2}{c}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*b^2*d^3*x^16+1/13*(2*a*b*d^3+3*b^2*c*d^2)*x^13+1/10*(a^2*d^3+6*a*b*c*d^2+3*b^2*c^2*d)*x^10+1/7*(3*a^2*c*d
^2+6*a*b*c^2*d+b^2*c^3)*x^7+1/4*(3*a^2*c^2*d+2*a*b*c^3)*x^4+a^2*c^3*x

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Maxima [A]  time = 0.956539, size = 167, normalized size = 1.37 \begin{align*} \frac{1}{16} \, b^{2} d^{3} x^{16} + \frac{1}{13} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{13} + \frac{1}{10} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{10} + \frac{1}{7} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{7} + a^{2} c^{3} x + \frac{1}{4} \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*b^2*d^3*x^16 + 1/13*(3*b^2*c*d^2 + 2*a*b*d^3)*x^13 + 1/10*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^10 + 1/
7*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^7 + a^2*c^3*x + 1/4*(2*a*b*c^3 + 3*a^2*c^2*d)*x^4

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Fricas [A]  time = 1.31728, size = 313, normalized size = 2.57 \begin{align*} \frac{1}{16} x^{16} d^{3} b^{2} + \frac{3}{13} x^{13} d^{2} c b^{2} + \frac{2}{13} x^{13} d^{3} b a + \frac{3}{10} x^{10} d c^{2} b^{2} + \frac{3}{5} x^{10} d^{2} c b a + \frac{1}{10} x^{10} d^{3} a^{2} + \frac{1}{7} x^{7} c^{3} b^{2} + \frac{6}{7} x^{7} d c^{2} b a + \frac{3}{7} x^{7} d^{2} c a^{2} + \frac{1}{2} x^{4} c^{3} b a + \frac{3}{4} x^{4} d c^{2} a^{2} + x c^{3} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*x^16*d^3*b^2 + 3/13*x^13*d^2*c*b^2 + 2/13*x^13*d^3*b*a + 3/10*x^10*d*c^2*b^2 + 3/5*x^10*d^2*c*b*a + 1/10*
x^10*d^3*a^2 + 1/7*x^7*c^3*b^2 + 6/7*x^7*d*c^2*b*a + 3/7*x^7*d^2*c*a^2 + 1/2*x^4*c^3*b*a + 3/4*x^4*d*c^2*a^2 +
x*c^3*a^2

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Sympy [A]  time = 0.086971, size = 139, normalized size = 1.14 \begin{align*} a^{2} c^{3} x + \frac{b^{2} d^{3} x^{16}}{16} + x^{13} \left (\frac{2 a b d^{3}}{13} + \frac{3 b^{2} c d^{2}}{13}\right ) + x^{10} \left (\frac{a^{2} d^{3}}{10} + \frac{3 a b c d^{2}}{5} + \frac{3 b^{2} c^{2} d}{10}\right ) + x^{7} \left (\frac{3 a^{2} c d^{2}}{7} + \frac{6 a b c^{2} d}{7} + \frac{b^{2} c^{3}}{7}\right ) + x^{4} \left (\frac{3 a^{2} c^{2} d}{4} + \frac{a b c^{3}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*c**3*x + b**2*d**3*x**16/16 + x**13*(2*a*b*d**3/13 + 3*b**2*c*d**2/13) + x**10*(a**2*d**3/10 + 3*a*b*c*d*
*2/5 + 3*b**2*c**2*d/10) + x**7*(3*a**2*c*d**2/7 + 6*a*b*c**2*d/7 + b**2*c**3/7) + x**4*(3*a**2*c**2*d/4 + a*b
*c**3/2)

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Giac [A]  time = 1.07199, size = 178, normalized size = 1.46 \begin{align*} \frac{1}{16} \, b^{2} d^{3} x^{16} + \frac{3}{13} \, b^{2} c d^{2} x^{13} + \frac{2}{13} \, a b d^{3} x^{13} + \frac{3}{10} \, b^{2} c^{2} d x^{10} + \frac{3}{5} \, a b c d^{2} x^{10} + \frac{1}{10} \, a^{2} d^{3} x^{10} + \frac{1}{7} \, b^{2} c^{3} x^{7} + \frac{6}{7} \, a b c^{2} d x^{7} + \frac{3}{7} \, a^{2} c d^{2} x^{7} + \frac{1}{2} \, a b c^{3} x^{4} + \frac{3}{4} \, a^{2} c^{2} d x^{4} + a^{2} c^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*b^2*d^3*x^16 + 3/13*b^2*c*d^2*x^13 + 2/13*a*b*d^3*x^13 + 3/10*b^2*c^2*d*x^10 + 3/5*a*b*c*d^2*x^10 + 1/10*
a^2*d^3*x^10 + 1/7*b^2*c^3*x^7 + 6/7*a*b*c^2*d*x^7 + 3/7*a^2*c*d^2*x^7 + 1/2*a*b*c^3*x^4 + 3/4*a^2*c^2*d*x^4 +
a^2*c^3*x