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

Optimal. Leaf size=122 \[ \frac{1}{13} d x^{13} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{9} c x^9 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 x+\frac{1}{5} a c^2 x^5 (3 a d+2 b c)+\frac{1}{17} b d^2 x^{17} (2 a d+3 b c)+\frac{1}{21} b^2 d^3 x^{21} \]

[Out]

a^2*c^3*x + (a*c^2*(2*b*c + 3*a*d)*x^5)/5 + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^9)/9 + (d*(3*b^2*c^2 + 6*a*
b*c*d + a^2*d^2)*x^13)/13 + (b*d^2*(3*b*c + 2*a*d)*x^17)/17 + (b^2*d^3*x^21)/21

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Rubi [A]  time = 0.0766227, 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} \[ \frac{1}{13} d x^{13} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{9} c x^9 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 x+\frac{1}{5} a c^2 x^5 (3 a d+2 b c)+\frac{1}{17} b d^2 x^{17} (2 a d+3 b c)+\frac{1}{21} b^2 d^3 x^{21} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^4)^2*(c + d*x^4)^3,x]

[Out]

a^2*c^3*x + (a*c^2*(2*b*c + 3*a*d)*x^5)/5 + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^9)/9 + (d*(3*b^2*c^2 + 6*a*
b*c*d + a^2*d^2)*x^13)/13 + (b*d^2*(3*b*c + 2*a*d)*x^17)/17 + (b^2*d^3*x^21)/21

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^4\right )^2 \left (c+d x^4\right )^3 \, dx &=\int \left (a^2 c^3+a c^2 (2 b c+3 a d) x^4+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^8+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{12}+b d^2 (3 b c+2 a d) x^{16}+b^2 d^3 x^{20}\right ) \, dx\\ &=a^2 c^3 x+\frac{1}{5} a c^2 (2 b c+3 a d) x^5+\frac{1}{9} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^9+\frac{1}{13} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{13}+\frac{1}{17} b d^2 (3 b c+2 a d) x^{17}+\frac{1}{21} b^2 d^3 x^{21}\\ \end{align*}

Mathematica [A]  time = 0.0222812, size = 122, normalized size = 1. \[ \frac{1}{13} d x^{13} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{9} c x^9 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 x+\frac{1}{5} a c^2 x^5 (3 a d+2 b c)+\frac{1}{17} b d^2 x^{17} (2 a d+3 b c)+\frac{1}{21} b^2 d^3 x^{21} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*c^3*x + (a*c^2*(2*b*c + 3*a*d)*x^5)/5 + (c*(b^2*c^2 + 6*a*b*c*d + 3*a^2*d^2)*x^9)/9 + (d*(3*b^2*c^2 + 6*a*
b*c*d + a^2*d^2)*x^13)/13 + (b*d^2*(3*b*c + 2*a*d)*x^17)/17 + (b^2*d^3*x^21)/21

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Maple [A]  time = 0.001, size = 125, normalized size = 1. \begin{align*}{\frac{{b}^{2}{d}^{3}{x}^{21}}{21}}+{\frac{ \left ( 2\,ab{d}^{3}+3\,{b}^{2}c{d}^{2} \right ){x}^{17}}{17}}+{\frac{ \left ({a}^{2}{d}^{3}+6\,abc{d}^{2}+3\,{b}^{2}{c}^{2}d \right ){x}^{13}}{13}}+{\frac{ \left ( 3\,{a}^{2}c{d}^{2}+6\,ab{c}^{2}d+{b}^{2}{c}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ( 3\,{a}^{2}{c}^{2}d+2\,ab{c}^{3} \right ){x}^{5}}{5}}+{a}^{2}{c}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*b^2*d^3*x^21+1/17*(2*a*b*d^3+3*b^2*c*d^2)*x^17+1/13*(a^2*d^3+6*a*b*c*d^2+3*b^2*c^2*d)*x^13+1/9*(3*a^2*c*d
^2+6*a*b*c^2*d+b^2*c^3)*x^9+1/5*(3*a^2*c^2*d+2*a*b*c^3)*x^5+a^2*c^3*x

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Maxima [A]  time = 0.95612, size = 167, normalized size = 1.37 \begin{align*} \frac{1}{21} \, b^{2} d^{3} x^{21} + \frac{1}{17} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{17} + \frac{1}{13} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{13} + \frac{1}{9} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{9} + a^{2} c^{3} x + \frac{1}{5} \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*b^2*d^3*x^21 + 1/17*(3*b^2*c*d^2 + 2*a*b*d^3)*x^17 + 1/13*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*x^13 + 1/
9*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*x^9 + a^2*c^3*x + 1/5*(2*a*b*c^3 + 3*a^2*c^2*d)*x^5

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Fricas [A]  time = 1.02337, size = 315, normalized size = 2.58 \begin{align*} \frac{1}{21} x^{21} d^{3} b^{2} + \frac{3}{17} x^{17} d^{2} c b^{2} + \frac{2}{17} x^{17} d^{3} b a + \frac{3}{13} x^{13} d c^{2} b^{2} + \frac{6}{13} x^{13} d^{2} c b a + \frac{1}{13} x^{13} d^{3} a^{2} + \frac{1}{9} x^{9} c^{3} b^{2} + \frac{2}{3} x^{9} d c^{2} b a + \frac{1}{3} x^{9} d^{2} c a^{2} + \frac{2}{5} x^{5} c^{3} b a + \frac{3}{5} x^{5} 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^4+a)^2*(d*x^4+c)^3,x, algorithm="fricas")

[Out]

1/21*x^21*d^3*b^2 + 3/17*x^17*d^2*c*b^2 + 2/17*x^17*d^3*b*a + 3/13*x^13*d*c^2*b^2 + 6/13*x^13*d^2*c*b*a + 1/13
*x^13*d^3*a^2 + 1/9*x^9*c^3*b^2 + 2/3*x^9*d*c^2*b*a + 1/3*x^9*d^2*c*a^2 + 2/5*x^5*c^3*b*a + 3/5*x^5*d*c^2*a^2
+ x*c^3*a^2

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Sympy [A]  time = 0.085057, size = 139, normalized size = 1.14 \begin{align*} a^{2} c^{3} x + \frac{b^{2} d^{3} x^{21}}{21} + x^{17} \left (\frac{2 a b d^{3}}{17} + \frac{3 b^{2} c d^{2}}{17}\right ) + x^{13} \left (\frac{a^{2} d^{3}}{13} + \frac{6 a b c d^{2}}{13} + \frac{3 b^{2} c^{2} d}{13}\right ) + x^{9} \left (\frac{a^{2} c d^{2}}{3} + \frac{2 a b c^{2} d}{3} + \frac{b^{2} c^{3}}{9}\right ) + x^{5} \left (\frac{3 a^{2} c^{2} d}{5} + \frac{2 a b c^{3}}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*c**3*x + b**2*d**3*x**21/21 + x**17*(2*a*b*d**3/17 + 3*b**2*c*d**2/17) + x**13*(a**2*d**3/13 + 6*a*b*c*d*
*2/13 + 3*b**2*c**2*d/13) + x**9*(a**2*c*d**2/3 + 2*a*b*c**2*d/3 + b**2*c**3/9) + x**5*(3*a**2*c**2*d/5 + 2*a*
b*c**3/5)

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Giac [A]  time = 1.10903, size = 178, normalized size = 1.46 \begin{align*} \frac{1}{21} \, b^{2} d^{3} x^{21} + \frac{3}{17} \, b^{2} c d^{2} x^{17} + \frac{2}{17} \, a b d^{3} x^{17} + \frac{3}{13} \, b^{2} c^{2} d x^{13} + \frac{6}{13} \, a b c d^{2} x^{13} + \frac{1}{13} \, a^{2} d^{3} x^{13} + \frac{1}{9} \, b^{2} c^{3} x^{9} + \frac{2}{3} \, a b c^{2} d x^{9} + \frac{1}{3} \, a^{2} c d^{2} x^{9} + \frac{2}{5} \, a b c^{3} x^{5} + \frac{3}{5} \, a^{2} c^{2} d x^{5} + a^{2} c^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*b^2*d^3*x^21 + 3/17*b^2*c*d^2*x^17 + 2/17*a*b*d^3*x^17 + 3/13*b^2*c^2*d*x^13 + 6/13*a*b*c*d^2*x^13 + 1/13
*a^2*d^3*x^13 + 1/9*b^2*c^3*x^9 + 2/3*a*b*c^2*d*x^9 + 1/3*a^2*c*d^2*x^9 + 2/5*a*b*c^3*x^5 + 3/5*a^2*c^2*d*x^5
+ a^2*c^3*x