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

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 = 0.172716, 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 \[ \frac{1}{13} d x^{13} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{9} c x^9 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 x+\frac{1}{5} a c^2 x^5 (3 a d+2 b c)+\frac{1}{17} b d^2 x^{17} (2 a d+3 b c)+\frac{1}{21} b^2 d^3 x^{21} \]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a c^{2} x^{5} \left (3 a d + 2 b c\right )}{5} + \frac{b^{2} d^{3} x^{21}}{21} + \frac{b d^{2} x^{17} \left (2 a d + 3 b c\right )}{17} + c^{3} \int a^{2}\, dx + \frac{c x^{9} \left (3 a^{2} d^{2} + 6 a b c d + b^{2} c^{2}\right )}{9} + \frac{d x^{13} \left (a^{2} d^{2} + 6 a b c d + 3 b^{2} c^{2}\right )}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**2*(d*x**4+c)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0428412, 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 = 0.002, size = 125, normalized size = 1. \[{\frac{{b}^{2}{d}^{3}{x}^{21}}{21}}+{\frac{ \left ( 2\,ab{d}^{3}+3\,{b}^{2}c{d}^{2} \right ){x}^{17}}{17}}+{\frac{ \left ({a}^{2}{d}^{3}+6\,abc{d}^{2}+3\,{b}^{2}{c}^{2}d \right ){x}^{13}}{13}}+{\frac{ \left ( 3\,{a}^{2}c{d}^{2}+6\,ab{c}^{2}d+{b}^{2}{c}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ( 3\,{a}^{2}{c}^{2}d+2\,ab{c}^{3} \right ){x}^{5}}{5}}+{a}^{2}{c}^{3}x \]

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 = 1.35818, size = 167, normalized size = 1.37 \[ \frac{1}{21} \, b^{2} d^{3} x^{21} + \frac{1}{17} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{17} + \frac{1}{13} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{13} + \frac{1}{9} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{9} + a^{2} c^{3} x + \frac{1}{5} \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{5} \]

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 = 0.191059, size = 1, normalized size = 0.01 \[ \frac{1}{21} x^{21} d^{3} b^{2} + \frac{3}{17} x^{17} d^{2} c b^{2} + \frac{2}{17} x^{17} d^{3} b a + \frac{3}{13} x^{13} d c^{2} b^{2} + \frac{6}{13} x^{13} d^{2} c b a + \frac{1}{13} x^{13} d^{3} a^{2} + \frac{1}{9} x^{9} c^{3} b^{2} + \frac{2}{3} x^{9} d c^{2} b a + \frac{1}{3} x^{9} d^{2} c a^{2} + \frac{2}{5} x^{5} c^{3} b a + \frac{3}{5} x^{5} d c^{2} a^{2} + x c^{3} a^{2} \]

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 = 0.156153, size = 139, normalized size = 1.14 \[ a^{2} c^{3} x + \frac{b^{2} d^{3} x^{21}}{21} + x^{17} \left (\frac{2 a b d^{3}}{17} + \frac{3 b^{2} c d^{2}}{17}\right ) + x^{13} \left (\frac{a^{2} d^{3}}{13} + \frac{6 a b c d^{2}}{13} + \frac{3 b^{2} c^{2} d}{13}\right ) + x^{9} \left (\frac{a^{2} c d^{2}}{3} + \frac{2 a b c^{2} d}{3} + \frac{b^{2} c^{3}}{9}\right ) + x^{5} \left (\frac{3 a^{2} c^{2} d}{5} + \frac{2 a b c^{3}}{5}\right ) \]

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/XCAS [A]  time = 0.212041, size = 178, normalized size = 1.46 \[ \frac{1}{21} \, b^{2} d^{3} x^{21} + \frac{3}{17} \, b^{2} c d^{2} x^{17} + \frac{2}{17} \, a b d^{3} x^{17} + \frac{3}{13} \, b^{2} c^{2} d x^{13} + \frac{6}{13} \, a b c d^{2} x^{13} + \frac{1}{13} \, a^{2} d^{3} x^{13} + \frac{1}{9} \, b^{2} c^{3} x^{9} + \frac{2}{3} \, a b c^{2} d x^{9} + \frac{1}{3} \, a^{2} c d^{2} x^{9} + \frac{2}{5} \, a b c^{3} x^{5} + \frac{3}{5} \, a^{2} c^{2} d x^{5} + a^{2} c^{3} x \]

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^1
3 + 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