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

Optimal. Leaf size=94 \[ \frac{1}{3} c^3 x^3 (4 a d+b c)+\frac{2}{5} c^2 d x^5 (3 a d+2 b c)+\frac{1}{9} d^3 x^9 (a d+4 b c)+\frac{2}{7} c d^2 x^7 (2 a d+3 b c)+a c^4 x+\frac{1}{11} b d^4 x^{11} \]

[Out]

a*c^4*x + (c^3*(b*c + 4*a*d)*x^3)/3 + (2*c^2*d*(2*b*c + 3*a*d)*x^5)/5 + (2*c*d^2
*(3*b*c + 2*a*d)*x^7)/7 + (d^3*(4*b*c + a*d)*x^9)/9 + (b*d^4*x^11)/11

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Rubi [A]  time = 0.145135, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{1}{3} c^3 x^3 (4 a d+b c)+\frac{2}{5} c^2 d x^5 (3 a d+2 b c)+\frac{1}{9} d^3 x^9 (a d+4 b c)+\frac{2}{7} c d^2 x^7 (2 a d+3 b c)+a c^4 x+\frac{1}{11} b d^4 x^{11} \]

Antiderivative was successfully verified.

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

[Out]

a*c^4*x + (c^3*(b*c + 4*a*d)*x^3)/3 + (2*c^2*d*(2*b*c + 3*a*d)*x^5)/5 + (2*c*d^2
*(3*b*c + 2*a*d)*x^7)/7 + (d^3*(4*b*c + a*d)*x^9)/9 + (b*d^4*x^11)/11

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b d^{4} x^{11}}{11} + c^{4} \int a\, dx + \frac{c^{3} x^{3} \left (4 a d + b c\right )}{3} + \frac{2 c^{2} d x^{5} \left (3 a d + 2 b c\right )}{5} + \frac{2 c d^{2} x^{7} \left (2 a d + 3 b c\right )}{7} + \frac{d^{3} x^{9} \left (a d + 4 b c\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*d**4*x**11/11 + c**4*Integral(a, x) + c**3*x**3*(4*a*d + b*c)/3 + 2*c**2*d*x**
5*(3*a*d + 2*b*c)/5 + 2*c*d**2*x**7*(2*a*d + 3*b*c)/7 + d**3*x**9*(a*d + 4*b*c)/
9

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Mathematica [A]  time = 0.034363, size = 94, normalized size = 1. \[ \frac{1}{3} c^3 x^3 (4 a d+b c)+\frac{2}{5} c^2 d x^5 (3 a d+2 b c)+\frac{1}{9} d^3 x^9 (a d+4 b c)+\frac{2}{7} c d^2 x^7 (2 a d+3 b c)+a c^4 x+\frac{1}{11} b d^4 x^{11} \]

Antiderivative was successfully verified.

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

[Out]

a*c^4*x + (c^3*(b*c + 4*a*d)*x^3)/3 + (2*c^2*d*(2*b*c + 3*a*d)*x^5)/5 + (2*c*d^2
*(3*b*c + 2*a*d)*x^7)/7 + (d^3*(4*b*c + a*d)*x^9)/9 + (b*d^4*x^11)/11

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Maple [A]  time = 0.002, size = 97, normalized size = 1. \[{\frac{b{d}^{4}{x}^{11}}{11}}+{\frac{ \left ( a{d}^{4}+4\,bc{d}^{3} \right ){x}^{9}}{9}}+{\frac{ \left ( 4\,ac{d}^{3}+6\,b{c}^{2}{d}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,a{c}^{2}{d}^{2}+4\,b{c}^{3}d \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,a{c}^{3}d+b{c}^{4} \right ){x}^{3}}{3}}+a{c}^{4}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)*(d*x^2+c)^4,x)

[Out]

1/11*b*d^4*x^11+1/9*(a*d^4+4*b*c*d^3)*x^9+1/7*(4*a*c*d^3+6*b*c^2*d^2)*x^7+1/5*(6
*a*c^2*d^2+4*b*c^3*d)*x^5+1/3*(4*a*c^3*d+b*c^4)*x^3+a*c^4*x

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Maxima [A]  time = 1.34763, size = 130, normalized size = 1.38 \[ \frac{1}{11} \, b d^{4} x^{11} + \frac{1}{9} \,{\left (4 \, b c d^{3} + a d^{4}\right )} x^{9} + \frac{2}{7} \,{\left (3 \, b c^{2} d^{2} + 2 \, a c d^{3}\right )} x^{7} + a c^{4} x + \frac{2}{5} \,{\left (2 \, b c^{3} d + 3 \, a c^{2} d^{2}\right )} x^{5} + \frac{1}{3} \,{\left (b c^{4} + 4 \, a c^{3} d\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*b*d^4*x^11 + 1/9*(4*b*c*d^3 + a*d^4)*x^9 + 2/7*(3*b*c^2*d^2 + 2*a*c*d^3)*x^
7 + a*c^4*x + 2/5*(2*b*c^3*d + 3*a*c^2*d^2)*x^5 + 1/3*(b*c^4 + 4*a*c^3*d)*x^3

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Fricas [A]  time = 0.178747, size = 1, normalized size = 0.01 \[ \frac{1}{11} x^{11} d^{4} b + \frac{4}{9} x^{9} d^{3} c b + \frac{1}{9} x^{9} d^{4} a + \frac{6}{7} x^{7} d^{2} c^{2} b + \frac{4}{7} x^{7} d^{3} c a + \frac{4}{5} x^{5} d c^{3} b + \frac{6}{5} x^{5} d^{2} c^{2} a + \frac{1}{3} x^{3} c^{4} b + \frac{4}{3} x^{3} d c^{3} a + x c^{4} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*x^11*d^4*b + 4/9*x^9*d^3*c*b + 1/9*x^9*d^4*a + 6/7*x^7*d^2*c^2*b + 4/7*x^7*
d^3*c*a + 4/5*x^5*d*c^3*b + 6/5*x^5*d^2*c^2*a + 1/3*x^3*c^4*b + 4/3*x^3*d*c^3*a
+ x*c^4*a

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Sympy [A]  time = 0.147961, size = 107, normalized size = 1.14 \[ a c^{4} x + \frac{b d^{4} x^{11}}{11} + x^{9} \left (\frac{a d^{4}}{9} + \frac{4 b c d^{3}}{9}\right ) + x^{7} \left (\frac{4 a c d^{3}}{7} + \frac{6 b c^{2} d^{2}}{7}\right ) + x^{5} \left (\frac{6 a c^{2} d^{2}}{5} + \frac{4 b c^{3} d}{5}\right ) + x^{3} \left (\frac{4 a c^{3} d}{3} + \frac{b c^{4}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*c**4*x + b*d**4*x**11/11 + x**9*(a*d**4/9 + 4*b*c*d**3/9) + x**7*(4*a*c*d**3/7
+ 6*b*c**2*d**2/7) + x**5*(6*a*c**2*d**2/5 + 4*b*c**3*d/5) + x**3*(4*a*c**3*d/3
+ b*c**4/3)

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GIAC/XCAS [A]  time = 0.226178, size = 132, normalized size = 1.4 \[ \frac{1}{11} \, b d^{4} x^{11} + \frac{4}{9} \, b c d^{3} x^{9} + \frac{1}{9} \, a d^{4} x^{9} + \frac{6}{7} \, b c^{2} d^{2} x^{7} + \frac{4}{7} \, a c d^{3} x^{7} + \frac{4}{5} \, b c^{3} d x^{5} + \frac{6}{5} \, a c^{2} d^{2} x^{5} + \frac{1}{3} \, b c^{4} x^{3} + \frac{4}{3} \, a c^{3} d x^{3} + a c^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*b*d^4*x^11 + 4/9*b*c*d^3*x^9 + 1/9*a*d^4*x^9 + 6/7*b*c^2*d^2*x^7 + 4/7*a*c*
d^3*x^7 + 4/5*b*c^3*d*x^5 + 6/5*a*c^2*d^2*x^5 + 1/3*b*c^4*x^3 + 4/3*a*c^3*d*x^3
+ a*c^4*x