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

Optimal. Leaf size=42 \[ \frac{\left (a+b x^2\right )^3 (b c-a d)}{6 b^2}+\frac{d \left (a+b x^2\right )^4}{8 b^2} \]

[Out]

((b*c - a*d)*(a + b*x^2)^3)/(6*b^2) + (d*(a + b*x^2)^4)/(8*b^2)

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Rubi [A]  time = 0.151151, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{\left (a+b x^2\right )^3 (b c-a d)}{6 b^2}+\frac{d \left (a+b x^2\right )^4}{8 b^2} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)*(a + b*x^2)^3)/(6*b^2) + (d*(a + b*x^2)^4)/(8*b^2)

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Rubi in Sympy [A]  time = 15.6108, size = 34, normalized size = 0.81 \[ \frac{d \left (a + b x^{2}\right )^{4}}{8 b^{2}} - \frac{\left (a + b x^{2}\right )^{3} \left (a d - b c\right )}{6 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*(a + b*x**2)**4/(8*b**2) - (a + b*x**2)**3*(a*d - b*c)/(6*b**2)

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Mathematica [A]  time = 0.0207506, size = 51, normalized size = 1.21 \[ \frac{1}{24} x^2 \left (12 a^2 c+4 b x^4 (2 a d+b c)+6 a x^2 (a d+2 b c)+3 b^2 d x^6\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(12*a^2*c + 6*a*(2*b*c + a*d)*x^2 + 4*b*(b*c + 2*a*d)*x^4 + 3*b^2*d*x^6))/2
4

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Maple [A]  time = 0.001, size = 52, normalized size = 1.2 \[{\frac{{b}^{2}d{x}^{8}}{8}}+{\frac{ \left ( 2\,abd+{b}^{2}c \right ){x}^{6}}{6}}+{\frac{ \left ({a}^{2}d+2\,abc \right ){x}^{4}}{4}}+{\frac{{a}^{2}c{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^2+a)^2*(d*x^2+c),x)

[Out]

1/8*b^2*d*x^8+1/6*(2*a*b*d+b^2*c)*x^6+1/4*(a^2*d+2*a*b*c)*x^4+1/2*a^2*c*x^2

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Maxima [A]  time = 1.32345, size = 69, normalized size = 1.64 \[ \frac{1}{8} \, b^{2} d x^{8} + \frac{1}{6} \,{\left (b^{2} c + 2 \, a b d\right )} x^{6} + \frac{1}{2} \, a^{2} c x^{2} + \frac{1}{4} \,{\left (2 \, a b c + a^{2} d\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*b^2*d*x^8 + 1/6*(b^2*c + 2*a*b*d)*x^6 + 1/2*a^2*c*x^2 + 1/4*(2*a*b*c + a^2*d
)*x^4

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Fricas [A]  time = 0.194868, size = 1, normalized size = 0.02 \[ \frac{1}{8} x^{8} d b^{2} + \frac{1}{6} x^{6} c b^{2} + \frac{1}{3} x^{6} d b a + \frac{1}{2} x^{4} c b a + \frac{1}{4} x^{4} d a^{2} + \frac{1}{2} x^{2} c a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*x^8*d*b^2 + 1/6*x^6*c*b^2 + 1/3*x^6*d*b*a + 1/2*x^4*c*b*a + 1/4*x^4*d*a^2 +
1/2*x^2*c*a^2

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Sympy [A]  time = 0.115064, size = 53, normalized size = 1.26 \[ \frac{a^{2} c x^{2}}{2} + \frac{b^{2} d x^{8}}{8} + x^{6} \left (\frac{a b d}{3} + \frac{b^{2} c}{6}\right ) + x^{4} \left (\frac{a^{2} d}{4} + \frac{a b c}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*c*x**2/2 + b**2*d*x**8/8 + x**6*(a*b*d/3 + b**2*c/6) + x**4*(a**2*d/4 + a*b
*c/2)

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GIAC/XCAS [A]  time = 0.220245, size = 72, normalized size = 1.71 \[ \frac{1}{8} \, b^{2} d x^{8} + \frac{1}{6} \, b^{2} c x^{6} + \frac{1}{3} \, a b d x^{6} + \frac{1}{2} \, a b c x^{4} + \frac{1}{4} \, a^{2} d x^{4} + \frac{1}{2} \, a^{2} c x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*b^2*d*x^8 + 1/6*b^2*c*x^6 + 1/3*a*b*d*x^6 + 1/2*a*b*c*x^4 + 1/4*a^2*d*x^4 +
1/2*a^2*c*x^2