3.146 \(\int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x} \, dx\)

Optimal. Leaf size=43 \[ a^2 c \log (x)+a b c x^2+\frac{d \left (a+b x^2\right )^3}{6 b}+\frac{1}{4} b^2 c x^4 \]

[Out]

a*b*c*x^2 + (b^2*c*x^4)/4 + (d*(a + b*x^2)^3)/(6*b) + a^2*c*Log[x]

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Rubi [A]  time = 0.0796131, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ a^2 c \log (x)+a b c x^2+\frac{d \left (a+b x^2\right )^3}{6 b}+\frac{1}{4} b^2 c x^4 \]

Antiderivative was successfully verified.

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

[Out]

a*b*c*x^2 + (b^2*c*x^4)/4 + (d*(a + b*x^2)^3)/(6*b) + a^2*c*Log[x]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*c*log(x**2)/2 + a*b*c*x**2 + b**2*c*Integral(x, (x, x**2))/2 + d*(a + b*x**
2)**3/(6*b)

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Mathematica [A]  time = 0.0238551, size = 51, normalized size = 1.19 \[ a^2 c \log (x)+\frac{1}{4} b x^4 (2 a d+b c)+\frac{1}{2} a x^2 (a d+2 b c)+\frac{1}{6} b^2 d x^6 \]

Antiderivative was successfully verified.

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

[Out]

(a*(2*b*c + a*d)*x^2)/2 + (b*(b*c + 2*a*d)*x^4)/4 + (b^2*d*x^6)/6 + a^2*c*Log[x]

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Maple [A]  time = 0.003, size = 51, normalized size = 1.2 \[{\frac{{b}^{2}d{x}^{6}}{6}}+{\frac{{x}^{4}abd}{2}}+{\frac{{b}^{2}c{x}^{4}}{4}}+{\frac{{x}^{2}{a}^{2}d}{2}}+abc{x}^{2}+{a}^{2}c\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)/x,x)

[Out]

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

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Maxima [A]  time = 1.34861, size = 70, normalized size = 1.63 \[ \frac{1}{6} \, b^{2} d x^{6} + \frac{1}{4} \,{\left (b^{2} c + 2 \, a b d\right )} x^{4} + \frac{1}{2} \, a^{2} c \log \left (x^{2}\right ) + \frac{1}{2} \,{\left (2 \, a b c + a^{2} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.218032, size = 66, normalized size = 1.53 \[ \frac{1}{6} \, b^{2} d x^{6} + \frac{1}{4} \,{\left (b^{2} c + 2 \, a b d\right )} x^{4} + a^{2} c \log \left (x\right ) + \frac{1}{2} \,{\left (2 \, a b c + a^{2} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.22489, size = 49, normalized size = 1.14 \[ a^{2} c \log{\left (x \right )} + \frac{b^{2} d x^{6}}{6} + x^{4} \left (\frac{a b d}{2} + \frac{b^{2} c}{4}\right ) + x^{2} \left (\frac{a^{2} d}{2} + a b c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.231116, size = 72, normalized size = 1.67 \[ \frac{1}{6} \, b^{2} d x^{6} + \frac{1}{4} \, b^{2} c x^{4} + \frac{1}{2} \, a b d x^{4} + a b c x^{2} + \frac{1}{2} \, a^{2} d x^{2} + \frac{1}{2} \, a^{2} c{\rm ln}\left (x^{2}\right ) \]

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]

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