3.269 \(\int \frac{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )}{x} \, dx\)

Optimal. Leaf size=74 \[ \frac{1}{4} x^4 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{2} d x^2 (2 a e+b d)+a d^2 \log (x)+\frac{1}{6} e x^6 (b e+2 c d)+\frac{1}{8} c e^2 x^8 \]

[Out]

(d*(b*d + 2*a*e)*x^2)/2 + ((c*d^2 + e*(2*b*d + a*e))*x^4)/4 + (e*(2*c*d + b*e)*x
^6)/6 + (c*e^2*x^8)/8 + a*d^2*Log[x]

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Rubi [A]  time = 0.194187, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{1}{4} x^4 \left (e (a e+2 b d)+c d^2\right )+\frac{1}{2} d x^2 (2 a e+b d)+a d^2 \log (x)+\frac{1}{6} e x^6 (b e+2 c d)+\frac{1}{8} c e^2 x^8 \]

Antiderivative was successfully verified.

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

[Out]

(d*(b*d + 2*a*e)*x^2)/2 + ((c*d^2 + e*(2*b*d + a*e))*x^4)/4 + (e*(2*c*d + b*e)*x
^6)/6 + (c*e^2*x^8)/8 + a*d^2*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a d^{2} \log{\left (x^{2} \right )}}{2} + \frac{c e^{2} x^{8}}{8} + \frac{e x^{6} \left (b e + 2 c d\right )}{6} + \left (a e + \frac{b d}{2}\right ) \int ^{x^{2}} d\, dx + \left (\frac{a e^{2}}{2} + b d e + \frac{c d^{2}}{2}\right ) \int ^{x^{2}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*d**2*log(x**2)/2 + c*e**2*x**8/8 + e*x**6*(b*e + 2*c*d)/6 + (a*e + b*d/2)*Inte
gral(d, (x, x**2)) + (a*e**2/2 + b*d*e + c*d**2/2)*Integral(x, (x, x**2))

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Mathematica [A]  time = 0.0387653, size = 74, normalized size = 1. \[ \frac{1}{4} x^4 \left (a e^2+2 b d e+c d^2\right )+\frac{1}{2} d x^2 (2 a e+b d)+a d^2 \log (x)+\frac{1}{6} e x^6 (b e+2 c d)+\frac{1}{8} c e^2 x^8 \]

Antiderivative was successfully verified.

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

[Out]

(d*(b*d + 2*a*e)*x^2)/2 + ((c*d^2 + 2*b*d*e + a*e^2)*x^4)/4 + (e*(2*c*d + b*e)*x
^6)/6 + (c*e^2*x^8)/8 + a*d^2*Log[x]

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Maple [A]  time = 0.004, size = 77, normalized size = 1. \[{\frac{c{e}^{2}{x}^{8}}{8}}+{\frac{{x}^{6}b{e}^{2}}{6}}+{\frac{{x}^{6}cde}{3}}+{\frac{{x}^{4}a{e}^{2}}{4}}+{\frac{{x}^{4}bde}{2}}+{\frac{{x}^{4}c{d}^{2}}{4}}+{x}^{2}ade+{\frac{{x}^{2}b{d}^{2}}{2}}+a{d}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*c*e^2*x^8+1/6*x^6*b*e^2+1/3*x^6*c*d*e+1/4*x^4*a*e^2+1/2*x^4*b*d*e+1/4*x^4*c*
d^2+x^2*a*d*e+1/2*x^2*b*d^2+a*d^2*ln(x)

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Maxima [A]  time = 0.700413, size = 99, normalized size = 1.34 \[ \frac{1}{8} \, c e^{2} x^{8} + \frac{1}{6} \,{\left (2 \, c d e + b e^{2}\right )} x^{6} + \frac{1}{4} \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{4} + \frac{1}{2} \, a d^{2} \log \left (x^{2}\right ) + \frac{1}{2} \,{\left (b d^{2} + 2 \, a d e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.252087, size = 95, normalized size = 1.28 \[ \frac{1}{8} \, c e^{2} x^{8} + \frac{1}{6} \,{\left (2 \, c d e + b e^{2}\right )} x^{6} + \frac{1}{4} \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{4} + a d^{2} \log \left (x\right ) + \frac{1}{2} \,{\left (b d^{2} + 2 \, a d e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.26214, size = 73, normalized size = 0.99 \[ a d^{2} \log{\left (x \right )} + \frac{c e^{2} x^{8}}{8} + x^{6} \left (\frac{b e^{2}}{6} + \frac{c d e}{3}\right ) + x^{4} \left (\frac{a e^{2}}{4} + \frac{b d e}{2} + \frac{c d^{2}}{4}\right ) + x^{2} \left (a d e + \frac{b d^{2}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.269057, size = 107, normalized size = 1.45 \[ \frac{1}{8} \, c x^{8} e^{2} + \frac{1}{3} \, c d x^{6} e + \frac{1}{6} \, b x^{6} e^{2} + \frac{1}{4} \, c d^{2} x^{4} + \frac{1}{2} \, b d x^{4} e + \frac{1}{4} \, a x^{4} e^{2} + \frac{1}{2} \, b d^{2} x^{2} + a d x^{2} e + \frac{1}{2} \, a d^{2}{\rm ln}\left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*c*x^8*e^2 + 1/3*c*d*x^6*e + 1/6*b*x^6*e^2 + 1/4*c*d^2*x^4 + 1/2*b*d*x^4*e +
1/4*a*x^4*e^2 + 1/2*b*d^2*x^2 + a*d*x^2*e + 1/2*a*d^2*ln(x^2)