3.271 \(\int \frac{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )}{x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.230301, 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 \[ \frac{1}{2} x^2 \left (e (a e+2 b d)+c d^2\right )+d \log (x) (2 a e+b d)-\frac{a d^2}{2 x^2}+\frac{1}{4} e x^4 (b e+2 c d)+\frac{1}{6} c e^2 x^6 \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x^2)^2*(a + b*x^2 + c*x^4))/x^3,x]

[Out]

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

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

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**3,x)

[Out]

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

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Mathematica [A]  time = 0.0828625, size = 71, normalized size = 0.96 \[ \frac{1}{12} \left (6 x^2 \left (e (a e+2 b d)+c d^2\right )+12 d \log (x) (2 a e+b d)-\frac{6 a d^2}{x^2}+3 e x^4 (b e+2 c d)+2 c e^2 x^6\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x^2)^2*(a + b*x^2 + c*x^4))/x^3,x]

[Out]

((-6*a*d^2)/x^2 + 6*(c*d^2 + e*(2*b*d + a*e))*x^2 + 3*e*(2*c*d + b*e)*x^4 + 2*c*
e^2*x^6 + 12*d*(b*d + 2*a*e)*Log[x])/12

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Maple [A]  time = 0.011, size = 76, normalized size = 1. \[{\frac{c{e}^{2}{x}^{6}}{6}}+{\frac{{x}^{4}b{e}^{2}}{4}}+{\frac{{x}^{4}cde}{2}}+{\frac{{x}^{2}a{e}^{2}}{2}}+{x}^{2}bde+{\frac{{x}^{2}c{d}^{2}}{2}}+2\,\ln \left ( x \right ) ade+\ln \left ( x \right ) b{d}^{2}-{\frac{a{d}^{2}}{2\,{x}^{2}}} \]

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^3,x)

[Out]

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

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Maxima [A]  time = 0.691421, size = 99, normalized size = 1.34 \[ \frac{1}{6} \, c e^{2} x^{6} + \frac{1}{4} \,{\left (2 \, c d e + b e^{2}\right )} x^{4} + \frac{1}{2} \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{2} + \frac{1}{2} \,{\left (b d^{2} + 2 \, a d e\right )} \log \left (x^{2}\right ) - \frac{a d^{2}}{2 \, x^{2}} \]

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^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.259345, size = 103, normalized size = 1.39 \[ \frac{2 \, c e^{2} x^{8} + 3 \,{\left (2 \, c d e + b e^{2}\right )} x^{6} + 6 \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{4} + 12 \,{\left (b d^{2} + 2 \, a d e\right )} x^{2} \log \left (x\right ) - 6 \, a d^{2}}{12 \, x^{2}} \]

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^3,x, algorithm="fricas")

[Out]

1/12*(2*c*e^2*x^8 + 3*(2*c*d*e + b*e^2)*x^6 + 6*(c*d^2 + 2*b*d*e + a*e^2)*x^4 +
12*(b*d^2 + 2*a*d*e)*x^2*log(x) - 6*a*d^2)/x^2

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Sympy [A]  time = 1.59322, size = 71, normalized size = 0.96 \[ - \frac{a d^{2}}{2 x^{2}} + \frac{c e^{2} x^{6}}{6} + d \left (2 a e + b d\right ) \log{\left (x \right )} + x^{4} \left (\frac{b e^{2}}{4} + \frac{c d e}{2}\right ) + x^{2} \left (\frac{a e^{2}}{2} + b d e + \frac{c d^{2}}{2}\right ) \]

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**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.270727, size = 131, normalized size = 1.77 \[ \frac{1}{6} \, c x^{6} e^{2} + \frac{1}{2} \, c d x^{4} e + \frac{1}{4} \, b x^{4} e^{2} + \frac{1}{2} \, c d^{2} x^{2} + b d x^{2} e + \frac{1}{2} \, a x^{2} e^{2} + \frac{1}{2} \,{\left (b d^{2} + 2 \, a d e\right )}{\rm ln}\left (x^{2}\right ) - \frac{b d^{2} x^{2} + 2 \, a d x^{2} e + a d^{2}}{2 \, x^{2}} \]

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^3,x, algorithm="giac")

[Out]

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