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

Optimal. Leaf size=71 \[ \frac{1}{3} x^3 \left (e (a e+2 b d)+c d^2\right )+d x (2 a e+b d)-\frac{a d^2}{x}+\frac{1}{5} e x^5 (b e+2 c d)+\frac{1}{7} c e^2 x^7 \]

[Out]

-((a*d^2)/x) + d*(b*d + 2*a*e)*x + ((c*d^2 + e*(2*b*d + a*e))*x^3)/3 + (e*(2*c*d
+ b*e)*x^5)/5 + (c*e^2*x^7)/7

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Rubi [A]  time = 0.12503, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ \frac{1}{3} x^3 \left (e (a e+2 b d)+c d^2\right )+d x (2 a e+b d)-\frac{a d^2}{x}+\frac{1}{5} e x^5 (b e+2 c d)+\frac{1}{7} c e^2 x^7 \]

Antiderivative was successfully verified.

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

[Out]

-((a*d^2)/x) + d*(b*d + 2*a*e)*x + ((c*d^2 + e*(2*b*d + a*e))*x^3)/3 + (e*(2*c*d
+ b*e)*x^5)/5 + (c*e^2*x^7)/7

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

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**2,x)

[Out]

-a*d**2/x + c*e**2*x**7/7 + e*x**5*(b*e + 2*c*d)/5 + x**3*(a*e**2/3 + 2*b*d*e/3
+ c*d**2/3) + d*(2*a*e + b*d)*Integral(b, x)/b

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Mathematica [A]  time = 0.0559375, size = 71, normalized size = 1. \[ \frac{1}{3} x^3 \left (a e^2+2 b d e+c d^2\right )+d x (2 a e+b d)-\frac{a d^2}{x}+\frac{1}{5} e x^5 (b e+2 c d)+\frac{1}{7} c e^2 x^7 \]

Antiderivative was successfully verified.

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

[Out]

-((a*d^2)/x) + d*(b*d + 2*a*e)*x + ((c*d^2 + 2*b*d*e + a*e^2)*x^3)/3 + (e*(2*c*d
+ b*e)*x^5)/5 + (c*e^2*x^7)/7

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Maple [A]  time = 0.006, size = 75, normalized size = 1.1 \[{\frac{c{e}^{2}{x}^{7}}{7}}+{\frac{{x}^{5}b{e}^{2}}{5}}+{\frac{2\,{x}^{5}cde}{5}}+{\frac{{x}^{3}a{e}^{2}}{3}}+{\frac{2\,{x}^{3}bde}{3}}+{\frac{{x}^{3}c{d}^{2}}{3}}+2\,xade+xb{d}^{2}-{\frac{a{d}^{2}}{x}} \]

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^2,x)

[Out]

1/7*c*e^2*x^7+1/5*x^5*b*e^2+2/5*x^5*c*d*e+1/3*x^3*a*e^2+2/3*x^3*b*d*e+1/3*x^3*c*
d^2+2*x*a*d*e+x*b*d^2-a*d^2/x

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Maxima [A]  time = 0.700418, size = 93, normalized size = 1.31 \[ \frac{1}{7} \, c e^{2} x^{7} + \frac{1}{5} \,{\left (2 \, c d e + b e^{2}\right )} x^{5} + \frac{1}{3} \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{3} - \frac{a d^{2}}{x} +{\left (b d^{2} + 2 \, a d e\right )} x \]

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^2,x, algorithm="maxima")

[Out]

1/7*c*e^2*x^7 + 1/5*(2*c*d*e + b*e^2)*x^5 + 1/3*(c*d^2 + 2*b*d*e + a*e^2)*x^3 -
a*d^2/x + (b*d^2 + 2*a*d*e)*x

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Fricas [A]  time = 0.245399, size = 100, normalized size = 1.41 \[ \frac{15 \, c e^{2} x^{8} + 21 \,{\left (2 \, c d e + b e^{2}\right )} x^{6} + 35 \,{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{4} - 105 \, a d^{2} + 105 \,{\left (b d^{2} + 2 \, a d e\right )} x^{2}}{105 \, x} \]

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^2,x, algorithm="fricas")

[Out]

1/105*(15*c*e^2*x^8 + 21*(2*c*d*e + b*e^2)*x^6 + 35*(c*d^2 + 2*b*d*e + a*e^2)*x^
4 - 105*a*d^2 + 105*(b*d^2 + 2*a*d*e)*x^2)/x

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Sympy [A]  time = 1.25525, size = 73, normalized size = 1.03 \[ - \frac{a d^{2}}{x} + \frac{c e^{2} x^{7}}{7} + x^{5} \left (\frac{b e^{2}}{5} + \frac{2 c d e}{5}\right ) + x^{3} \left (\frac{a e^{2}}{3} + \frac{2 b d e}{3} + \frac{c d^{2}}{3}\right ) + x \left (2 a d e + b d^{2}\right ) \]

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**2,x)

[Out]

-a*d**2/x + c*e**2*x**7/7 + x**5*(b*e**2/5 + 2*c*d*e/5) + x**3*(a*e**2/3 + 2*b*d
*e/3 + c*d**2/3) + x*(2*a*d*e + b*d**2)

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GIAC/XCAS [A]  time = 0.266909, size = 100, normalized size = 1.41 \[ \frac{1}{7} \, c x^{7} e^{2} + \frac{2}{5} \, c d x^{5} e + \frac{1}{5} \, b x^{5} e^{2} + \frac{1}{3} \, c d^{2} x^{3} + \frac{2}{3} \, b d x^{3} e + \frac{1}{3} \, a x^{3} e^{2} + b d^{2} x + 2 \, a d x e - \frac{a d^{2}}{x} \]

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^2,x, algorithm="giac")

[Out]

1/7*c*x^7*e^2 + 2/5*c*d*x^5*e + 1/5*b*x^5*e^2 + 1/3*c*d^2*x^3 + 2/3*b*d*x^3*e +
1/3*a*x^3*e^2 + b*d^2*x + 2*a*d*x*e - a*d^2/x