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

Optimal. Leaf size=124 \[ \frac{x \left (7 c d^2-e (a e+3 b d)\right )}{8 d e^3 \left (d+e x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (15 c d^2-e (a e+3 b d)\right )}{8 d^{3/2} e^{7/2}}+\frac{x^3 \left (a+\frac{d (c d-b e)}{e^2}\right )}{4 d \left (d+e x^2\right )^2}+\frac{c x}{e^3} \]

[Out]

(c*x)/e^3 + ((a + (d*(c*d - b*e))/e^2)*x^3)/(4*d*(d + e*x^2)^2) + ((7*c*d^2 - e*
(3*b*d + a*e))*x)/(8*d*e^3*(d + e*x^2)) - ((15*c*d^2 - e*(3*b*d + a*e))*ArcTan[(
Sqrt[e]*x)/Sqrt[d]])/(8*d^(3/2)*e^(7/2))

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Rubi [A]  time = 0.367433, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{x \left (7 c d^2-e (a e+3 b d)\right )}{8 d e^3 \left (d+e x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (15 c d^2-e (a e+3 b d)\right )}{8 d^{3/2} e^{7/2}}+\frac{x^3 \left (a+\frac{d (c d-b e)}{e^2}\right )}{4 d \left (d+e x^2\right )^2}+\frac{c x}{e^3} \]

Antiderivative was successfully verified.

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

[Out]

(c*x)/e^3 + ((a + (d*(c*d - b*e))/e^2)*x^3)/(4*d*(d + e*x^2)^2) + ((7*c*d^2 - e*
(3*b*d + a*e))*x)/(8*d*e^3*(d + e*x^2)) - ((15*c*d^2 - e*(3*b*d + a*e))*ArcTan[(
Sqrt[e]*x)/Sqrt[d]])/(8*d^(3/2)*e^(7/2))

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Rubi in Sympy [A]  time = 54.6233, size = 114, normalized size = 0.92 \[ \frac{c x}{e^{3}} - \frac{x \left (a e^{2} - b d e + c d^{2}\right )}{4 e^{3} \left (d + e x^{2}\right )^{2}} + \frac{x \left (a e^{2} - 5 b d e + 9 c d^{2}\right )}{8 d e^{3} \left (d + e x^{2}\right )} + \frac{\left (a e^{2} + 3 b d e - 15 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{8 d^{\frac{3}{2}} e^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*x/e**3 - x*(a*e**2 - b*d*e + c*d**2)/(4*e**3*(d + e*x**2)**2) + x*(a*e**2 - 5*
b*d*e + 9*c*d**2)/(8*d*e**3*(d + e*x**2)) + (a*e**2 + 3*b*d*e - 15*c*d**2)*atan(
sqrt(e)*x/sqrt(d))/(8*d**(3/2)*e**(7/2))

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Mathematica [A]  time = 0.184947, size = 122, normalized size = 0.98 \[ \frac{x \left (a e^2-5 b d e+9 c d^2\right )}{8 d e^3 \left (d+e x^2\right )}-\frac{x \left (a e^2-b d e+c d^2\right )}{4 e^3 \left (d+e x^2\right )^2}-\frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (-a e^2-3 b d e+15 c d^2\right )}{8 d^{3/2} e^{7/2}}+\frac{c x}{e^3} \]

Antiderivative was successfully verified.

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

[Out]

(c*x)/e^3 - ((c*d^2 - b*d*e + a*e^2)*x)/(4*e^3*(d + e*x^2)^2) + ((9*c*d^2 - 5*b*
d*e + a*e^2)*x)/(8*d*e^3*(d + e*x^2)) - ((15*c*d^2 - 3*b*d*e - a*e^2)*ArcTan[(Sq
rt[e]*x)/Sqrt[d]])/(8*d^(3/2)*e^(7/2))

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Maple [A]  time = 0.015, size = 179, normalized size = 1.4 \[{\frac{cx}{{e}^{3}}}+{\frac{{x}^{3}a}{8\, \left ( e{x}^{2}+d \right ) ^{2}d}}-{\frac{5\,b{x}^{3}}{8\,e \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{9\,cd{x}^{3}}{8\,{e}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{ax}{8\,e \left ( e{x}^{2}+d \right ) ^{2}}}-{\frac{3\,bxd}{8\,{e}^{2} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{7\,c{d}^{2}x}{8\,{e}^{3} \left ( e{x}^{2}+d \right ) ^{2}}}+{\frac{a}{8\,de}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{3\,b}{8\,{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{15\,cd}{8\,{e}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*x/e^3+1/8/(e*x^2+d)^2/d*x^3*a-5/8/e/(e*x^2+d)^2*x^3*b+9/8/e^2/(e*x^2+d)^2*c*d*
x^3-1/8/e/(e*x^2+d)^2*a*x-3/8/e^2/(e*x^2+d)^2*x*b*d+7/8/e^3/(e*x^2+d)^2*c*d^2*x+
1/8/e/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*a+3/8/e^2/(d*e)^(1/2)*arctan(x*e/(d*
e)^(1/2))*b-15/8/e^3*d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)*x^2/(e*x^2 + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.269027, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (15 \, c d^{4} - 3 \, b d^{3} e - a d^{2} e^{2} +{\left (15 \, c d^{2} e^{2} - 3 \, b d e^{3} - a e^{4}\right )} x^{4} + 2 \,{\left (15 \, c d^{3} e - 3 \, b d^{2} e^{2} - a d e^{3}\right )} x^{2}\right )} \log \left (\frac{2 \, d e x +{\left (e x^{2} - d\right )} \sqrt{-d e}}{e x^{2} + d}\right ) - 2 \,{\left (8 \, c d e^{2} x^{5} +{\left (25 \, c d^{2} e - 5 \, b d e^{2} + a e^{3}\right )} x^{3} +{\left (15 \, c d^{3} - 3 \, b d^{2} e - a d e^{2}\right )} x\right )} \sqrt{-d e}}{16 \,{\left (d e^{5} x^{4} + 2 \, d^{2} e^{4} x^{2} + d^{3} e^{3}\right )} \sqrt{-d e}}, -\frac{{\left (15 \, c d^{4} - 3 \, b d^{3} e - a d^{2} e^{2} +{\left (15 \, c d^{2} e^{2} - 3 \, b d e^{3} - a e^{4}\right )} x^{4} + 2 \,{\left (15 \, c d^{3} e - 3 \, b d^{2} e^{2} - a d e^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{d e} x}{d}\right ) -{\left (8 \, c d e^{2} x^{5} +{\left (25 \, c d^{2} e - 5 \, b d e^{2} + a e^{3}\right )} x^{3} +{\left (15 \, c d^{3} - 3 \, b d^{2} e - a d e^{2}\right )} x\right )} \sqrt{d e}}{8 \,{\left (d e^{5} x^{4} + 2 \, d^{2} e^{4} x^{2} + d^{3} e^{3}\right )} \sqrt{d e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)*x^2/(e*x^2 + d)^3,x, algorithm="fricas")

[Out]

[-1/16*((15*c*d^4 - 3*b*d^3*e - a*d^2*e^2 + (15*c*d^2*e^2 - 3*b*d*e^3 - a*e^4)*x
^4 + 2*(15*c*d^3*e - 3*b*d^2*e^2 - a*d*e^3)*x^2)*log((2*d*e*x + (e*x^2 - d)*sqrt
(-d*e))/(e*x^2 + d)) - 2*(8*c*d*e^2*x^5 + (25*c*d^2*e - 5*b*d*e^2 + a*e^3)*x^3 +
(15*c*d^3 - 3*b*d^2*e - a*d*e^2)*x)*sqrt(-d*e))/((d*e^5*x^4 + 2*d^2*e^4*x^2 + d
^3*e^3)*sqrt(-d*e)), -1/8*((15*c*d^4 - 3*b*d^3*e - a*d^2*e^2 + (15*c*d^2*e^2 - 3
*b*d*e^3 - a*e^4)*x^4 + 2*(15*c*d^3*e - 3*b*d^2*e^2 - a*d*e^3)*x^2)*arctan(sqrt(
d*e)*x/d) - (8*c*d*e^2*x^5 + (25*c*d^2*e - 5*b*d*e^2 + a*e^3)*x^3 + (15*c*d^3 -
3*b*d^2*e - a*d*e^2)*x)*sqrt(d*e))/((d*e^5*x^4 + 2*d^2*e^4*x^2 + d^3*e^3)*sqrt(d
*e))]

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Sympy [A]  time = 11.1834, size = 201, normalized size = 1.62 \[ \frac{c x}{e^{3}} - \frac{\sqrt{- \frac{1}{d^{3} e^{7}}} \left (a e^{2} + 3 b d e - 15 c d^{2}\right ) \log{\left (- d^{2} e^{3} \sqrt{- \frac{1}{d^{3} e^{7}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{d^{3} e^{7}}} \left (a e^{2} + 3 b d e - 15 c d^{2}\right ) \log{\left (d^{2} e^{3} \sqrt{- \frac{1}{d^{3} e^{7}}} + x \right )}}{16} + \frac{x^{3} \left (a e^{3} - 5 b d e^{2} + 9 c d^{2} e\right ) + x \left (- a d e^{2} - 3 b d^{2} e + 7 c d^{3}\right )}{8 d^{3} e^{3} + 16 d^{2} e^{4} x^{2} + 8 d e^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*x/e**3 - sqrt(-1/(d**3*e**7))*(a*e**2 + 3*b*d*e - 15*c*d**2)*log(-d**2*e**3*sq
rt(-1/(d**3*e**7)) + x)/16 + sqrt(-1/(d**3*e**7))*(a*e**2 + 3*b*d*e - 15*c*d**2)
*log(d**2*e**3*sqrt(-1/(d**3*e**7)) + x)/16 + (x**3*(a*e**3 - 5*b*d*e**2 + 9*c*d
**2*e) + x*(-a*d*e**2 - 3*b*d**2*e + 7*c*d**3))/(8*d**3*e**3 + 16*d**2*e**4*x**2
+ 8*d*e**5*x**4)

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GIAC/XCAS [A]  time = 0.271772, size = 144, normalized size = 1.16 \[ c x e^{\left (-3\right )} - \frac{{\left (15 \, c d^{2} - 3 \, b d e - a e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{7}{2}\right )}}{8 \, d^{\frac{3}{2}}} + \frac{{\left (9 \, c d^{2} x^{3} e - 5 \, b d x^{3} e^{2} + 7 \, c d^{3} x + a x^{3} e^{3} - 3 \, b d^{2} x e - a d x e^{2}\right )} e^{\left (-3\right )}}{8 \,{\left (x^{2} e + d\right )}^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)*x^2/(e*x^2 + d)^3,x, algorithm="giac")

[Out]

c*x*e^(-3) - 1/8*(15*c*d^2 - 3*b*d*e - a*e^2)*arctan(x*e^(1/2)/sqrt(d))*e^(-7/2)
/d^(3/2) + 1/8*(9*c*d^2*x^3*e - 5*b*d*x^3*e^2 + 7*c*d^3*x + a*x^3*e^3 - 3*b*d^2*
x*e - a*d*x*e^2)*e^(-3)/((x^2*e + d)^2*d)