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

Optimal. Leaf size=148 \[ \frac{d e^2 \log \left (a+c x^4\right )}{4 \left (a e^2+c d^2\right )^2}-\frac{d e^2 \log \left (d+e x^2\right )}{2 \left (a e^2+c d^2\right )^2}-\frac{e \left (c d^2-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 \sqrt{a} \sqrt{c} \left (a e^2+c d^2\right )^2}-\frac{d-e x^2}{4 \left (a+c x^4\right ) \left (a e^2+c d^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.387509, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{d e^2 \log \left (a+c x^4\right )}{4 \left (a e^2+c d^2\right )^2}-\frac{d e^2 \log \left (d+e x^2\right )}{2 \left (a e^2+c d^2\right )^2}-\frac{e \left (c d^2-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 \sqrt{a} \sqrt{c} \left (a e^2+c d^2\right )^2}-\frac{d-e x^2}{4 \left (a+c x^4\right ) \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 67.4309, size = 129, normalized size = 0.87 \[ \frac{d e^{2} \log{\left (a + c x^{4} \right )}}{4 \left (a e^{2} + c d^{2}\right )^{2}} - \frac{d e^{2} \log{\left (d + e x^{2} \right )}}{2 \left (a e^{2} + c d^{2}\right )^{2}} - \frac{d - e x^{2}}{4 \left (a + c x^{4}\right ) \left (a e^{2} + c d^{2}\right )} + \frac{e \left (a e^{2} - c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{4 \sqrt{a} \sqrt{c} \left (a e^{2} + c d^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*e**2*log(a + c*x**4)/(4*(a*e**2 + c*d**2)**2) - d*e**2*log(d + e*x**2)/(2*(a*e
**2 + c*d**2)**2) - (d - e*x**2)/(4*(a + c*x**4)*(a*e**2 + c*d**2)) + e*(a*e**2
- c*d**2)*atan(sqrt(c)*x**2/sqrt(a))/(4*sqrt(a)*sqrt(c)*(a*e**2 + c*d**2)**2)

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Mathematica [A]  time = 0.258924, size = 114, normalized size = 0.77 \[ \frac{\frac{e \left (a e^2-c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{c}}+\frac{\left (e x^2-d\right ) \left (a e^2+c d^2\right )}{a+c x^4}+d e^2 \log \left (a+c x^4\right )-2 d e^2 \log \left (d+e x^2\right )}{4 \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.021, size = 247, normalized size = 1.7 \[{\frac{{x}^{2}{e}^{3}a}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{{x}^{2}{d}^{2}ec}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}-{\frac{{e}^{2}da}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}-{\frac{c{d}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{{e}^{2}d\ln \left ( c{x}^{4}+a \right ) }{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}+{\frac{{e}^{3}a}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{{d}^{2}ec}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{{e}^{2}d\ln \left ( e{x}^{2}+d \right ) }{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4/(a*e^2+c*d^2)^2/(c*x^4+a)*x^2*e^3*a+1/4/(a*e^2+c*d^2)^2/(c*x^4+a)*x^2*d^2*e*
c-1/4/(a*e^2+c*d^2)^2/(c*x^4+a)*e^2*d*a-1/4/(a*e^2+c*d^2)^2/(c*x^4+a)*c*d^3+1/4*
d*e^2*ln(c*x^4+a)/(a*e^2+c*d^2)^2+1/4/(a*e^2+c*d^2)^2*e^3/(a*c)^(1/2)*arctan(c*x
^2/(a*c)^(1/2))*a-1/4/(a*e^2+c*d^2)^2*e/(a*c)^(1/2)*arctan(c*x^2/(a*c)^(1/2))*c*
d^2-1/2*d*e^2*ln(e*x^2+d)/(a*e^2+c*d^2)^2

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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(x^3/((c*x^4 + a)^2*(e*x^2 + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.82016, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (a c d^{2} e - a^{2} e^{3} +{\left (c^{2} d^{2} e - a c e^{3}\right )} x^{4}\right )} \log \left (\frac{2 \, a c x^{2} +{\left (c x^{4} - a\right )} \sqrt{-a c}}{c x^{4} + a}\right ) + 2 \,{\left (c d^{3} + a d e^{2} -{\left (c d^{2} e + a e^{3}\right )} x^{2} -{\left (c d e^{2} x^{4} + a d e^{2}\right )} \log \left (c x^{4} + a\right ) + 2 \,{\left (c d e^{2} x^{4} + a d e^{2}\right )} \log \left (e x^{2} + d\right )\right )} \sqrt{-a c}}{8 \,{\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4} +{\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} x^{4}\right )} \sqrt{-a c}}, \frac{{\left (a c d^{2} e - a^{2} e^{3} +{\left (c^{2} d^{2} e - a c e^{3}\right )} x^{4}\right )} \arctan \left (\frac{a}{\sqrt{a c} x^{2}}\right ) -{\left (c d^{3} + a d e^{2} -{\left (c d^{2} e + a e^{3}\right )} x^{2} -{\left (c d e^{2} x^{4} + a d e^{2}\right )} \log \left (c x^{4} + a\right ) + 2 \,{\left (c d e^{2} x^{4} + a d e^{2}\right )} \log \left (e x^{2} + d\right )\right )} \sqrt{a c}}{4 \,{\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4} +{\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} x^{4}\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275832, size = 254, normalized size = 1.72 \[ \frac{d e^{2}{\rm ln}\left (c x^{4} + a\right )}{4 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} - \frac{d e^{3}{\rm ln}\left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}} - \frac{{\left (c d^{2} e - a e^{3}\right )} \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{4 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{a c}} - \frac{c d^{3} -{\left (c d^{2} e + a e^{3}\right )} x^{2} + a d e^{2}}{4 \,{\left (c x^{4} + a\right )}{\left (c d^{2} + a e^{2}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*d*e^2*ln(c*x^4 + a)/(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4) - 1/2*d*e^3*ln(abs(x
^2*e + d))/(c^2*d^4*e + 2*a*c*d^2*e^3 + a^2*e^5) - 1/4*(c*d^2*e - a*e^3)*arctan(
c*x^2/sqrt(a*c))/((c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*sqrt(a*c)) - 1/4*(c*d^3 -
(c*d^2*e + a*e^3)*x^2 + a*d*e^2)/((c*x^4 + a)*(c*d^2 + a*e^2)^2)