3.229 \(\int \frac{x^9}{\left (d+e x^2\right ) \left (a+c x^4\right )} \, dx\)

Optimal. Leaf size=134 \[ \frac{a^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 c^{3/2} \left (a e^2+c d^2\right )}-\frac{a^2 e \log \left (a+c x^4\right )}{4 c^2 \left (a e^2+c d^2\right )}+\frac{d^4 \log \left (d+e x^2\right )}{2 e^3 \left (a e^2+c d^2\right )}-\frac{d x^2}{2 c e^2}+\frac{x^4}{4 c e} \]

[Out]

-(d*x^2)/(2*c*e^2) + x^4/(4*c*e) + (a^(3/2)*d*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*
c^(3/2)*(c*d^2 + a*e^2)) + (d^4*Log[d + e*x^2])/(2*e^3*(c*d^2 + a*e^2)) - (a^2*e
*Log[a + c*x^4])/(4*c^2*(c*d^2 + a*e^2))

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Rubi [A]  time = 0.322171, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{a^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 c^{3/2} \left (a e^2+c d^2\right )}-\frac{a^2 e \log \left (a+c x^4\right )}{4 c^2 \left (a e^2+c d^2\right )}+\frac{d^4 \log \left (d+e x^2\right )}{2 e^3 \left (a e^2+c d^2\right )}-\frac{d x^2}{2 c e^2}+\frac{x^4}{4 c e} \]

Antiderivative was successfully verified.

[In]  Int[x^9/((d + e*x^2)*(a + c*x^4)),x]

[Out]

-(d*x^2)/(2*c*e^2) + x^4/(4*c*e) + (a^(3/2)*d*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*
c^(3/2)*(c*d^2 + a*e^2)) + (d^4*Log[d + e*x^2])/(2*e^3*(c*d^2 + a*e^2)) - (a^2*e
*Log[a + c*x^4])/(4*c^2*(c*d^2 + a*e^2))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{\frac{3}{2}} d \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 c^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )} - \frac{a^{2} e \log{\left (a + c x^{4} \right )}}{4 c^{2} \left (a e^{2} + c d^{2}\right )} + \frac{d^{4} \log{\left (d + e x^{2} \right )}}{2 e^{3} \left (a e^{2} + c d^{2}\right )} + \frac{\int ^{x^{2}} x\, dx}{2 c e} - \frac{\int ^{x^{2}} d\, dx}{2 c e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(e*x**2+d)/(c*x**4+a),x)

[Out]

a**(3/2)*d*atan(sqrt(c)*x**2/sqrt(a))/(2*c**(3/2)*(a*e**2 + c*d**2)) - a**2*e*lo
g(a + c*x**4)/(4*c**2*(a*e**2 + c*d**2)) + d**4*log(d + e*x**2)/(2*e**3*(a*e**2
+ c*d**2)) + Integral(x, (x, x**2))/(2*c*e) - Integral(d, (x, x**2))/(2*c*e**2)

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Mathematica [A]  time = 0.107434, size = 134, normalized size = 1. \[ \frac{a^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 c^{3/2} \left (a e^2+c d^2\right )}-\frac{a^2 e \log \left (a+c x^4\right )}{4 c^2 \left (a e^2+c d^2\right )}+\frac{d^4 \log \left (d+e x^2\right )}{2 e^3 \left (a e^2+c d^2\right )}-\frac{d x^2}{2 c e^2}+\frac{x^4}{4 c e} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/((d + e*x^2)*(a + c*x^4)),x]

[Out]

-(d*x^2)/(2*c*e^2) + x^4/(4*c*e) + (a^(3/2)*d*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*
c^(3/2)*(c*d^2 + a*e^2)) + (d^4*Log[d + e*x^2])/(2*e^3*(c*d^2 + a*e^2)) - (a^2*e
*Log[a + c*x^4])/(4*c^2*(c*d^2 + a*e^2))

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Maple [A]  time = 0.016, size = 122, normalized size = 0.9 \[{\frac{{x}^{4}}{4\,ce}}-{\frac{d{x}^{2}}{2\,{e}^{2}c}}-{\frac{{a}^{2}e\ln \left ( c{x}^{4}+a \right ) }{4\,{c}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) }}+{\frac{{a}^{2}d}{ \left ( 2\,a{e}^{2}+2\,c{d}^{2} \right ) c}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{{d}^{4}\ln \left ( e{x}^{2}+d \right ) }{2\,{e}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(e*x^2+d)/(c*x^4+a),x)

[Out]

1/4*x^4/c/e-1/2*d*x^2/e^2/c-1/4*a^2*e*ln(c*x^4+a)/c^2/(a*e^2+c*d^2)+1/2*a^2/(a*e
^2+c*d^2)/c*d/(a*c)^(1/2)*arctan(c*x^2/(a*c)^(1/2))+1/2*d^4*ln(e*x^2+d)/e^3/(a*e
^2+c*d^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^9/((c*x^4 + a)*(e*x^2 + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 5.19803, size = 1, normalized size = 0.01 \[ \left [\frac{a c d e^{3} \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{4} + 2 \, c x^{2} \sqrt{-\frac{a}{c}} - a}{c x^{4} + a}\right ) - a^{2} e^{4} \log \left (c x^{4} + a\right ) + 2 \, c^{2} d^{4} \log \left (e x^{2} + d\right ) +{\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{4} - 2 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x^{2}}{4 \,{\left (c^{3} d^{2} e^{3} + a c^{2} e^{5}\right )}}, \frac{2 \, a c d e^{3} \sqrt{\frac{a}{c}} \arctan \left (\frac{x^{2}}{\sqrt{\frac{a}{c}}}\right ) - a^{2} e^{4} \log \left (c x^{4} + a\right ) + 2 \, c^{2} d^{4} \log \left (e x^{2} + d\right ) +{\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{4} - 2 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x^{2}}{4 \,{\left (c^{3} d^{2} e^{3} + a c^{2} e^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/((c*x^4 + a)*(e*x^2 + d)),x, algorithm="fricas")

[Out]

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

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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**9/(e*x**2+d)/(c*x**4+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.275585, size = 163, normalized size = 1.22 \[ \frac{d^{4}{\rm ln}\left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c d^{2} e^{3} + a e^{5}\right )}} - \frac{a^{2} e{\rm ln}\left (c x^{4} + a\right )}{4 \,{\left (c^{3} d^{2} + a c^{2} e^{2}\right )}} + \frac{a^{2} d \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{2 \,{\left (c^{2} d^{2} + a c e^{2}\right )} \sqrt{a c}} + \frac{{\left (c x^{4} e - 2 \, c d x^{2}\right )} e^{\left (-2\right )}}{4 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/((c*x^4 + a)*(e*x^2 + d)),x, algorithm="giac")

[Out]

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