3.230 \(\int \frac{x^7}{\left (d+e x^2\right ) \left (a+c x^4\right )} \, dx\)

Optimal. Leaf size=118 \[ -\frac{a^{3/2} e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 c^{3/2} \left (a e^2+c d^2\right )}-\frac{a d \log \left (a+c x^4\right )}{4 c \left (a e^2+c d^2\right )}-\frac{d^3 \log \left (d+e x^2\right )}{2 e^2 \left (a e^2+c d^2\right )}+\frac{x^2}{2 c e} \]

[Out]

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

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Rubi [A]  time = 0.277518, antiderivative size = 118, 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 \[ -\frac{a^{3/2} e \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 c^{3/2} \left (a e^2+c d^2\right )}-\frac{a d \log \left (a+c x^4\right )}{4 c \left (a e^2+c d^2\right )}-\frac{d^3 \log \left (d+e x^2\right )}{2 e^2 \left (a e^2+c d^2\right )}+\frac{x^2}{2 c e} \]

Antiderivative was successfully verified.

[In]  Int[x^7/((d + e*x^2)*(a + c*x^4)),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{\frac{3}{2}} e \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 c^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )} - \frac{a d \log{\left (a + c x^{4} \right )}}{4 c \left (a e^{2} + c d^{2}\right )} - \frac{d^{3} \log{\left (d + e x^{2} \right )}}{2 e^{2} \left (a e^{2} + c d^{2}\right )} + \frac{\int ^{x^{2}} \frac{1}{c}\, dx}{2 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(e*x**2+d)/(c*x**4+a),x)

[Out]

-a**(3/2)*e*atan(sqrt(c)*x**2/sqrt(a))/(2*c**(3/2)*(a*e**2 + c*d**2)) - a*d*log(
a + c*x**4)/(4*c*(a*e**2 + c*d**2)) - d**3*log(d + e*x**2)/(2*e**2*(a*e**2 + c*d
**2)) + Integral(1/c, (x, x**2))/(2*e)

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Mathematica [A]  time = 0.159741, size = 99, normalized size = 0.84 \[ \frac{-\frac{2 a^{3/2} e^3 \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{c^{3/2}}+\frac{e \left (2 x^2 \left (a e^2+c d^2\right )-a d e \log \left (a+c x^4\right )\right )}{c}-2 d^3 \log \left (d+e x^2\right )}{4 e^2 \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/((d + e*x^2)*(a + c*x^4)),x]

[Out]

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

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Maple [A]  time = 0.012, size = 108, normalized size = 0.9 \[{\frac{{x}^{2}}{2\,ce}}-{\frac{ad\ln \left ( c{x}^{4}+a \right ) }{4\, \left ( a{e}^{2}+c{d}^{2} \right ) c}}-{\frac{{a}^{2}e}{ \left ( 2\,a{e}^{2}+2\,c{d}^{2} \right ) c}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{{d}^{3}\ln \left ( e{x}^{2}+d \right ) }{2\,{e}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(e*x^2+d)/(c*x^4+a),x)

[Out]

1/2*x^2/c/e-1/4*a*d*ln(c*x^4+a)/c/(a*e^2+c*d^2)-1/2*a^2/(a*e^2+c*d^2)/c*e/(a*c)^
(1/2)*arctan(c*x^2/(a*c)^(1/2))-1/2*d^3*ln(e*x^2+d)/e^2/(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^7/((c*x^4 + a)*(e*x^2 + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.83586, size = 1, normalized size = 0.01 \[ \left [\frac{a e^{3} \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{4} - 2 \, c x^{2} \sqrt{-\frac{a}{c}} - a}{c x^{4} + a}\right ) - a d e^{2} \log \left (c x^{4} + a\right ) - 2 \, c d^{3} \log \left (e x^{2} + d\right ) + 2 \,{\left (c d^{2} e + a e^{3}\right )} x^{2}}{4 \,{\left (c^{2} d^{2} e^{2} + a c e^{4}\right )}}, -\frac{2 \, a e^{3} \sqrt{\frac{a}{c}} \arctan \left (\frac{x^{2}}{\sqrt{\frac{a}{c}}}\right ) + a d e^{2} \log \left (c x^{4} + a\right ) + 2 \, c d^{3} \log \left (e x^{2} + d\right ) - 2 \,{\left (c d^{2} e + a e^{3}\right )} x^{2}}{4 \,{\left (c^{2} d^{2} e^{2} + a c e^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/((c*x^4 + a)*(e*x^2 + d)),x, algorithm="fricas")

[Out]

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

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.27395, size = 142, normalized size = 1.2 \[ -\frac{d^{3}{\rm ln}\left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c d^{2} e^{2} + a e^{4}\right )}} - \frac{a^{2} \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right ) e}{2 \,{\left (c^{2} d^{2} + a c e^{2}\right )} \sqrt{a c}} + \frac{x^{2} e^{\left (-1\right )}}{2 \, c} - \frac{a d{\rm ln}\left (c x^{4} + a\right )}{4 \,{\left (c^{2} d^{2} + a c e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/((c*x^4 + a)*(e*x^2 + d)),x, algorithm="giac")

[Out]

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