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

Optimal. Leaf size=150 \[ \frac{\sqrt{a} e \left (a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 c^{3/2} \left (a e^2+c d^2\right )^2}+\frac{a \left (d-e x^2\right )}{4 c \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac{d^3 \log \left (a+c x^4\right )}{4 \left (a e^2+c d^2\right )^2}-\frac{d^3 \log \left (d+e x^2\right )}{2 \left (a e^2+c d^2\right )^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.485226, antiderivative size = 150, 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 \[ \frac{\sqrt{a} e \left (a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 c^{3/2} \left (a e^2+c d^2\right )^2}+\frac{a \left (d-e x^2\right )}{4 c \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac{d^3 \log \left (a+c x^4\right )}{4 \left (a e^2+c d^2\right )^2}-\frac{d^3 \log \left (d+e x^2\right )}{2 \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 70.3934, size = 170, normalized size = 1.13 \[ - \frac{\sqrt{a} e \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{4 c^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )} + \frac{\sqrt{a} e \left (a e^{2} + 2 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 c^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )^{2}} + \frac{a \left (d - e x^{2}\right )}{4 c \left (a + c x^{4}\right ) \left (a e^{2} + c d^{2}\right )} + \frac{d^{3} \log{\left (a + c x^{4} \right )}}{4 \left (a e^{2} + c d^{2}\right )^{2}} - \frac{d^{3} \log{\left (d + e x^{2} \right )}}{2 \left (a e^{2} + c d^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.190637, size = 142, normalized size = 0.95 \[ \frac{\sqrt{a} e \left (a+c x^4\right ) \left (a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )+\sqrt{c} \left (-2 c d^3 \left (a+c x^4\right ) \log \left (d+e x^2\right )+c d^3 \left (a+c x^4\right ) \log \left (a+c x^4\right )+a \left (d-e x^2\right ) \left (a e^2+c d^2\right )\right )}{4 c^{3/2} \left (a+c x^4\right ) \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.023, size = 262, normalized size = 1.8 \[ -{\frac{{a}^{2}{e}^{3}{x}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) c}}-{\frac{ae{x}^{2}{d}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{d{a}^{2}{e}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) c}}+{\frac{a{d}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{{d}^{3}\ln \left ( \left ( c{x}^{4}+a \right ) c \right ) }{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}+{\frac{{a}^{2}{e}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}c}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,{d}^{2}ea}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{{d}^{3}\ln \left ( e{x}^{2}+d \right ) }{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)^2*(e*x^2 + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 7.11036, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, a c d^{3} + 2 \, a^{2} d e^{2} - 2 \,{\left (a c d^{2} e + a^{2} e^{3}\right )} x^{2} +{\left (3 \, a c d^{2} e + a^{2} e^{3} +{\left (3 \, c^{2} d^{2} e + a c e^{3}\right )} x^{4}\right )} \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{4} + 2 \, c x^{2} \sqrt{-\frac{a}{c}} - a}{c x^{4} + a}\right ) + 2 \,{\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (c x^{4} + a\right ) - 4 \,{\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (e x^{2} + d\right )}{8 \,{\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4} +{\left (c^{4} d^{4} + 2 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )} x^{4}\right )}}, \frac{a c d^{3} + a^{2} d e^{2} -{\left (a c d^{2} e + a^{2} e^{3}\right )} x^{2} +{\left (3 \, a c d^{2} e + a^{2} e^{3} +{\left (3 \, c^{2} d^{2} e + a c e^{3}\right )} x^{4}\right )} \sqrt{\frac{a}{c}} \arctan \left (\frac{x^{2}}{\sqrt{\frac{a}{c}}}\right ) +{\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (c x^{4} + a\right ) - 2 \,{\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (e x^{2} + d\right )}{4 \,{\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4} +{\left (c^{4} d^{4} + 2 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )} x^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.274248, size = 301, normalized size = 2.01 \[ -\frac{d^{3} e{\rm ln}\left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}} + \frac{d^{3}{\rm ln}\left (c x^{4} + a\right )}{4 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac{{\left (3 \, a c d^{2} e + a^{2} e^{3}\right )} \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{4 \,{\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} \sqrt{a c}} - \frac{c^{2} d^{3} x^{4} + a c d^{2} x^{2} e + a^{2} x^{2} e^{3} - a^{2} d e^{2}}{4 \,{\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )}{\left (c x^{4} + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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