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

Optimal. Leaf size=169 \[ -\frac{\sqrt{a} d \left (a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 c^{3/2} \left (a e^2+c d^2\right )^2}+\frac{a e \left (a e^2+2 c d^2\right ) \log \left (a+c x^4\right )}{4 c^2 \left (a e^2+c d^2\right )^2}+\frac{a \left (a e+c d x^2\right )}{4 c^2 \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac{d^4 \log \left (d+e x^2\right )}{2 e \left (a e^2+c d^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.630753, antiderivative size = 169, 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 \[ -\frac{\sqrt{a} d \left (a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 c^{3/2} \left (a e^2+c d^2\right )^2}+\frac{a e \left (a e^2+2 c d^2\right ) \log \left (a+c x^4\right )}{4 c^2 \left (a e^2+c d^2\right )^2}+\frac{a \left (a e+c d x^2\right )}{4 c^2 \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac{d^4 \log \left (d+e x^2\right )}{2 e \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 67.401, size = 192, normalized size = 1.14 \[ \frac{\sqrt{a} d \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{4 c^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )} - \frac{\sqrt{a} d \left (a e^{2} + 2 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 c^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )^{2}} + \frac{a e \left (a e^{2} + 2 c d^{2}\right ) \log{\left (a + c x^{4} \right )}}{4 c^{2} \left (a e^{2} + c d^{2}\right )^{2}} + \frac{a \left (a e + c d x^{2}\right )}{4 c^{2} \left (a + c x^{4}\right ) \left (a e^{2} + c d^{2}\right )} + \frac{d^{4} \log{\left (d + e x^{2} \right )}}{2 e \left (a e^{2} + c d^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.365544, size = 135, normalized size = 0.8 \[ \frac{-\frac{\sqrt{a} d \left (a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{c^{3/2}}+\frac{a e \left (a e^2+2 c d^2\right ) \log \left (a+c x^4\right )}{c^2}+\frac{a \left (a e^2+c d^2\right ) \left (a e+c d x^2\right )}{c^2 \left (a+c x^4\right )}+\frac{2 d^4 \log \left (d+e x^2\right )}{e}}{4 \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.033, size = 309, normalized size = 1.8 \[{\frac{{a}^{2}d{x}^{2}{e}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) c}}+{\frac{a{x}^{2}{d}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{{a}^{3}{e}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ){c}^{2}}}+{\frac{{a}^{2}e{d}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) c}}+{\frac{{a}^{2}\ln \left ( \left ( c{x}^{4}+a \right ) c \right ){e}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}{c}^{2}}}+{\frac{a\ln \left ( \left ( c{x}^{4}+a \right ) c \right ){d}^{2}e}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}c}}-{\frac{d{a}^{2}{e}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}c}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{3\,a{d}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{{d}^{4}\ln \left ( e{x}^{2}+d \right ) }{2\,e \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 15.3437, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, a^{2} c d^{2} e^{2} + 2 \, a^{3} e^{4} + 2 \,{\left (a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x^{2} +{\left (3 \, a c^{2} d^{3} e + a^{2} c d e^{3} +{\left (3 \, c^{3} d^{3} e + a c^{2} d e^{3}\right )} x^{4}\right )} \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{4} - 2 \, c x^{2} \sqrt{-\frac{a}{c}} - a}{c x^{4} + a}\right ) + 2 \,{\left (2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4} +{\left (2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} x^{4}\right )} \log \left (c x^{4} + a\right ) + 4 \,{\left (c^{3} d^{4} x^{4} + a c^{2} d^{4}\right )} \log \left (e x^{2} + d\right )}{8 \,{\left (a c^{4} d^{4} e + 2 \, a^{2} c^{3} d^{2} e^{3} + a^{3} c^{2} e^{5} +{\left (c^{5} d^{4} e + 2 \, a c^{4} d^{2} e^{3} + a^{2} c^{3} e^{5}\right )} x^{4}\right )}}, \frac{a^{2} c d^{2} e^{2} + a^{3} e^{4} +{\left (a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x^{2} -{\left (3 \, a c^{2} d^{3} e + a^{2} c d e^{3} +{\left (3 \, c^{3} d^{3} e + a c^{2} d e^{3}\right )} x^{4}\right )} \sqrt{\frac{a}{c}} \arctan \left (\frac{x^{2}}{\sqrt{\frac{a}{c}}}\right ) +{\left (2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4} +{\left (2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} x^{4}\right )} \log \left (c x^{4} + a\right ) + 2 \,{\left (c^{3} d^{4} x^{4} + a c^{2} d^{4}\right )} \log \left (e x^{2} + d\right )}{4 \,{\left (a c^{4} d^{4} e + 2 \, a^{2} c^{3} d^{2} e^{3} + a^{3} c^{2} e^{5} +{\left (c^{5} d^{4} e + 2 \, a c^{4} d^{2} e^{3} + a^{2} c^{3} e^{5}\right )} x^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(2*a^2*c*d^2*e^2 + 2*a^3*e^4 + 2*(a*c^2*d^3*e + a^2*c*d*e^3)*x^2 + (3*a*c^2
*d^3*e + a^2*c*d*e^3 + (3*c^3*d^3*e + a*c^2*d*e^3)*x^4)*sqrt(-a/c)*log((c*x^4 -
2*c*x^2*sqrt(-a/c) - a)/(c*x^4 + a)) + 2*(2*a^2*c*d^2*e^2 + a^3*e^4 + (2*a*c^2*d
^2*e^2 + a^2*c*e^4)*x^4)*log(c*x^4 + a) + 4*(c^3*d^4*x^4 + a*c^2*d^4)*log(e*x^2
+ d))/(a*c^4*d^4*e + 2*a^2*c^3*d^2*e^3 + a^3*c^2*e^5 + (c^5*d^4*e + 2*a*c^4*d^2*
e^3 + a^2*c^3*e^5)*x^4), 1/4*(a^2*c*d^2*e^2 + a^3*e^4 + (a*c^2*d^3*e + a^2*c*d*e
^3)*x^2 - (3*a*c^2*d^3*e + a^2*c*d*e^3 + (3*c^3*d^3*e + a*c^2*d*e^3)*x^4)*sqrt(a
/c)*arctan(x^2/sqrt(a/c)) + (2*a^2*c*d^2*e^2 + a^3*e^4 + (2*a*c^2*d^2*e^2 + a^2*
c*e^4)*x^4)*log(c*x^4 + a) + 2*(c^3*d^4*x^4 + a*c^2*d^4)*log(e*x^2 + d))/(a*c^4*
d^4*e + 2*a^2*c^3*d^2*e^3 + a^3*c^2*e^5 + (c^5*d^4*e + 2*a*c^4*d^2*e^3 + a^2*c^3
*e^5)*x^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**9/(e*x**2+d)/(c*x**4+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.276606, size = 339, normalized size = 2.01 \[ \frac{d^{4}{\rm ln}\left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}} + \frac{{\left (2 \, a c d^{2} e + a^{2} e^{3}\right )}{\rm ln}\left (c x^{4} + a\right )}{4 \,{\left (c^{4} d^{4} + 2 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )}} - \frac{{\left (3 \, a c d^{3} + a^{2} d e^{2}\right )} \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{4 \,{\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} \sqrt{a c}} - \frac{2 \, a c d^{2} x^{4} e - a c d^{3} x^{2} + a^{2} x^{4} e^{3} - a^{2} d x^{2} e^{2} + a^{2} d^{2} e}{4 \,{\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )}{\left (c x^{4} + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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