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

Optimal. Leaf size=230 \[ -\frac{\left (a^2 c e-a b^2 e-2 a b c d+b^3 d\right ) \log \left (a+b x^2+c x^4\right )}{4 c^3 \left (a e^2-b d e+c d^2\right )}-\frac{\left (3 a^2 b c e+2 a^2 c^2 d-a b^3 e-4 a b^2 c d+b^4 d\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^3 \sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac{d^4 \log \left (d+e x^2\right )}{2 e^3 \left (a e^2-b d e+c d^2\right )}-\frac{x^2 (b e+c d)}{2 c^2 e^2}+\frac{x^4}{4 c e} \]

[Out]

-((c*d + b*e)*x^2)/(2*c^2*e^2) + x^4/(4*c*e) - ((b^4*d - 4*a*b^2*c*d + 2*a^2*c^2
*d - a*b^3*e + 3*a^2*b*c*e)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^3*Sqr
t[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)) + (d^4*Log[d + e*x^2])/(2*e^3*(c*d^2 - b
*d*e + a*e^2)) - ((b^3*d - 2*a*b*c*d - a*b^2*e + a^2*c*e)*Log[a + b*x^2 + c*x^4]
)/(4*c^3*(c*d^2 - b*d*e + a*e^2))

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Rubi [A]  time = 0.926173, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{\left (a^2 c e-a b^2 e-2 a b c d+b^3 d\right ) \log \left (a+b x^2+c x^4\right )}{4 c^3 \left (a e^2-b d e+c d^2\right )}-\frac{\left (3 a^2 b c e+2 a^2 c^2 d-a b^3 e-4 a b^2 c d+b^4 d\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^3 \sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac{d^4 \log \left (d+e x^2\right )}{2 e^3 \left (a e^2-b d e+c d^2\right )}-\frac{x^2 (b e+c d)}{2 c^2 e^2}+\frac{x^4}{4 c e} \]

Antiderivative was successfully verified.

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

[Out]

-((c*d + b*e)*x^2)/(2*c^2*e^2) + x^4/(4*c*e) - ((b^4*d - 4*a*b^2*c*d + 2*a^2*c^2
*d - a*b^3*e + 3*a^2*b*c*e)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^3*Sqr
t[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)) + (d^4*Log[d + e*x^2])/(2*e^3*(c*d^2 - b
*d*e + a*e^2)) - ((b^3*d - 2*a*b*c*d - a*b^2*e + a^2*c*e)*Log[a + b*x^2 + c*x^4]
)/(4*c^3*(c*d^2 - b*d*e + a*e^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.410466, size = 228, normalized size = 0.99 \[ \frac{1}{4} \left (\frac{\left (-a^2 c e+a b^2 e+2 a b c d+b^3 (-d)\right ) \log \left (a+b x^2+c x^4\right )}{c^3 \left (e (a e-b d)+c d^2\right )}-\frac{2 \left (3 a^2 b c e+2 a^2 c^2 d-a b^3 e-4 a b^2 c d+b^4 d\right ) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{c^3 \sqrt{4 a c-b^2} \left (e (b d-a e)-c d^2\right )}+\frac{2 d^4 \log \left (d+e x^2\right )}{e^3 \left (e (a e-b d)+c d^2\right )}-\frac{2 x^2 (b e+c d)}{c^2 e^2}+\frac{x^4}{c e}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.019, size = 538, normalized size = 2.3 \[{\frac{{x}^{4}}{4\,ce}}-{\frac{b{x}^{2}}{2\,{c}^{2}e}}-{\frac{d{x}^{2}}{2\,{e}^{2}c}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){a}^{2}e}{4\, \left ( a{e}^{2}-bde+c{d}^{2} \right ){c}^{2}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) a{b}^{2}e}{4\,{c}^{3} \left ( a{e}^{2}-bde+c{d}^{2} \right ) }}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) abd}{2\, \left ( a{e}^{2}-bde+c{d}^{2} \right ){c}^{2}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{3}d}{4\,{c}^{3} \left ( a{e}^{2}-bde+c{d}^{2} \right ) }}+{\frac{3\,{a}^{2}be}{2\, \left ( a{e}^{2}-bde+c{d}^{2} \right ){c}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{a}^{2}d}{c \left ( a{e}^{2}-bde+c{d}^{2} \right ) }\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-2\,{\frac{a{b}^{2}d}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ){c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{a{b}^{3}e}{2\,{c}^{3} \left ( a{e}^{2}-bde+c{d}^{2} \right ) }\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{4}d}{2\,{c}^{3} \left ( a{e}^{2}-bde+c{d}^{2} \right ) }\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{d}^{4}\ln \left ( e{x}^{2}+d \right ) }{2\,{e}^{3} \left ( a{e}^{2}-bde+c{d}^{2} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [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/((c*x^4 + b*x^2 + a)*(e*x^2 + d)),x, algorithm="fricas")

[Out]

Timed out

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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+b*x**2+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.301296, size = 319, normalized size = 1.39 \[ \frac{d^{4}{\rm ln}\left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c d^{2} e^{3} - b d e^{4} + a e^{5}\right )}} - \frac{{\left (b^{3} d - 2 \, a b c d - a b^{2} e + a^{2} c e\right )}{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \,{\left (c^{4} d^{2} - b c^{3} d e + a c^{3} e^{2}\right )}} + \frac{{\left (b^{4} d - 4 \, a b^{2} c d + 2 \, a^{2} c^{2} d - a b^{3} e + 3 \, a^{2} b c e\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (c^{4} d^{2} - b c^{3} d e + a c^{3} e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\left (c x^{4} e - 2 \, c d x^{2} - 2 \, b x^{2} e\right )} e^{\left (-2\right )}}{4 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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