3.981 \(\int \frac{x^9}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=190 \[ \frac{3 \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{16 c^{7/2}}-\frac{\left (b \left (15 b^2-52 a c\right )-2 c x^2 \left (5 b^2-12 a c\right )\right ) \sqrt{a+b x^2+c x^4}}{8 c^3 \left (b^2-4 a c\right )}-\frac{b x^4 \sqrt{a+b x^2+c x^4}}{c \left (b^2-4 a c\right )}+\frac{x^6 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}} \]

[Out]

(x^6*(2*a + b*x^2))/((b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4]) - (b*x^4*Sqrt[a + b*
x^2 + c*x^4])/(c*(b^2 - 4*a*c)) - ((b*(15*b^2 - 52*a*c) - 2*c*(5*b^2 - 12*a*c)*x
^2)*Sqrt[a + b*x^2 + c*x^4])/(8*c^3*(b^2 - 4*a*c)) + (3*(5*b^2 - 4*a*c)*ArcTanh[
(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(16*c^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.537616, antiderivative size = 190, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{3 \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{16 c^{7/2}}-\frac{\left (b \left (15 b^2-52 a c\right )-2 c x^2 \left (5 b^2-12 a c\right )\right ) \sqrt{a+b x^2+c x^4}}{8 c^3 \left (b^2-4 a c\right )}-\frac{b x^4 \sqrt{a+b x^2+c x^4}}{c \left (b^2-4 a c\right )}+\frac{x^6 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^9/(a + b*x^2 + c*x^4)^(3/2),x]

[Out]

(x^6*(2*a + b*x^2))/((b^2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4]) - (b*x^4*Sqrt[a + b*
x^2 + c*x^4])/(c*(b^2 - 4*a*c)) - ((b*(15*b^2 - 52*a*c) - 2*c*(5*b^2 - 12*a*c)*x
^2)*Sqrt[a + b*x^2 + c*x^4])/(8*c^3*(b^2 - 4*a*c)) + (3*(5*b^2 - 4*a*c)*ArcTanh[
(b + 2*c*x^2)/(2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4])])/(16*c^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 40.9247, size = 177, normalized size = 0.93 \[ - \frac{b x^{4} \sqrt{a + b x^{2} + c x^{4}}}{c \left (- 4 a c + b^{2}\right )} + \frac{x^{6} \left (2 a + b x^{2}\right )}{\left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}} - \frac{\left (b \left (- 39 a c + \frac{45 b^{2}}{4}\right ) - \frac{3 c x^{2} \left (- 12 a c + 5 b^{2}\right )}{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}{6 c^{3} \left (- 4 a c + b^{2}\right )} + \frac{3 \left (- 4 a c + 5 b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{16 c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(c*x**4+b*x**2+a)**(3/2),x)

[Out]

-b*x**4*sqrt(a + b*x**2 + c*x**4)/(c*(-4*a*c + b**2)) + x**6*(2*a + b*x**2)/((-4
*a*c + b**2)*sqrt(a + b*x**2 + c*x**4)) - (b*(-39*a*c + 45*b**2/4) - 3*c*x**2*(-
12*a*c + 5*b**2)/2)*sqrt(a + b*x**2 + c*x**4)/(6*c**3*(-4*a*c + b**2)) + 3*(-4*a
*c + 5*b**2)*atanh((b + 2*c*x**2)/(2*sqrt(c)*sqrt(a + b*x**2 + c*x**4)))/(16*c**
(7/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.235533, size = 147, normalized size = 0.77 \[ \frac{\sqrt{a+b x^2+c x^4} \left (-\frac{8 \left (a^2 c \left (2 c x^2-3 b\right )+a b^2 \left (b-4 c x^2\right )+b^4 x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-7 b+2 c x^2\right )}{8 c^3}+\frac{3 \left (5 b^2-4 a c\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right )}{16 c^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(a + b*x^2 + c*x^4)^(3/2),x]

[Out]

(Sqrt[a + b*x^2 + c*x^4]*(-7*b + 2*c*x^2 - (8*(b^4*x^2 + a*b^2*(b - 4*c*x^2) + a
^2*c*(-3*b + 2*c*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4))))/(8*c^3) + (3*(5*b^
2 - 4*a*c)*Log[b + 2*c*x^2 + 2*Sqrt[c]*Sqrt[a + b*x^2 + c*x^4]])/(16*c^(7/2))

_______________________________________________________________________________________

Maple [B]  time = 0.023, size = 354, normalized size = 1.9 \[{\frac{{x}^{6}}{4\,c}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{5\,b{x}^{4}}{8\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{15\,{b}^{2}{x}^{2}}{16\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{15\,{b}^{3}}{32\,{c}^{4}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{15\,{b}^{4}{x}^{2}}{16\,{c}^{3} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{15\,{b}^{5}}{32\,{c}^{4} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{15\,{b}^{2}}{16}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{7}{2}}}}-{\frac{13\,ab}{8\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{13\,a{b}^{2}{x}^{2}}{4\, \left ( 4\,ac-{b}^{2} \right ){c}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{13\,a{b}^{3}}{8\,{c}^{3} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{3\,a{x}^{2}}{4\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{3\,a}{4}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(c*x^4+b*x^2+a)^(3/2),x)

[Out]

1/4*x^6/c/(c*x^4+b*x^2+a)^(1/2)-5/8*b/c^2*x^4/(c*x^4+b*x^2+a)^(1/2)-15/16*b^2/c^
3*x^2/(c*x^4+b*x^2+a)^(1/2)+15/32*b^3/c^4/(c*x^4+b*x^2+a)^(1/2)+15/16*b^4/c^3/(4
*a*c-b^2)/(c*x^4+b*x^2+a)^(1/2)*x^2+15/32*b^5/c^4/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(1
/2)+15/16*b^2/c^(7/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))-13/8*b/c^3
*a/(c*x^4+b*x^2+a)^(1/2)-13/4*b^2/c^2*a/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(1/2)*x^2-13
/8*b^3/c^3*a/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(1/2)+3/4/c^2*a*x^2/(c*x^4+b*x^2+a)^(1/
2)-3/4/c^(5/2)*a*ln((1/2*b+c*x^2)/c^(1/2)+(c*x^4+b*x^2+a)^(1/2))

_______________________________________________________________________________________

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^9/(c*x^4 + b*x^2 + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.357921, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{6} - 5 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{4} - 15 \, a b^{3} + 52 \, a^{2} b c -{\left (15 \, b^{4} - 62 \, a b^{2} c + 24 \, a^{2} c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{c} - 3 \,{\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} +{\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{4} +{\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{2}\right )} \log \left (4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c^{2} x^{2} + b c\right )} -{\left (8 \, c^{2} x^{4} + 8 \, b c x^{2} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{32 \,{\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} +{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} +{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{2}\right )} \sqrt{c}}, \frac{2 \,{\left (2 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{6} - 5 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{4} - 15 \, a b^{3} + 52 \, a^{2} b c -{\left (15 \, b^{4} - 62 \, a b^{2} c + 24 \, a^{2} c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{-c} + 3 \,{\left (5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} +{\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{4} +{\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{2}\right )} \arctan \left (\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{4} + b x^{2} + a} c}\right )}{16 \,{\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} +{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} +{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{2}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(c*x^4 + b*x^2 + a)^(3/2),x, algorithm="fricas")

[Out]

[1/32*(4*(2*(b^2*c^2 - 4*a*c^3)*x^6 - 5*(b^3*c - 4*a*b*c^2)*x^4 - 15*a*b^3 + 52*
a^2*b*c - (15*b^4 - 62*a*b^2*c + 24*a^2*c^2)*x^2)*sqrt(c*x^4 + b*x^2 + a)*sqrt(c
) - 3*(5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2 + (5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^
3)*x^4 + (5*b^5 - 24*a*b^3*c + 16*a^2*b*c^2)*x^2)*log(4*sqrt(c*x^4 + b*x^2 + a)*
(2*c^2*x^2 + b*c) - (8*c^2*x^4 + 8*b*c*x^2 + b^2 + 4*a*c)*sqrt(c)))/((a*b^2*c^3
- 4*a^2*c^4 + (b^2*c^4 - 4*a*c^5)*x^4 + (b^3*c^3 - 4*a*b*c^4)*x^2)*sqrt(c)), 1/1
6*(2*(2*(b^2*c^2 - 4*a*c^3)*x^6 - 5*(b^3*c - 4*a*b*c^2)*x^4 - 15*a*b^3 + 52*a^2*
b*c - (15*b^4 - 62*a*b^2*c + 24*a^2*c^2)*x^2)*sqrt(c*x^4 + b*x^2 + a)*sqrt(-c) +
3*(5*a*b^4 - 24*a^2*b^2*c + 16*a^3*c^2 + (5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*
x^4 + (5*b^5 - 24*a*b^3*c + 16*a^2*b*c^2)*x^2)*arctan(1/2*(2*c*x^2 + b)*sqrt(-c)
/(sqrt(c*x^4 + b*x^2 + a)*c)))/((a*b^2*c^3 - 4*a^2*c^4 + (b^2*c^4 - 4*a*c^5)*x^4
+ (b^3*c^3 - 4*a*b*c^4)*x^2)*sqrt(-c))]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{9}}{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(c*x**4+b*x**2+a)**(3/2),x)

[Out]

Integral(x**9/(a + b*x**2 + c*x**4)**(3/2), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.334651, size = 490, normalized size = 2.58 \[ \frac{{\left ({\left (\frac{2 \,{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{2}}{b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}} - \frac{5 \,{\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )}}{b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}}\right )} x^{2} - \frac{15 \, b^{6} c - 122 \, a b^{4} c^{2} + 272 \, a^{2} b^{2} c^{3} - 96 \, a^{3} c^{4}}{b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}}\right )} x^{2} - \frac{15 \, a b^{5} c - 112 \, a^{2} b^{3} c^{2} + 208 \, a^{3} b c^{3}}{b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}}}{8 \, \sqrt{c x^{4} + b x^{2} + a}} - \frac{3 \,{\left (5 \, b^{6} c - 44 \, a b^{4} c^{2} + 112 \, a^{2} b^{2} c^{3} - 64 \, a^{3} c^{4}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} \sqrt{c} - b \right |}\right )}{16 \,{\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(c*x^4 + b*x^2 + a)^(3/2),x, algorithm="giac")

[Out]

1/8*(((2*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^2/(b^4*c^4 - 8*a*b^2*c^5 + 16*a^
2*c^6) - 5*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)/(b^4*c^4 - 8*a*b^2*c^5 + 16*a^
2*c^6))*x^2 - (15*b^6*c - 122*a*b^4*c^2 + 272*a^2*b^2*c^3 - 96*a^3*c^4)/(b^4*c^4
- 8*a*b^2*c^5 + 16*a^2*c^6))*x^2 - (15*a*b^5*c - 112*a^2*b^3*c^2 + 208*a^3*b*c^
3)/(b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6))/sqrt(c*x^4 + b*x^2 + a) - 3/16*(5*b^6*c
- 44*a*b^4*c^2 + 112*a^2*b^2*c^3 - 64*a^3*c^4)*ln(abs(-2*(sqrt(c)*x^2 - sqrt(c*
x^4 + b*x^2 + a))*sqrt(c) - b))/((b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*sqrt(c))