3.982 \(\int \frac{x^7}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=134 \[ \frac{\left (-8 a c+3 b^2-2 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{2 c^2 \left (b^2-4 a c\right )}+\frac{x^4 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{3 b \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 c^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.254923, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\left (-8 a c+3 b^2-2 b c x^2\right ) \sqrt{a+b x^2+c x^4}}{2 c^2 \left (b^2-4 a c\right )}+\frac{x^4 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}-\frac{3 b \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 c^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(a + b*x^2 + c*x^4)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 25.3105, size = 124, normalized size = 0.93 \[ - \frac{3 b \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{4 c^{\frac{5}{2}}} + \frac{x^{4} \left (2 a + b x^{2}\right )}{\left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}} + \frac{\sqrt{a + b x^{2} + c x^{4}} \left (- 8 a c + 3 b^{2} - 2 b c x^{2}\right )}{2 c^{2} \left (- 4 a c + b^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(c*x**4+b*x**2+a)**(3/2),x)

[Out]

-3*b*atanh((b + 2*c*x**2)/(2*sqrt(c)*sqrt(a + b*x**2 + c*x**4)))/(4*c**(5/2)) +
x**4*(2*a + b*x**2)/((-4*a*c + b**2)*sqrt(a + b*x**2 + c*x**4)) + sqrt(a + b*x**
2 + c*x**4)*(-8*a*c + 3*b**2 - 2*b*c*x**2)/(2*c**2*(-4*a*c + b**2))

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Mathematica [A]  time = 0.181717, size = 127, normalized size = 0.95 \[ \frac{1}{2} \sqrt{a+b x^2+c x^4} \left (\frac{2 \left (2 a^2 c-a b^2+3 a b c x^2+b^3 \left (-x^2\right )\right )}{c^2 \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )}+\frac{1}{c^2}\right )-\frac{3 b \log \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right )}{4 c^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(a + b*x^2 + c*x^4)^(3/2),x]

[Out]

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

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Maple [B]  time = 0.019, size = 264, normalized size = 2. \[{\frac{{x}^{4}}{2\,c}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{3\,b{x}^{2}}{4\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{3\,{b}^{2}}{8\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{3\,{b}^{3}{x}^{2}}{4\, \left ( 4\,ac-{b}^{2} \right ){c}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{3\,{b}^{4}}{8\,{c}^{3} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{3\,b}{4}\ln \left ({1 \left ({\frac{b}{2}}+c{x}^{2} \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{4}+b{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{a}{{c}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+2\,{\frac{ab{x}^{2}}{ \left ( 4\,ac-{b}^{2} \right ) c\sqrt{c{x}^{4}+b{x}^{2}+a}}}+{\frac{a{b}^{2}}{ \left ( 4\,ac-{b}^{2} \right ){c}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(c*x^4+b*x^2+a)^(3/2),x)

[Out]

1/2*x^4/c/(c*x^4+b*x^2+a)^(1/2)+3/4*b/c^2*x^2/(c*x^4+b*x^2+a)^(1/2)-3/8*b^2/c^3/
(c*x^4+b*x^2+a)^(1/2)-3/4*b^3/c^2/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(1/2)*x^2-3/8*b^4/
c^3/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(1/2)-3/4*b/c^(5/2)*ln((1/2*b+c*x^2)/c^(1/2)+(c*
x^4+b*x^2+a)^(1/2))+1/c^2*a/(c*x^4+b*x^2+a)^(1/2)+2/c*a*b/(4*a*c-b^2)/(c*x^4+b*x
^2+a)^(1/2)*x^2+1/c^2*a*b^2/(4*a*c-b^2)/(c*x^4+b*x^2+a)^(1/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.325895, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + 3 \, a b^{2} - 8 \, a^{2} c +{\left (3 \, b^{3} - 10 \, a b c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{c} + 3 \,{\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{4} + a b^{3} - 4 \, a^{2} b c +{\left (b^{4} - 4 \, a b^{2} c\right )} x^{2}\right )} \log \left (4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c^{2} x^{2} + b c\right )} -{\left (8 \, c^{2} x^{4} + 8 \, b c x^{2} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{8 \,{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} +{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{4} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2}\right )} \sqrt{c}}, \frac{2 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + 3 \, a b^{2} - 8 \, a^{2} c +{\left (3 \, b^{3} - 10 \, a b c\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{-c} - 3 \,{\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{4} + a b^{3} - 4 \, a^{2} b c +{\left (b^{4} - 4 \, a b^{2} c\right )} x^{2}\right )} \arctan \left (\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{4} + b x^{2} + a} c}\right )}{4 \,{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} +{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{4} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(c*x^4 + b*x^2 + a)^(3/2),x, algorithm="fricas")

[Out]

[1/8*(4*((b^2*c - 4*a*c^2)*x^4 + 3*a*b^2 - 8*a^2*c + (3*b^3 - 10*a*b*c)*x^2)*sqr
t(c*x^4 + b*x^2 + a)*sqrt(c) + 3*((b^3*c - 4*a*b*c^2)*x^4 + a*b^3 - 4*a^2*b*c +
(b^4 - 4*a*b^2*c)*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^2 - 4*a^2*c^3 + (b^2*c^3 - 4*
a*c^4)*x^4 + (b^3*c^2 - 4*a*b*c^3)*x^2)*sqrt(c)), 1/4*(2*((b^2*c - 4*a*c^2)*x^4
+ 3*a*b^2 - 8*a^2*c + (3*b^3 - 10*a*b*c)*x^2)*sqrt(c*x^4 + b*x^2 + a)*sqrt(-c) -
3*((b^3*c - 4*a*b*c^2)*x^4 + a*b^3 - 4*a^2*b*c + (b^4 - 4*a*b^2*c)*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^2 - 4*a^2*c^3
+ (b^2*c^3 - 4*a*c^4)*x^4 + (b^3*c^2 - 4*a*b*c^3)*x^2)*sqrt(-c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{7}}{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(c*x**4+b*x**2+a)**(3/2),x)

[Out]

Integral(x**7/(a + b*x**2 + c*x**4)**(3/2), x)

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GIAC/XCAS [A]  time = 0.331666, size = 359, normalized size = 2.68 \[ \frac{{\left (\frac{{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{2}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \, b^{5} - 22 \, a b^{3} c + 40 \, a^{2} b c^{2}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x^{2} + \frac{3 \, a b^{4} - 20 \, a^{2} b^{2} c + 32 \, a^{3} c^{2}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}}{2 \, \sqrt{c x^{4} + b x^{2} + a}} + \frac{3 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x^{2} - \sqrt{c x^{4} + b x^{2} + a}\right )} \sqrt{c} - b \right |}\right )}{4 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(c*x^4 + b*x^2 + a)^(3/2),x, algorithm="giac")

[Out]

1/2*(((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*x^2/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4
) + (3*b^5 - 22*a*b^3*c + 40*a^2*b*c^2)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x^
2 + (3*a*b^4 - 20*a^2*b^2*c + 32*a^3*c^2)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))/
sqrt(c*x^4 + b*x^2 + a) + 3/4*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*ln(abs(-2*(sqrt(c
)*x^2 - sqrt(c*x^4 + b*x^2 + a))*sqrt(c) - b))/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*
c^4)*sqrt(c))