3.968 \(\int \frac{x^7}{\sqrt{a+b x^2-c x^4}} \, dx\)

Optimal. Leaf size=124 \[ -\frac{b \left (12 a c+5 b^2\right ) \tan ^{-1}\left (\frac{b-2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2-c x^4}}\right )}{32 c^{7/2}}-\frac{\left (16 a c+15 b^2+10 b c x^2\right ) \sqrt{a+b x^2-c x^4}}{48 c^3}-\frac{x^4 \sqrt{a+b x^2-c x^4}}{6 c} \]

[Out]

-(x^4*Sqrt[a + b*x^2 - c*x^4])/(6*c) - ((15*b^2 + 16*a*c + 10*b*c*x^2)*Sqrt[a +
b*x^2 - c*x^4])/(48*c^3) - (b*(5*b^2 + 12*a*c)*ArcTan[(b - 2*c*x^2)/(2*Sqrt[c]*S
qrt[a + b*x^2 - c*x^4])])/(32*c^(7/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.264882, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{b \left (12 a c+5 b^2\right ) \tan ^{-1}\left (\frac{b-2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2-c x^4}}\right )}{32 c^{7/2}}-\frac{\left (16 a c+15 b^2+10 b c x^2\right ) \sqrt{a+b x^2-c x^4}}{48 c^3}-\frac{x^4 \sqrt{a+b x^2-c x^4}}{6 c} \]

Antiderivative was successfully verified.

[In]  Int[x^7/Sqrt[a + b*x^2 - c*x^4],x]

[Out]

-(x^4*Sqrt[a + b*x^2 - c*x^4])/(6*c) - ((15*b^2 + 16*a*c + 10*b*c*x^2)*Sqrt[a +
b*x^2 - c*x^4])/(48*c^3) - (b*(5*b^2 + 12*a*c)*ArcTan[(b - 2*c*x^2)/(2*Sqrt[c]*S
qrt[a + b*x^2 - c*x^4])])/(32*c^(7/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.4444, size = 116, normalized size = 0.94 \[ - \frac{b \left (12 a c + 5 b^{2}\right ) \operatorname{atan}{\left (\frac{b - 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} - c x^{4}}} \right )}}{32 c^{\frac{7}{2}}} - \frac{x^{4} \sqrt{a + b x^{2} - c x^{4}}}{6 c} - \frac{\sqrt{a + b x^{2} - c x^{4}} \left (4 a c + \frac{15 b^{2}}{4} + \frac{5 b c x^{2}}{2}\right )}{12 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(-c*x**4+b*x**2+a)**(1/2),x)

[Out]

-b*(12*a*c + 5*b**2)*atan((b - 2*c*x**2)/(2*sqrt(c)*sqrt(a + b*x**2 - c*x**4)))/
(32*c**(7/2)) - x**4*sqrt(a + b*x**2 - c*x**4)/(6*c) - sqrt(a + b*x**2 - c*x**4)
*(4*a*c + 15*b**2/4 + 5*b*c*x**2/2)/(12*c**3)

_______________________________________________________________________________________

Mathematica [C]  time = 0.240083, size = 112, normalized size = 0.9 \[ \frac{-2 \sqrt{c} \sqrt{a+b x^2-c x^4} \left (8 c \left (2 a+c x^4\right )+15 b^2+10 b c x^2\right )+3 i \left (12 a b c+5 b^3\right ) \log \left (2 \sqrt{a+b x^2-c x^4}+\frac{i \left (b-2 c x^2\right )}{\sqrt{c}}\right )}{96 c^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/Sqrt[a + b*x^2 - c*x^4],x]

[Out]

(-2*Sqrt[c]*Sqrt[a + b*x^2 - c*x^4]*(15*b^2 + 10*b*c*x^2 + 8*c*(2*a + c*x^4)) +
(3*I)*(5*b^3 + 12*a*b*c)*Log[(I*(b - 2*c*x^2))/Sqrt[c] + 2*Sqrt[a + b*x^2 - c*x^
4]])/(96*c^(7/2))

_______________________________________________________________________________________

Maple [A]  time = 0.028, size = 168, normalized size = 1.4 \[ -{\frac{{x}^{4}}{6\,c}\sqrt{-c{x}^{4}+b{x}^{2}+a}}-{\frac{5\,b{x}^{2}}{24\,{c}^{2}}\sqrt{-c{x}^{4}+b{x}^{2}+a}}-{\frac{5\,{b}^{2}}{16\,{c}^{3}}\sqrt{-c{x}^{4}+b{x}^{2}+a}}+{\frac{5\,{b}^{3}}{32}\arctan \left ({1\sqrt{c} \left ({x}^{2}-{\frac{b}{2\,c}} \right ){\frac{1}{\sqrt{-c{x}^{4}+b{x}^{2}+a}}}} \right ){c}^{-{\frac{7}{2}}}}+{\frac{3\,ab}{8}\arctan \left ({1\sqrt{c} \left ({x}^{2}-{\frac{b}{2\,c}} \right ){\frac{1}{\sqrt{-c{x}^{4}+b{x}^{2}+a}}}} \right ){c}^{-{\frac{5}{2}}}}-{\frac{a}{3\,{c}^{2}}\sqrt{-c{x}^{4}+b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(-c*x^4+b*x^2+a)^(1/2),x)

[Out]

-1/6*x^4*(-c*x^4+b*x^2+a)^(1/2)/c-5/24*b/c^2*x^2*(-c*x^4+b*x^2+a)^(1/2)-5/16*b^2
/c^3*(-c*x^4+b*x^2+a)^(1/2)+5/32*b^3/c^(7/2)*arctan(c^(1/2)*(x^2-1/2*b/c)/(-c*x^
4+b*x^2+a)^(1/2))+3/8*b/c^(5/2)*a*arctan(c^(1/2)*(x^2-1/2*b/c)/(-c*x^4+b*x^2+a)^
(1/2))-1/3/c^2*a*(-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^7/sqrt(-c*x^4 + b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.300477, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (8 \, c^{2} x^{4} + 10 \, b c x^{2} + 15 \, b^{2} + 16 \, a c\right )} \sqrt{-c x^{4} + b x^{2} + a} \sqrt{-c} - 3 \,{\left (5 \, b^{3} + 12 \, a b c\right )} \log \left (4 \, \sqrt{-c x^{4} + b x^{2} + a}{\left (2 \, c^{2} x^{2} - b c\right )} +{\left (8 \, c^{2} x^{4} - 8 \, b c x^{2} + b^{2} - 4 \, a c\right )} \sqrt{-c}\right )}{192 \, \sqrt{-c} c^{3}}, -\frac{2 \,{\left (8 \, c^{2} x^{4} + 10 \, b c x^{2} + 15 \, b^{2} + 16 \, a c\right )} \sqrt{-c x^{4} + b x^{2} + a} \sqrt{c} - 3 \,{\left (5 \, b^{3} + 12 \, a b c\right )} \arctan \left (\frac{2 \, c x^{2} - b}{2 \, \sqrt{-c x^{4} + b x^{2} + a} \sqrt{c}}\right )}{96 \, c^{\frac{7}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/192*(4*(8*c^2*x^4 + 10*b*c*x^2 + 15*b^2 + 16*a*c)*sqrt(-c*x^4 + b*x^2 + a)*s
qrt(-c) - 3*(5*b^3 + 12*a*b*c)*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)))/(sqrt(-c)*c^3), -1/96*(2*(8*c
^2*x^4 + 10*b*c*x^2 + 15*b^2 + 16*a*c)*sqrt(-c*x^4 + b*x^2 + a)*sqrt(c) - 3*(5*b
^3 + 12*a*b*c)*arctan(1/2*(2*c*x^2 - b)/(sqrt(-c*x^4 + b*x^2 + a)*sqrt(c))))/c^(
7/2)]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{7}}{\sqrt{a + b x^{2} - c x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(-c*x**4+b*x**2+a)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.304953, size = 159, normalized size = 1.28 \[ -\frac{1}{48} \, \sqrt{-c x^{4} + b x^{2} + a}{\left (2 \, x^{2}{\left (\frac{4 \, x^{2}}{c} + \frac{5 \, b}{c^{2}}\right )} + \frac{15 \, b^{2} c + 16 \, a c^{2}}{c^{4}}\right )} - \frac{{\left (5 \, b^{3} c + 12 \, a b c^{2}\right )}{\rm ln}\left ({\left | 2 \,{\left (\sqrt{-c} x^{2} - \sqrt{-c x^{4} + b x^{2} + a}\right )} \sqrt{-c} + b \right |}\right )}{32 \, \sqrt{-c} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*sqrt(-c*x^4 + b*x^2 + a)*(2*x^2*(4*x^2/c + 5*b/c^2) + (15*b^2*c + 16*a*c^2
)/c^4) - 1/32*(5*b^3*c + 12*a*b*c^2)*ln(abs(2*(sqrt(-c)*x^2 - sqrt(-c*x^4 + b*x^
2 + a))*sqrt(-c) + b))/(sqrt(-c)*c^4)