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

Optimal. Leaf size=114 \[ -\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{16 c^{7/2}}+\frac{5 b^2 \sqrt{b x^2+c x^4}}{16 c^3}-\frac{5 b x^2 \sqrt{b x^2+c x^4}}{24 c^2}+\frac{x^4 \sqrt{b x^2+c x^4}}{6 c} \]

[Out]

(5*b^2*Sqrt[b*x^2 + c*x^4])/(16*c^3) - (5*b*x^2*Sqrt[b*x^2 + c*x^4])/(24*c^2) +
(x^4*Sqrt[b*x^2 + c*x^4])/(6*c) - (5*b^3*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^
4]])/(16*c^(7/2))

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Rubi [A]  time = 0.270571, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )}{16 c^{7/2}}+\frac{5 b^2 \sqrt{b x^2+c x^4}}{16 c^3}-\frac{5 b x^2 \sqrt{b x^2+c x^4}}{24 c^2}+\frac{x^4 \sqrt{b x^2+c x^4}}{6 c} \]

Antiderivative was successfully verified.

[In]  Int[x^7/Sqrt[b*x^2 + c*x^4],x]

[Out]

(5*b^2*Sqrt[b*x^2 + c*x^4])/(16*c^3) - (5*b*x^2*Sqrt[b*x^2 + c*x^4])/(24*c^2) +
(x^4*Sqrt[b*x^2 + c*x^4])/(6*c) - (5*b^3*ArcTanh[(Sqrt[c]*x^2)/Sqrt[b*x^2 + c*x^
4]])/(16*c^(7/2))

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Rubi in Sympy [A]  time = 23.1461, size = 104, normalized size = 0.91 \[ - \frac{5 b^{3} \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{b x^{2} + c x^{4}}} \right )}}{16 c^{\frac{7}{2}}} + \frac{5 b^{2} \sqrt{b x^{2} + c x^{4}}}{16 c^{3}} - \frac{5 b x^{2} \sqrt{b x^{2} + c x^{4}}}{24 c^{2}} + \frac{x^{4} \sqrt{b x^{2} + c x^{4}}}{6 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*b**3*atanh(sqrt(c)*x**2/sqrt(b*x**2 + c*x**4))/(16*c**(7/2)) + 5*b**2*sqrt(b*
x**2 + c*x**4)/(16*c**3) - 5*b*x**2*sqrt(b*x**2 + c*x**4)/(24*c**2) + x**4*sqrt(
b*x**2 + c*x**4)/(6*c)

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Mathematica [A]  time = 0.0764267, size = 103, normalized size = 0.9 \[ \frac{x \left (\sqrt{c} x \left (15 b^3+5 b^2 c x^2-2 b c^2 x^4+8 c^3 x^6\right )-15 b^3 \sqrt{b+c x^2} \log \left (\sqrt{c} \sqrt{b+c x^2}+c x\right )\right )}{48 c^{7/2} \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(Sqrt[c]*x*(15*b^3 + 5*b^2*c*x^2 - 2*b*c^2*x^4 + 8*c^3*x^6) - 15*b^3*Sqrt[b +
c*x^2]*Log[c*x + Sqrt[c]*Sqrt[b + c*x^2]]))/(48*c^(7/2)*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.013, size = 105, normalized size = 0.9 \[ -{\frac{x}{48}\sqrt{c{x}^{2}+b} \left ( -8\,{x}^{5}\sqrt{c{x}^{2}+b}{c}^{7/2}+10\,\sqrt{c{x}^{2}+b}{c}^{5/2}{x}^{3}b-15\,\sqrt{c{x}^{2}+b}{c}^{3/2}x{b}^{2}+15\,\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ){b}^{3}c \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{c}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(c*x^4+b*x^2)^(1/2),x)

[Out]

-1/48*x*(c*x^2+b)^(1/2)*(-8*x^5*(c*x^2+b)^(1/2)*c^(7/2)+10*(c*x^2+b)^(1/2)*c^(5/
2)*x^3*b-15*(c*x^2+b)^(1/2)*c^(3/2)*x*b^2+15*ln(x*c^(1/2)+(c*x^2+b)^(1/2))*b^3*c
)/(c*x^4+b*x^2)^(1/2)/c^(9/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283934, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, b^{3} \sqrt{c} \log \left (-{\left (2 \, c x^{2} + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{4} + b x^{2}} c\right ) + 2 \,{\left (8 \, c^{3} x^{4} - 10 \, b c^{2} x^{2} + 15 \, b^{2} c\right )} \sqrt{c x^{4} + b x^{2}}}{96 \, c^{4}}, \frac{15 \, b^{3} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + b x^{2}}}\right ) +{\left (8 \, c^{3} x^{4} - 10 \, b c^{2} x^{2} + 15 \, b^{2} c\right )} \sqrt{c x^{4} + b x^{2}}}{48 \, c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(15*b^3*sqrt(c)*log(-(2*c*x^2 + b)*sqrt(c) + 2*sqrt(c*x^4 + b*x^2)*c) + 2*
(8*c^3*x^4 - 10*b*c^2*x^2 + 15*b^2*c)*sqrt(c*x^4 + b*x^2))/c^4, 1/48*(15*b^3*sqr
t(-c)*arctan(sqrt(-c)*x^2/sqrt(c*x^4 + b*x^2)) + (8*c^3*x^4 - 10*b*c^2*x^2 + 15*
b^2*c)*sqrt(c*x^4 + b*x^2))/c^4]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{7}}{\sqrt{c x^{4} + b x^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^7/sqrt(c*x^4 + b*x^2), x)