3.2022 \(\int \frac{x^7}{\sqrt{a+\frac{b}{x^3}}} \, dx\)

Optimal. Leaf size=294 \[ \frac{91 b^2 x^2 \sqrt{a+\frac{b}{x^3}}}{320 a^3}-\frac{13 b x^5 \sqrt{a+\frac{b}{x^3}}}{80 a^2}+\frac{91 \sqrt{2+\sqrt{3}} b^{8/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{320 \sqrt [4]{3} a^3 \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}+\frac{x^8 \sqrt{a+\frac{b}{x^3}}}{8 a} \]

[Out]

(91*b^2*Sqrt[a + b/x^3]*x^2)/(320*a^3) - (13*b*Sqrt[a + b/x^3]*x^5)/(80*a^2) + (
Sqrt[a + b/x^3]*x^8)/(8*a) + (91*Sqrt[2 + Sqrt[3]]*b^(8/3)*(a^(1/3) + b^(1/3)/x)
*Sqrt[(a^(2/3) + b^(2/3)/x^2 - (a^(1/3)*b^(1/3))/x)/((1 + Sqrt[3])*a^(1/3) + b^(
1/3)/x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)/x)/((1 + Sqrt[3])*a
^(1/3) + b^(1/3)/x)], -7 - 4*Sqrt[3]])/(320*3^(1/4)*a^3*Sqrt[a + b/x^3]*Sqrt[(a^
(1/3)*(a^(1/3) + b^(1/3)/x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)^2])

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Rubi [A]  time = 0.457128, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{91 b^2 x^2 \sqrt{a+\frac{b}{x^3}}}{320 a^3}-\frac{13 b x^5 \sqrt{a+\frac{b}{x^3}}}{80 a^2}+\frac{91 \sqrt{2+\sqrt{3}} b^{8/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{320 \sqrt [4]{3} a^3 \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}+\frac{x^8 \sqrt{a+\frac{b}{x^3}}}{8 a} \]

Antiderivative was successfully verified.

[In]  Int[x^7/Sqrt[a + b/x^3],x]

[Out]

(91*b^2*Sqrt[a + b/x^3]*x^2)/(320*a^3) - (13*b*Sqrt[a + b/x^3]*x^5)/(80*a^2) + (
Sqrt[a + b/x^3]*x^8)/(8*a) + (91*Sqrt[2 + Sqrt[3]]*b^(8/3)*(a^(1/3) + b^(1/3)/x)
*Sqrt[(a^(2/3) + b^(2/3)/x^2 - (a^(1/3)*b^(1/3))/x)/((1 + Sqrt[3])*a^(1/3) + b^(
1/3)/x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)/x)/((1 + Sqrt[3])*a
^(1/3) + b^(1/3)/x)], -7 - 4*Sqrt[3]])/(320*3^(1/4)*a^3*Sqrt[a + b/x^3]*Sqrt[(a^
(1/3)*(a^(1/3) + b^(1/3)/x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)^2])

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Rubi in Sympy [A]  time = 24.0791, size = 250, normalized size = 0.85 \[ \frac{x^{8} \sqrt{a + \frac{b}{x^{3}}}}{8 a} - \frac{13 b x^{5} \sqrt{a + \frac{b}{x^{3}}}}{80 a^{2}} + \frac{91 \cdot 3^{\frac{3}{4}} b^{\frac{8}{3}} \sqrt{\frac{a^{\frac{2}{3}} - \frac{\sqrt [3]{a} \sqrt [3]{b}}{x} + \frac{b^{\frac{2}{3}}}{x^{2}}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{960 a^{3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{a + \frac{b}{x^{3}}}} + \frac{91 b^{2} x^{2} \sqrt{a + \frac{b}{x^{3}}}}{320 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(a+b/x**3)**(1/2),x)

[Out]

x**8*sqrt(a + b/x**3)/(8*a) - 13*b*x**5*sqrt(a + b/x**3)/(80*a**2) + 91*3**(3/4)
*b**(8/3)*sqrt((a**(2/3) - a**(1/3)*b**(1/3)/x + b**(2/3)/x**2)/(a**(1/3)*(1 + s
qrt(3)) + b**(1/3)/x)**2)*sqrt(sqrt(3) + 2)*(a**(1/3) + b**(1/3)/x)*elliptic_f(a
sin((-a**(1/3)*(-1 + sqrt(3)) + b**(1/3)/x)/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)/x
)), -7 - 4*sqrt(3))/(960*a**3*sqrt(a**(1/3)*(a**(1/3) + b**(1/3)/x)/(a**(1/3)*(1
+ sqrt(3)) + b**(1/3)/x)**2)*sqrt(a + b/x**3)) + 91*b**2*x**2*sqrt(a + b/x**3)/
(320*a**3)

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Mathematica [C]  time = 0.578193, size = 199, normalized size = 0.68 \[ \frac{3 \sqrt [3]{-b} \left (40 a^3 x^9-12 a^2 b x^6+39 a b^2 x^3+91 b^3\right )+91 i 3^{3/4} \sqrt [3]{a} b^3 x \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-b}}{\sqrt [3]{a} x}-1\right )} \sqrt{\frac{\frac{(-b)^{2/3}}{a^{2/3}}+\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}+x^2}{x^2}} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b}}{\sqrt [3]{a} x}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{960 a^3 \sqrt [3]{-b} x \sqrt{a+\frac{b}{x^3}}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^7/Sqrt[a + b/x^3],x]

[Out]

(3*(-b)^(1/3)*(91*b^3 + 39*a*b^2*x^3 - 12*a^2*b*x^6 + 40*a^3*x^9) + (91*I)*3^(3/
4)*a^(1/3)*b^3*Sqrt[(-1)^(5/6)*(-1 + (-b)^(1/3)/(a^(1/3)*x))]*x*Sqrt[((-b)^(2/3)
/a^(2/3) + ((-b)^(1/3)*x)/a^(1/3) + x^2)/x^2]*EllipticF[ArcSin[Sqrt[-(-1)^(5/6)
- (I*(-b)^(1/3))/(a^(1/3)*x)]/3^(1/4)], (-1)^(1/3)])/(960*a^3*(-b)^(1/3)*Sqrt[a
+ b/x^3]*x)

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Maple [B]  time = 0.042, size = 2233, normalized size = 7.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(a+b/x^3)^(1/2),x)

[Out]

1/320/((a*x^3+b)/x^3)^(1/2)/x*(a*x^3+b)/(-a^2*b)^(1/3)/a^4*(40*I*(1/a^2*x*(-a*x+
(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2
*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)*(-a^2*b)^(1/3)*(a*x^4+b*x)^(1/2)*3^(1/2)*
x^6*a^3+182*I*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I
*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)
))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-
a^2*b)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^
(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*3
^(1/2)*x^2*a^2*b^3-364*I*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3))
)^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a
^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-
1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*
x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3
))^(1/2))*(-a^2*b)^(1/3)*3^(1/2)*x*a*b^3-120*x^6*(a*x^4+b*x)^(1/2)*a^3*(-a^2*b)^
(1/3)*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1
/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)+182*I*(-(I*3^(1/2)-3
)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*
x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b
)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*Ellipti
cF((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)
*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*(-a^2*b)^(2/3)*3^(1/2)*b^3-18
2*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a
^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I
*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)
))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2
),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*x^2*a^2*b^3-5
2*I*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3
))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)*(-a^2*b)^(1/3)*(a*x^4+
b*x)^(1/2)*3^(1/2)*x^3*a^2*b+364*(-a^2*b)^(1/3)*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1
)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/
(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^
2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)
*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*
3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*x*a*b^3-182*(-a^2*b)^(2/3)*(-(I*3^(1/2)-3)*x*a/
(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^
2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3
)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticF((-(
I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^
(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*b^3+156*b*x^3*(a*x^4+b*x)^(1/2)*a^2
*(-a^2*b)^(1/3)*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(
-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)+91*I*(1/a^
2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(
1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)*(-a^2*b)^(1/3)*(a*x^4+b*x)^(1/2
)*3^(1/2)*a*b^2-273*b^2*(a*x^4+b*x)^(1/2)*a*(-a^2*b)^(1/3)*(1/a^2*x*(-a*x+(-a^2*
b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1
/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2))/(x*(a*x^3+b))^(1/2)/(I*3^(1/2)-3)/(1/a^2*x*(-a
*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-
a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{7}}{\sqrt{a + \frac{b}{x^{3}}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/sqrt(a + b/x^3),x, algorithm="maxima")

[Out]

integrate(x^7/sqrt(a + b/x^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{7}}{\sqrt{\frac{a x^{3} + b}{x^{3}}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/sqrt(a + b/x^3),x, algorithm="fricas")

[Out]

integral(x^7/sqrt((a*x^3 + b)/x^3), x)

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Sympy [A]  time = 6.3758, size = 46, normalized size = 0.16 \[ - \frac{x^{8} \Gamma \left (- \frac{8}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{8}{3}, \frac{1}{2} \\ - \frac{5}{3} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{3}}} \right )}}{3 \sqrt{a} \Gamma \left (- \frac{5}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(a+b/x**3)**(1/2),x)

[Out]

-x**8*gamma(-8/3)*hyper((-8/3, 1/2), (-5/3,), b*exp_polar(I*pi)/(a*x**3))/(3*sqr
t(a)*gamma(-5/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{7}}{\sqrt{a + \frac{b}{x^{3}}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/sqrt(a + b/x^3),x, algorithm="giac")

[Out]

integrate(x^7/sqrt(a + b/x^3), x)