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3.2042 \(\int \frac{x^7}{\left (a+\frac{b}{x^3}\right )^{3/2}} \, dx\)

Optimal. Leaf size=315 \[ \frac{1729 b^2 x^2 \sqrt{a+\frac{b}{x^3}}}{960 a^4}-\frac{247 b x^5 \sqrt{a+\frac{b}{x^3}}}{240 a^3}+\frac{19 x^8 \sqrt{a+\frac{b}{x^3}}}{24 a^2}+\frac{1729 \sqrt{2+\sqrt{3}} b^{8/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{960 \sqrt [4]{3} a^4 \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}-\frac{2 x^8}{3 a \sqrt{a+\frac{b}{x^3}}} \]

[Out]

(1729*b^2*Sqrt[a + b/x^3]*x^2)/(960*a^4) - (247*b*Sqrt[a + b/x^3]*x^5)/(240*a^3)
 - (2*x^8)/(3*a*Sqrt[a + b/x^3]) + (19*Sqrt[a + b/x^3]*x^8)/(24*a^2) + (1729*Sqr
t[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 - Sq
rt[3])*a^(1/3) + b^(1/3)/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)], -7 - 4*Sqrt[3]
])/(960*3^(1/4)*a^4*Sqrt[a + b/x^3]*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)/x))/((1 + S
qrt[3])*a^(1/3) + b^(1/3)/x)^2])

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Rubi [A]  time = 0.538178, antiderivative size = 315, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{1729 b^2 x^2 \sqrt{a+\frac{b}{x^3}}}{960 a^4}-\frac{247 b x^5 \sqrt{a+\frac{b}{x^3}}}{240 a^3}+\frac{19 x^8 \sqrt{a+\frac{b}{x^3}}}{24 a^2}+\frac{1729 \sqrt{2+\sqrt{3}} b^{8/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{960 \sqrt [4]{3} a^4 \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}-\frac{2 x^8}{3 a \sqrt{a+\frac{b}{x^3}}} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(a + b/x^3)^(3/2),x]

[Out]

(1729*b^2*Sqrt[a + b/x^3]*x^2)/(960*a^4) - (247*b*Sqrt[a + b/x^3]*x^5)/(240*a^3)
 - (2*x^8)/(3*a*Sqrt[a + b/x^3]) + (19*Sqrt[a + b/x^3]*x^8)/(24*a^2) + (1729*Sqr
t[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 - Sq
rt[3])*a^(1/3) + b^(1/3)/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)], -7 - 4*Sqrt[3]
])/(960*3^(1/4)*a^4*Sqrt[a + b/x^3]*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)/x))/((1 + S
qrt[3])*a^(1/3) + b^(1/3)/x)^2])

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Rubi in Sympy [A]  time = 31.0789, size = 272, normalized size = 0.86 \[ - \frac{2 x^{8}}{3 a \sqrt{a + \frac{b}{x^{3}}}} + \frac{19 x^{8} \sqrt{a + \frac{b}{x^{3}}}}{24 a^{2}} - \frac{247 b x^{5} \sqrt{a + \frac{b}{x^{3}}}}{240 a^{3}} + \frac{1729 \cdot 3^{\frac{3}{4}} b^{\frac{8}{3}} \sqrt{\frac{a^{\frac{2}{3}} - \frac{\sqrt [3]{a} \sqrt [3]{b}}{x} + \frac{b^{\frac{2}{3}}}{x^{2}}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{2880 a^{4} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{a + \frac{b}{x^{3}}}} + \frac{1729 b^{2} x^{2} \sqrt{a + \frac{b}{x^{3}}}}{960 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**8/(3*a*sqrt(a + b/x**3)) + 19*x**8*sqrt(a + b/x**3)/(24*a**2) - 247*b*x**5
*sqrt(a + b/x**3)/(240*a**3) + 1729*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 + sqrt(3)) + b**(1/3)/x)**2)*sqrt(sqrt(
3) + 2)*(a**(1/3) + b**(1/3)/x)*elliptic_f(asin((-a**(1/3)*(-1 + sqrt(3)) + b**(
1/3)/x)/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)/x)), -7 - 4*sqrt(3))/(2880*a**4*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)) + 1729*b**2*x**2*sqrt(a + b/x**3)/(960*a**4)

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Mathematica [C]  time = 0.589697, size = 199, normalized size = 0.63 \[ \frac{3 \sqrt [3]{-b} \left (120 a^3 x^9-228 a^2 b x^6+741 a b^2 x^3+1729 b^3\right )+1729 i 3^{3/4} \sqrt [3]{a} b^3 x \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-b}}{\sqrt [3]{a} x}-1\right )} \sqrt{\frac{\frac{(-b)^{2/3}}{a^{2/3}}+\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}+x^2}{x^2}} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b}}{\sqrt [3]{a} x}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{2880 a^4 \sqrt [3]{-b} x \sqrt{a+\frac{b}{x^3}}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(3*(-b)^(1/3)*(1729*b^3 + 741*a*b^2*x^3 - 228*a^2*b*x^6 + 120*a^3*x^9) + (1729*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)])/(2880*a^4*(-b)^(1/3)
*Sqrt[a + b/x^3]*x)

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Maple [B]  time = 0.053, size = 2540, normalized size = 8.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(a+b/x^3)^(3/2),x)

[Out]

1/960/((a*x^3+b)/x^3)^(3/2)/x^5*(a*x^3+b)/(-a^2*b)^(1/3)/a^5*(120*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*(a*x^3+b))^(1/2)*x^6*a^3+3458*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*(a*x^3+b))^(1/2)*x^2*a^2*b^3-6916*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)*Ell
ipticF((-(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*x^3+b))^(1/2)*x*a*b^3-360*(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)*(x*(a*x^3+b))^(1/2)*x^6*a^3+3458*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)*E
llipticF((-(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)*
(x*(a*x^3+b))^(1/2)*b^3-3458*(-(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*x^3+b))^(1/2)*x^2*a^2*b^3-348*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*(a*x^3+b))
^(1/2)*x^3*a^2*b+6916*(-(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)*(x*(a*x^3+b))^(1/2)*x*a*b^3-3458*(-(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)^(2/3)*(x*(a*x^3+b))^(1/2)*b^3+1
044*(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)*(x*(a*x^3+b))^(1/2)*x^3*a^2*b+1089*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*(a*x^3+b))^(1/
2)*a*b^2+640*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)*3^(1/2)*x*a*b^3-3267*(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)*(x*(a*x^3+b))^(1/2)*a*b^2-1920*(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)*x*a*b^3)/(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. \[ \int \frac{x^{7}}{{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(a + b/x^3)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^7/(a + b/x^3)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{10}}{{\left (a x^{3} + b\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^10/((a*x^3 + b)*sqrt((a*x^3 + b)/x^3)), x)

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Sympy [A]  time = 7.15815, size = 46, normalized size = 0.15 \[ - \frac{x^{8} \Gamma \left (- \frac{8}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{8}{3}, \frac{3}{2} \\ - \frac{5}{3} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{3}}} \right )}}{3 a^{\frac{3}{2}} \Gamma \left (- \frac{5}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{7}}{{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^7/(a + b/x^3)^(3/2), x)

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