3.292 \(\int \frac{x^5}{\sqrt{a x^2+b x^5}} \, dx\)

Optimal. Leaf size=514 \[ -\frac{8 \sqrt{2} a^{4/3} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt [4]{3} b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a x^2+b x^5}}+\frac{4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a x^2+b x^5}}-\frac{8 a x \left (a+b x^3\right )}{7 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{a x^2+b x^5}}+\frac{2 x \sqrt{a x^2+b x^5}}{7 b} \]

[Out]

(-8*a*x*(a + b*x^3))/(7*b^(5/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)*Sqrt[a*x^2 +
b*x^5]) + (2*x*Sqrt[a*x^2 + b*x^5])/(7*b) + (4*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(4/3
)*x*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 +
Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticE[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]])/(7*b^(5/3)*Sqrt[(a
^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a*x^2
+ b*x^5]) - (8*Sqrt[2]*a^(4/3)*x*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b
^(1/3)*x + b^(2/3)*x^2)/((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]])/(7*3^(1/4)*b^(5/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3]
)*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a*x^2 + b*x^5])

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Rubi [A]  time = 0.599516, antiderivative size = 514, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{8 \sqrt{2} a^{4/3} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 \sqrt [4]{3} b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a x^2+b x^5}}+\frac{4 \sqrt [4]{3} \sqrt{2-\sqrt{3}} a^{4/3} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{7 b^{5/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a x^2+b x^5}}-\frac{8 a x \left (a+b x^3\right )}{7 b^{5/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{a x^2+b x^5}}+\frac{2 x \sqrt{a x^2+b x^5}}{7 b} \]

Antiderivative was successfully verified.

[In]  Int[x^5/Sqrt[a*x^2 + b*x^5],x]

[Out]

(-8*a*x*(a + b*x^3))/(7*b^(5/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)*Sqrt[a*x^2 +
b*x^5]) + (2*x*Sqrt[a*x^2 + b*x^5])/(7*b) + (4*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(4/3
)*x*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 +
Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticE[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]])/(7*b^(5/3)*Sqrt[(a
^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a*x^2
+ b*x^5]) - (8*Sqrt[2]*a^(4/3)*x*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b
^(1/3)*x + b^(2/3)*x^2)/((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]])/(7*3^(1/4)*b^(5/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3]
)*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a*x^2 + b*x^5])

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Rubi in Sympy [A]  time = 52.5044, size = 462, normalized size = 0.9 \[ \frac{4 \sqrt [4]{3} a^{\frac{4}{3}} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \sqrt{a x^{2} + b x^{5}} E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{7 b^{\frac{5}{3}} x \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \left (a + b x^{3}\right )} - \frac{8 \sqrt{2} \cdot 3^{\frac{3}{4}} a^{\frac{4}{3}} \sqrt{\frac{a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right ) \sqrt{a x^{2} + b x^{5}} F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{b} x}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x} \right )}\middle | -7 - 4 \sqrt{3}\right )}{21 b^{\frac{5}{3}} x \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \sqrt [3]{b} x\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )^{2}}} \left (a + b x^{3}\right )} - \frac{8 a \sqrt{a x^{2} + b x^{5}}}{7 b^{\frac{5}{3}} x \left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \sqrt [3]{b} x\right )} + \frac{2 x \sqrt{a x^{2} + b x^{5}}}{7 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x**5+a*x**2)**(1/2),x)

[Out]

4*3**(1/4)*a**(4/3)*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)*s
qrt(a*x**2 + b*x**5)*elliptic_e(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))/(7*b**(5/3)*x*sqrt(a**(1/3)
*(a**(1/3) + b**(1/3)*x)/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*x)**2)*(a + b*x**3))
- 8*sqrt(2)*3**(3/4)*a**(4/3)*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)*(a**(1/3) + b**(1/3)*x)*sqrt(a*x*
*2 + b*x**5)*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))/(21*b**(5/3)*x*sqrt(a**(1/3)*(a**(1
/3) + b**(1/3)*x)/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*x)**2)*(a + b*x**3)) - 8*a*
sqrt(a*x**2 + b*x**5)/(7*b**(5/3)*x*(a**(1/3)*(1 + sqrt(3)) + b**(1/3)*x)) + 2*x
*sqrt(a*x**2 + b*x**5)/(7*b)

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Mathematica [C]  time = 3.29281, size = 228, normalized size = 0.44 \[ \frac{2 x \left (3 x^2 \left (a+b x^3\right )-\frac{4 \sqrt [6]{-1} 3^{3/4} a^{5/3} \sqrt{\frac{(-1)^{5/6} \left (\sqrt [3]{-b} x-\sqrt [3]{a}\right )}{\sqrt [3]{a}}} \sqrt{\frac{(-b)^{2/3} x^2}{a^{2/3}}+\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}+1} \left (\sqrt [3]{-1} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )-i \sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b} x}{\sqrt [3]{a}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{(-b)^{2/3}}\right )}{21 b \sqrt{x^2 \left (a+b x^3\right )}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^5/Sqrt[a*x^2 + b*x^5],x]

[Out]

(2*x*(3*x^2*(a + b*x^3) - (4*(-1)^(1/6)*3^(3/4)*a^(5/3)*Sqrt[((-1)^(5/6)*(-a^(1/
3) + (-b)^(1/3)*x))/a^(1/3)]*Sqrt[1 + ((-b)^(1/3)*x)/a^(1/3) + ((-b)^(2/3)*x^2)/
a^(2/3)]*((-I)*Sqrt[3]*EllipticE[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-b)^(1/3)*x)/a^(1
/3)]/3^(1/4)], (-1)^(1/3)] + (-1)^(1/3)*EllipticF[ArcSin[Sqrt[-(-1)^(5/6) - (I*(
-b)^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)]))/(-b)^(2/3)))/(21*b*Sqrt[x^2*(a + b
*x^3)])

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Maple [A]  time = 0.011, size = 676, normalized size = 1.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^5+a*x^2)^(1/2),x)

[Out]

-2/21*x*(3*I*(-a*b^2)^(2/3)*3^(1/2)*(-2*(-b*x+(-a*b^2)^(1/3))/(-a*b^2)^(1/3)/(I*
3^(1/2)-3))^(1/2)*(-I*(I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))*3^(1/2)/(-
a*b^2)^(1/3))^(1/2)*EllipticE(1/6*3^(1/2)*2^(1/2)*(-I*(I*3^(1/2)*(-a*b^2)^(1/3)-
2*b*x-(-a*b^2)^(1/3))*3^(1/2)/(-a*b^2)^(1/3))^(1/2),2^(1/2)*(I*3^(1/2)/(I*3^(1/2
)-3))^(1/2))*(-I*(I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3))*3^(1/2)/(-a*b^2
)^(1/3))^(1/2)*a-2*I*(-a*b^2)^(2/3)*3^(1/2)*(-2*(-b*x+(-a*b^2)^(1/3))/(-a*b^2)^(
1/3)/(I*3^(1/2)-3))^(1/2)*(-I*(I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(1/3))*3^
(1/2)/(-a*b^2)^(1/3))^(1/2)*EllipticF(1/6*3^(1/2)*2^(1/2)*(-I*(I*3^(1/2)*(-a*b^2
)^(1/3)-2*b*x-(-a*b^2)^(1/3))*3^(1/2)/(-a*b^2)^(1/3))^(1/2),2^(1/2)*(I*3^(1/2)/(
I*3^(1/2)-3))^(1/2))*(-I*(I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3))*3^(1/2)
/(-a*b^2)^(1/3))^(1/2)*a-3*b^3*x^5+3*(-a*b^2)^(2/3)*(-2*(-b*x+(-a*b^2)^(1/3))/(-
a*b^2)^(1/3)/(I*3^(1/2)-3))^(1/2)*(-I*(I*3^(1/2)*(-a*b^2)^(1/3)+2*b*x+(-a*b^2)^(
1/3))*3^(1/2)/(-a*b^2)^(1/3))^(1/2)*EllipticE(1/6*3^(1/2)*2^(1/2)*(-I*(I*3^(1/2)
*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3))*3^(1/2)/(-a*b^2)^(1/3))^(1/2),2^(1/2)*(I*3
^(1/2)/(I*3^(1/2)-3))^(1/2))*(-I*(I*3^(1/2)*(-a*b^2)^(1/3)-2*b*x-(-a*b^2)^(1/3))
*3^(1/2)/(-a*b^2)^(1/3))^(1/2)*a-3*a*b^2*x^2)/(b*x^5+a*x^2)^(1/2)/b^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\sqrt{b x^{5} + a x^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(b*x^5 + a*x^2),x, algorithm="maxima")

[Out]

integrate(x^5/sqrt(b*x^5 + a*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{5}}{\sqrt{b x^{5} + a x^{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(b*x^5 + a*x^2),x, algorithm="fricas")

[Out]

integral(x^5/sqrt(b*x^5 + a*x^2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\sqrt{x^{2} \left (a + b x^{3}\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x**5+a*x**2)**(1/2),x)

[Out]

Integral(x**5/sqrt(x**2*(a + b*x**3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\sqrt{b x^{5} + a x^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(b*x^5 + a*x^2),x, algorithm="giac")

[Out]

integrate(x^5/sqrt(b*x^5 + a*x^2), x)