3.149 \(\int \frac{x^4}{\sqrt{b \sqrt [3]{x}+a x}} \, dx\)

Optimal. Leaf size=304 \[ -\frac{5525 b^{27/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{14421 a^{29/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{11050 b^6 \sqrt{a x+b \sqrt [3]{x}}}{14421 a^7}-\frac{2210 b^5 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{4807 a^6}+\frac{15470 b^4 x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{43263 a^5}-\frac{1190 b^3 x^2 \sqrt{a x+b \sqrt [3]{x}}}{3933 a^4}+\frac{350 b^2 x^{8/3} \sqrt{a x+b \sqrt [3]{x}}}{1311 a^3}-\frac{50 b x^{10/3} \sqrt{a x+b \sqrt [3]{x}}}{207 a^2}+\frac{2 x^4 \sqrt{a x+b \sqrt [3]{x}}}{9 a} \]

[Out]

(11050*b^6*Sqrt[b*x^(1/3) + a*x])/(14421*a^7) - (2210*b^5*x^(2/3)*Sqrt[b*x^(1/3)
+ a*x])/(4807*a^6) + (15470*b^4*x^(4/3)*Sqrt[b*x^(1/3) + a*x])/(43263*a^5) - (1
190*b^3*x^2*Sqrt[b*x^(1/3) + a*x])/(3933*a^4) + (350*b^2*x^(8/3)*Sqrt[b*x^(1/3)
+ a*x])/(1311*a^3) - (50*b*x^(10/3)*Sqrt[b*x^(1/3) + a*x])/(207*a^2) + (2*x^4*Sq
rt[b*x^(1/3) + a*x])/(9*a) - (5525*b^(27/4)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[(b
+ a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x^(1/3))^2]*x^(1/6)*EllipticF[2*ArcTan[(a^(1/4)*
x^(1/6))/b^(1/4)], 1/2])/(14421*a^(29/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi [A]  time = 0.900077, antiderivative size = 304, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{5525 b^{27/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{14421 a^{29/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{11050 b^6 \sqrt{a x+b \sqrt [3]{x}}}{14421 a^7}-\frac{2210 b^5 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{4807 a^6}+\frac{15470 b^4 x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{43263 a^5}-\frac{1190 b^3 x^2 \sqrt{a x+b \sqrt [3]{x}}}{3933 a^4}+\frac{350 b^2 x^{8/3} \sqrt{a x+b \sqrt [3]{x}}}{1311 a^3}-\frac{50 b x^{10/3} \sqrt{a x+b \sqrt [3]{x}}}{207 a^2}+\frac{2 x^4 \sqrt{a x+b \sqrt [3]{x}}}{9 a} \]

Antiderivative was successfully verified.

[In]  Int[x^4/Sqrt[b*x^(1/3) + a*x],x]

[Out]

(11050*b^6*Sqrt[b*x^(1/3) + a*x])/(14421*a^7) - (2210*b^5*x^(2/3)*Sqrt[b*x^(1/3)
+ a*x])/(4807*a^6) + (15470*b^4*x^(4/3)*Sqrt[b*x^(1/3) + a*x])/(43263*a^5) - (1
190*b^3*x^2*Sqrt[b*x^(1/3) + a*x])/(3933*a^4) + (350*b^2*x^(8/3)*Sqrt[b*x^(1/3)
+ a*x])/(1311*a^3) - (50*b*x^(10/3)*Sqrt[b*x^(1/3) + a*x])/(207*a^2) + (2*x^4*Sq
rt[b*x^(1/3) + a*x])/(9*a) - (5525*b^(27/4)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[(b
+ a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x^(1/3))^2]*x^(1/6)*EllipticF[2*ArcTan[(a^(1/4)*
x^(1/6))/b^(1/4)], 1/2])/(14421*a^(29/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi in Sympy [A]  time = 81.7957, size = 292, normalized size = 0.96 \[ \frac{2 x^{4} \sqrt{a x + b \sqrt [3]{x}}}{9 a} - \frac{50 b x^{\frac{10}{3}} \sqrt{a x + b \sqrt [3]{x}}}{207 a^{2}} + \frac{350 b^{2} x^{\frac{8}{3}} \sqrt{a x + b \sqrt [3]{x}}}{1311 a^{3}} - \frac{1190 b^{3} x^{2} \sqrt{a x + b \sqrt [3]{x}}}{3933 a^{4}} + \frac{15470 b^{4} x^{\frac{4}{3}} \sqrt{a x + b \sqrt [3]{x}}}{43263 a^{5}} - \frac{2210 b^{5} x^{\frac{2}{3}} \sqrt{a x + b \sqrt [3]{x}}}{4807 a^{6}} + \frac{11050 b^{6} \sqrt{a x + b \sqrt [3]{x}}}{14421 a^{7}} - \frac{5525 b^{\frac{27}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{14421 a^{\frac{29}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**(1/3)+a*x)**(1/2),x)

[Out]

2*x**4*sqrt(a*x + b*x**(1/3))/(9*a) - 50*b*x**(10/3)*sqrt(a*x + b*x**(1/3))/(207
*a**2) + 350*b**2*x**(8/3)*sqrt(a*x + b*x**(1/3))/(1311*a**3) - 1190*b**3*x**2*s
qrt(a*x + b*x**(1/3))/(3933*a**4) + 15470*b**4*x**(4/3)*sqrt(a*x + b*x**(1/3))/(
43263*a**5) - 2210*b**5*x**(2/3)*sqrt(a*x + b*x**(1/3))/(4807*a**6) + 11050*b**6
*sqrt(a*x + b*x**(1/3))/(14421*a**7) - 5525*b**(27/4)*sqrt((a*x**(2/3) + b)/(sqr
t(a)*x**(1/3) + sqrt(b))**2)*(sqrt(a)*x**(1/3) + sqrt(b))*sqrt(a*x + b*x**(1/3))
*elliptic_f(2*atan(a**(1/4)*x**(1/6)/b**(1/4)), 1/2)/(14421*a**(29/4)*x**(1/6)*(
a*x**(2/3) + b))

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Mathematica [C]  time = 0.11924, size = 155, normalized size = 0.51 \[ \frac{2 \sqrt [3]{x} \left (4807 a^7 x^{14/3}-418 a^6 b x^4+550 a^5 b^2 x^{10/3}-770 a^4 b^3 x^{8/3}+1190 a^3 b^4 x^2-2210 a^2 b^5 x^{4/3}+16575 b^7 \sqrt{\frac{b}{a x^{2/3}}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b}{a x^{2/3}}\right )+6630 a b^6 x^{2/3}+16575 b^7\right )}{43263 a^7 \sqrt{a x+b \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/Sqrt[b*x^(1/3) + a*x],x]

[Out]

(2*x^(1/3)*(16575*b^7 + 6630*a*b^6*x^(2/3) - 2210*a^2*b^5*x^(4/3) + 1190*a^3*b^4
*x^2 - 770*a^4*b^3*x^(8/3) + 550*a^5*b^2*x^(10/3) - 418*a^6*b*x^4 + 4807*a^7*x^(
14/3) + 16575*b^7*Sqrt[1 + b/(a*x^(2/3))]*Hypergeometric2F1[1/4, 1/2, 5/4, -(b/(
a*x^(2/3)))]))/(43263*a^7*Sqrt[b*x^(1/3) + a*x])

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Maple [A]  time = 0.054, size = 196, normalized size = 0.6 \[ -{\frac{1}{43263\,{a}^{8}} \left ( -1100\,{x}^{11/3}{a}^{6}{b}^{2}+836\,{x}^{13/3}{a}^{7}b+1540\,{a}^{5}{b}^{3}{x}^{3}+4420\,{x}^{5/3}{a}^{3}{b}^{5}-2380\,{x}^{7/3}{a}^{4}{b}^{4}-9614\,{a}^{8}{x}^{5}+16575\,{b}^{7}\sqrt{-ab}\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) -13260\,{a}^{2}{b}^{6}x-33150\,\sqrt [3]{x}a{b}^{7} \right ){\frac{1}{\sqrt{\sqrt [3]{x} \left ( b+a{x}^{{\frac{2}{3}}} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^(1/3)+a*x)^(1/2),x)

[Out]

-1/43263*(-1100*x^(11/3)*a^6*b^2+836*x^(13/3)*a^7*b+1540*a^5*b^3*x^3+4420*x^(5/3
)*a^3*b^5-2380*x^(7/3)*a^4*b^4-9614*a^8*x^5+16575*b^7*(-a*b)^(1/2)*((a*x^(1/3)+(
-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-2*(a*x^(1/3)-(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2
)*(-x^(1/3)/(-a*b)^(1/2)*a)^(1/2)*EllipticF(((a*x^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/
2))^(1/2),1/2*2^(1/2))-13260*a^2*b^6*x-33150*x^(1/3)*a*b^7)/(x^(1/3)*(b+a*x^(2/3
)))^(1/2)/a^8

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{a x + b x^{\frac{1}{3}}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(a*x + b*x^(1/3)),x, algorithm="maxima")

[Out]

integrate(x^4/sqrt(a*x + b*x^(1/3)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{4}}{\sqrt{a x + b x^{\frac{1}{3}}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(a*x + b*x^(1/3)),x, algorithm="fricas")

[Out]

integral(x^4/sqrt(a*x + b*x^(1/3)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**(1/3)+a*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{a x + b x^{\frac{1}{3}}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(a*x + b*x^(1/3)),x, algorithm="giac")

[Out]

integrate(x^4/sqrt(a*x + b*x^(1/3)), x)