3.140 \(\int x^2 \left (b \sqrt [3]{x}+a x\right )^{3/2} \, dx\)

Optimal. Leaf size=298 \[ -\frac{884 b^{27/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{100947 a^{21/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{1768 b^6 \sqrt{a x+b \sqrt [3]{x}}}{100947 a^5}-\frac{1768 b^5 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{168245 a^4}+\frac{1768 b^4 x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{216315 a^3}-\frac{136 b^3 x^2 \sqrt{a x+b \sqrt [3]{x}}}{19665 a^2}+\frac{8 b^2 x^{8/3} \sqrt{a x+b \sqrt [3]{x}}}{1311 a}+\frac{4}{69} b x^{10/3} \sqrt{a x+b \sqrt [3]{x}}+\frac{2}{9} x^3 \left (a x+b \sqrt [3]{x}\right )^{3/2} \]

[Out]

(1768*b^6*Sqrt[b*x^(1/3) + a*x])/(100947*a^5) - (1768*b^5*x^(2/3)*Sqrt[b*x^(1/3)
+ a*x])/(168245*a^4) + (1768*b^4*x^(4/3)*Sqrt[b*x^(1/3) + a*x])/(216315*a^3) -
(136*b^3*x^2*Sqrt[b*x^(1/3) + a*x])/(19665*a^2) + (8*b^2*x^(8/3)*Sqrt[b*x^(1/3)
+ a*x])/(1311*a) + (4*b*x^(10/3)*Sqrt[b*x^(1/3) + a*x])/69 + (2*x^3*(b*x^(1/3) +
a*x)^(3/2))/9 - (884*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])/(100947*a^(21/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi [A]  time = 0.880356, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ -\frac{884 b^{27/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{100947 a^{21/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{1768 b^6 \sqrt{a x+b \sqrt [3]{x}}}{100947 a^5}-\frac{1768 b^5 x^{2/3} \sqrt{a x+b \sqrt [3]{x}}}{168245 a^4}+\frac{1768 b^4 x^{4/3} \sqrt{a x+b \sqrt [3]{x}}}{216315 a^3}-\frac{136 b^3 x^2 \sqrt{a x+b \sqrt [3]{x}}}{19665 a^2}+\frac{8 b^2 x^{8/3} \sqrt{a x+b \sqrt [3]{x}}}{1311 a}+\frac{4}{69} b x^{10/3} \sqrt{a x+b \sqrt [3]{x}}+\frac{2}{9} x^3 \left (a x+b \sqrt [3]{x}\right )^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

(1768*b^6*Sqrt[b*x^(1/3) + a*x])/(100947*a^5) - (1768*b^5*x^(2/3)*Sqrt[b*x^(1/3)
+ a*x])/(168245*a^4) + (1768*b^4*x^(4/3)*Sqrt[b*x^(1/3) + a*x])/(216315*a^3) -
(136*b^3*x^2*Sqrt[b*x^(1/3) + a*x])/(19665*a^2) + (8*b^2*x^(8/3)*Sqrt[b*x^(1/3)
+ a*x])/(1311*a) + (4*b*x^(10/3)*Sqrt[b*x^(1/3) + a*x])/69 + (2*x^3*(b*x^(1/3) +
a*x)^(3/2))/9 - (884*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])/(100947*a^(21/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi in Sympy [A]  time = 87.1221, size = 286, normalized size = 0.96 \[ \frac{4 b x^{\frac{10}{3}} \sqrt{a x + b \sqrt [3]{x}}}{69} + \frac{2 x^{3} \left (a x + b \sqrt [3]{x}\right )^{\frac{3}{2}}}{9} + \frac{8 b^{2} x^{\frac{8}{3}} \sqrt{a x + b \sqrt [3]{x}}}{1311 a} - \frac{136 b^{3} x^{2} \sqrt{a x + b \sqrt [3]{x}}}{19665 a^{2}} + \frac{1768 b^{4} x^{\frac{4}{3}} \sqrt{a x + b \sqrt [3]{x}}}{216315 a^{3}} - \frac{1768 b^{5} x^{\frac{2}{3}} \sqrt{a x + b \sqrt [3]{x}}}{168245 a^{4}} + \frac{1768 b^{6} \sqrt{a x + b \sqrt [3]{x}}}{100947 a^{5}} - \frac{884 b^{\frac{27}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{100947 a^{\frac{21}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*b*x**(10/3)*sqrt(a*x + b*x**(1/3))/69 + 2*x**3*(a*x + b*x**(1/3))**(3/2)/9 + 8
*b**2*x**(8/3)*sqrt(a*x + b*x**(1/3))/(1311*a) - 136*b**3*x**2*sqrt(a*x + b*x**(
1/3))/(19665*a**2) + 1768*b**4*x**(4/3)*sqrt(a*x + b*x**(1/3))/(216315*a**3) - 1
768*b**5*x**(2/3)*sqrt(a*x + b*x**(1/3))/(168245*a**4) + 1768*b**6*sqrt(a*x + b*
x**(1/3))/(100947*a**5) - 884*b**(27/4)*sqrt((a*x**(2/3) + b)/(sqrt(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)/(100947*a**(21/4)*x**(1/6)*(a*x**(2/3) +
b))

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Mathematica [C]  time = 0.112439, size = 155, normalized size = 0.52 \[ \frac{2 \sqrt [3]{x} \left (168245 a^7 x^{14/3}+380380 a^6 b x^4+216755 a^5 b^2 x^{10/3}-616 a^4 b^3 x^{8/3}+952 a^3 b^4 x^2-1768 a^2 b^5 x^{4/3}+13260 b^7 \sqrt{\frac{b}{a x^{2/3}}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b}{a x^{2/3}}\right )+5304 a b^6 x^{2/3}+13260 b^7\right )}{1514205 a^5 \sqrt{a x+b \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(1/3)*(13260*b^7 + 5304*a*b^6*x^(2/3) - 1768*a^2*b^5*x^(4/3) + 952*a^3*b^4*
x^2 - 616*a^4*b^3*x^(8/3) + 216755*a^5*b^2*x^(10/3) + 380380*a^6*b*x^4 + 168245*
a^7*x^(14/3) + 13260*b^7*Sqrt[1 + b/(a*x^(2/3))]*Hypergeometric2F1[1/4, 1/2, 5/4
, -(b/(a*x^(2/3)))]))/(1514205*a^5*Sqrt[b*x^(1/3) + a*x])

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Maple [A]  time = 0.042, size = 196, normalized size = 0.7 \[{\frac{2}{1514205\,{a}^{6}} \left ( 216755\,{x}^{11/3}{a}^{6}{b}^{2}+380380\,{x}^{13/3}{a}^{7}b-616\,{a}^{5}{b}^{3}{x}^{3}-1768\,{x}^{5/3}{a}^{3}{b}^{5}+952\,{x}^{7/3}{a}^{4}{b}^{4}+168245\,{a}^{8}{x}^{5}-6630\,{b}^{7}\sqrt{-ab}\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) +5304\,{a}^{2}{b}^{6}x+13260\,\sqrt [3]{x}a{b}^{7} \right ){\frac{1}{\sqrt{\sqrt [3]{x} \left ( b+a{x}^{{\frac{2}{3}}} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/1514205*(216755*x^(11/3)*a^6*b^2+380380*x^(13/3)*a^7*b-616*a^5*b^3*x^3-1768*x^
(5/3)*a^3*b^5+952*x^(7/3)*a^4*b^4+168245*a^8*x^5-6630*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))+5304*a^2*b^6*x+13260*x^(1/3)*a*b^7)/a^6/(x^(1/3)*(b+a
*x^(2/3)))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + b*x^(1/3))^(3/2)*x^2,x, algorithm="maxima")

[Out]

integrate((a*x + b*x^(1/3))^(3/2)*x^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (a x^{3} + b x^{\frac{7}{3}}\right )} \sqrt{a x + b x^{\frac{1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + b*x^(1/3))^(3/2)*x^2,x, algorithm="fricas")

[Out]

integral((a*x^3 + b*x^(7/3))*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**2*(b*x**(1/3)+a*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + b*x^(1/3))^(3/2)*x^2,x, algorithm="giac")

[Out]

integrate((a*x + b*x^(1/3))^(3/2)*x^2, x)