3.148 \(\int \frac{\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=301 \[ -\frac{884 a^{27/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{100947 b^{21/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{1768 a^6 \sqrt{a x+b \sqrt [3]{x}}}{100947 b^5 x^{2/3}}+\frac{1768 a^5 \sqrt{a x+b \sqrt [3]{x}}}{168245 b^4 x^{4/3}}-\frac{1768 a^4 \sqrt{a x+b \sqrt [3]{x}}}{216315 b^3 x^2}+\frac{136 a^3 \sqrt{a x+b \sqrt [3]{x}}}{19665 b^2 x^{8/3}}-\frac{8 a^2 \sqrt{a x+b \sqrt [3]{x}}}{1311 b x^{10/3}}-\frac{2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{9 x^5}-\frac{4 a \sqrt{a x+b \sqrt [3]{x}}}{69 x^4} \]

[Out]

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

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Rubi [A]  time = 0.854703, antiderivative size = 301, 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 \[ -\frac{884 a^{27/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{100947 b^{21/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{1768 a^6 \sqrt{a x+b \sqrt [3]{x}}}{100947 b^5 x^{2/3}}+\frac{1768 a^5 \sqrt{a x+b \sqrt [3]{x}}}{168245 b^4 x^{4/3}}-\frac{1768 a^4 \sqrt{a x+b \sqrt [3]{x}}}{216315 b^3 x^2}+\frac{136 a^3 \sqrt{a x+b \sqrt [3]{x}}}{19665 b^2 x^{8/3}}-\frac{8 a^2 \sqrt{a x+b \sqrt [3]{x}}}{1311 b x^{10/3}}-\frac{2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{9 x^5}-\frac{4 a \sqrt{a x+b \sqrt [3]{x}}}{69 x^4} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^(1/3) + a*x)^(3/2)/x^6,x]

[Out]

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

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Rubi in Sympy [A]  time = 79.4054, size = 289, normalized size = 0.96 \[ - \frac{884 a^{\frac{27}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{100947 b^{\frac{21}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} - \frac{1768 a^{6} \sqrt{a x + b \sqrt [3]{x}}}{100947 b^{5} x^{\frac{2}{3}}} + \frac{1768 a^{5} \sqrt{a x + b \sqrt [3]{x}}}{168245 b^{4} x^{\frac{4}{3}}} - \frac{1768 a^{4} \sqrt{a x + b \sqrt [3]{x}}}{216315 b^{3} x^{2}} + \frac{136 a^{3} \sqrt{a x + b \sqrt [3]{x}}}{19665 b^{2} x^{\frac{8}{3}}} - \frac{8 a^{2} \sqrt{a x + b \sqrt [3]{x}}}{1311 b x^{\frac{10}{3}}} - \frac{4 a \sqrt{a x + b \sqrt [3]{x}}}{69 x^{4}} - \frac{2 \left (a x + b \sqrt [3]{x}\right )^{\frac{3}{2}}}{9 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**(1/3)+a*x)**(3/2)/x**6,x)

[Out]

-884*a**(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*b**(21/4)*x**(1/6)*(a*x**(2/3) + b)) - 1768*a**6*sqrt(a*x
+ b*x**(1/3))/(100947*b**5*x**(2/3)) + 1768*a**5*sqrt(a*x + b*x**(1/3))/(168245*
b**4*x**(4/3)) - 1768*a**4*sqrt(a*x + b*x**(1/3))/(216315*b**3*x**2) + 136*a**3*
sqrt(a*x + b*x**(1/3))/(19665*b**2*x**(8/3)) - 8*a**2*sqrt(a*x + b*x**(1/3))/(13
11*b*x**(10/3)) - 4*a*sqrt(a*x + b*x**(1/3))/(69*x**4) - 2*(a*x + b*x**(1/3))**(
3/2)/(9*x**5)

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Mathematica [C]  time = 0.113067, size = 160, normalized size = 0.53 \[ -\frac{2 \left (-13260 a^7 x^{14/3} \sqrt{\frac{b}{a x^{2/3}}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b}{a x^{2/3}}\right )+13260 a^7 x^{14/3}+5304 a^6 b x^4-1768 a^5 b^2 x^{10/3}+952 a^4 b^3 x^{8/3}-616 a^3 b^4 x^2+216755 a^2 b^5 x^{4/3}+380380 a b^6 x^{2/3}+168245 b^7\right )}{1514205 b^5 x^{13/3} \sqrt{a x+b \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^(1/3) + a*x)^(3/2)/x^6,x]

[Out]

(-2*(168245*b^7 + 380380*a*b^6*x^(2/3) + 216755*a^2*b^5*x^(4/3) - 616*a^3*b^4*x^
2 + 952*a^4*b^3*x^(8/3) - 1768*a^5*b^2*x^(10/3) + 5304*a^6*b*x^4 + 13260*a^7*x^(
14/3) - 13260*a^7*Sqrt[1 + b/(a*x^(2/3))]*x^(14/3)*Hypergeometric2F1[1/4, 1/2, 5
/4, -(b/(a*x^(2/3)))]))/(1514205*b^5*x^(13/3)*Sqrt[b*x^(1/3) + a*x])

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Maple [A]  time = 0.044, size = 201, normalized size = 0.7 \[ -{\frac{2}{1514205\,{b}^{5}} \left ( 6630\,{a}^{6}\sqrt{-ab}\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ){x}^{{\frac{26}{3}}}-1768\,{x}^{{\frac{23}{3}}}{a}^{5}{b}^{2}+5304\,{x}^{{\frac{25}{3}}}{a}^{6}b+952\,{x}^{7}{a}^{4}{b}^{3}+216755\,{x}^{{\frac{17}{3}}}{a}^{2}{b}^{5}-616\,{x}^{{\frac{19}{3}}}{a}^{3}{b}^{4}+380380\,{x}^{5}a{b}^{6}+13260\,{x}^{9}{a}^{7}+168245\,{x}^{13/3}{b}^{7} \right ){\frac{1}{\sqrt{\sqrt [3]{x} \left ( b+a{x}^{{\frac{2}{3}}} \right ) }}}{x}^{-{\frac{26}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^(1/3)+a*x)^(3/2)/x^6,x)

[Out]

-2/1514205*(6630*a^6*(-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))*x^(26/3)-1
768*x^(23/3)*a^5*b^2+5304*x^(25/3)*a^6*b+952*x^7*a^4*b^3+216755*x^(17/3)*a^2*b^5
-616*x^(19/3)*a^3*b^4+380380*x^5*a*b^6+13260*x^9*a^7+168245*x^(13/3)*b^7)/b^5/(x
^(1/3)*(b+a*x^(2/3)))^(1/2)/x^(26/3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}}}{x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + b*x^(1/3))^(3/2)/x^6,x, algorithm="maxima")

[Out]

integrate((a*x + b*x^(1/3))^(3/2)/x^6, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}}}{x^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + b*x^(1/3))^(3/2)/x^6,x, algorithm="fricas")

[Out]

integral((a*x + b*x^(1/3))^(3/2)/x^6, 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((b*x**(1/3)+a*x)**(3/2)/x**6,x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + b*x^(1/3))^(3/2)/x^6,x, algorithm="giac")

[Out]

Exception raised: RuntimeError