3.147 \(\int \frac{\left (b \sqrt [3]{x}+a x\right )^{3/2}}{x^5} \, dx\)

Optimal. Leaf size=438 \[ -\frac{44 a^{21/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1105 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{88 a^{21/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1105 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{88 a^{11/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{1105 b^4 \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}+\frac{88 a^5 \sqrt{a x+b \sqrt [3]{x}}}{1105 b^4 \sqrt [3]{x}}-\frac{88 a^4 \sqrt{a x+b \sqrt [3]{x}}}{3315 b^3 x}+\frac{88 a^3 \sqrt{a x+b \sqrt [3]{x}}}{4641 b^2 x^{5/3}}-\frac{24 a^2 \sqrt{a x+b \sqrt [3]{x}}}{1547 b x^{7/3}}-\frac{2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{7 x^4}-\frac{12 a \sqrt{a x+b \sqrt [3]{x}}}{119 x^3} \]

[Out]

(-88*a^(11/2)*(b + a*x^(2/3))*x^(1/3))/(1105*b^4*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqr
t[b*x^(1/3) + a*x]) - (12*a*Sqrt[b*x^(1/3) + a*x])/(119*x^3) - (24*a^2*Sqrt[b*x^
(1/3) + a*x])/(1547*b*x^(7/3)) + (88*a^3*Sqrt[b*x^(1/3) + a*x])/(4641*b^2*x^(5/3
)) - (88*a^4*Sqrt[b*x^(1/3) + a*x])/(3315*b^3*x) + (88*a^5*Sqrt[b*x^(1/3) + a*x]
)/(1105*b^4*x^(1/3)) - (2*(b*x^(1/3) + a*x)^(3/2))/(7*x^4) + (88*a^(21/4)*(Sqrt[
b] + Sqrt[a]*x^(1/3))*Sqrt[(b + a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x^(1/3))^2]*x^(1/6
)*EllipticE[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(1105*b^(15/4)*Sqrt[b*x^(
1/3) + a*x]) - (44*a^(21/4)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[(b + a*x^(2/3))/(Sq
rt[b] + Sqrt[a]*x^(1/3))^2]*x^(1/6)*EllipticF[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)
], 1/2])/(1105*b^(15/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi [A]  time = 1.15219, antiderivative size = 438, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421 \[ -\frac{44 a^{21/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1105 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}+\frac{88 a^{21/4} \sqrt [6]{x} \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{\frac{a x^{2/3}+b}{\left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{1105 b^{15/4} \sqrt{a x+b \sqrt [3]{x}}}-\frac{88 a^{11/2} \sqrt [3]{x} \left (a x^{2/3}+b\right )}{1105 b^4 \left (\sqrt{a} \sqrt [3]{x}+\sqrt{b}\right ) \sqrt{a x+b \sqrt [3]{x}}}+\frac{88 a^5 \sqrt{a x+b \sqrt [3]{x}}}{1105 b^4 \sqrt [3]{x}}-\frac{88 a^4 \sqrt{a x+b \sqrt [3]{x}}}{3315 b^3 x}+\frac{88 a^3 \sqrt{a x+b \sqrt [3]{x}}}{4641 b^2 x^{5/3}}-\frac{24 a^2 \sqrt{a x+b \sqrt [3]{x}}}{1547 b x^{7/3}}-\frac{2 \left (a x+b \sqrt [3]{x}\right )^{3/2}}{7 x^4}-\frac{12 a \sqrt{a x+b \sqrt [3]{x}}}{119 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^(1/3) + a*x)^(3/2)/x^5,x]

[Out]

(-88*a^(11/2)*(b + a*x^(2/3))*x^(1/3))/(1105*b^4*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqr
t[b*x^(1/3) + a*x]) - (12*a*Sqrt[b*x^(1/3) + a*x])/(119*x^3) - (24*a^2*Sqrt[b*x^
(1/3) + a*x])/(1547*b*x^(7/3)) + (88*a^3*Sqrt[b*x^(1/3) + a*x])/(4641*b^2*x^(5/3
)) - (88*a^4*Sqrt[b*x^(1/3) + a*x])/(3315*b^3*x) + (88*a^5*Sqrt[b*x^(1/3) + a*x]
)/(1105*b^4*x^(1/3)) - (2*(b*x^(1/3) + a*x)^(3/2))/(7*x^4) + (88*a^(21/4)*(Sqrt[
b] + Sqrt[a]*x^(1/3))*Sqrt[(b + a*x^(2/3))/(Sqrt[b] + Sqrt[a]*x^(1/3))^2]*x^(1/6
)*EllipticE[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)], 1/2])/(1105*b^(15/4)*Sqrt[b*x^(
1/3) + a*x]) - (44*a^(21/4)*(Sqrt[b] + Sqrt[a]*x^(1/3))*Sqrt[(b + a*x^(2/3))/(Sq
rt[b] + Sqrt[a]*x^(1/3))^2]*x^(1/6)*EllipticF[2*ArcTan[(a^(1/4)*x^(1/6))/b^(1/4)
], 1/2])/(1105*b^(15/4)*Sqrt[b*x^(1/3) + a*x])

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Rubi in Sympy [A]  time = 109.694, size = 406, normalized size = 0.93 \[ \frac{88 a^{\frac{21}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{1105 b^{\frac{15}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} - \frac{44 a^{\frac{21}{4}} \sqrt{\frac{a x^{\frac{2}{3}} + b}{\left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )^{2}}} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right ) \sqrt{a x + b \sqrt [3]{x}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{1105 b^{\frac{15}{4}} \sqrt [6]{x} \left (a x^{\frac{2}{3}} + b\right )} - \frac{88 a^{\frac{11}{2}} \sqrt{a x + b \sqrt [3]{x}}}{1105 b^{4} \left (\sqrt{a} \sqrt [3]{x} + \sqrt{b}\right )} + \frac{88 a^{5} \sqrt{a x + b \sqrt [3]{x}}}{1105 b^{4} \sqrt [3]{x}} - \frac{88 a^{4} \sqrt{a x + b \sqrt [3]{x}}}{3315 b^{3} x} + \frac{88 a^{3} \sqrt{a x + b \sqrt [3]{x}}}{4641 b^{2} x^{\frac{5}{3}}} - \frac{24 a^{2} \sqrt{a x + b \sqrt [3]{x}}}{1547 b x^{\frac{7}{3}}} - \frac{12 a \sqrt{a x + b \sqrt [3]{x}}}{119 x^{3}} - \frac{2 \left (a x + b \sqrt [3]{x}\right )^{\frac{3}{2}}}{7 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**(1/3)+a*x)**(3/2)/x**5,x)

[Out]

88*a**(21/4)*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_e(2*atan(a**(1/4)*x**(1/6)/b**(
1/4)), 1/2)/(1105*b**(15/4)*x**(1/6)*(a*x**(2/3) + b)) - 44*a**(21/4)*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)/(1105*b**(1
5/4)*x**(1/6)*(a*x**(2/3) + b)) - 88*a**(11/2)*sqrt(a*x + b*x**(1/3))/(1105*b**4
*(sqrt(a)*x**(1/3) + sqrt(b))) + 88*a**5*sqrt(a*x + b*x**(1/3))/(1105*b**4*x**(1
/3)) - 88*a**4*sqrt(a*x + b*x**(1/3))/(3315*b**3*x) + 88*a**3*sqrt(a*x + b*x**(1
/3))/(4641*b**2*x**(5/3)) - 24*a**2*sqrt(a*x + b*x**(1/3))/(1547*b*x**(7/3)) - 1
2*a*sqrt(a*x + b*x**(1/3))/(119*x**3) - 2*(a*x + b*x**(1/3))**(3/2)/(7*x**4)

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Mathematica [C]  time = 0.0984591, size = 145, normalized size = 0.33 \[ -\frac{2 \left (924 a^6 x^4 \sqrt{\frac{b}{a x^{2/3}}+1} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-\frac{b}{a x^{2/3}}\right )-924 a^6 x^4-616 a^5 b x^{10/3}+88 a^4 b^2 x^{8/3}-40 a^3 b^3 x^2+4665 a^2 b^4 x^{4/3}+7800 a b^5 x^{2/3}+3315 b^6\right )}{23205 b^4 x^{10/3} \sqrt{a x+b \sqrt [3]{x}}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^(1/3) + a*x)^(3/2)/x^5,x]

[Out]

(-2*(3315*b^6 + 7800*a*b^5*x^(2/3) + 4665*a^2*b^4*x^(4/3) - 40*a^3*b^3*x^2 + 88*
a^4*b^2*x^(8/3) - 616*a^5*b*x^(10/3) - 924*a^6*x^4 + 924*a^6*Sqrt[1 + b/(a*x^(2/
3))]*x^4*Hypergeometric2F1[-1/4, 1/2, 3/4, -(b/(a*x^(2/3)))]))/(23205*b^4*x^(10/
3)*Sqrt[b*x^(1/3) + a*x])

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Maple [A]  time = 0.043, size = 411, normalized size = 0.9 \[ -{\frac{2}{23205\,{b}^{4}{x}^{7}} \left ( 924\,{a}^{5}b\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{x}^{{\frac{20}{3}}}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{\it EllipticE} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) -462\,{a}^{5}b\sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-2\,{\frac{a\sqrt [3]{x}-\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{a\sqrt [3]{x}}{\sqrt{-ab}}}}{x}^{{\frac{20}{3}}}\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{\it EllipticF} \left ( \sqrt{{\frac{a\sqrt [3]{x}+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) -924\,\sqrt{b\sqrt [3]{x}+ax}{x}^{{\frac{22}{3}}}{a}^{6}-924\,\sqrt{b\sqrt [3]{x}+ax}{x}^{{\frac{20}{3}}}{a}^{5}b+88\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{6}{a}^{4}{b}^{2}+308\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{{\frac{20}{3}}}{a}^{5}b+4665\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{14/3}{a}^{2}{b}^{4}-40\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{16/3}{a}^{3}{b}^{3}+7800\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{4}a{b}^{5}+3315\,\sqrt{\sqrt [3]{x} \left ( b+a{x}^{2/3} \right ) }{x}^{10/3}{b}^{6} \right ) \left ( b+a{x}^{{\frac{2}{3}}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^(1/3)+a*x)^(3/2)/x^5,x)

[Out]

-2/23205*(924*a^5*b*((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)*x^(20/3)*(x^(
1/3)*(b+a*x^(2/3)))^(1/2)*EllipticE(((a*x^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2
),1/2*2^(1/2))-462*a^5*b*((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)*x^(20/3)
*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*EllipticF(((a*x^(1/3)+(-a*b)^(1/2))/(-a*b)^(1/2))
^(1/2),1/2*2^(1/2))-924*(b*x^(1/3)+a*x)^(1/2)*x^(22/3)*a^6-924*(b*x^(1/3)+a*x)^(
1/2)*x^(20/3)*a^5*b+88*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*x^6*a^4*b^2+308*(x^(1/3)*(b
+a*x^(2/3)))^(1/2)*x^(20/3)*a^5*b+4665*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*x^(14/3)*a^
2*b^4-40*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*x^(16/3)*a^3*b^3+7800*(x^(1/3)*(b+a*x^(2/
3)))^(1/2)*x^4*a*b^5+3315*(x^(1/3)*(b+a*x^(2/3)))^(1/2)*x^(10/3)*b^6)/b^4/x^7/(b
+a*x^(2/3))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}}}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + b*x^(1/3))^(3/2)/x^5,x, algorithm="maxima")

[Out]

integrate((a*x + b*x^(1/3))^(3/2)/x^5, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (a x + b x^{\frac{1}{3}}\right )}^{\frac{3}{2}}}{x^{5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x + b*x^(1/3))^(3/2)/x^5,x, algorithm="fricas")

[Out]

integral((a*x + b*x^(1/3))^(3/2)/x^5, 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**5,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^5,x, algorithm="giac")

[Out]

Exception raised: RuntimeError