3.175 \(\int \frac{\sqrt{b x^{2/3}+a x}}{x^5} \, dx\)

Optimal. Leaf size=354 \[ -\frac{12597 a^{11} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{262144 b^{21/2}}+\frac{12597 a^{10} \sqrt{a x+b x^{2/3}}}{262144 b^{10} x^{2/3}}-\frac{4199 a^9 \sqrt{a x+b x^{2/3}}}{131072 b^9 x}+\frac{4199 a^8 \sqrt{a x+b x^{2/3}}}{163840 b^8 x^{4/3}}-\frac{12597 a^7 \sqrt{a x+b x^{2/3}}}{573440 b^7 x^{5/3}}+\frac{4199 a^6 \sqrt{a x+b x^{2/3}}}{215040 b^6 x^2}-\frac{4199 a^5 \sqrt{a x+b x^{2/3}}}{236544 b^5 x^{7/3}}+\frac{323 a^4 \sqrt{a x+b x^{2/3}}}{19712 b^4 x^{8/3}}-\frac{323 a^3 \sqrt{a x+b x^{2/3}}}{21120 b^3 x^3}+\frac{19 a^2 \sqrt{a x+b x^{2/3}}}{1320 b^2 x^{10/3}}-\frac{3 a \sqrt{a x+b x^{2/3}}}{220 b x^{11/3}}-\frac{3 \sqrt{a x+b x^{2/3}}}{11 x^4} \]

[Out]

(-3*Sqrt[b*x^(2/3) + a*x])/(11*x^4) - (3*a*Sqrt[b*x^(2/3) + a*x])/(220*b*x^(11/3
)) + (19*a^2*Sqrt[b*x^(2/3) + a*x])/(1320*b^2*x^(10/3)) - (323*a^3*Sqrt[b*x^(2/3
) + a*x])/(21120*b^3*x^3) + (323*a^4*Sqrt[b*x^(2/3) + a*x])/(19712*b^4*x^(8/3))
- (4199*a^5*Sqrt[b*x^(2/3) + a*x])/(236544*b^5*x^(7/3)) + (4199*a^6*Sqrt[b*x^(2/
3) + a*x])/(215040*b^6*x^2) - (12597*a^7*Sqrt[b*x^(2/3) + a*x])/(573440*b^7*x^(5
/3)) + (4199*a^8*Sqrt[b*x^(2/3) + a*x])/(163840*b^8*x^(4/3)) - (4199*a^9*Sqrt[b*
x^(2/3) + a*x])/(131072*b^9*x) + (12597*a^10*Sqrt[b*x^(2/3) + a*x])/(262144*b^10
*x^(2/3)) - (12597*a^11*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(26214
4*b^(21/2))

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Rubi [A]  time = 1.13445, antiderivative size = 354, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{12597 a^{11} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{262144 b^{21/2}}+\frac{12597 a^{10} \sqrt{a x+b x^{2/3}}}{262144 b^{10} x^{2/3}}-\frac{4199 a^9 \sqrt{a x+b x^{2/3}}}{131072 b^9 x}+\frac{4199 a^8 \sqrt{a x+b x^{2/3}}}{163840 b^8 x^{4/3}}-\frac{12597 a^7 \sqrt{a x+b x^{2/3}}}{573440 b^7 x^{5/3}}+\frac{4199 a^6 \sqrt{a x+b x^{2/3}}}{215040 b^6 x^2}-\frac{4199 a^5 \sqrt{a x+b x^{2/3}}}{236544 b^5 x^{7/3}}+\frac{323 a^4 \sqrt{a x+b x^{2/3}}}{19712 b^4 x^{8/3}}-\frac{323 a^3 \sqrt{a x+b x^{2/3}}}{21120 b^3 x^3}+\frac{19 a^2 \sqrt{a x+b x^{2/3}}}{1320 b^2 x^{10/3}}-\frac{3 a \sqrt{a x+b x^{2/3}}}{220 b x^{11/3}}-\frac{3 \sqrt{a x+b x^{2/3}}}{11 x^4} \]

Antiderivative was successfully verified.

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

[Out]

(-3*Sqrt[b*x^(2/3) + a*x])/(11*x^4) - (3*a*Sqrt[b*x^(2/3) + a*x])/(220*b*x^(11/3
)) + (19*a^2*Sqrt[b*x^(2/3) + a*x])/(1320*b^2*x^(10/3)) - (323*a^3*Sqrt[b*x^(2/3
) + a*x])/(21120*b^3*x^3) + (323*a^4*Sqrt[b*x^(2/3) + a*x])/(19712*b^4*x^(8/3))
- (4199*a^5*Sqrt[b*x^(2/3) + a*x])/(236544*b^5*x^(7/3)) + (4199*a^6*Sqrt[b*x^(2/
3) + a*x])/(215040*b^6*x^2) - (12597*a^7*Sqrt[b*x^(2/3) + a*x])/(573440*b^7*x^(5
/3)) + (4199*a^8*Sqrt[b*x^(2/3) + a*x])/(163840*b^8*x^(4/3)) - (4199*a^9*Sqrt[b*
x^(2/3) + a*x])/(131072*b^9*x) + (12597*a^10*Sqrt[b*x^(2/3) + a*x])/(262144*b^10
*x^(2/3)) - (12597*a^11*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(26214
4*b^(21/2))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{4199 a^{11} \int \frac{1}{x^{\frac{2}{3}} \sqrt{a x + b x^{\frac{2}{3}}}}\, dx}{524288 b^{10}} + \frac{12597 a^{10} \sqrt{a x + b x^{\frac{2}{3}}}}{262144 b^{10} x^{\frac{2}{3}}} - \frac{4199 a^{9} \sqrt{a x + b x^{\frac{2}{3}}}}{131072 b^{9} x} + \frac{4199 a^{8} \sqrt{a x + b x^{\frac{2}{3}}}}{163840 b^{8} x^{\frac{4}{3}}} - \frac{12597 a^{7} \sqrt{a x + b x^{\frac{2}{3}}}}{573440 b^{7} x^{\frac{5}{3}}} + \frac{4199 a^{6} \sqrt{a x + b x^{\frac{2}{3}}}}{215040 b^{6} x^{2}} - \frac{4199 a^{5} \sqrt{a x + b x^{\frac{2}{3}}}}{236544 b^{5} x^{\frac{7}{3}}} + \frac{323 a^{4} \sqrt{a x + b x^{\frac{2}{3}}}}{19712 b^{4} x^{\frac{8}{3}}} - \frac{323 a^{3} \sqrt{a x + b x^{\frac{2}{3}}}}{21120 b^{3} x^{3}} + \frac{19 a^{2} \sqrt{a x + b x^{\frac{2}{3}}}}{1320 b^{2} x^{\frac{10}{3}}} - \frac{3 a \sqrt{a x + b x^{\frac{2}{3}}}}{220 b x^{\frac{11}{3}}} - \frac{3 \sqrt{a x + b x^{\frac{2}{3}}}}{11 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4199*a**11*Integral(1/(x**(2/3)*sqrt(a*x + b*x**(2/3))), x)/(524288*b**10) + 125
97*a**10*sqrt(a*x + b*x**(2/3))/(262144*b**10*x**(2/3)) - 4199*a**9*sqrt(a*x + b
*x**(2/3))/(131072*b**9*x) + 4199*a**8*sqrt(a*x + b*x**(2/3))/(163840*b**8*x**(4
/3)) - 12597*a**7*sqrt(a*x + b*x**(2/3))/(573440*b**7*x**(5/3)) + 4199*a**6*sqrt
(a*x + b*x**(2/3))/(215040*b**6*x**2) - 4199*a**5*sqrt(a*x + b*x**(2/3))/(236544
*b**5*x**(7/3)) + 323*a**4*sqrt(a*x + b*x**(2/3))/(19712*b**4*x**(8/3)) - 323*a*
*3*sqrt(a*x + b*x**(2/3))/(21120*b**3*x**3) + 19*a**2*sqrt(a*x + b*x**(2/3))/(13
20*b**2*x**(10/3)) - 3*a*sqrt(a*x + b*x**(2/3))/(220*b*x**(11/3)) - 3*sqrt(a*x +
b*x**(2/3))/(11*x**4)

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Mathematica [A]  time = 0.355985, size = 190, normalized size = 0.54 \[ \frac{\sqrt{b} \sqrt{a x+b x^{2/3}} \left (14549535 a^{10} x^{10/3}-9699690 a^9 b x^3+7759752 a^8 b^2 x^{8/3}-6651216 a^7 b^3 x^{7/3}+5912192 a^6 b^4 x^2-5374720 a^5 b^5 x^{5/3}+4961280 a^4 b^6 x^{4/3}-4630528 a^3 b^7 x+4358144 a^2 b^8 x^{2/3}-4128768 a b^9 \sqrt [3]{x}-82575360 b^{10}\right )-14549535 a^{11} x^4 \tanh ^{-1}\left (\frac{\sqrt{a x+b x^{2/3}}}{\sqrt{b} \sqrt [3]{x}}\right )}{302776320 b^{21/2} x^4} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*Sqrt[b*x^(2/3) + a*x]*(-82575360*b^10 - 4128768*a*b^9*x^(1/3) + 4358144
*a^2*b^8*x^(2/3) - 4630528*a^3*b^7*x + 4961280*a^4*b^6*x^(4/3) - 5374720*a^5*b^5
*x^(5/3) + 5912192*a^6*b^4*x^2 - 6651216*a^7*b^3*x^(7/3) + 7759752*a^8*b^2*x^(8/
3) - 9699690*a^9*b*x^3 + 14549535*a^10*x^(10/3)) - 14549535*a^11*x^4*ArcTanh[Sqr
t[b*x^(2/3) + a*x]/(Sqrt[b]*x^(1/3))])/(302776320*b^(21/2)*x^4)

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Maple [A]  time = 0.024, size = 209, normalized size = 0.6 \[{\frac{1}{302776320\,{x}^{4}}\sqrt{b{x}^{{\frac{2}{3}}}+ax} \left ( 14549535\, \left ( b+a\sqrt [3]{x} \right ) ^{21/2}{b}^{21/2}-155195040\, \left ( b+a\sqrt [3]{x} \right ) ^{19/2}{b}^{23/2}[phone]\, \left ( b+a\sqrt [3]{x} \right ) ^{17/2}{b}^{{\frac{25}{2}}}-2163862272\, \left ( b+a\sqrt [3]{x} \right ) ^{15/2}{b}^{{\frac{27}{2}}}[phone]\, \left ( b+a\sqrt [3]{x} \right ) ^{13/2}{b}^{{\frac{29}{2}}}-5503713280\, \left ( b+a\sqrt [3]{x} \right ) ^{11/2}{b}^{{\frac{31}{2}}}[phone]\, \left ( b+a\sqrt [3]{x} \right ) ^{9/2}{b}^{{\frac{33}{2}}}-3424523520\, \left ( b+a\sqrt [3]{x} \right ) ^{7/2}{b}^{{\frac{35}{2}}}[phone]\, \left ( b+a\sqrt [3]{x} \right ) ^{5/2}{b}^{{\frac{37}{2}}}-450357600\, \left ( b+a\sqrt [3]{x} \right ) ^{3/2}{b}^{{\frac{39}{2}}}-14549535\,{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ){b}^{10}{a}^{11}{x}^{11/3}-14549535\,\sqrt{b+a\sqrt [3]{x}}{b}^{{\frac{41}{2}}} \right ){\frac{1}{\sqrt{b+a\sqrt [3]{x}}}}{b}^{-{\frac{41}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/302776320*(b*x^(2/3)+a*x)^(1/2)*(14549535*(b+a*x^(1/3))^(21/2)*b^(21/2)-155195
040*(b+a*x^(1/3))^(19/2)*b^(23/2)[phone]*(b+a*x^(1/3))^(17/2)*b^(25/2)-216386
2272*(b+a*x^(1/3))^(15/2)*b^(27/2)[phone]*(b+a*x^(1/3))^(13/2)*b^(29/2)-5503
713280*(b+a*x^(1/3))^(11/2)*b^(31/2)[phone]*(b+a*x^(1/3))^(9/2)*b^(33/2)-342
4523520*(b+a*x^(1/3))^(7/2)*b^(35/2)[phone]*(b+a*x^(1/3))^(5/2)*b^(37/2)-450
357600*(b+a*x^(1/3))^(3/2)*b^(39/2)-14549535*arctanh((b+a*x^(1/3))^(1/2)/b^(1/2)
)*b^10*a^11*x^(11/3)-14549535*(b+a*x^(1/3))^(1/2)*b^(41/2))/x^4/(b+a*x^(1/3))^(1
/2)/b^(41/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.509795, size = 313, normalized size = 0.88 \[ \frac{{\left (\frac{14549535 \, a^{12} \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{10}} + \frac{14549535 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{21}{2}} a^{12} - 155195040 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} a^{12} b + 749786037 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} a^{12} b^{2} - 2163862272 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} a^{12} b^{3} + 4139920070 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} a^{12} b^{4} - 5503713280 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{12} b^{5} + 5174056250 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{12} b^{6} - 3424523520 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{12} b^{7} + 1551313995 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{12} b^{8} - 450357600 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{12} b^{9} - 14549535 \, \sqrt{a x^{\frac{1}{3}} + b} a^{12} b^{10}}{a^{11} b^{10} x^{\frac{11}{3}}}\right )}{\rm sign}\left (x^{\frac{1}{3}}\right )}{302776320 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/302776320*(14549535*a^12*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^10)
+ (14549535*(a*x^(1/3) + b)^(21/2)*a^12 - 155195040*(a*x^(1/3) + b)^(19/2)*a^12*
b + 749786037*(a*x^(1/3) + b)^(17/2)*a^12*b^2 - 2163862272*(a*x^(1/3) + b)^(15/2
)*a^12*b^3 + 4139920070*(a*x^(1/3) + b)^(13/2)*a^12*b^4 - 5503713280*(a*x^(1/3)
+ b)^(11/2)*a^12*b^5 + 5174056250*(a*x^(1/3) + b)^(9/2)*a^12*b^6 - 3424523520*(a
*x^(1/3) + b)^(7/2)*a^12*b^7 + 1551313995*(a*x^(1/3) + b)^(5/2)*a^12*b^8 - 45035
7600*(a*x^(1/3) + b)^(3/2)*a^12*b^9 - 14549535*sqrt(a*x^(1/3) + b)*a^12*b^10)/(a
^11*b^10*x^(11/3)))*sign(x^(1/3))/a