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

Optimal. Leaf size=379 \[ -\frac{12597 a^{12} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{2097152 b^{21/2}}+\frac{12597 a^{11} \sqrt{a x+b x^{2/3}}}{2097152 b^{10} x^{2/3}}-\frac{4199 a^{10} \sqrt{a x+b x^{2/3}}}{1048576 b^9 x}+\frac{4199 a^9 \sqrt{a x+b x^{2/3}}}{1310720 b^8 x^{4/3}}-\frac{12597 a^8 \sqrt{a x+b x^{2/3}}}{4587520 b^7 x^{5/3}}+\frac{4199 a^7 \sqrt{a x+b x^{2/3}}}{1720320 b^6 x^2}-\frac{4199 a^6 \sqrt{a x+b x^{2/3}}}{1892352 b^5 x^{7/3}}+\frac{323 a^5 \sqrt{a x+b x^{2/3}}}{157696 b^4 x^{8/3}}-\frac{323 a^4 \sqrt{a x+b x^{2/3}}}{168960 b^3 x^3}+\frac{19 a^3 \sqrt{a x+b x^{2/3}}}{10560 b^2 x^{10/3}}-\frac{3 a^2 \sqrt{a x+b x^{2/3}}}{1760 b x^{11/3}}-\frac{\left (a x+b x^{2/3}\right )^{3/2}}{4 x^5}-\frac{3 a \sqrt{a x+b x^{2/3}}}{88 x^4} \]

[Out]

(-3*a*Sqrt[b*x^(2/3) + a*x])/(88*x^4) - (3*a^2*Sqrt[b*x^(2/3) + a*x])/(1760*b*x^
(11/3)) + (19*a^3*Sqrt[b*x^(2/3) + a*x])/(10560*b^2*x^(10/3)) - (323*a^4*Sqrt[b*
x^(2/3) + a*x])/(168960*b^3*x^3) + (323*a^5*Sqrt[b*x^(2/3) + a*x])/(157696*b^4*x
^(8/3)) - (4199*a^6*Sqrt[b*x^(2/3) + a*x])/(1892352*b^5*x^(7/3)) + (4199*a^7*Sqr
t[b*x^(2/3) + a*x])/(1720320*b^6*x^2) - (12597*a^8*Sqrt[b*x^(2/3) + a*x])/(45875
20*b^7*x^(5/3)) + (4199*a^9*Sqrt[b*x^(2/3) + a*x])/(1310720*b^8*x^(4/3)) - (4199
*a^10*Sqrt[b*x^(2/3) + a*x])/(1048576*b^9*x) + (12597*a^11*Sqrt[b*x^(2/3) + a*x]
)/(2097152*b^10*x^(2/3)) - (b*x^(2/3) + a*x)^(3/2)/(4*x^5) - (12597*a^12*ArcTanh
[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(2097152*b^(21/2))

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Rubi [A]  time = 1.22452, antiderivative size = 379, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{12597 a^{12} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{2097152 b^{21/2}}+\frac{12597 a^{11} \sqrt{a x+b x^{2/3}}}{2097152 b^{10} x^{2/3}}-\frac{4199 a^{10} \sqrt{a x+b x^{2/3}}}{1048576 b^9 x}+\frac{4199 a^9 \sqrt{a x+b x^{2/3}}}{1310720 b^8 x^{4/3}}-\frac{12597 a^8 \sqrt{a x+b x^{2/3}}}{4587520 b^7 x^{5/3}}+\frac{4199 a^7 \sqrt{a x+b x^{2/3}}}{1720320 b^6 x^2}-\frac{4199 a^6 \sqrt{a x+b x^{2/3}}}{1892352 b^5 x^{7/3}}+\frac{323 a^5 \sqrt{a x+b x^{2/3}}}{157696 b^4 x^{8/3}}-\frac{323 a^4 \sqrt{a x+b x^{2/3}}}{168960 b^3 x^3}+\frac{19 a^3 \sqrt{a x+b x^{2/3}}}{10560 b^2 x^{10/3}}-\frac{3 a^2 \sqrt{a x+b x^{2/3}}}{1760 b x^{11/3}}-\frac{\left (a x+b x^{2/3}\right )^{3/2}}{4 x^5}-\frac{3 a \sqrt{a x+b x^{2/3}}}{88 x^4} \]

Antiderivative was successfully verified.

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

[Out]

(-3*a*Sqrt[b*x^(2/3) + a*x])/(88*x^4) - (3*a^2*Sqrt[b*x^(2/3) + a*x])/(1760*b*x^
(11/3)) + (19*a^3*Sqrt[b*x^(2/3) + a*x])/(10560*b^2*x^(10/3)) - (323*a^4*Sqrt[b*
x^(2/3) + a*x])/(168960*b^3*x^3) + (323*a^5*Sqrt[b*x^(2/3) + a*x])/(157696*b^4*x
^(8/3)) - (4199*a^6*Sqrt[b*x^(2/3) + a*x])/(1892352*b^5*x^(7/3)) + (4199*a^7*Sqr
t[b*x^(2/3) + a*x])/(1720320*b^6*x^2) - (12597*a^8*Sqrt[b*x^(2/3) + a*x])/(45875
20*b^7*x^(5/3)) + (4199*a^9*Sqrt[b*x^(2/3) + a*x])/(1310720*b^8*x^(4/3)) - (4199
*a^10*Sqrt[b*x^(2/3) + a*x])/(1048576*b^9*x) + (12597*a^11*Sqrt[b*x^(2/3) + a*x]
)/(2097152*b^10*x^(2/3)) - (b*x^(2/3) + a*x)^(3/2)/(4*x^5) - (12597*a^12*ArcTanh
[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(2097152*b^(21/2))

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{4199 a^{11} \int \frac{1}{x \sqrt{a x + b x^{\frac{2}{3}}}}\, dx}{2097152 b^{9}} - \frac{4199 a^{10} \sqrt{a x + b x^{\frac{2}{3}}}}{1048576 b^{9} x} + \frac{4199 a^{9} \sqrt{a x + b x^{\frac{2}{3}}}}{1310720 b^{8} x^{\frac{4}{3}}} - \frac{12597 a^{8} \sqrt{a x + b x^{\frac{2}{3}}}}{4587520 b^{7} x^{\frac{5}{3}}} + \frac{4199 a^{7} \sqrt{a x + b x^{\frac{2}{3}}}}{1720320 b^{6} x^{2}} - \frac{4199 a^{6} \sqrt{a x + b x^{\frac{2}{3}}}}{1892352 b^{5} x^{\frac{7}{3}}} + \frac{323 a^{5} \sqrt{a x + b x^{\frac{2}{3}}}}{157696 b^{4} x^{\frac{8}{3}}} - \frac{323 a^{4} \sqrt{a x + b x^{\frac{2}{3}}}}{168960 b^{3} x^{3}} + \frac{19 a^{3} \sqrt{a x + b x^{\frac{2}{3}}}}{10560 b^{2} x^{\frac{10}{3}}} - \frac{3 a^{2} \sqrt{a x + b x^{\frac{2}{3}}}}{1760 b x^{\frac{11}{3}}} - \frac{3 a \sqrt{a x + b x^{\frac{2}{3}}}}{88 x^{4}} - \frac{\left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}{4 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4199*a**11*Integral(1/(x*sqrt(a*x + b*x**(2/3))), x)/(2097152*b**9) - 4199*a**1
0*sqrt(a*x + b*x**(2/3))/(1048576*b**9*x) + 4199*a**9*sqrt(a*x + b*x**(2/3))/(13
10720*b**8*x**(4/3)) - 12597*a**8*sqrt(a*x + b*x**(2/3))/(4587520*b**7*x**(5/3))
+ 4199*a**7*sqrt(a*x + b*x**(2/3))/(1720320*b**6*x**2) - 4199*a**6*sqrt(a*x + b
*x**(2/3))/(1892352*b**5*x**(7/3)) + 323*a**5*sqrt(a*x + b*x**(2/3))/(157696*b**
4*x**(8/3)) - 323*a**4*sqrt(a*x + b*x**(2/3))/(168960*b**3*x**3) + 19*a**3*sqrt(
a*x + b*x**(2/3))/(10560*b**2*x**(10/3)) - 3*a**2*sqrt(a*x + b*x**(2/3))/(1760*b
*x**(11/3)) - 3*a*sqrt(a*x + b*x**(2/3))/(88*x**4) - (a*x + b*x**(2/3))**(3/2)/(
4*x**5)

_______________________________________________________________________________________

Mathematica [A]  time = 0.475617, size = 201, normalized size = 0.53 \[ \frac{\sqrt{a x+b x^{2/3}} \left (14549535 a^{11} x^{11/3}-9699690 a^{10} b x^{10/3}+7759752 a^9 b^2 x^3-6651216 a^8 b^3 x^{8/3}+5912192 a^7 b^4 x^{7/3}-5374720 a^6 b^5 x^2+4961280 a^5 b^6 x^{5/3}-4630528 a^4 b^7 x^{4/3}+4358144 a^3 b^8 x-4128768 a^2 b^9 x^{2/3}-688128000 a b^{10} \sqrt [3]{x}-605552640 b^{11}\right )}{2422210560 b^{10} x^{13/3}}-\frac{12597 a^{12} \tanh ^{-1}\left (\frac{\sqrt{a x+b x^{2/3}}}{\sqrt{b} \sqrt [3]{x}}\right )}{2097152 b^{21/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b*x^(2/3) + a*x]*(-605552640*b^11 - 688128000*a*b^10*x^(1/3) - 4128768*a^2
*b^9*x^(2/3) + 4358144*a^3*b^8*x - 4630528*a^4*b^7*x^(4/3) + 4961280*a^5*b^6*x^(
5/3) - 5374720*a^6*b^5*x^2 + 5912192*a^7*b^4*x^(7/3) - 6651216*a^8*b^3*x^(8/3) +
7759752*a^9*b^2*x^3 - 9699690*a^10*b*x^(10/3) + 14549535*a^11*x^(11/3)))/(24222
10560*b^10*x^(13/3)) - (12597*a^12*ArcTanh[Sqrt[b*x^(2/3) + a*x]/(Sqrt[b]*x^(1/3
))])/(2097152*b^(21/2))

_______________________________________________________________________________________

Maple [A]  time = 0.028, size = 223, normalized size = 0.6 \[{\frac{1}{2422210560\,{x}^{5}} \left ( b{x}^{{\frac{2}{3}}}+ax \right ) ^{{\frac{3}{2}}} \left ( 14549535\, \left ( b+a\sqrt [3]{x} \right ) ^{23/2}{b}^{21/2}-169744575\, \left ( b+a\sqrt [3]{x} \right ) ^{21/2}{b}^{23/2}[phone]\, \left ( b+a\sqrt [3]{x} \right ) ^{19/2}{b}^{{\frac{25}{2}}}-2913648309\, \left ( b+a\sqrt [3]{x} \right ) ^{17/2}{b}^{{\frac{27}{2}}}[phone]\, \left ( b+a\sqrt [3]{x} \right ) ^{15/2}{b}^{{\frac{29}{2}}}-9643633350\, \left ( b+a\sqrt [3]{x} \right ) ^{13/2}{b}^{{\frac{31}{2}}}[phone]\, \left ( b+a\sqrt [3]{x} \right ) ^{11/2}{b}^{{\frac{33}{2}}}-8598579770\, \left ( b+a\sqrt [3]{x} \right ) ^{9/2}{b}^{{\frac{35}{2}}}[phone]\, \left ( b+a\sqrt [3]{x} \right ) ^{7/2}{b}^{{\frac{37}{2}}}-2001671595\, \left ( b+a\sqrt [3]{x} \right ) ^{5/2}{b}^{{\frac{39}{2}}}-169744575\, \left ( b+a\sqrt [3]{x} \right ) ^{3/2}{b}^{{\frac{41}{2}}}[phone]\,\sqrt{b+a\sqrt [3]{x}}{b}^{{\frac{43}{2}}}-14549535\,{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ){b}^{10}{a}^{12}{x}^{4} \right ) \left ( b+a\sqrt [3]{x} \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{41}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2422210560*(b*x^(2/3)+a*x)^(3/2)*(14549535*(b+a*x^(1/3))^(23/2)*b^(21/2)-16974
4575*(b+a*x^(1/3))^(21/2)*b^(23/2)[phone]*(b+a*x^(1/3))^(19/2)*b^(25/2)-29136
48309*(b+a*x^(1/3))^(17/2)*b^(27/2)[phone]*(b+a*x^(1/3))^(15/2)*b^(29/2)-964
3633350*(b+a*x^(1/3))^(13/2)*b^(31/2)[phone]*(b+a*x^(1/3))^(11/2)*b^(33/2)-
8598579770*(b+a*x^(1/3))^(9/2)*b^(35/2)[phone]*(b+a*x^(1/3))^(7/2)*b^(37/2)-
2001671595*(b+a*x^(1/3))^(5/2)*b^(39/2)-169744575*(b+a*x^(1/3))^(3/2)*b^(41/2)+1
4549535*(b+a*x^(1/3))^(1/2)*b^(43/2)-14549535*arctanh((b+a*x^(1/3))^(1/2)/b^(1/2
))*b^10*a^12*x^4)/x^5/(b+a*x^(1/3))^(3/2)/b^(41/2)

_______________________________________________________________________________________

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((a*x + b*x^(2/3))^(3/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

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((a*x + b*x^(2/3))^(3/2)/x^6,x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

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)**(3/2)/x**6,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.542489, size = 401, normalized size = 1.06 \[ \frac{\frac{14549535 \, a^{13} \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right ){\rm sign}\left (x^{\frac{1}{3}}\right )}{\sqrt{-b} b^{10}} + \frac{14549535 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{23}{2}} a^{13}{\rm sign}\left (x^{\frac{1}{3}}\right ) - 169744575 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{21}{2}} a^{13} b{\rm sign}\left (x^{\frac{1}{3}}\right ) + 904981077 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} a^{13} b^{2}{\rm sign}\left (x^{\frac{1}{3}}\right ) - 2913648309 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} a^{13} b^{3}{\rm sign}\left (x^{\frac{1}{3}}\right ) + 6303782342 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} a^{13} b^{4}{\rm sign}\left (x^{\frac{1}{3}}\right ) - 9643633350 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} a^{13} b^{5}{\rm sign}\left (x^{\frac{1}{3}}\right ) + 10677769530 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{13} b^{6}{\rm sign}\left (x^{\frac{1}{3}}\right ) - 8598579770 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{13} b^{7}{\rm sign}\left (x^{\frac{1}{3}}\right ) + 4975837515 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{13} b^{8}{\rm sign}\left (x^{\frac{1}{3}}\right ) - 2001671595 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{13} b^{9}{\rm sign}\left (x^{\frac{1}{3}}\right ) - 169744575 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{13} b^{10}{\rm sign}\left (x^{\frac{1}{3}}\right ) + 14549535 \, \sqrt{a x^{\frac{1}{3}} + b} a^{13} b^{11}{\rm sign}\left (x^{\frac{1}{3}}\right )}{a^{12} b^{10} x^{4}}}{2422210560 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2422210560*(14549535*a^13*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))*sign(x^(1/3))/(
sqrt(-b)*b^10) + (14549535*(a*x^(1/3) + b)^(23/2)*a^13*sign(x^(1/3)) - 169744575
*(a*x^(1/3) + b)^(21/2)*a^13*b*sign(x^(1/3)) + 904981077*(a*x^(1/3) + b)^(19/2)*
a^13*b^2*sign(x^(1/3)) - 2913648309*(a*x^(1/3) + b)^(17/2)*a^13*b^3*sign(x^(1/3)
) + 6303782342*(a*x^(1/3) + b)^(15/2)*a^13*b^4*sign(x^(1/3)) - 9643633350*(a*x^(
1/3) + b)^(13/2)*a^13*b^5*sign(x^(1/3)) + 10677769530*(a*x^(1/3) + b)^(11/2)*a^1
3*b^6*sign(x^(1/3)) - 8598579770*(a*x^(1/3) + b)^(9/2)*a^13*b^7*sign(x^(1/3)) +
4975837515*(a*x^(1/3) + b)^(7/2)*a^13*b^8*sign(x^(1/3)) - 2001671595*(a*x^(1/3)
+ b)^(5/2)*a^13*b^9*sign(x^(1/3)) - 169744575*(a*x^(1/3) + b)^(3/2)*a^13*b^10*si
gn(x^(1/3)) + 14549535*sqrt(a*x^(1/3) + b)*a^13*b^11*sign(x^(1/3)))/(a^12*b^10*x
^4))/a