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

Optimal. Leaf size=371 \[ \frac{8388608 b^{12} \left (a x+b x^{2/3}\right )^{3/2}}{152108775 a^{13} x}-\frac{4194304 b^{11} \left (a x+b x^{2/3}\right )^{3/2}}{50702925 a^{12} x^{2/3}}+\frac{1048576 b^{10} \left (a x+b x^{2/3}\right )^{3/2}}{10140585 a^{11} \sqrt [3]{x}}-\frac{524288 b^9 \left (a x+b x^{2/3}\right )^{3/2}}{4345965 a^{10}}+\frac{65536 b^8 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{482885 a^9}-\frac{360448 b^7 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{2414425 a^8}+\frac{90112 b^6 x \left (a x+b x^{2/3}\right )^{3/2}}{557175 a^7}-\frac{45056 b^5 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{260015 a^6}+\frac{2816 b^4 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{15295 a^5}-\frac{1408 b^3 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7245 a^4}+\frac{352 b^2 x^{7/3} \left (a x+b x^{2/3}\right )^{3/2}}{1725 a^3}-\frac{16 b x^{8/3} \left (a x+b x^{2/3}\right )^{3/2}}{75 a^2}+\frac{2 x^3 \left (a x+b x^{2/3}\right )^{3/2}}{9 a} \]

[Out]

(-524288*b^9*(b*x^(2/3) + a*x)^(3/2))/(4345965*a^10) + (8388608*b^12*(b*x^(2/3)
+ a*x)^(3/2))/(152108775*a^13*x) - (4194304*b^11*(b*x^(2/3) + a*x)^(3/2))/(50702
925*a^12*x^(2/3)) + (1048576*b^10*(b*x^(2/3) + a*x)^(3/2))/(10140585*a^11*x^(1/3
)) + (65536*b^8*x^(1/3)*(b*x^(2/3) + a*x)^(3/2))/(482885*a^9) - (360448*b^7*x^(2
/3)*(b*x^(2/3) + a*x)^(3/2))/(2414425*a^8) + (90112*b^6*x*(b*x^(2/3) + a*x)^(3/2
))/(557175*a^7) - (45056*b^5*x^(4/3)*(b*x^(2/3) + a*x)^(3/2))/(260015*a^6) + (28
16*b^4*x^(5/3)*(b*x^(2/3) + a*x)^(3/2))/(15295*a^5) - (1408*b^3*x^2*(b*x^(2/3) +
a*x)^(3/2))/(7245*a^4) + (352*b^2*x^(7/3)*(b*x^(2/3) + a*x)^(3/2))/(1725*a^3) -
(16*b*x^(8/3)*(b*x^(2/3) + a*x)^(3/2))/(75*a^2) + (2*x^3*(b*x^(2/3) + a*x)^(3/2
))/(9*a)

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Rubi [A]  time = 1.08863, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{8388608 b^{12} \left (a x+b x^{2/3}\right )^{3/2}}{152108775 a^{13} x}-\frac{4194304 b^{11} \left (a x+b x^{2/3}\right )^{3/2}}{50702925 a^{12} x^{2/3}}+\frac{1048576 b^{10} \left (a x+b x^{2/3}\right )^{3/2}}{10140585 a^{11} \sqrt [3]{x}}-\frac{524288 b^9 \left (a x+b x^{2/3}\right )^{3/2}}{4345965 a^{10}}+\frac{65536 b^8 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{482885 a^9}-\frac{360448 b^7 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{2414425 a^8}+\frac{90112 b^6 x \left (a x+b x^{2/3}\right )^{3/2}}{557175 a^7}-\frac{45056 b^5 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{260015 a^6}+\frac{2816 b^4 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{15295 a^5}-\frac{1408 b^3 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7245 a^4}+\frac{352 b^2 x^{7/3} \left (a x+b x^{2/3}\right )^{3/2}}{1725 a^3}-\frac{16 b x^{8/3} \left (a x+b x^{2/3}\right )^{3/2}}{75 a^2}+\frac{2 x^3 \left (a x+b x^{2/3}\right )^{3/2}}{9 a} \]

Antiderivative was successfully verified.

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

[Out]

(-524288*b^9*(b*x^(2/3) + a*x)^(3/2))/(4345965*a^10) + (8388608*b^12*(b*x^(2/3)
+ a*x)^(3/2))/(152108775*a^13*x) - (4194304*b^11*(b*x^(2/3) + a*x)^(3/2))/(50702
925*a^12*x^(2/3)) + (1048576*b^10*(b*x^(2/3) + a*x)^(3/2))/(10140585*a^11*x^(1/3
)) + (65536*b^8*x^(1/3)*(b*x^(2/3) + a*x)^(3/2))/(482885*a^9) - (360448*b^7*x^(2
/3)*(b*x^(2/3) + a*x)^(3/2))/(2414425*a^8) + (90112*b^6*x*(b*x^(2/3) + a*x)^(3/2
))/(557175*a^7) - (45056*b^5*x^(4/3)*(b*x^(2/3) + a*x)^(3/2))/(260015*a^6) + (28
16*b^4*x^(5/3)*(b*x^(2/3) + a*x)^(3/2))/(15295*a^5) - (1408*b^3*x^2*(b*x^(2/3) +
a*x)^(3/2))/(7245*a^4) + (352*b^2*x^(7/3)*(b*x^(2/3) + a*x)^(3/2))/(1725*a^3) -
(16*b*x^(8/3)*(b*x^(2/3) + a*x)^(3/2))/(75*a^2) + (2*x^3*(b*x^(2/3) + a*x)^(3/2
))/(9*a)

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Rubi in Sympy [A]  time = 113.452, size = 352, normalized size = 0.95 \[ \frac{2 x^{3} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}{9 a} - \frac{16 b x^{\frac{8}{3}} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}{75 a^{2}} + \frac{352 b^{2} x^{\frac{7}{3}} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}{1725 a^{3}} - \frac{1408 b^{3} x^{2} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}{7245 a^{4}} + \frac{2816 b^{4} x^{\frac{5}{3}} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}{15295 a^{5}} - \frac{45056 b^{5} x^{\frac{4}{3}} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}{260015 a^{6}} + \frac{90112 b^{6} x \left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}{557175 a^{7}} - \frac{360448 b^{7} x^{\frac{2}{3}} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}{2414425 a^{8}} + \frac{65536 b^{8} \sqrt [3]{x} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}{482885 a^{9}} - \frac{524288 b^{9} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}{4345965 a^{10}} + \frac{1048576 b^{10} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}{10140585 a^{11} \sqrt [3]{x}} - \frac{4194304 b^{11} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}{50702925 a^{12} x^{\frac{2}{3}}} + \frac{8388608 b^{12} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}{152108775 a^{13} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**3*(a*x + b*x**(2/3))**(3/2)/(9*a) - 16*b*x**(8/3)*(a*x + b*x**(2/3))**(3/2)
/(75*a**2) + 352*b**2*x**(7/3)*(a*x + b*x**(2/3))**(3/2)/(1725*a**3) - 1408*b**3
*x**2*(a*x + b*x**(2/3))**(3/2)/(7245*a**4) + 2816*b**4*x**(5/3)*(a*x + b*x**(2/
3))**(3/2)/(15295*a**5) - 45056*b**5*x**(4/3)*(a*x + b*x**(2/3))**(3/2)/(260015*
a**6) + 90112*b**6*x*(a*x + b*x**(2/3))**(3/2)/(557175*a**7) - 360448*b**7*x**(2
/3)*(a*x + b*x**(2/3))**(3/2)/(2414425*a**8) + 65536*b**8*x**(1/3)*(a*x + b*x**(
2/3))**(3/2)/(482885*a**9) - 524288*b**9*(a*x + b*x**(2/3))**(3/2)/(4345965*a**1
0) + 1048576*b**10*(a*x + b*x**(2/3))**(3/2)/(10140585*a**11*x**(1/3)) - 4194304
*b**11*(a*x + b*x**(2/3))**(3/2)/(50702925*a**12*x**(2/3)) + 8388608*b**12*(a*x
+ b*x**(2/3))**(3/2)/(152108775*a**13*x)

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Mathematica [A]  time = 0.0892474, size = 185, normalized size = 0.5 \[ \frac{2 \sqrt{a x+b x^{2/3}} \left (16900975 a^{13} x^{13/3}+676039 a^{12} b x^4-705432 a^{11} b^2 x^{11/3}+739024 a^{10} b^3 x^{10/3}-777920 a^9 b^4 x^3+823680 a^8 b^5 x^{8/3}-878592 a^7 b^6 x^{7/3}+946176 a^6 b^7 x^2-1032192 a^5 b^8 x^{5/3}+1146880 a^4 b^9 x^{4/3}-1310720 a^3 b^{10} x+1572864 a^2 b^{11} x^{2/3}-2097152 a b^{12} \sqrt [3]{x}+4194304 b^{13}\right )}{152108775 a^{13} \sqrt [3]{x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[b*x^(2/3) + a*x]*(4194304*b^13 - 2097152*a*b^12*x^(1/3) + 1572864*a^2*b^
11*x^(2/3) - 1310720*a^3*b^10*x + 1146880*a^4*b^9*x^(4/3) - 1032192*a^5*b^8*x^(5
/3) + 946176*a^6*b^7*x^2 - 878592*a^7*b^6*x^(7/3) + 823680*a^8*b^5*x^(8/3) - 777
920*a^9*b^4*x^3 + 739024*a^10*b^3*x^(10/3) - 705432*a^11*b^2*x^(11/3) + 676039*a
^12*b*x^4 + 16900975*a^13*x^(13/3)))/(152108775*a^13*x^(1/3))

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Maple [A]  time = 0.016, size = 156, normalized size = 0.4 \[ -{\frac{2}{152108775\,{a}^{13}}\sqrt{b{x}^{{\frac{2}{3}}}+ax} \left ( b+a\sqrt [3]{x} \right ) \left ( 16224936\,{x}^{11/3}{a}^{11}b-15519504\,{x}^{10/3}{a}^{10}{b}^{2}-14002560\,{x}^{8/3}{a}^{8}{b}^{4}[phone]\,{x}^{7/3}{a}^{7}{b}^{5}[phone]\,{x}^{5/3}{a}^{5}{b}^{7}-10321920\,{x}^{4/3}{a}^{4}{b}^{8}-16900975\,{x}^{4}{a}^{12}[phone]\,{x}^{3}{a}^{9}{b}^{3}-7864320\,{x}^{2/3}{a}^{2}{b}^{10}-12300288\,{x}^{2}{a}^{6}{b}^{6}+6291456\,\sqrt [3]{x}a{b}^{11}+9175040\,x{a}^{3}{b}^{9}-4194304\,{b}^{12} \right ){\frac{1}{\sqrt [3]{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/152108775*(b*x^(2/3)+a*x)^(1/2)*(b+a*x^(1/3))*(16224936*x^(11/3)*a^11*b-15519
504*x^(10/3)*a^10*b^2-14002560*x^(8/3)*a^8*b^4+13178880*x^(7/3)*a^7*b^5+11354112
*x^(5/3)*a^5*b^7-10321920*x^(4/3)*a^4*b^8-16900975*x^4*a^12+14780480*x^3*a^9*b^3
-7864320*x^(2/3)*a^2*b^10-12300288*x^2*a^6*b^6+6291456*x^(1/3)*a*b^11+9175040*x*
a^3*b^9-4194304*b^12)/x^(1/3)/a^13

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Maxima [A]  time = 1.44924, size = 293, normalized size = 0.79 \[ \frac{2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{27}{2}}}{9 \, a^{13}} - \frac{72 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{25}{2}} b}{25 \, a^{13}} + \frac{396 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{23}{2}} b^{2}}{23 \, a^{13}} - \frac{440 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{21}{2}} b^{3}}{7 \, a^{13}} + \frac{2970 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} b^{4}}{19 \, a^{13}} - \frac{4752 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} b^{5}}{17 \, a^{13}} + \frac{1848 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} b^{6}}{5 \, a^{13}} - \frac{4752 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} b^{7}}{13 \, a^{13}} + \frac{270 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} b^{8}}{a^{13}} - \frac{440 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} b^{9}}{3 \, a^{13}} + \frac{396 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} b^{10}}{7 \, a^{13}} - \frac{72 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} b^{11}}{5 \, a^{13}} + \frac{2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} b^{12}}{a^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*(a*x^(1/3) + b)^(27/2)/a^13 - 72/25*(a*x^(1/3) + b)^(25/2)*b/a^13 + 396/23*(
a*x^(1/3) + b)^(23/2)*b^2/a^13 - 440/7*(a*x^(1/3) + b)^(21/2)*b^3/a^13 + 2970/19
*(a*x^(1/3) + b)^(19/2)*b^4/a^13 - 4752/17*(a*x^(1/3) + b)^(17/2)*b^5/a^13 + 184
8/5*(a*x^(1/3) + b)^(15/2)*b^6/a^13 - 4752/13*(a*x^(1/3) + b)^(13/2)*b^7/a^13 +
270*(a*x^(1/3) + b)^(11/2)*b^8/a^13 - 440/3*(a*x^(1/3) + b)^(9/2)*b^9/a^13 + 396
/7*(a*x^(1/3) + b)^(7/2)*b^10/a^13 - 72/5*(a*x^(1/3) + b)^(5/2)*b^11/a^13 + 2*(a
*x^(1/3) + b)^(3/2)*b^12/a^13

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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^3,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{3} \sqrt{a x + b x^{\frac{2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**3*sqrt(a*x + b*x**(2/3)), x)

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GIAC/XCAS [A]  time = 0.23287, size = 323, normalized size = 0.87 \[ -\frac{8388608 \, b^{\frac{27}{2}}{\rm sign}\left (x^{\frac{1}{3}}\right )}{152108775 \, a^{13}} + \frac{2 \,{\left (16900975 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{27}{2}} a^{312} - 219036636 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{25}{2}} a^{312} b + 1309458150 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{23}{2}} a^{312} b^{2} - 4780561500 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{21}{2}} a^{312} b^{3} + 11888501625 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} a^{312} b^{4} - 21259438200 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} a^{312} b^{5} + 28109701620 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} a^{312} b^{6} - 27800803800 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} a^{312} b^{7} + 20534684625 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{312} b^{8} - 11154643500 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{312} b^{9} + 4302505350 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{312} b^{10} - 1095183180 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{312} b^{11} + 152108775 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{312} b^{12}\right )}{\rm sign}\left (x^{\frac{1}{3}}\right )}{152108775 \, a^{325}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8388608/152108775*b^(27/2)*sign(x^(1/3))/a^13 + 2/152108775*(16900975*(a*x^(1/3
) + b)^(27/2)*a^312 - 219036636*(a*x^(1/3) + b)^(25/2)*a^312*b + 1309458150*(a*x
^(1/3) + b)^(23/2)*a^312*b^2 - 4780561500*(a*x^(1/3) + b)^(21/2)*a^312*b^3 + 118
88501625*(a*x^(1/3) + b)^(19/2)*a^312*b^4 - 21259438200*(a*x^(1/3) + b)^(17/2)*a
^312*b^5 + 28109701620*(a*x^(1/3) + b)^(15/2)*a^312*b^6 - 27800803800*(a*x^(1/3)
+ b)^(13/2)*a^312*b^7 + 20534684625*(a*x^(1/3) + b)^(11/2)*a^312*b^8 - 11154643
500*(a*x^(1/3) + b)^(9/2)*a^312*b^9 + 4302505350*(a*x^(1/3) + b)^(7/2)*a^312*b^1
0 - 1095183180*(a*x^(1/3) + b)^(5/2)*a^312*b^11 + 152108775*(a*x^(1/3) + b)^(3/2
)*a^312*b^12)*sign(x^(1/3))/a^325