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

Optimal. Leaf size=313 \[ \frac{524288 b^{10} \sqrt{a x+b x^{2/3}}}{323323 a^{11} \sqrt [3]{x}}-\frac{262144 b^9 \sqrt{a x+b x^{2/3}}}{323323 a^{10}}+\frac{196608 b^8 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{323323 a^9}-\frac{163840 b^7 x^{2/3} \sqrt{a x+b x^{2/3}}}{323323 a^8}+\frac{20480 b^6 x \sqrt{a x+b x^{2/3}}}{46189 a^7}-\frac{18432 b^5 x^{4/3} \sqrt{a x+b x^{2/3}}}{46189 a^6}+\frac{1536 b^4 x^{5/3} \sqrt{a x+b x^{2/3}}}{4199 a^5}-\frac{768 b^3 x^2 \sqrt{a x+b x^{2/3}}}{2261 a^4}+\frac{720 b^2 x^{7/3} \sqrt{a x+b x^{2/3}}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{a x+b x^{2/3}}}{133 a^2}+\frac{2 x^3 \sqrt{a x+b x^{2/3}}}{7 a} \]

[Out]

(-262144*b^9*Sqrt[b*x^(2/3) + a*x])/(323323*a^10) + (524288*b^10*Sqrt[b*x^(2/3)
+ a*x])/(323323*a^11*x^(1/3)) + (196608*b^8*x^(1/3)*Sqrt[b*x^(2/3) + a*x])/(3233
23*a^9) - (163840*b^7*x^(2/3)*Sqrt[b*x^(2/3) + a*x])/(323323*a^8) + (20480*b^6*x
*Sqrt[b*x^(2/3) + a*x])/(46189*a^7) - (18432*b^5*x^(4/3)*Sqrt[b*x^(2/3) + a*x])/
(46189*a^6) + (1536*b^4*x^(5/3)*Sqrt[b*x^(2/3) + a*x])/(4199*a^5) - (768*b^3*x^2
*Sqrt[b*x^(2/3) + a*x])/(2261*a^4) + (720*b^2*x^(7/3)*Sqrt[b*x^(2/3) + a*x])/(22
61*a^3) - (40*b*x^(8/3)*Sqrt[b*x^(2/3) + a*x])/(133*a^2) + (2*x^3*Sqrt[b*x^(2/3)
+ a*x])/(7*a)

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Rubi [A]  time = 0.900092, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{524288 b^{10} \sqrt{a x+b x^{2/3}}}{323323 a^{11} \sqrt [3]{x}}-\frac{262144 b^9 \sqrt{a x+b x^{2/3}}}{323323 a^{10}}+\frac{196608 b^8 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{323323 a^9}-\frac{163840 b^7 x^{2/3} \sqrt{a x+b x^{2/3}}}{323323 a^8}+\frac{20480 b^6 x \sqrt{a x+b x^{2/3}}}{46189 a^7}-\frac{18432 b^5 x^{4/3} \sqrt{a x+b x^{2/3}}}{46189 a^6}+\frac{1536 b^4 x^{5/3} \sqrt{a x+b x^{2/3}}}{4199 a^5}-\frac{768 b^3 x^2 \sqrt{a x+b x^{2/3}}}{2261 a^4}+\frac{720 b^2 x^{7/3} \sqrt{a x+b x^{2/3}}}{2261 a^3}-\frac{40 b x^{8/3} \sqrt{a x+b x^{2/3}}}{133 a^2}+\frac{2 x^3 \sqrt{a x+b x^{2/3}}}{7 a} \]

Antiderivative was successfully verified.

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

[Out]

(-262144*b^9*Sqrt[b*x^(2/3) + a*x])/(323323*a^10) + (524288*b^10*Sqrt[b*x^(2/3)
+ a*x])/(323323*a^11*x^(1/3)) + (196608*b^8*x^(1/3)*Sqrt[b*x^(2/3) + a*x])/(3233
23*a^9) - (163840*b^7*x^(2/3)*Sqrt[b*x^(2/3) + a*x])/(323323*a^8) + (20480*b^6*x
*Sqrt[b*x^(2/3) + a*x])/(46189*a^7) - (18432*b^5*x^(4/3)*Sqrt[b*x^(2/3) + a*x])/
(46189*a^6) + (1536*b^4*x^(5/3)*Sqrt[b*x^(2/3) + a*x])/(4199*a^5) - (768*b^3*x^2
*Sqrt[b*x^(2/3) + a*x])/(2261*a^4) + (720*b^2*x^(7/3)*Sqrt[b*x^(2/3) + a*x])/(22
61*a^3) - (40*b*x^(8/3)*Sqrt[b*x^(2/3) + a*x])/(133*a^2) + (2*x^3*Sqrt[b*x^(2/3)
+ a*x])/(7*a)

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Rubi in Sympy [A]  time = 87.221, size = 298, normalized size = 0.95 \[ \frac{2 x^{3} \sqrt{a x + b x^{\frac{2}{3}}}}{7 a} - \frac{40 b x^{\frac{8}{3}} \sqrt{a x + b x^{\frac{2}{3}}}}{133 a^{2}} + \frac{720 b^{2} x^{\frac{7}{3}} \sqrt{a x + b x^{\frac{2}{3}}}}{2261 a^{3}} - \frac{768 b^{3} x^{2} \sqrt{a x + b x^{\frac{2}{3}}}}{2261 a^{4}} + \frac{1536 b^{4} x^{\frac{5}{3}} \sqrt{a x + b x^{\frac{2}{3}}}}{4199 a^{5}} - \frac{18432 b^{5} x^{\frac{4}{3}} \sqrt{a x + b x^{\frac{2}{3}}}}{46189 a^{6}} + \frac{20480 b^{6} x \sqrt{a x + b x^{\frac{2}{3}}}}{46189 a^{7}} - \frac{163840 b^{7} x^{\frac{2}{3}} \sqrt{a x + b x^{\frac{2}{3}}}}{323323 a^{8}} + \frac{196608 b^{8} \sqrt [3]{x} \sqrt{a x + b x^{\frac{2}{3}}}}{323323 a^{9}} - \frac{262144 b^{9} \sqrt{a x + b x^{\frac{2}{3}}}}{323323 a^{10}} + \frac{524288 b^{10} \sqrt{a x + b x^{\frac{2}{3}}}}{323323 a^{11} \sqrt [3]{x}} \]

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*sqrt(a*x + b*x**(2/3))/(7*a) - 40*b*x**(8/3)*sqrt(a*x + b*x**(2/3))/(133*
a**2) + 720*b**2*x**(7/3)*sqrt(a*x + b*x**(2/3))/(2261*a**3) - 768*b**3*x**2*sqr
t(a*x + b*x**(2/3))/(2261*a**4) + 1536*b**4*x**(5/3)*sqrt(a*x + b*x**(2/3))/(419
9*a**5) - 18432*b**5*x**(4/3)*sqrt(a*x + b*x**(2/3))/(46189*a**6) + 20480*b**6*x
*sqrt(a*x + b*x**(2/3))/(46189*a**7) - 163840*b**7*x**(2/3)*sqrt(a*x + b*x**(2/3
))/(323323*a**8) + 196608*b**8*x**(1/3)*sqrt(a*x + b*x**(2/3))/(323323*a**9) - 2
62144*b**9*sqrt(a*x + b*x**(2/3))/(323323*a**10) + 524288*b**10*sqrt(a*x + b*x**
(2/3))/(323323*a**11*x**(1/3))

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Mathematica [A]  time = 0.072648, size = 148, normalized size = 0.47 \[ \frac{2 \sqrt{a x+b x^{2/3}} \left (46189 a^{10} x^{10/3}-48620 a^9 b x^3+51480 a^8 b^2 x^{8/3}-54912 a^7 b^3 x^{7/3}+59136 a^6 b^4 x^2-64512 a^5 b^5 x^{5/3}+71680 a^4 b^6 x^{4/3}-81920 a^3 b^7 x+98304 a^2 b^8 x^{2/3}-131072 a b^9 \sqrt [3]{x}+262144 b^{10}\right )}{323323 a^{11} \sqrt [3]{x}} \]

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]*(262144*b^10 - 131072*a*b^9*x^(1/3) + 98304*a^2*b^8*x^(
2/3) - 81920*a^3*b^7*x + 71680*a^4*b^6*x^(4/3) - 64512*a^5*b^5*x^(5/3) + 59136*a
^6*b^4*x^2 - 54912*a^7*b^3*x^(7/3) + 51480*a^8*b^2*x^(8/3) - 48620*a^9*b*x^3 + 4
6189*a^10*x^(10/3)))/(323323*a^11*x^(1/3))

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Maple [A]  time = 0.007, size = 134, normalized size = 0.4 \[{\frac{2}{323323\,{a}^{11}}\sqrt [3]{x} \left ( b+a\sqrt [3]{x} \right ) \left ( 46189\,{x}^{10/3}{a}^{10}-48620\,{x}^{3}{a}^{9}b+51480\,{a}^{8}{b}^{2}{x}^{8/3}-54912\,{x}^{7/3}{a}^{7}{b}^{3}+59136\,{a}^{6}{b}^{4}{x}^{2}-64512\,{a}^{5}{b}^{5}{x}^{5/3}+71680\,{x}^{4/3}{a}^{4}{b}^{6}-81920\,{a}^{3}{b}^{7}x+98304\,{a}^{2}{b}^{8}{x}^{2/3}-131072\,\sqrt [3]{x}a{b}^{9}+262144\,{b}^{10} \right ){\frac{1}{\sqrt{b{x}^{{\frac{2}{3}}}+ax}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/323323*x^(1/3)*(b+a*x^(1/3))*(46189*x^(10/3)*a^10-48620*x^3*a^9*b+51480*a^8*b^
2*x^(8/3)-54912*x^(7/3)*a^7*b^3+59136*a^6*b^4*x^2-64512*a^5*b^5*x^(5/3)+71680*x^
(4/3)*a^4*b^6-81920*a^3*b^7*x+98304*a^2*b^8*x^(2/3)-131072*x^(1/3)*a*b^9+262144*
b^10)/(b*x^(2/3)+a*x)^(1/2)/a^11

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Maxima [A]  time = 1.45045, size = 247, normalized size = 0.79 \[ \frac{2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{21}{2}}}{7 \, a^{11}} - \frac{60 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} b}{19 \, a^{11}} + \frac{270 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} b^{2}}{17 \, a^{11}} - \frac{48 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} b^{3}}{a^{11}} + \frac{1260 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} b^{4}}{13 \, a^{11}} - \frac{1512 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} b^{5}}{11 \, a^{11}} + \frac{140 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} b^{6}}{a^{11}} - \frac{720 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} b^{7}}{7 \, a^{11}} + \frac{54 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} b^{8}}{a^{11}} - \frac{20 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} b^{9}}{a^{11}} + \frac{6 \, \sqrt{a x^{\frac{1}{3}} + b} b^{10}}{a^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*(a*x^(1/3) + b)^(21/2)/a^11 - 60/19*(a*x^(1/3) + b)^(19/2)*b/a^11 + 270/17*(
a*x^(1/3) + b)^(17/2)*b^2/a^11 - 48*(a*x^(1/3) + b)^(15/2)*b^3/a^11 + 1260/13*(a
*x^(1/3) + b)^(13/2)*b^4/a^11 - 1512/11*(a*x^(1/3) + b)^(11/2)*b^5/a^11 + 140*(a
*x^(1/3) + b)^(9/2)*b^6/a^11 - 720/7*(a*x^(1/3) + b)^(7/2)*b^7/a^11 + 54*(a*x^(1
/3) + b)^(5/2)*b^8/a^11 - 20*(a*x^(1/3) + b)^(3/2)*b^9/a^11 + 6*sqrt(a*x^(1/3) +
b)*b^10/a^11

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

[Out]

Timed out

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

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 = 0.226699, size = 279, normalized size = 0.89 \[ -\frac{524288 \, b^{\frac{21}{2}}{\rm sign}\left (x^{\frac{1}{3}}\right )}{323323 \, a^{11}} + \frac{2 \,{\left (46189 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{21}{2}} a^{200} - 510510 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} a^{200} b + 2567565 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} a^{200} b^{2} - 7759752 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} a^{200} b^{3} + 15668730 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} a^{200} b^{4} - 22221108 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{200} b^{5} + 22632610 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{200} b^{6} - 16628040 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{200} b^{7} + 8729721 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{200} b^{8} - 3233230 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{200} b^{9} + 969969 \, \sqrt{a x^{\frac{1}{3}} + b} a^{200} b^{10}\right )}}{323323 \, a^{211}{\rm sign}\left (x^{\frac{1}{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-524288/323323*b^(21/2)*sign(x^(1/3))/a^11 + 2/323323*(46189*(a*x^(1/3) + b)^(21
/2)*a^200 - 510510*(a*x^(1/3) + b)^(19/2)*a^200*b + 2567565*(a*x^(1/3) + b)^(17/
2)*a^200*b^2 - 7759752*(a*x^(1/3) + b)^(15/2)*a^200*b^3 + 15668730*(a*x^(1/3) +
b)^(13/2)*a^200*b^4 - 22221108*(a*x^(1/3) + b)^(11/2)*a^200*b^5 + 22632610*(a*x^
(1/3) + b)^(9/2)*a^200*b^6 - 16628040*(a*x^(1/3) + b)^(7/2)*a^200*b^7 + 8729721*
(a*x^(1/3) + b)^(5/2)*a^200*b^8 - 3233230*(a*x^(1/3) + b)^(3/2)*a^200*b^9 + 9699
69*sqrt(a*x^(1/3) + b)*a^200*b^10)/(a^211*sign(x^(1/3)))