3.194 \(\int \frac{x^4}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx\)

Optimal. Leaf size=336 \[ \frac{1048576 b^{10} \sqrt{a x+b x^{2/3}}}{29393 a^{12} \sqrt [3]{x}}-\frac{524288 b^9 \sqrt{a x+b x^{2/3}}}{29393 a^{11}}+\frac{393216 b^8 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{29393 a^{10}}-\frac{327680 b^7 x^{2/3} \sqrt{a x+b x^{2/3}}}{29393 a^9}+\frac{40960 b^6 x \sqrt{a x+b x^{2/3}}}{4199 a^8}-\frac{36864 b^5 x^{4/3} \sqrt{a x+b x^{2/3}}}{4199 a^7}+\frac{33792 b^4 x^{5/3} \sqrt{a x+b x^{2/3}}}{4199 a^6}-\frac{16896 b^3 x^2 \sqrt{a x+b x^{2/3}}}{2261 a^5}+\frac{15840 b^2 x^{7/3} \sqrt{a x+b x^{2/3}}}{2261 a^4}-\frac{880 b x^{8/3} \sqrt{a x+b x^{2/3}}}{133 a^3}+\frac{44 x^3 \sqrt{a x+b x^{2/3}}}{7 a^2}-\frac{6 x^4}{a \sqrt{a x+b x^{2/3}}} \]

[Out]

(-6*x^4)/(a*Sqrt[b*x^(2/3) + a*x]) - (524288*b^9*Sqrt[b*x^(2/3) + a*x])/(29393*a
^11) + (1048576*b^10*Sqrt[b*x^(2/3) + a*x])/(29393*a^12*x^(1/3)) + (393216*b^8*x
^(1/3)*Sqrt[b*x^(2/3) + a*x])/(29393*a^10) - (327680*b^7*x^(2/3)*Sqrt[b*x^(2/3)
+ a*x])/(29393*a^9) + (40960*b^6*x*Sqrt[b*x^(2/3) + a*x])/(4199*a^8) - (36864*b^
5*x^(4/3)*Sqrt[b*x^(2/3) + a*x])/(4199*a^7) + (33792*b^4*x^(5/3)*Sqrt[b*x^(2/3)
+ a*x])/(4199*a^6) - (16896*b^3*x^2*Sqrt[b*x^(2/3) + a*x])/(2261*a^5) + (15840*b
^2*x^(7/3)*Sqrt[b*x^(2/3) + a*x])/(2261*a^4) - (880*b*x^(8/3)*Sqrt[b*x^(2/3) + a
*x])/(133*a^3) + (44*x^3*Sqrt[b*x^(2/3) + a*x])/(7*a^2)

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Rubi [A]  time = 1.00667, antiderivative size = 336, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{1048576 b^{10} \sqrt{a x+b x^{2/3}}}{29393 a^{12} \sqrt [3]{x}}-\frac{524288 b^9 \sqrt{a x+b x^{2/3}}}{29393 a^{11}}+\frac{393216 b^8 \sqrt [3]{x} \sqrt{a x+b x^{2/3}}}{29393 a^{10}}-\frac{327680 b^7 x^{2/3} \sqrt{a x+b x^{2/3}}}{29393 a^9}+\frac{40960 b^6 x \sqrt{a x+b x^{2/3}}}{4199 a^8}-\frac{36864 b^5 x^{4/3} \sqrt{a x+b x^{2/3}}}{4199 a^7}+\frac{33792 b^4 x^{5/3} \sqrt{a x+b x^{2/3}}}{4199 a^6}-\frac{16896 b^3 x^2 \sqrt{a x+b x^{2/3}}}{2261 a^5}+\frac{15840 b^2 x^{7/3} \sqrt{a x+b x^{2/3}}}{2261 a^4}-\frac{880 b x^{8/3} \sqrt{a x+b x^{2/3}}}{133 a^3}+\frac{44 x^3 \sqrt{a x+b x^{2/3}}}{7 a^2}-\frac{6 x^4}{a \sqrt{a x+b x^{2/3}}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(b*x^(2/3) + a*x)^(3/2),x]

[Out]

(-6*x^4)/(a*Sqrt[b*x^(2/3) + a*x]) - (524288*b^9*Sqrt[b*x^(2/3) + a*x])/(29393*a
^11) + (1048576*b^10*Sqrt[b*x^(2/3) + a*x])/(29393*a^12*x^(1/3)) + (393216*b^8*x
^(1/3)*Sqrt[b*x^(2/3) + a*x])/(29393*a^10) - (327680*b^7*x^(2/3)*Sqrt[b*x^(2/3)
+ a*x])/(29393*a^9) + (40960*b^6*x*Sqrt[b*x^(2/3) + a*x])/(4199*a^8) - (36864*b^
5*x^(4/3)*Sqrt[b*x^(2/3) + a*x])/(4199*a^7) + (33792*b^4*x^(5/3)*Sqrt[b*x^(2/3)
+ a*x])/(4199*a^6) - (16896*b^3*x^2*Sqrt[b*x^(2/3) + a*x])/(2261*a^5) + (15840*b
^2*x^(7/3)*Sqrt[b*x^(2/3) + a*x])/(2261*a^4) - (880*b*x^(8/3)*Sqrt[b*x^(2/3) + a
*x])/(133*a^3) + (44*x^3*Sqrt[b*x^(2/3) + a*x])/(7*a^2)

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Rubi in Sympy [A]  time = 99.2973, size = 320, normalized size = 0.95 \[ - \frac{6 x^{4}}{a \sqrt{a x + b x^{\frac{2}{3}}}} + \frac{44 x^{3} \sqrt{a x + b x^{\frac{2}{3}}}}{7 a^{2}} - \frac{880 b x^{\frac{8}{3}} \sqrt{a x + b x^{\frac{2}{3}}}}{133 a^{3}} + \frac{15840 b^{2} x^{\frac{7}{3}} \sqrt{a x + b x^{\frac{2}{3}}}}{2261 a^{4}} - \frac{16896 b^{3} x^{2} \sqrt{a x + b x^{\frac{2}{3}}}}{2261 a^{5}} + \frac{33792 b^{4} x^{\frac{5}{3}} \sqrt{a x + b x^{\frac{2}{3}}}}{4199 a^{6}} - \frac{36864 b^{5} x^{\frac{4}{3}} \sqrt{a x + b x^{\frac{2}{3}}}}{4199 a^{7}} + \frac{40960 b^{6} x \sqrt{a x + b x^{\frac{2}{3}}}}{4199 a^{8}} - \frac{327680 b^{7} x^{\frac{2}{3}} \sqrt{a x + b x^{\frac{2}{3}}}}{29393 a^{9}} + \frac{393216 b^{8} \sqrt [3]{x} \sqrt{a x + b x^{\frac{2}{3}}}}{29393 a^{10}} - \frac{524288 b^{9} \sqrt{a x + b x^{\frac{2}{3}}}}{29393 a^{11}} + \frac{1048576 b^{10} \sqrt{a x + b x^{\frac{2}{3}}}}{29393 a^{12} \sqrt [3]{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**(2/3)+a*x)**(3/2),x)

[Out]

-6*x**4/(a*sqrt(a*x + b*x**(2/3))) + 44*x**3*sqrt(a*x + b*x**(2/3))/(7*a**2) - 8
80*b*x**(8/3)*sqrt(a*x + b*x**(2/3))/(133*a**3) + 15840*b**2*x**(7/3)*sqrt(a*x +
b*x**(2/3))/(2261*a**4) - 16896*b**3*x**2*sqrt(a*x + b*x**(2/3))/(2261*a**5) +
33792*b**4*x**(5/3)*sqrt(a*x + b*x**(2/3))/(4199*a**6) - 36864*b**5*x**(4/3)*sqr
t(a*x + b*x**(2/3))/(4199*a**7) + 40960*b**6*x*sqrt(a*x + b*x**(2/3))/(4199*a**8
) - 327680*b**7*x**(2/3)*sqrt(a*x + b*x**(2/3))/(29393*a**9) + 393216*b**8*x**(1
/3)*sqrt(a*x + b*x**(2/3))/(29393*a**10) - 524288*b**9*sqrt(a*x + b*x**(2/3))/(2
9393*a**11) + 1048576*b**10*sqrt(a*x + b*x**(2/3))/(29393*a**12*x**(1/3))

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Mathematica [A]  time = 0.0933845, size = 172, normalized size = 0.51 \[ \frac{2 \sqrt{a x+b x^{2/3}} \left (4199 a^{11} x^{11/3}-4862 a^{10} b x^{10/3}+5720 a^9 b^2 x^3-6864 a^8 b^3 x^{8/3}+8448 a^7 b^4 x^{7/3}-10752 a^6 b^5 x^2+14336 a^5 b^6 x^{5/3}-20480 a^4 b^7 x^{4/3}+32768 a^3 b^8 x-65536 a^2 b^9 x^{2/3}+262144 a b^{10} \sqrt [3]{x}+524288 b^{11}\right )}{29393 a^{12} \sqrt [3]{x} \left (a \sqrt [3]{x}+b\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(b*x^(2/3) + a*x)^(3/2),x]

[Out]

(2*Sqrt[b*x^(2/3) + a*x]*(524288*b^11 + 262144*a*b^10*x^(1/3) - 65536*a^2*b^9*x^
(2/3) + 32768*a^3*b^8*x - 20480*a^4*b^7*x^(4/3) + 14336*a^5*b^6*x^(5/3) - 10752*
a^6*b^5*x^2 + 8448*a^7*b^4*x^(7/3) - 6864*a^8*b^3*x^(8/3) + 5720*a^9*b^2*x^3 - 4
862*a^10*b*x^(10/3) + 4199*a^11*x^(11/3)))/(29393*a^12*(b + a*x^(1/3))*x^(1/3))

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Maple [A]  time = 0.01, size = 143, normalized size = 0.4 \[{\frac{2\,x}{29393\,{a}^{12}} \left ( b+a\sqrt [3]{x} \right ) \left ( 4199\,{x}^{11/3}{a}^{11}-4862\,{x}^{10/3}{a}^{10}b+5720\,{x}^{3}{a}^{9}{b}^{2}-6864\,{x}^{8/3}{a}^{8}{b}^{3}+8448\,{x}^{7/3}{a}^{7}{b}^{4}-10752\,{x}^{2}{a}^{6}{b}^{5}+14336\,{x}^{5/3}{a}^{5}{b}^{6}-20480\,{x}^{4/3}{a}^{4}{b}^{7}+32768\,x{a}^{3}{b}^{8}-65536\,{x}^{2/3}{a}^{2}{b}^{9}+262144\,\sqrt [3]{x}a{b}^{10}+524288\,{b}^{11} \right ) \left ( b{x}^{{\frac{2}{3}}}+ax \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^(2/3)+a*x)^(3/2),x)

[Out]

2/29393*x*(b+a*x^(1/3))*(4199*x^(11/3)*a^11-4862*x^(10/3)*a^10*b+5720*x^3*a^9*b^
2-6864*x^(8/3)*a^8*b^3+8448*x^(7/3)*a^7*b^4-10752*x^2*a^6*b^5+14336*x^(5/3)*a^5*
b^6-20480*x^(4/3)*a^4*b^7+32768*x*a^3*b^8-65536*x^(2/3)*a^2*b^9+262144*x^(1/3)*a
*b^10+524288*b^11)/(b*x^(2/3)+a*x)^(3/2)/a^12

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Maxima [A]  time = 1.49012, size = 270, normalized size = 0.8 \[ \frac{2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{21}{2}}}{7 \, a^{12}} - \frac{66 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} b}{19 \, a^{12}} + \frac{330 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} b^{2}}{17 \, a^{12}} - \frac{66 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} b^{3}}{a^{12}} + \frac{1980 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} b^{4}}{13 \, a^{12}} - \frac{252 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} b^{5}}{a^{12}} + \frac{308 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} b^{6}}{a^{12}} - \frac{1980 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} b^{7}}{7 \, a^{12}} + \frac{198 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} b^{8}}{a^{12}} - \frac{110 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} b^{9}}{a^{12}} + \frac{66 \, \sqrt{a x^{\frac{1}{3}} + b} b^{10}}{a^{12}} + \frac{6 \, b^{11}}{\sqrt{a x^{\frac{1}{3}} + b} a^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(a*x + b*x^(2/3))^(3/2),x, algorithm="maxima")

[Out]

2/7*(a*x^(1/3) + b)^(21/2)/a^12 - 66/19*(a*x^(1/3) + b)^(19/2)*b/a^12 + 330/17*(
a*x^(1/3) + b)^(17/2)*b^2/a^12 - 66*(a*x^(1/3) + b)^(15/2)*b^3/a^12 + 1980/13*(a
*x^(1/3) + b)^(13/2)*b^4/a^12 - 252*(a*x^(1/3) + b)^(11/2)*b^5/a^12 + 308*(a*x^(
1/3) + b)^(9/2)*b^6/a^12 - 1980/7*(a*x^(1/3) + b)^(7/2)*b^7/a^12 + 198*(a*x^(1/3
) + b)^(5/2)*b^8/a^12 - 110*(a*x^(1/3) + b)^(3/2)*b^9/a^12 + 66*sqrt(a*x^(1/3) +
b)*b^10/a^12 + 6*b^11/(sqrt(a*x^(1/3) + b)*a^12)

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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^4/(a*x + b*x^(2/3))^(3/2),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(x**4/(b*x**(2/3)+a*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.239522, size = 311, normalized size = 0.93 \[ -\frac{1048576 \, b^{\frac{21}{2}}{\rm sign}\left (x^{\frac{1}{3}}\right )}{29393 \, a^{12}} + \frac{6 \, b^{11}}{\sqrt{a x^{\frac{1}{3}} + b} a^{12}{\rm sign}\left (x^{\frac{1}{3}}\right )} + \frac{2 \,{\left (4199 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{21}{2}} a^{240} - 51051 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} a^{240} b + 285285 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} a^{240} b^{2} - 969969 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} a^{240} b^{3} + 2238390 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} a^{240} b^{4} - 3703518 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{240} b^{5} + 4526522 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{240} b^{6} - 4157010 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{240} b^{7} + 2909907 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{240} b^{8} - 1616615 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{240} b^{9} + 969969 \, \sqrt{a x^{\frac{1}{3}} + b} a^{240} b^{10}\right )}}{29393 \, a^{252}{\rm sign}\left (x^{\frac{1}{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(a*x + b*x^(2/3))^(3/2),x, algorithm="giac")

[Out]

-1048576/29393*b^(21/2)*sign(x^(1/3))/a^12 + 6*b^11/(sqrt(a*x^(1/3) + b)*a^12*si
gn(x^(1/3))) + 2/29393*(4199*(a*x^(1/3) + b)^(21/2)*a^240 - 51051*(a*x^(1/3) + b
)^(19/2)*a^240*b + 285285*(a*x^(1/3) + b)^(17/2)*a^240*b^2 - 969969*(a*x^(1/3) +
b)^(15/2)*a^240*b^3 + 2238390*(a*x^(1/3) + b)^(13/2)*a^240*b^4 - 3703518*(a*x^(
1/3) + b)^(11/2)*a^240*b^5 + 4526522*(a*x^(1/3) + b)^(9/2)*a^240*b^6 - 4157010*(
a*x^(1/3) + b)^(7/2)*a^240*b^7 + 2909907*(a*x^(1/3) + b)^(5/2)*a^240*b^8 - 16166
15*(a*x^(1/3) + b)^(3/2)*a^240*b^9 + 969969*sqrt(a*x^(1/3) + b)*a^240*b^10)/(a^2
52*sign(x^(1/3)))