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

Optimal. Leaf size=255 \[ \frac{65536 b^8 \left (a x+b x^{2/3}\right )^{5/2}}{4849845 a^9 x^{5/3}}-\frac{32768 b^7 \left (a x+b x^{2/3}\right )^{5/2}}{969969 a^8 x^{4/3}}+\frac{8192 b^6 \left (a x+b x^{2/3}\right )^{5/2}}{138567 a^7 x}-\frac{4096 b^5 \left (a x+b x^{2/3}\right )^{5/2}}{46189 a^6 x^{2/3}}+\frac{512 b^4 \left (a x+b x^{2/3}\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}-\frac{256 b^3 \left (a x+b x^{2/3}\right )^{5/2}}{1615 a^4}+\frac{64 b^2 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{5/2}}{323 a^3}-\frac{32 b x^{2/3} \left (a x+b x^{2/3}\right )^{5/2}}{133 a^2}+\frac{2 x \left (a x+b x^{2/3}\right )^{5/2}}{7 a} \]

[Out]

(-256*b^3*(b*x^(2/3) + a*x)^(5/2))/(1615*a^4) + (65536*b^8*(b*x^(2/3) + a*x)^(5/
2))/(4849845*a^9*x^(5/3)) - (32768*b^7*(b*x^(2/3) + a*x)^(5/2))/(969969*a^8*x^(4
/3)) + (8192*b^6*(b*x^(2/3) + a*x)^(5/2))/(138567*a^7*x) - (4096*b^5*(b*x^(2/3)
+ a*x)^(5/2))/(46189*a^6*x^(2/3)) + (512*b^4*(b*x^(2/3) + a*x)^(5/2))/(4199*a^5*
x^(1/3)) + (64*b^2*x^(1/3)*(b*x^(2/3) + a*x)^(5/2))/(323*a^3) - (32*b*x^(2/3)*(b
*x^(2/3) + a*x)^(5/2))/(133*a^2) + (2*x*(b*x^(2/3) + a*x)^(5/2))/(7*a)

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Rubi [A]  time = 0.651473, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{65536 b^8 \left (a x+b x^{2/3}\right )^{5/2}}{4849845 a^9 x^{5/3}}-\frac{32768 b^7 \left (a x+b x^{2/3}\right )^{5/2}}{969969 a^8 x^{4/3}}+\frac{8192 b^6 \left (a x+b x^{2/3}\right )^{5/2}}{138567 a^7 x}-\frac{4096 b^5 \left (a x+b x^{2/3}\right )^{5/2}}{46189 a^6 x^{2/3}}+\frac{512 b^4 \left (a x+b x^{2/3}\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}-\frac{256 b^3 \left (a x+b x^{2/3}\right )^{5/2}}{1615 a^4}+\frac{64 b^2 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{5/2}}{323 a^3}-\frac{32 b x^{2/3} \left (a x+b x^{2/3}\right )^{5/2}}{133 a^2}+\frac{2 x \left (a x+b x^{2/3}\right )^{5/2}}{7 a} \]

Antiderivative was successfully verified.

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

[Out]

(-256*b^3*(b*x^(2/3) + a*x)^(5/2))/(1615*a^4) + (65536*b^8*(b*x^(2/3) + a*x)^(5/
2))/(4849845*a^9*x^(5/3)) - (32768*b^7*(b*x^(2/3) + a*x)^(5/2))/(969969*a^8*x^(4
/3)) + (8192*b^6*(b*x^(2/3) + a*x)^(5/2))/(138567*a^7*x) - (4096*b^5*(b*x^(2/3)
+ a*x)^(5/2))/(46189*a^6*x^(2/3)) + (512*b^4*(b*x^(2/3) + a*x)^(5/2))/(4199*a^5*
x^(1/3)) + (64*b^2*x^(1/3)*(b*x^(2/3) + a*x)^(5/2))/(323*a^3) - (32*b*x^(2/3)*(b
*x^(2/3) + a*x)^(5/2))/(133*a^2) + (2*x*(b*x^(2/3) + a*x)^(5/2))/(7*a)

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Rubi in Sympy [A]  time = 64.3438, size = 240, normalized size = 0.94 \[ \frac{2 x \left (a x + b x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{7 a} - \frac{32 b x^{\frac{2}{3}} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{133 a^{2}} + \frac{64 b^{2} \sqrt [3]{x} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{323 a^{3}} - \frac{256 b^{3} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{1615 a^{4}} + \frac{512 b^{4} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{4199 a^{5} \sqrt [3]{x}} - \frac{4096 b^{5} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{46189 a^{6} x^{\frac{2}{3}}} + \frac{8192 b^{6} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{138567 a^{7} x} - \frac{32768 b^{7} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{969969 a^{8} x^{\frac{4}{3}}} + \frac{65536 b^{8} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{4849845 a^{9} x^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x*(a*x + b*x**(2/3))**(5/2)/(7*a) - 32*b*x**(2/3)*(a*x + b*x**(2/3))**(5/2)/(1
33*a**2) + 64*b**2*x**(1/3)*(a*x + b*x**(2/3))**(5/2)/(323*a**3) - 256*b**3*(a*x
+ b*x**(2/3))**(5/2)/(1615*a**4) + 512*b**4*(a*x + b*x**(2/3))**(5/2)/(4199*a**
5*x**(1/3)) - 4096*b**5*(a*x + b*x**(2/3))**(5/2)/(46189*a**6*x**(2/3)) + 8192*b
**6*(a*x + b*x**(2/3))**(5/2)/(138567*a**7*x) - 32768*b**7*(a*x + b*x**(2/3))**(
5/2)/(969969*a**8*x**(4/3)) + 65536*b**8*(a*x + b*x**(2/3))**(5/2)/(4849845*a**9
*x**(5/3))

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Mathematica [A]  time = 0.0890247, size = 135, normalized size = 0.53 \[ \frac{2 \left (a \sqrt [3]{x}+b\right )^2 \sqrt{a x+b x^{2/3}} \left (692835 a^8 x^{8/3}-583440 a^7 b x^{7/3}+480480 a^6 b^2 x^2-384384 a^5 b^3 x^{5/3}+295680 a^4 b^4 x^{4/3}-215040 a^3 b^5 x+143360 a^2 b^6 x^{2/3}-81920 a b^7 \sqrt [3]{x}+32768 b^8\right )}{4849845 a^9 \sqrt [3]{x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b + a*x^(1/3))^2*Sqrt[b*x^(2/3) + a*x]*(32768*b^8 - 81920*a*b^7*x^(1/3) + 14
3360*a^2*b^6*x^(2/3) - 215040*a^3*b^5*x + 295680*a^4*b^4*x^(4/3) - 384384*a^5*b^
3*x^(5/3) + 480480*a^6*b^2*x^2 - 583440*a^7*b*x^(7/3) + 692835*a^8*x^(8/3)))/(48
49845*a^9*x^(1/3))

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Maple [A]  time = 0.007, size = 112, normalized size = 0.4 \[{\frac{2}{4849845\,{a}^{9}x} \left ( b{x}^{{\frac{2}{3}}}+ax \right ) ^{{\frac{3}{2}}} \left ( b+a\sqrt [3]{x} \right ) \left ( 692835\,{x}^{8/3}{a}^{8}-583440\,{x}^{7/3}{a}^{7}b+480480\,{a}^{6}{b}^{2}{x}^{2}-384384\,{x}^{5/3}{a}^{5}{b}^{3}+295680\,{x}^{4/3}{a}^{4}{b}^{4}-215040\,{a}^{3}{b}^{5}x+143360\,{x}^{2/3}{a}^{2}{b}^{6}-81920\,\sqrt [3]{x}a{b}^{7}+32768\,{b}^{8} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^(2/3)+a*x)^(3/2),x)

[Out]

2/4849845*(b*x^(2/3)+a*x)^(3/2)*(b+a*x^(1/3))*(692835*x^(8/3)*a^8-583440*x^(7/3)
*a^7*b+480480*a^6*b^2*x^2-384384*x^(5/3)*a^5*b^3+295680*x^(4/3)*a^4*b^4-215040*a
^3*b^5*x+143360*x^(2/3)*a^2*b^6-81920*x^(1/3)*a*b^7+32768*b^8)/a^9/x

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Maxima [A]  time = 1.51444, size = 171, normalized size = 0.67 \[ \frac{2 \,{\left (692835 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{21}{2}} - 6126120 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} b + 23963940 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} b^{2} - 54318264 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} b^{3} + 78343650 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} b^{4} - 74070360 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} b^{5} + 45265220 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} b^{6} - 16628040 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} b^{7} + 2909907 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} b^{8}\right )}}{4849845 \, a^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/4849845*(692835*(a*x^(1/3) + b)^(21/2) - 6126120*(a*x^(1/3) + b)^(19/2)*b + 23
963940*(a*x^(1/3) + b)^(17/2)*b^2 - 54318264*(a*x^(1/3) + b)^(15/2)*b^3 + 783436
50*(a*x^(1/3) + b)^(13/2)*b^4 - 74070360*(a*x^(1/3) + b)^(11/2)*b^5 + 45265220*(
a*x^(1/3) + b)^(9/2)*b^6 - 16628040*(a*x^(1/3) + b)^(7/2)*b^7 + 2909907*(a*x^(1/
3) + b)^(5/2)*b^8)/a^9

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.234992, size = 493, normalized size = 1.93 \[ \frac{2}{1616615} \,{\left (\frac{65536 \, b^{\frac{21}{2}}{\rm sign}\left (x^{\frac{1}{3}}\right )}{a^{10}} + \frac{{\left (230945 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{21}{2}} a^{180} - 2297295 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} a^{180} b + 10270260 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} a^{180} b^{2} - 27159132 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} a^{180} b^{3} + 47006190 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} a^{180} b^{4} - 55552770 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{180} b^{5} + 45265220 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{180} b^{6} - 24942060 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{180} b^{7} + 8729721 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{180} b^{8} - 1616615 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{180} b^{9}\right )}{\rm sign}\left (x^{\frac{1}{3}}\right )}{a^{190}}\right )} a - \frac{2}{692835} \,{\left (\frac{32768 \, b^{\frac{19}{2}}{\rm sign}\left (x^{\frac{1}{3}}\right )}{a^{9}} - \frac{{\left (109395 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} a^{144} - 978120 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} a^{144} b + 3879876 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} a^{144} b^{2} - 8953560 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} a^{144} b^{3} + 13226850 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{144} b^{4} - 12932920 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{144} b^{5} + 8314020 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{144} b^{6} - 3325608 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{144} b^{7} + 692835 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{144} b^{8}\right )}{\rm sign}\left (x^{\frac{1}{3}}\right )}{a^{153}}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/1616615*(65536*b^(21/2)*sign(x^(1/3))/a^10 + (230945*(a*x^(1/3) + b)^(21/2)*a^
180 - 2297295*(a*x^(1/3) + b)^(19/2)*a^180*b + 10270260*(a*x^(1/3) + b)^(17/2)*a
^180*b^2 - 27159132*(a*x^(1/3) + b)^(15/2)*a^180*b^3 + 47006190*(a*x^(1/3) + b)^
(13/2)*a^180*b^4 - 55552770*(a*x^(1/3) + b)^(11/2)*a^180*b^5 + 45265220*(a*x^(1/
3) + b)^(9/2)*a^180*b^6 - 24942060*(a*x^(1/3) + b)^(7/2)*a^180*b^7 + 8729721*(a*
x^(1/3) + b)^(5/2)*a^180*b^8 - 1616615*(a*x^(1/3) + b)^(3/2)*a^180*b^9)*sign(x^(
1/3))/a^190)*a - 2/692835*(32768*b^(19/2)*sign(x^(1/3))/a^9 - (109395*(a*x^(1/3)
+ b)^(19/2)*a^144 - 978120*(a*x^(1/3) + b)^(17/2)*a^144*b + 3879876*(a*x^(1/3)
+ b)^(15/2)*a^144*b^2 - 8953560*(a*x^(1/3) + b)^(13/2)*a^144*b^3 + 13226850*(a*x
^(1/3) + b)^(11/2)*a^144*b^4 - 12932920*(a*x^(1/3) + b)^(9/2)*a^144*b^5 + 831402
0*(a*x^(1/3) + b)^(7/2)*a^144*b^6 - 3325608*(a*x^(1/3) + b)^(5/2)*a^144*b^7 + 69
2835*(a*x^(1/3) + b)^(3/2)*a^144*b^8)*sign(x^(1/3))/a^153)*b