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

Optimal. Leaf size=343 \[ -\frac{1048576 b^{11} \left (a x+b x^{2/3}\right )^{5/2}}{152108775 a^{12} x^{5/3}}+\frac{524288 b^{10} \left (a x+b x^{2/3}\right )^{5/2}}{30421755 a^{11} x^{4/3}}-\frac{131072 b^9 \left (a x+b x^{2/3}\right )^{5/2}}{4345965 a^{10} x}+\frac{65536 b^8 \left (a x+b x^{2/3}\right )^{5/2}}{1448655 a^9 x^{2/3}}-\frac{90112 b^7 \left (a x+b x^{2/3}\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}+\frac{45056 b^6 \left (a x+b x^{2/3}\right )^{5/2}}{557175 a^7}-\frac{11264 b^5 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{5/2}}{111435 a^6}+\frac{5632 b^4 x^{2/3} \left (a x+b x^{2/3}\right )^{5/2}}{45885 a^5}-\frac{352 b^3 x \left (a x+b x^{2/3}\right )^{5/2}}{2415 a^4}+\frac{176 b^2 x^{4/3} \left (a x+b x^{2/3}\right )^{5/2}}{1035 a^3}-\frac{44 b x^{5/3} \left (a x+b x^{2/3}\right )^{5/2}}{225 a^2}+\frac{2 x^2 \left (a x+b x^{2/3}\right )^{5/2}}{9 a} \]

[Out]

(45056*b^6*(b*x^(2/3) + a*x)^(5/2))/(557175*a^7) - (1048576*b^11*(b*x^(2/3) + a*
x)^(5/2))/(152108775*a^12*x^(5/3)) + (524288*b^10*(b*x^(2/3) + a*x)^(5/2))/(3042
1755*a^11*x^(4/3)) - (131072*b^9*(b*x^(2/3) + a*x)^(5/2))/(4345965*a^10*x) + (65
536*b^8*(b*x^(2/3) + a*x)^(5/2))/(1448655*a^9*x^(2/3)) - (90112*b^7*(b*x^(2/3) +
a*x)^(5/2))/(1448655*a^8*x^(1/3)) - (11264*b^5*x^(1/3)*(b*x^(2/3) + a*x)^(5/2))
/(111435*a^6) + (5632*b^4*x^(2/3)*(b*x^(2/3) + a*x)^(5/2))/(45885*a^5) - (352*b^
3*x*(b*x^(2/3) + a*x)^(5/2))/(2415*a^4) + (176*b^2*x^(4/3)*(b*x^(2/3) + a*x)^(5/
2))/(1035*a^3) - (44*b*x^(5/3)*(b*x^(2/3) + a*x)^(5/2))/(225*a^2) + (2*x^2*(b*x^
(2/3) + a*x)^(5/2))/(9*a)

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Rubi [A]  time = 0.952827, antiderivative size = 343, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{1048576 b^{11} \left (a x+b x^{2/3}\right )^{5/2}}{152108775 a^{12} x^{5/3}}+\frac{524288 b^{10} \left (a x+b x^{2/3}\right )^{5/2}}{30421755 a^{11} x^{4/3}}-\frac{131072 b^9 \left (a x+b x^{2/3}\right )^{5/2}}{4345965 a^{10} x}+\frac{65536 b^8 \left (a x+b x^{2/3}\right )^{5/2}}{1448655 a^9 x^{2/3}}-\frac{90112 b^7 \left (a x+b x^{2/3}\right )^{5/2}}{1448655 a^8 \sqrt [3]{x}}+\frac{45056 b^6 \left (a x+b x^{2/3}\right )^{5/2}}{557175 a^7}-\frac{11264 b^5 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{5/2}}{111435 a^6}+\frac{5632 b^4 x^{2/3} \left (a x+b x^{2/3}\right )^{5/2}}{45885 a^5}-\frac{352 b^3 x \left (a x+b x^{2/3}\right )^{5/2}}{2415 a^4}+\frac{176 b^2 x^{4/3} \left (a x+b x^{2/3}\right )^{5/2}}{1035 a^3}-\frac{44 b x^{5/3} \left (a x+b x^{2/3}\right )^{5/2}}{225 a^2}+\frac{2 x^2 \left (a x+b x^{2/3}\right )^{5/2}}{9 a} \]

Antiderivative was successfully verified.

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

[Out]

(45056*b^6*(b*x^(2/3) + a*x)^(5/2))/(557175*a^7) - (1048576*b^11*(b*x^(2/3) + a*
x)^(5/2))/(152108775*a^12*x^(5/3)) + (524288*b^10*(b*x^(2/3) + a*x)^(5/2))/(3042
1755*a^11*x^(4/3)) - (131072*b^9*(b*x^(2/3) + a*x)^(5/2))/(4345965*a^10*x) + (65
536*b^8*(b*x^(2/3) + a*x)^(5/2))/(1448655*a^9*x^(2/3)) - (90112*b^7*(b*x^(2/3) +
a*x)^(5/2))/(1448655*a^8*x^(1/3)) - (11264*b^5*x^(1/3)*(b*x^(2/3) + a*x)^(5/2))
/(111435*a^6) + (5632*b^4*x^(2/3)*(b*x^(2/3) + a*x)^(5/2))/(45885*a^5) - (352*b^
3*x*(b*x^(2/3) + a*x)^(5/2))/(2415*a^4) + (176*b^2*x^(4/3)*(b*x^(2/3) + a*x)^(5/
2))/(1035*a^3) - (44*b*x^(5/3)*(b*x^(2/3) + a*x)^(5/2))/(225*a^2) + (2*x^2*(b*x^
(2/3) + a*x)^(5/2))/(9*a)

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Rubi in Sympy [A]  time = 100.082, size = 325, normalized size = 0.95 \[ \frac{2 x^{2} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{9 a} - \frac{44 b x^{\frac{5}{3}} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{225 a^{2}} + \frac{176 b^{2} x^{\frac{4}{3}} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{1035 a^{3}} - \frac{352 b^{3} x \left (a x + b x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{2415 a^{4}} + \frac{5632 b^{4} x^{\frac{2}{3}} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{45885 a^{5}} - \frac{11264 b^{5} \sqrt [3]{x} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{111435 a^{6}} + \frac{45056 b^{6} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{557175 a^{7}} - \frac{90112 b^{7} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{1448655 a^{8} \sqrt [3]{x}} + \frac{65536 b^{8} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{1448655 a^{9} x^{\frac{2}{3}}} - \frac{131072 b^{9} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{4345965 a^{10} x} + \frac{524288 b^{10} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{30421755 a^{11} x^{\frac{4}{3}}} - \frac{1048576 b^{11} \left (a x + b x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{152108775 a^{12} x^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**2*(a*x + b*x**(2/3))**(5/2)/(9*a) - 44*b*x**(5/3)*(a*x + b*x**(2/3))**(5/2)
/(225*a**2) + 176*b**2*x**(4/3)*(a*x + b*x**(2/3))**(5/2)/(1035*a**3) - 352*b**3
*x*(a*x + b*x**(2/3))**(5/2)/(2415*a**4) + 5632*b**4*x**(2/3)*(a*x + b*x**(2/3))
**(5/2)/(45885*a**5) - 11264*b**5*x**(1/3)*(a*x + b*x**(2/3))**(5/2)/(111435*a**
6) + 45056*b**6*(a*x + b*x**(2/3))**(5/2)/(557175*a**7) - 90112*b**7*(a*x + b*x*
*(2/3))**(5/2)/(1448655*a**8*x**(1/3)) + 65536*b**8*(a*x + b*x**(2/3))**(5/2)/(1
448655*a**9*x**(2/3)) - 131072*b**9*(a*x + b*x**(2/3))**(5/2)/(4345965*a**10*x)
+ 524288*b**10*(a*x + b*x**(2/3))**(5/2)/(30421755*a**11*x**(4/3)) - 1048576*b**
11*(a*x + b*x**(2/3))**(5/2)/(152108775*a**12*x**(5/3))

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Mathematica [A]  time = 0.114692, size = 172, normalized size = 0.5 \[ \frac{2 \left (a \sqrt [3]{x}+b\right )^2 \sqrt{a x+b x^{2/3}} \left (16900975 a^{11} x^{11/3}-14872858 a^{10} b x^{10/3}[phone] a^9 b^2 x^3-11085360 a^8 b^3 x^{8/3}+9335040 a^7 b^4 x^{7/3}-7687680 a^6 b^5 x^2+6150144 a^5 b^6 x^{5/3}-4730880 a^4 b^7 x^{4/3}+3440640 a^3 b^8 x-2293760 a^2 b^9 x^{2/3}+1310720 a b^{10} \sqrt [3]{x}-524288 b^{11}\right )}{152108775 a^{12} \sqrt [3]{x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b + a*x^(1/3))^2*Sqrt[b*x^(2/3) + a*x]*(-524288*b^11 + 1310720*a*b^10*x^(1/3
) - 2293760*a^2*b^9*x^(2/3) + 3440640*a^3*b^8*x - 4730880*a^4*b^7*x^(4/3) + 6150
144*a^5*b^6*x^(5/3) - 7687680*a^6*b^5*x^2 + 9335040*a^7*b^4*x^(7/3) - 11085360*a
^8*b^3*x^(8/3) + 12932920*a^9*b^2*x^3 - 14872858*a^10*b*x^(10/3) + 16900975*a^11
*x^(11/3)))/(152108775*a^12*x^(1/3))

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Maple [A]  time = 0.007, size = 145, normalized size = 0.4 \[{\frac{2}{152108775\,x{a}^{12}} \left ( b{x}^{{\frac{2}{3}}}+ax \right ) ^{{\frac{3}{2}}} \left ( b+a\sqrt [3]{x} \right ) \left ( 16900975\,{x}^{11/3}{a}^{11}-14872858\,{x}^{10/3}{a}^{10}b+12932920\,{x}^{3}{a}^{9}{b}^{2}-11085360\,{x}^{8/3}{a}^{8}{b}^{3}+9335040\,{x}^{7/3}{a}^{7}{b}^{4}-7687680\,{x}^{2}{a}^{6}{b}^{5}+6150144\,{x}^{5/3}{a}^{5}{b}^{6}-4730880\,{x}^{4/3}{a}^{4}{b}^{7}+3440640\,x{a}^{3}{b}^{8}-2293760\,{x}^{2/3}{a}^{2}{b}^{9}+1310720\,\sqrt [3]{x}a{b}^{10}-524288\,{b}^{11} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/152108775*(b*x^(2/3)+a*x)^(3/2)*(b+a*x^(1/3))*(16900975*x^(11/3)*a^11-14872858
*x^(10/3)*a^10*b+12932920*x^3*a^9*b^2-11085360*x^(8/3)*a^8*b^3+9335040*x^(7/3)*a
^7*b^4-7687680*x^2*a^6*b^5+6150144*x^(5/3)*a^5*b^6-4730880*x^(4/3)*a^4*b^7+34406
40*x*a^3*b^8-2293760*x^(2/3)*a^2*b^9+1310720*x^(1/3)*a*b^10-524288*b^11)/x/a^12

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Maxima [A]  time = 1.53639, size = 228, normalized size = 0.66 \[ \frac{2 \,{\left (16900975 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{27}{2}} - 200783583 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{25}{2}} b + 1091215125 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{23}{2}} b^{2} - 3585421125 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{21}{2}} b^{3} + 7925667750 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{19}{2}} b^{4} - 12401338950 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} b^{5} + 14054850810 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} b^{6} - 11583668250 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} b^{7} + 6844894875 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} b^{8} - 2788660875 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} b^{9} + 717084225 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} b^{10} - 91265265 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} b^{11}\right )}}{152108775 \, a^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/152108775*(16900975*(a*x^(1/3) + b)^(27/2) - 200783583*(a*x^(1/3) + b)^(25/2)*
b + 1091215125*(a*x^(1/3) + b)^(23/2)*b^2 - 3585421125*(a*x^(1/3) + b)^(21/2)*b^
3 + 7925667750*(a*x^(1/3) + b)^(19/2)*b^4 - 12401338950*(a*x^(1/3) + b)^(17/2)*b
^5 + 14054850810*(a*x^(1/3) + b)^(15/2)*b^6 - 11583668250*(a*x^(1/3) + b)^(13/2)
*b^7 + 6844894875*(a*x^(1/3) + b)^(11/2)*b^8 - 2788660875*(a*x^(1/3) + b)^(9/2)*
b^9 + 717084225*(a*x^(1/3) + b)^(7/2)*b^10 - 91265265*(a*x^(1/3) + b)^(5/2)*b^11
)/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((a*x + b*x^(2/3))^(3/2)*x^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**2*(b*x**(2/3)+a*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.249311, size = 630, normalized size = 1.84 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/152108775*(4194304*b^(27/2)*sign(x^(1/3))/a^13 - (16900975*(a*x^(1/3) + b)^(2
7/2)*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 + 11888501625*
(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 - 11154643500*(a*x^
(1/3) + b)^(9/2)*a^312*b^9 + 4302505350*(a*x^(1/3) + b)^(7/2)*a^312*b^10 - 10951
83180*(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)*a + 2/16900975*(524288*b^(25/2)*sign(x^(1/3))/a^12 + (
2028117*(a*x^(1/3) + b)^(25/2)*a^264 - 24249225*(a*x^(1/3) + b)^(23/2)*a^264*b +
132793375*(a*x^(1/3) + b)^(21/2)*a^264*b^2 - 440314875*(a*x^(1/3) + b)^(19/2)*a
^264*b^3 + 984233250*(a*x^(1/3) + b)^(17/2)*a^264*b^4 - 1561650090*(a*x^(1/3) +
b)^(15/2)*a^264*b^5 + 1801903950*(a*x^(1/3) + b)^(13/2)*a^264*b^6 - 1521087750*(
a*x^(1/3) + b)^(11/2)*a^264*b^7 + 929553625*(a*x^(1/3) + b)^(9/2)*a^264*b^8 - 39
8380125*(a*x^(1/3) + b)^(7/2)*a^264*b^9 + 111546435*(a*x^(1/3) + b)^(5/2)*a^264*
b^10 - 16900975*(a*x^(1/3) + b)^(3/2)*a^264*b^11)*sign(x^(1/3))/a^276)*b