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

Optimal. Leaf size=136 \[ \frac{256 a^4 \left (a x^2+b x^3\right )^{5/2}}{15015 b^5 x^5}-\frac{128 a^3 \left (a x^2+b x^3\right )^{5/2}}{3003 b^4 x^4}+\frac{32 a^2 \left (a x^2+b x^3\right )^{5/2}}{429 b^3 x^3}-\frac{16 a \left (a x^2+b x^3\right )^{5/2}}{143 b^2 x^2}+\frac{2 \left (a x^2+b x^3\right )^{5/2}}{13 b x} \]

[Out]

(256*a^4*(a*x^2 + b*x^3)^(5/2))/(15015*b^5*x^5) - (128*a^3*(a*x^2 + b*x^3)^(5/2)
)/(3003*b^4*x^4) + (32*a^2*(a*x^2 + b*x^3)^(5/2))/(429*b^3*x^3) - (16*a*(a*x^2 +
b*x^3)^(5/2))/(143*b^2*x^2) + (2*(a*x^2 + b*x^3)^(5/2))/(13*b*x)

_______________________________________________________________________________________

Rubi [A]  time = 0.278172, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{256 a^4 \left (a x^2+b x^3\right )^{5/2}}{15015 b^5 x^5}-\frac{128 a^3 \left (a x^2+b x^3\right )^{5/2}}{3003 b^4 x^4}+\frac{32 a^2 \left (a x^2+b x^3\right )^{5/2}}{429 b^3 x^3}-\frac{16 a \left (a x^2+b x^3\right )^{5/2}}{143 b^2 x^2}+\frac{2 \left (a x^2+b x^3\right )^{5/2}}{13 b x} \]

Antiderivative was successfully verified.

[In]  Int[x*(a*x^2 + b*x^3)^(3/2),x]

[Out]

(256*a^4*(a*x^2 + b*x^3)^(5/2))/(15015*b^5*x^5) - (128*a^3*(a*x^2 + b*x^3)^(5/2)
)/(3003*b^4*x^4) + (32*a^2*(a*x^2 + b*x^3)^(5/2))/(429*b^3*x^3) - (16*a*(a*x^2 +
b*x^3)^(5/2))/(143*b^2*x^2) + (2*(a*x^2 + b*x^3)^(5/2))/(13*b*x)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 28.9365, size = 126, normalized size = 0.93 \[ \frac{256 a^{4} \left (a x^{2} + b x^{3}\right )^{\frac{5}{2}}}{15015 b^{5} x^{5}} - \frac{128 a^{3} \left (a x^{2} + b x^{3}\right )^{\frac{5}{2}}}{3003 b^{4} x^{4}} + \frac{32 a^{2} \left (a x^{2} + b x^{3}\right )^{\frac{5}{2}}}{429 b^{3} x^{3}} - \frac{16 a \left (a x^{2} + b x^{3}\right )^{\frac{5}{2}}}{143 b^{2} x^{2}} + \frac{2 \left (a x^{2} + b x^{3}\right )^{\frac{5}{2}}}{13 b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

256*a**4*(a*x**2 + b*x**3)**(5/2)/(15015*b**5*x**5) - 128*a**3*(a*x**2 + b*x**3)
**(5/2)/(3003*b**4*x**4) + 32*a**2*(a*x**2 + b*x**3)**(5/2)/(429*b**3*x**3) - 16
*a*(a*x**2 + b*x**3)**(5/2)/(143*b**2*x**2) + 2*(a*x**2 + b*x**3)**(5/2)/(13*b*x
)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0394142, size = 69, normalized size = 0.51 \[ \frac{2 x (a+b x)^3 \left (128 a^4-320 a^3 b x+560 a^2 b^2 x^2-840 a b^3 x^3+1155 b^4 x^4\right )}{15015 b^5 \sqrt{x^2 (a+b x)}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(a + b*x)^3*(128*a^4 - 320*a^3*b*x + 560*a^2*b^2*x^2 - 840*a*b^3*x^3 + 1155
*b^4*x^4))/(15015*b^5*Sqrt[x^2*(a + b*x)])

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 68, normalized size = 0.5 \[{\frac{ \left ( 2\,bx+2\,a \right ) \left ( 1155\,{x}^{4}{b}^{4}-840\,a{b}^{3}{x}^{3}+560\,{a}^{2}{x}^{2}{b}^{2}-320\,x{a}^{3}b+128\,{a}^{4} \right ) }{15015\,{b}^{5}{x}^{3}} \left ( b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^3+a*x^2)^(3/2),x)

[Out]

2/15015*(b*x+a)*(1155*b^4*x^4-840*a*b^3*x^3+560*a^2*b^2*x^2-320*a^3*b*x+128*a^4)
*(b*x^3+a*x^2)^(3/2)/b^5/x^3

_______________________________________________________________________________________

Maxima [A]  time = 1.40243, size = 101, normalized size = 0.74 \[ \frac{2 \,{\left (1155 \, b^{6} x^{6} + 1470 \, a b^{5} x^{5} + 35 \, a^{2} b^{4} x^{4} - 40 \, a^{3} b^{3} x^{3} + 48 \, a^{4} b^{2} x^{2} - 64 \, a^{5} b x + 128 \, a^{6}\right )} \sqrt{b x + a}}{15015 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15015*(1155*b^6*x^6 + 1470*a*b^5*x^5 + 35*a^2*b^4*x^4 - 40*a^3*b^3*x^3 + 48*a^
4*b^2*x^2 - 64*a^5*b*x + 128*a^6)*sqrt(b*x + a)/b^5

_______________________________________________________________________________________

Fricas [A]  time = 0.226774, size = 113, normalized size = 0.83 \[ \frac{2 \,{\left (1155 \, b^{6} x^{6} + 1470 \, a b^{5} x^{5} + 35 \, a^{2} b^{4} x^{4} - 40 \, a^{3} b^{3} x^{3} + 48 \, a^{4} b^{2} x^{2} - 64 \, a^{5} b x + 128 \, a^{6}\right )} \sqrt{b x^{3} + a x^{2}}}{15015 \, b^{5} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15015*(1155*b^6*x^6 + 1470*a*b^5*x^5 + 35*a^2*b^4*x^4 - 40*a^3*b^3*x^3 + 48*a^
4*b^2*x^2 - 64*a^5*b*x + 128*a^6)*sqrt(b*x^3 + a*x^2)/(b^5*x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.223762, size = 255, normalized size = 1.88 \[ -\frac{256 \, a^{\frac{13}{2}}{\rm sign}\left (x\right )}{15015 \, b^{5}} + \frac{2 \,{\left (\frac{13 \,{\left (315 \,{\left (b x + a\right )}^{\frac{11}{2}} b^{40} - 1540 \,{\left (b x + a\right )}^{\frac{9}{2}} a b^{40} + 2970 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2} b^{40} - 2772 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{3} b^{40} + 1155 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4} b^{40}\right )} a{\rm sign}\left (x\right )}{b^{44}} + \frac{5 \,{\left (693 \,{\left (b x + a\right )}^{\frac{13}{2}} b^{60} - 4095 \,{\left (b x + a\right )}^{\frac{11}{2}} a b^{60} + 10010 \,{\left (b x + a\right )}^{\frac{9}{2}} a^{2} b^{60} - 12870 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{3} b^{60} + 9009 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{4} b^{60} - 3003 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{5} b^{60}\right )}{\rm sign}\left (x\right )}{b^{64}}\right )}}{45045 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-256/15015*a^(13/2)*sign(x)/b^5 + 2/45045*(13*(315*(b*x + a)^(11/2)*b^40 - 1540*
(b*x + a)^(9/2)*a*b^40 + 2970*(b*x + a)^(7/2)*a^2*b^40 - 2772*(b*x + a)^(5/2)*a^
3*b^40 + 1155*(b*x + a)^(3/2)*a^4*b^40)*a*sign(x)/b^44 + 5*(693*(b*x + a)^(13/2)
*b^60 - 4095*(b*x + a)^(11/2)*a*b^60 + 10010*(b*x + a)^(9/2)*a^2*b^60 - 12870*(b
*x + a)^(7/2)*a^3*b^60 + 9009*(b*x + a)^(5/2)*a^4*b^60 - 3003*(b*x + a)^(3/2)*a^
5*b^60)*sign(x)/b^64)/b