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

Optimal. Leaf size=161 \[ -\frac{512 a^5 \left (a x^2+b x^3\right )^{5/2}}{45045 b^6 x^5}+\frac{256 a^4 \left (a x^2+b x^3\right )^{5/2}}{9009 b^5 x^4}-\frac{64 a^3 \left (a x^2+b x^3\right )^{5/2}}{1287 b^4 x^3}+\frac{32 a^2 \left (a x^2+b x^3\right )^{5/2}}{429 b^3 x^2}-\frac{4 a \left (a x^2+b x^3\right )^{5/2}}{39 b^2 x}+\frac{2 \left (a x^2+b x^3\right )^{5/2}}{15 b} \]

[Out]

(2*(a*x^2 + b*x^3)^(5/2))/(15*b) - (512*a^5*(a*x^2 + b*x^3)^(5/2))/(45045*b^6*x^
5) + (256*a^4*(a*x^2 + b*x^3)^(5/2))/(9009*b^5*x^4) - (64*a^3*(a*x^2 + b*x^3)^(5
/2))/(1287*b^4*x^3) + (32*a^2*(a*x^2 + b*x^3)^(5/2))/(429*b^3*x^2) - (4*a*(a*x^2
+ b*x^3)^(5/2))/(39*b^2*x)

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Rubi [A]  time = 0.357242, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{512 a^5 \left (a x^2+b x^3\right )^{5/2}}{45045 b^6 x^5}+\frac{256 a^4 \left (a x^2+b x^3\right )^{5/2}}{9009 b^5 x^4}-\frac{64 a^3 \left (a x^2+b x^3\right )^{5/2}}{1287 b^4 x^3}+\frac{32 a^2 \left (a x^2+b x^3\right )^{5/2}}{429 b^3 x^2}-\frac{4 a \left (a x^2+b x^3\right )^{5/2}}{39 b^2 x}+\frac{2 \left (a x^2+b x^3\right )^{5/2}}{15 b} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a*x^2 + b*x^3)^(5/2))/(15*b) - (512*a^5*(a*x^2 + b*x^3)^(5/2))/(45045*b^6*x^
5) + (256*a^4*(a*x^2 + b*x^3)^(5/2))/(9009*b^5*x^4) - (64*a^3*(a*x^2 + b*x^3)^(5
/2))/(1287*b^4*x^3) + (32*a^2*(a*x^2 + b*x^3)^(5/2))/(429*b^3*x^2) - (4*a*(a*x^2
+ b*x^3)^(5/2))/(39*b^2*x)

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Rubi in Sympy [A]  time = 38.5697, size = 150, normalized size = 0.93 \[ - \frac{512 a^{5} \left (a x^{2} + b x^{3}\right )^{\frac{5}{2}}}{45045 b^{6} x^{5}} + \frac{256 a^{4} \left (a x^{2} + b x^{3}\right )^{\frac{5}{2}}}{9009 b^{5} x^{4}} - \frac{64 a^{3} \left (a x^{2} + b x^{3}\right )^{\frac{5}{2}}}{1287 b^{4} x^{3}} + \frac{32 a^{2} \left (a x^{2} + b x^{3}\right )^{\frac{5}{2}}}{429 b^{3} x^{2}} - \frac{4 a \left (a x^{2} + b x^{3}\right )^{\frac{5}{2}}}{39 b^{2} x} + \frac{2 \left (a x^{2} + b x^{3}\right )^{\frac{5}{2}}}{15 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-512*a**5*(a*x**2 + b*x**3)**(5/2)/(45045*b**6*x**5) + 256*a**4*(a*x**2 + b*x**3
)**(5/2)/(9009*b**5*x**4) - 64*a**3*(a*x**2 + b*x**3)**(5/2)/(1287*b**4*x**3) +
32*a**2*(a*x**2 + b*x**3)**(5/2)/(429*b**3*x**2) - 4*a*(a*x**2 + b*x**3)**(5/2)/
(39*b**2*x) + 2*(a*x**2 + b*x**3)**(5/2)/(15*b)

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Mathematica [A]  time = 0.0467646, size = 80, normalized size = 0.5 \[ \frac{2 x (a+b x)^3 \left (-256 a^5+640 a^4 b x-1120 a^3 b^2 x^2+1680 a^2 b^3 x^3-2310 a b^4 x^4+3003 b^5 x^5\right )}{45045 b^6 \sqrt{x^2 (a+b x)}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*(a + b*x)^3*(-256*a^5 + 640*a^4*b*x - 1120*a^3*b^2*x^2 + 1680*a^2*b^3*x^3 -
2310*a*b^4*x^4 + 3003*b^5*x^5))/(45045*b^6*Sqrt[x^2*(a + b*x)])

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Maple [A]  time = 0.009, size = 79, normalized size = 0.5 \[ -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( -3003\,{x}^{5}{b}^{5}+2310\,a{x}^{4}{b}^{4}-1680\,{a}^{2}{x}^{3}{b}^{3}+1120\,{x}^{2}{a}^{3}{b}^{2}-640\,{a}^{4}xb+256\,{a}^{5} \right ) }{45045\,{b}^{6}{x}^{3}} \left ( b{x}^{3}+a{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x^3+a*x^2)^(3/2),x)

[Out]

-2/45045*(b*x+a)*(-3003*b^5*x^5+2310*a*b^4*x^4-1680*a^2*b^3*x^3+1120*a^3*b^2*x^2
-640*a^4*b*x+256*a^5)*(b*x^3+a*x^2)^(3/2)/b^6/x^3

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Maxima [A]  time = 1.41135, size = 116, normalized size = 0.72 \[ \frac{2 \,{\left (3003 \, b^{7} x^{7} + 3696 \, a b^{6} x^{6} + 63 \, a^{2} b^{5} x^{5} - 70 \, a^{3} b^{4} x^{4} + 80 \, a^{4} b^{3} x^{3} - 96 \, a^{5} b^{2} x^{2} + 128 \, a^{6} b x - 256 \, a^{7}\right )} \sqrt{b x + a}}{45045 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(3003*b^7*x^7 + 3696*a*b^6*x^6 + 63*a^2*b^5*x^5 - 70*a^3*b^4*x^4 + 80*a^
4*b^3*x^3 - 96*a^5*b^2*x^2 + 128*a^6*b*x - 256*a^7)*sqrt(b*x + a)/b^6

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Fricas [A]  time = 0.216184, size = 128, normalized size = 0.8 \[ \frac{2 \,{\left (3003 \, b^{7} x^{7} + 3696 \, a b^{6} x^{6} + 63 \, a^{2} b^{5} x^{5} - 70 \, a^{3} b^{4} x^{4} + 80 \, a^{4} b^{3} x^{3} - 96 \, a^{5} b^{2} x^{2} + 128 \, a^{6} b x - 256 \, a^{7}\right )} \sqrt{b x^{3} + a x^{2}}}{45045 \, b^{6} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(3003*b^7*x^7 + 3696*a*b^6*x^6 + 63*a^2*b^5*x^5 - 70*a^3*b^4*x^4 + 80*a^
4*b^3*x^3 - 96*a^5*b^2*x^2 + 128*a^6*b*x - 256*a^7)*sqrt(b*x^3 + a*x^2)/(b^6*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.228395, size = 294, normalized size = 1.83 \[ \frac{512 \, a^{\frac{15}{2}}{\rm sign}\left (x\right )}{45045 \, b^{6}} + \frac{2 \,{\left (\frac{5 \,{\left (693 \,{\left (b x + a\right )}^{\frac{13}{2}} b^{60} - 4095 \,{\left (b x + a\right )}^{\frac{11}{2}} a b^{60} + 10010 \,{\left (b x + a\right )}^{\frac{9}{2}} a^{2} b^{60} - 12870 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{3} b^{60} + 9009 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{4} b^{60} - 3003 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{5} b^{60}\right )} a{\rm sign}\left (x\right )}{b^{65}} + \frac{{\left (3003 \,{\left (b x + a\right )}^{\frac{15}{2}} b^{84} - 20790 \,{\left (b x + a\right )}^{\frac{13}{2}} a b^{84} + 61425 \,{\left (b x + a\right )}^{\frac{11}{2}} a^{2} b^{84} - 100100 \,{\left (b x + a\right )}^{\frac{9}{2}} a^{3} b^{84} + 96525 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{4} b^{84} - 54054 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{5} b^{84} + 15015 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{6} b^{84}\right )}{\rm sign}\left (x\right )}{b^{89}}\right )}}{45045 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

512/45045*a^(15/2)*sign(x)/b^6 + 2/45045*(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)*a*sign(
x)/b^65 + (3003*(b*x + a)^(15/2)*b^84 - 20790*(b*x + a)^(13/2)*a*b^84 + 61425*(b
*x + a)^(11/2)*a^2*b^84 - 100100*(b*x + a)^(9/2)*a^3*b^84 + 96525*(b*x + a)^(7/2
)*a^4*b^84 - 54054*(b*x + a)^(5/2)*a^5*b^84 + 15015*(b*x + a)^(3/2)*a^6*b^84)*si
gn(x)/b^89)/b