3.83 \(\int x^{5/2} \left (b x+c x^2\right )^{3/2} \, dx\)

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

[Out]

(256*b^4*(b*x + c*x^2)^(5/2))/(15015*c^5*x^(5/2)) - (128*b^3*(b*x + c*x^2)^(5/2)
)/(3003*c^4*x^(3/2)) + (32*b^2*(b*x + c*x^2)^(5/2))/(429*c^3*Sqrt[x]) - (16*b*Sq
rt[x]*(b*x + c*x^2)^(5/2))/(143*c^2) + (2*x^(3/2)*(b*x + c*x^2)^(5/2))/(13*c)

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Rubi [A]  time = 0.16907, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{256 b^4 \left (b x+c x^2\right )^{5/2}}{15015 c^5 x^{5/2}}-\frac{128 b^3 \left (b x+c x^2\right )^{5/2}}{3003 c^4 x^{3/2}}+\frac{32 b^2 \left (b x+c x^2\right )^{5/2}}{429 c^3 \sqrt{x}}-\frac{16 b \sqrt{x} \left (b x+c x^2\right )^{5/2}}{143 c^2}+\frac{2 x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)*(b*x + c*x^2)^(3/2),x]

[Out]

(256*b^4*(b*x + c*x^2)^(5/2))/(15015*c^5*x^(5/2)) - (128*b^3*(b*x + c*x^2)^(5/2)
)/(3003*c^4*x^(3/2)) + (32*b^2*(b*x + c*x^2)^(5/2))/(429*c^3*Sqrt[x]) - (16*b*Sq
rt[x]*(b*x + c*x^2)^(5/2))/(143*c^2) + (2*x^(3/2)*(b*x + c*x^2)^(5/2))/(13*c)

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Rubi in Sympy [A]  time = 18.267, size = 128, normalized size = 0.94 \[ \frac{256 b^{4} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{15015 c^{5} x^{\frac{5}{2}}} - \frac{128 b^{3} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{3003 c^{4} x^{\frac{3}{2}}} + \frac{32 b^{2} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{429 c^{3} \sqrt{x}} - \frac{16 b \sqrt{x} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{143 c^{2}} + \frac{2 x^{\frac{3}{2}} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{13 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(c*x**2+b*x)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0444456, size = 64, normalized size = 0.47 \[ \frac{2 (x (b+c x))^{5/2} \left (128 b^4-320 b^3 c x+560 b^2 c^2 x^2-840 b c^3 x^3+1155 c^4 x^4\right )}{15015 c^5 x^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)*(b*x + c*x^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.009, size = 66, normalized size = 0.5 \[{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 1155\,{x}^{4}{c}^{4}-840\,b{x}^{3}{c}^{3}+560\,{b}^{2}{x}^{2}{c}^{2}-320\,{b}^{3}xc+128\,{b}^{4} \right ) }{15015\,{c}^{5}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(c*x^2+b*x)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.718444, size = 198, normalized size = 1.46 \[ \frac{2 \,{\left (5 \,{\left (693 \, c^{6} x^{6} + 63 \, b c^{5} x^{5} - 70 \, b^{2} c^{4} x^{4} + 80 \, b^{3} c^{3} x^{3} - 96 \, b^{4} c^{2} x^{2} + 128 \, b^{5} c x - 256 \, b^{6}\right )} x^{5} + 13 \,{\left (315 \, b c^{5} x^{6} + 35 \, b^{2} c^{4} x^{5} - 40 \, b^{3} c^{3} x^{4} + 48 \, b^{4} c^{2} x^{3} - 64 \, b^{5} c x^{2} + 128 \, b^{6} x\right )} x^{4}\right )} \sqrt{c x + b}}{45045 \, c^{5} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)*x^(5/2),x, algorithm="maxima")

[Out]

2/45045*(5*(693*c^6*x^6 + 63*b*c^5*x^5 - 70*b^2*c^4*x^4 + 80*b^3*c^3*x^3 - 96*b^
4*c^2*x^2 + 128*b^5*c*x - 256*b^6)*x^5 + 13*(315*b*c^5*x^6 + 35*b^2*c^4*x^5 - 40
*b^3*c^3*x^4 + 48*b^4*c^2*x^3 - 64*b^5*c*x^2 + 128*b^6*x)*x^4)*sqrt(c*x + b)/(c^
5*x^5)

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Fricas [A]  time = 0.220249, size = 130, normalized size = 0.96 \[ \frac{2 \,{\left (1155 \, c^{7} x^{8} + 2625 \, b c^{6} x^{7} + 1505 \, b^{2} c^{5} x^{6} - 5 \, b^{3} c^{4} x^{5} + 8 \, b^{4} c^{3} x^{4} - 16 \, b^{5} c^{2} x^{3} + 64 \, b^{6} c x^{2} + 128 \, b^{7} x\right )}}{15015 \, \sqrt{c x^{2} + b x} c^{5} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)*x^(5/2),x, algorithm="fricas")

[Out]

2/15015*(1155*c^7*x^8 + 2625*b*c^6*x^7 + 1505*b^2*c^5*x^6 - 5*b^3*c^4*x^5 + 8*b^
4*c^3*x^4 - 16*b^5*c^2*x^3 + 64*b^6*c*x^2 + 128*b^7*x)/(sqrt(c*x^2 + b*x)*c^5*sq
rt(x))

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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**(5/2)*(c*x**2+b*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21705, size = 213, normalized size = 1.57 \[ \frac{2}{9009} \, c{\left (\frac{256 \, b^{\frac{13}{2}}}{c^{6}} + \frac{693 \,{\left (c x + b\right )}^{\frac{13}{2}} - 4095 \,{\left (c x + b\right )}^{\frac{11}{2}} b + 10010 \,{\left (c x + b\right )}^{\frac{9}{2}} b^{2} - 12870 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{3} + 9009 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{4} - 3003 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{5}}{c^{6}}\right )} - \frac{2}{3465} \, b{\left (\frac{128 \, b^{\frac{11}{2}}}{c^{5}} - \frac{315 \,{\left (c x + b\right )}^{\frac{11}{2}} - 1540 \,{\left (c x + b\right )}^{\frac{9}{2}} b + 2970 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{2} - 2772 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{3} + 1155 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{4}}{c^{5}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)*x^(5/2),x, algorithm="giac")

[Out]

2/9009*c*(256*b^(13/2)/c^6 + (693*(c*x + b)^(13/2) - 4095*(c*x + b)^(11/2)*b + 1
0010*(c*x + b)^(9/2)*b^2 - 12870*(c*x + b)^(7/2)*b^3 + 9009*(c*x + b)^(5/2)*b^4
- 3003*(c*x + b)^(3/2)*b^5)/c^6) - 2/3465*b*(128*b^(11/2)/c^5 - (315*(c*x + b)^(
11/2) - 1540*(c*x + b)^(9/2)*b + 2970*(c*x + b)^(7/2)*b^2 - 2772*(c*x + b)^(5/2)
*b^3 + 1155*(c*x + b)^(3/2)*b^4)/c^5)