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

Optimal. Leaf size=108 \[ -\frac{32 b^3 \left (b x+c x^2\right )^{5/2}}{1155 c^4 x^{5/2}}+\frac{16 b^2 \left (b x+c x^2\right )^{5/2}}{231 c^3 x^{3/2}}-\frac{4 b \left (b x+c x^2\right )^{5/2}}{33 c^2 \sqrt{x}}+\frac{2 \sqrt{x} \left (b x+c x^2\right )^{5/2}}{11 c} \]

[Out]

(-32*b^3*(b*x + c*x^2)^(5/2))/(1155*c^4*x^(5/2)) + (16*b^2*(b*x + c*x^2)^(5/2))/
(231*c^3*x^(3/2)) - (4*b*(b*x + c*x^2)^(5/2))/(33*c^2*Sqrt[x]) + (2*Sqrt[x]*(b*x
+ c*x^2)^(5/2))/(11*c)

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Rubi [A]  time = 0.130372, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{32 b^3 \left (b x+c x^2\right )^{5/2}}{1155 c^4 x^{5/2}}+\frac{16 b^2 \left (b x+c x^2\right )^{5/2}}{231 c^3 x^{3/2}}-\frac{4 b \left (b x+c x^2\right )^{5/2}}{33 c^2 \sqrt{x}}+\frac{2 \sqrt{x} \left (b x+c x^2\right )^{5/2}}{11 c} \]

Antiderivative was successfully verified.

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

[Out]

(-32*b^3*(b*x + c*x^2)^(5/2))/(1155*c^4*x^(5/2)) + (16*b^2*(b*x + c*x^2)^(5/2))/
(231*c^3*x^(3/2)) - (4*b*(b*x + c*x^2)^(5/2))/(33*c^2*Sqrt[x]) + (2*Sqrt[x]*(b*x
+ c*x^2)^(5/2))/(11*c)

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Rubi in Sympy [A]  time = 13.5178, size = 100, normalized size = 0.93 \[ - \frac{32 b^{3} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{1155 c^{4} x^{\frac{5}{2}}} + \frac{16 b^{2} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{231 c^{3} x^{\frac{3}{2}}} - \frac{4 b \left (b x + c x^{2}\right )^{\frac{5}{2}}}{33 c^{2} \sqrt{x}} + \frac{2 \sqrt{x} \left (b x + c x^{2}\right )^{\frac{5}{2}}}{11 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-32*b**3*(b*x + c*x**2)**(5/2)/(1155*c**4*x**(5/2)) + 16*b**2*(b*x + c*x**2)**(5
/2)/(231*c**3*x**(3/2)) - 4*b*(b*x + c*x**2)**(5/2)/(33*c**2*sqrt(x)) + 2*sqrt(x
)*(b*x + c*x**2)**(5/2)/(11*c)

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Mathematica [A]  time = 0.0407198, size = 53, normalized size = 0.49 \[ \frac{2 (x (b+c x))^{5/2} \left (-16 b^3+40 b^2 c x-70 b c^2 x^2+105 c^3 x^3\right )}{1155 c^4 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(x*(b + c*x))^(5/2)*(-16*b^3 + 40*b^2*c*x - 70*b*c^2*x^2 + 105*c^3*x^3))/(115
5*c^4*x^(5/2))

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Maple [A]  time = 0.007, size = 55, normalized size = 0.5 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -105\,{x}^{3}{c}^{3}+70\,b{x}^{2}{c}^{2}-40\,{b}^{2}xc+16\,{b}^{3} \right ) }{1155\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/1155*(c*x+b)*(-105*c^3*x^3+70*b*c^2*x^2-40*b^2*c*x+16*b^3)*(c*x^2+b*x)^(3/2)/
c^4/x^(3/2)

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Maxima [A]  time = 0.720438, size = 167, normalized size = 1.55 \[ \frac{2 \,{\left ({\left (315 \, c^{5} x^{5} + 35 \, b c^{4} x^{4} - 40 \, b^{2} c^{3} x^{3} + 48 \, b^{3} c^{2} x^{2} - 64 \, b^{4} c x + 128 \, b^{5}\right )} x^{4} + 11 \,{\left (35 \, b c^{4} x^{5} + 5 \, b^{2} c^{3} x^{4} - 6 \, b^{3} c^{2} x^{3} + 8 \, b^{4} c x^{2} - 16 \, b^{5} x\right )} x^{3}\right )} \sqrt{c x + b}}{3465 \, c^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3465*((315*c^5*x^5 + 35*b*c^4*x^4 - 40*b^2*c^3*x^3 + 48*b^3*c^2*x^2 - 64*b^4*c
*x + 128*b^5)*x^4 + 11*(35*b*c^4*x^5 + 5*b^2*c^3*x^4 - 6*b^3*c^2*x^3 + 8*b^4*c*x
^2 - 16*b^5*x)*x^3)*sqrt(c*x + b)/(c^4*x^4)

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Fricas [A]  time = 0.220742, size = 115, normalized size = 1.06 \[ \frac{2 \,{\left (105 \, c^{6} x^{7} + 245 \, b c^{5} x^{6} + 145 \, b^{2} c^{4} x^{5} - b^{3} c^{3} x^{4} + 2 \, b^{4} c^{2} x^{3} - 8 \, b^{5} c x^{2} - 16 \, b^{6} x\right )}}{1155 \, \sqrt{c x^{2} + b x} c^{4} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/1155*(105*c^6*x^7 + 245*b*c^5*x^6 + 145*b^2*c^4*x^5 - b^3*c^3*x^4 + 2*b^4*c^2*
x^3 - 8*b^5*c*x^2 - 16*b^6*x)/(sqrt(c*x^2 + b*x)*c^4*sqrt(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**(3/2)*(c*x**2+b*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216215, size = 181, normalized size = 1.68 \[ -\frac{2}{3465} \, c{\left (\frac{128 \, b^{\frac{11}{2}}}{c^{5}} - \frac{315 \,{\left (c x + b\right )}^{\frac{11}{2}} - 1540 \,{\left (c x + b\right )}^{\frac{9}{2}} b + 2970 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{2} - 2772 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{3} + 1155 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{4}}{c^{5}}\right )} + \frac{2}{315} \, b{\left (\frac{16 \, b^{\frac{9}{2}}}{c^{4}} + \frac{35 \,{\left (c x + b\right )}^{\frac{9}{2}} - 135 \,{\left (c x + b\right )}^{\frac{7}{2}} b + 189 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{3}}{c^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3465*c*(128*b^(11/2)/c^5 - (315*(c*x + b)^(11/2) - 1540*(c*x + b)^(9/2)*b + 2
970*(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) + 2/315*b*(16*b^(9/2)/c^4 + (35*(c*x + b)^(9/2) - 135*(c*x + b)^(7/2)*b + 18
9*(c*x + b)^(5/2)*b^2 - 105*(c*x + b)^(3/2)*b^3)/c^4)