3.352 \(\int \frac{x^3}{(a+b x)^{5/2}} \, dx\)

Optimal. Leaf size=68 \[ \frac{2 a^3}{3 b^4 (a+b x)^{3/2}}-\frac{6 a^2}{b^4 \sqrt{a+b x}}-\frac{6 a \sqrt{a+b x}}{b^4}+\frac{2 (a+b x)^{3/2}}{3 b^4} \]

[Out]

(2*a^3)/(3*b^4*(a + b*x)^(3/2)) - (6*a^2)/(b^4*Sqrt[a + b*x]) - (6*a*Sqrt[a + b*
x])/b^4 + (2*(a + b*x)^(3/2))/(3*b^4)

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Rubi [A]  time = 0.0515173, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{2 a^3}{3 b^4 (a+b x)^{3/2}}-\frac{6 a^2}{b^4 \sqrt{a+b x}}-\frac{6 a \sqrt{a+b x}}{b^4}+\frac{2 (a+b x)^{3/2}}{3 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*x)^(5/2),x]

[Out]

(2*a^3)/(3*b^4*(a + b*x)^(3/2)) - (6*a^2)/(b^4*Sqrt[a + b*x]) - (6*a*Sqrt[a + b*
x])/b^4 + (2*(a + b*x)^(3/2))/(3*b^4)

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Rubi in Sympy [A]  time = 10.9518, size = 65, normalized size = 0.96 \[ \frac{2 a^{3}}{3 b^{4} \left (a + b x\right )^{\frac{3}{2}}} - \frac{6 a^{2}}{b^{4} \sqrt{a + b x}} - \frac{6 a \sqrt{a + b x}}{b^{4}} + \frac{2 \left (a + b x\right )^{\frac{3}{2}}}{3 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**3/(3*b**4*(a + b*x)**(3/2)) - 6*a**2/(b**4*sqrt(a + b*x)) - 6*a*sqrt(a + b*
x)/b**4 + 2*(a + b*x)**(3/2)/(3*b**4)

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Mathematica [A]  time = 0.03258, size = 45, normalized size = 0.66 \[ \frac{2 \left (-16 a^3-24 a^2 b x-6 a b^2 x^2+b^3 x^3\right )}{3 b^4 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-16*a^3 - 24*a^2*b*x - 6*a*b^2*x^2 + b^3*x^3))/(3*b^4*(a + b*x)^(3/2))

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Maple [A]  time = 0.009, size = 43, normalized size = 0.6 \[ -{\frac{-2\,{b}^{3}{x}^{3}+12\,a{b}^{2}{x}^{2}+48\,{a}^{2}bx+32\,{a}^{3}}{3\,{b}^{4}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(b*x+a)^(5/2),x)

[Out]

-2/3/(b*x+a)^(3/2)*(-b^3*x^3+6*a*b^2*x^2+24*a^2*b*x+16*a^3)/b^4

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Maxima [A]  time = 1.61575, size = 76, normalized size = 1.12 \[ \frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}}}{3 \, b^{4}} - \frac{6 \, \sqrt{b x + a} a}{b^{4}} - \frac{6 \, a^{2}}{\sqrt{b x + a} b^{4}} + \frac{2 \, a^{3}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(b*x + a)^(3/2)/b^4 - 6*sqrt(b*x + a)*a/b^4 - 6*a^2/(sqrt(b*x + a)*b^4) + 2/
3*a^3/((b*x + a)^(3/2)*b^4)

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Fricas [A]  time = 0.216634, size = 69, normalized size = 1.01 \[ \frac{2 \,{\left (b^{3} x^{3} - 6 \, a b^{2} x^{2} - 24 \, a^{2} b x - 16 \, a^{3}\right )}}{3 \,{\left (b^{5} x + a b^{4}\right )} \sqrt{b x + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(b^3*x^3 - 6*a*b^2*x^2 - 24*a^2*b*x - 16*a^3)/((b^5*x + a*b^4)*sqrt(b*x + a)
)

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Sympy [A]  time = 3.6753, size = 163, normalized size = 2.4 \[ \begin{cases} - \frac{32 a^{3}}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} - \frac{48 a^{2} b x}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} - \frac{12 a b^{2} x^{2}}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} + \frac{2 b^{3} x^{3}}{3 a b^{4} \sqrt{a + b x} + 3 b^{5} x \sqrt{a + b x}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(b*x+a)**(5/2),x)

[Out]

Piecewise((-32*a**3/(3*a*b**4*sqrt(a + b*x) + 3*b**5*x*sqrt(a + b*x)) - 48*a**2*
b*x/(3*a*b**4*sqrt(a + b*x) + 3*b**5*x*sqrt(a + b*x)) - 12*a*b**2*x**2/(3*a*b**4
*sqrt(a + b*x) + 3*b**5*x*sqrt(a + b*x)) + 2*b**3*x**3/(3*a*b**4*sqrt(a + b*x) +
3*b**5*x*sqrt(a + b*x)), Ne(b, 0)), (x**4/(4*a**(5/2)), True))

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GIAC/XCAS [A]  time = 0.20662, size = 80, normalized size = 1.18 \[ -\frac{2 \,{\left (9 \,{\left (b x + a\right )} a^{2} - a^{3}\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{4}} + \frac{2 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} b^{8} - 9 \, \sqrt{b x + a} a b^{8}\right )}}{3 \, b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(9*(b*x + a)*a^2 - a^3)/((b*x + a)^(3/2)*b^4) + 2/3*((b*x + a)^(3/2)*b^8 -
9*sqrt(b*x + a)*a*b^8)/b^12