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

Optimal. Leaf size=34 \[ \frac{2 (a+b x)^{7/2}}{7 b^2}-\frac{2 a (a+b x)^{5/2}}{5 b^2} \]

[Out]

(-2*a*(a + b*x)^(5/2))/(5*b^2) + (2*(a + b*x)^(7/2))/(7*b^2)

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Rubi [A]  time = 0.0248236, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{2 (a+b x)^{7/2}}{7 b^2}-\frac{2 a (a+b x)^{5/2}}{5 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x)^(3/2),x]

[Out]

(-2*a*(a + b*x)^(5/2))/(5*b^2) + (2*(a + b*x)^(7/2))/(7*b^2)

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Rubi in Sympy [A]  time = 4.94087, size = 31, normalized size = 0.91 \[ - \frac{2 a \left (a + b x\right )^{\frac{5}{2}}}{5 b^{2}} + \frac{2 \left (a + b x\right )^{\frac{7}{2}}}{7 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*(a + b*x)**(5/2)/(5*b**2) + 2*(a + b*x)**(7/2)/(7*b**2)

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Mathematica [A]  time = 0.0227082, size = 24, normalized size = 0.71 \[ \frac{2 (a+b x)^{5/2} (5 b x-2 a)}{35 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a + b*x)^(3/2),x]

[Out]

(2*(a + b*x)^(5/2)*(-2*a + 5*b*x))/(35*b^2)

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Maple [A]  time = 0.004, size = 21, normalized size = 0.6 \[ -{\frac{-10\,bx+4\,a}{35\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)^(3/2),x)

[Out]

-2/35*(b*x+a)^(5/2)*(-5*b*x+2*a)/b^2

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Maxima [A]  time = 1.34162, size = 35, normalized size = 1.03 \[ \frac{2 \,{\left (b x + a\right )}^{\frac{7}{2}}}{7 \, b^{2}} - \frac{2 \,{\left (b x + a\right )}^{\frac{5}{2}} a}{5 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*(b*x + a)^(7/2)/b^2 - 2/5*(b*x + a)^(5/2)*a/b^2

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Fricas [A]  time = 0.219235, size = 55, normalized size = 1.62 \[ \frac{2 \,{\left (5 \, b^{3} x^{3} + 8 \, a b^{2} x^{2} + a^{2} b x - 2 \, a^{3}\right )} \sqrt{b x + a}}{35 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*(5*b^3*x^3 + 8*a*b^2*x^2 + a^2*b*x - 2*a^3)*sqrt(b*x + a)/b^2

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Sympy [A]  time = 1.71991, size = 80, normalized size = 2.35 \[ \begin{cases} - \frac{4 a^{3} \sqrt{a + b x}}{35 b^{2}} + \frac{2 a^{2} x \sqrt{a + b x}}{35 b} + \frac{16 a x^{2} \sqrt{a + b x}}{35} + \frac{2 b x^{3} \sqrt{a + b x}}{7} & \text{for}\: b \neq 0 \\\frac{a^{\frac{3}{2}} x^{2}}{2} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x+a)**(3/2),x)

[Out]

Piecewise((-4*a**3*sqrt(a + b*x)/(35*b**2) + 2*a**2*x*sqrt(a + b*x)/(35*b) + 16*
a*x**2*sqrt(a + b*x)/35 + 2*b*x**3*sqrt(a + b*x)/7, Ne(b, 0)), (a**(3/2)*x**2/2,
True))

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GIAC/XCAS [A]  time = 0.218844, size = 104, normalized size = 3.06 \[ \frac{2 \,{\left (\frac{7 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} a}{b} + \frac{15 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{12} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a b^{12} + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} b^{12}}{b^{13}}\right )}}{105 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*(7*(3*(b*x + a)^(5/2) - 5*(b*x + a)^(3/2)*a)*a/b + (15*(b*x + a)^(7/2)*b^1
2 - 42*(b*x + a)^(5/2)*a*b^12 + 35*(b*x + a)^(3/2)*a^2*b^12)/b^13)/b