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

Optimal. Leaf size=52 \[ \frac{2 \left (b x+c x^2\right )^{5/2}}{7 c x^{3/2}}-\frac{4 b \left (b x+c x^2\right )^{5/2}}{35 c^2 x^{5/2}} \]

[Out]

(-4*b*(b*x + c*x^2)^(5/2))/(35*c^2*x^(5/2)) + (2*(b*x + c*x^2)^(5/2))/(7*c*x^(3/
2))

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Rubi [A]  time = 0.0598186, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2 \left (b x+c x^2\right )^{5/2}}{7 c x^{3/2}}-\frac{4 b \left (b x+c x^2\right )^{5/2}}{35 c^2 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*b*(b*x + c*x^2)^(5/2))/(35*c^2*x^(5/2)) + (2*(b*x + c*x^2)^(5/2))/(7*c*x^(3/
2))

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Rubi in Sympy [A]  time = 6.00553, size = 46, normalized size = 0.88 \[ - \frac{4 b \left (b x + c x^{2}\right )^{\frac{5}{2}}}{35 c^{2} x^{\frac{5}{2}}} + \frac{2 \left (b x + c x^{2}\right )^{\frac{5}{2}}}{7 c x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(3/2)/x**(1/2),x)

[Out]

-4*b*(b*x + c*x**2)**(5/2)/(35*c**2*x**(5/2)) + 2*(b*x + c*x**2)**(5/2)/(7*c*x**
(3/2))

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Mathematica [A]  time = 0.0310224, size = 31, normalized size = 0.6 \[ \frac{2 (x (b+c x))^{5/2} (5 c x-2 b)}{35 c^2 x^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 33, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -5\,cx+2\,b \right ) }{35\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(3/2)/x^(1/2),x)

[Out]

-2/35*(c*x+b)*(-5*c*x+2*b)*(c*x^2+b*x)^(3/2)/c^2/x^(3/2)

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Maxima [A]  time = 0.712897, size = 104, normalized size = 2. \[ \frac{2 \,{\left ({\left (15 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} - 4 \, b^{2} c x + 8 \, b^{3}\right )} x^{2} + 7 \,{\left (3 \, b c^{2} x^{3} + b^{2} c x^{2} - 2 \, b^{3} x\right )} x\right )} \sqrt{c x + b}}{105 \, c^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/105*((15*c^3*x^3 + 3*b*c^2*x^2 - 4*b^2*c*x + 8*b^3)*x^2 + 7*(3*b*c^2*x^3 + b^2
*c*x^2 - 2*b^3*x)*x)*sqrt(c*x + b)/(c^2*x^2)

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Fricas [A]  time = 0.22249, size = 85, normalized size = 1.63 \[ \frac{2 \,{\left (5 \, c^{4} x^{5} + 13 \, b c^{3} x^{4} + 9 \, b^{2} c^{2} x^{3} - b^{3} c x^{2} - 2 \, b^{4} x\right )}}{35 \, \sqrt{c x^{2} + b x} c^{2} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*(5*c^4*x^5 + 13*b*c^3*x^4 + 9*b^2*c^2*x^3 - b^3*c*x^2 - 2*b^4*x)/(sqrt(c*x^
2 + b*x)*c^2*sqrt(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(3/2)/x**(1/2),x)

[Out]

Integral((x*(b + c*x))**(3/2)/sqrt(x), x)

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GIAC/XCAS [A]  time = 0.212348, size = 116, normalized size = 2.23 \[ -\frac{2}{105} \, c{\left (\frac{8 \, b^{\frac{7}{2}}}{c^{3}} - \frac{15 \,{\left (c x + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2}}{c^{3}}\right )} + \frac{2}{15} \, b{\left (\frac{2 \, b^{\frac{5}{2}}}{c^{2}} + \frac{3 \,{\left (c x + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x + b\right )}^{\frac{3}{2}} b}{c^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105*c*(8*b^(7/2)/c^3 - (15*(c*x + b)^(7/2) - 42*(c*x + b)^(5/2)*b + 35*(c*x +
b)^(3/2)*b^2)/c^3) + 2/15*b*(2*b^(5/2)/c^2 + (3*(c*x + b)^(5/2) - 5*(c*x + b)^(
3/2)*b)/c^2)