\(\int \frac {x^{3/2}}{(b x+c x^2)^{3/2}} \, dx\) [107]

Optimal result

Integrand size = 19, antiderivative size = 23 \[ \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {x}}{c \sqrt {b x+c x^2}} \]

[Out]

-2*x^(1/2)/c/(c*x^2+b*x)^(1/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {662} \[ \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {x}}{c \sqrt {b x+c x^2}} \]

[In]

Int[x^(3/2)/(b*x + c*x^2)^(3/2),x]

[Out]

(-2*Sqrt[x])/(c*Sqrt[b*x + c*x^2])

Rule 662

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c
*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + p, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {x}}{c \sqrt {b x+c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {x}}{c \sqrt {x (b+c x)}} \]

[In]

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

[Out]

(-2*Sqrt[x])/(c*Sqrt[x*(b + c*x)])

Maple [A] (verified)

Time = 2.11 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09

method	result	size
gosper	\(-\frac {2 \left (c x +b \right ) x^{\frac {3}{2}}}{c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}\)	\(25\)
default	\(-\frac {2 \sqrt {x \left (c x +b \right )}}{\sqrt {x}\, \left (c x +b \right ) c}\)	\(25\)

[In]

int(x^(3/2)/(c*x^2+b*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {c x^{2} + b x} \sqrt {x}}{c^{2} x^{2} + b c x} \]

[In]

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

[Out]

-2*sqrt(c*x^2 + b*x)*sqrt(x)/(c^2*x^2 + b*c*x)

Sympy [F]

\[ \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {x^{\frac {3}{2}}}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int { \frac {x^{\frac {3}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate(x^(3/2)/(c*x^2 + b*x)^(3/2), x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2}{\sqrt {c x + b} c} + \frac {2}{\sqrt {b} c} \]

[In]

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

[Out]

-2/(sqrt(c*x + b)*c) + 2/(sqrt(b)*c)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {x^{3/2}}{{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \]

[In]

int(x^(3/2)/(b*x + c*x^2)^(3/2),x)

[Out]

int(x^(3/2)/(b*x + c*x^2)^(3/2), x)