3.4.25 \(\int \frac {1}{(b x+c x^2)^{3/2}} \, dx\) [325]

Optimal. Leaf size=24 \[ -\frac {2 (b+2 c x)}{b^2 \sqrt {b x+c x^2}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {627} \begin {gather*} -\frac {2 (b+2 c x)}{b^2 \sqrt {b x+c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x + c*x^2)^(-3/2),x]

[Out]

(-2*(b + 2*c*x))/(b^2*Sqrt[b*x + c*x^2])

Rule 627

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {1}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (b+2 c x)}{b^2 \sqrt {b x+c x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 22, normalized size = 0.92 \begin {gather*} -\frac {2 (b+2 c x)}{b^2 \sqrt {x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x + c*x^2)^(-3/2),x]

[Out]

(-2*(b + 2*c*x))/(b^2*Sqrt[x*(b + c*x)])

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Maple [A]
time = 0.40, size = 23, normalized size = 0.96

method	result	size
default	\(-\frac {2 \left (2 c x +b \right )}{b^{2} \sqrt {c \,x^{2}+b x}}\)	\(23\)
gosper	\(-\frac {2 x \left (c x +b \right ) \left (2 c x +b \right )}{b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}\)	\(29\)
trager	\(-\frac {2 \left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{\left (c x +b \right ) b^{2} x}\)	\(33\)
risch	\(-\frac {2 \left (c x +b \right )}{b^{2} \sqrt {x \left (c x +b \right )}}-\frac {2 c x}{\sqrt {x \left (c x +b \right )}\, b^{2}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x^2+b*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 35, normalized size = 1.46 \begin {gather*} -\frac {4 \, c x}{\sqrt {c x^{2} + b x} b^{2}} - \frac {2}{\sqrt {c x^{2} + b x} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x)^(3/2),x, algorithm="maxima")

[Out]

-4*c*x/(sqrt(c*x^2 + b*x)*b^2) - 2/(sqrt(c*x^2 + b*x)*b)

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Fricas [A]
time = 1.40, size = 35, normalized size = 1.46 \begin {gather*} -\frac {2 \, \sqrt {c x^{2} + b x} {\left (2 \, c x + b\right )}}{b^{2} c x^{2} + b^{3} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((b*x + c*x**2)**(-3/2), x)

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Giac [A]
time = 1.25, size = 24, normalized size = 1.00 \begin {gather*} -\frac {2 \, {\left (\frac {2 \, c x}{b^{2}} + \frac {1}{b}\right )}}{\sqrt {c x^{2} + b x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x^2+b*x)^(3/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.17, size = 24, normalized size = 1.00 \begin {gather*} -\frac {2\,b+4\,c\,x}{b^2\,\sqrt {c\,x^2+b\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x + c*x^2)^(3/2),x)

[Out]

-(2*b + 4*c*x)/(b^2*(b*x + c*x^2)^(1/2))

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