3.9.97 \(\int \frac {\sqrt {c x^2}}{x (a+b x)^2} \, dx\) [897]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 32} \begin {gather*} -\frac {\sqrt {c x^2}}{b x (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*x^2]/(x*(a + b*x)^2),x]

[Out]

-(Sqrt[c*x^2]/(b*x*(a + b*x)))

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 23, normalized size = 0.96 \begin {gather*} -\frac {c x}{b \sqrt {c x^2} (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((c*x)/(b*Sqrt[c*x^2]*(a + b*x)))

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 1.85, size = 39, normalized size = 1.62 \begin {gather*} \text {Piecewise}\left [\left \{\left \{-\frac {\sqrt {c x^2}}{b x \left (a+b x\right )},b\text {!=}0\right \}\right \},\frac {\sqrt {c x^2}}{a^2}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[Sqrt[c*x^2]/(x*(a + b*x)^2),x]')

[Out]

Piecewise[{{-Sqrt[c x ^ 2] / (b x (a + b x)), b != 0}}, Sqrt[c x ^ 2] / a ^ 2]

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Maple [A]
time = 0.12, size = 23, normalized size = 0.96

method	result	size
gosper	\(-\frac {\sqrt {c \,x^{2}}}{b x \left (b x +a \right )}\)	\(23\)
default	\(-\frac {\sqrt {c \,x^{2}}}{b x \left (b x +a \right )}\)	\(23\)
risch	\(-\frac {\sqrt {c \,x^{2}}}{b x \left (b x +a \right )}\)	\(23\)
trager	\(\frac {\left (-1+x \right ) \sqrt {c \,x^{2}}}{\left (b x +a \right ) \left (a +b \right ) x}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 16, normalized size = 0.67 \begin {gather*} -\frac {\sqrt {c}}{b^{2} x + a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(c)/(b^2*x + a*b)

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Fricas [A]
time = 0.29, size = 23, normalized size = 0.96 \begin {gather*} -\frac {\sqrt {c x^{2}}}{b^{2} x^{2} + a b x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(c*x^2)/(b^2*x^2 + a*b*x)

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Sympy [A]
time = 0.32, size = 32, normalized size = 1.33 \begin {gather*} \begin {cases} - \frac {\sqrt {c x^{2}}}{a b x + b^{2} x^{2}} & \text {for}\: b \neq 0 \\\frac {\sqrt {c x^{2}}}{a^{2}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-sqrt(c*x**2)/(a*b*x + b**2*x**2), Ne(b, 0)), (sqrt(c*x**2)/a**2, True))

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Giac [A]
time = 0.00, size = 26, normalized size = 1.08 \begin {gather*} \sqrt {c} \left (\frac {\mathrm {sign}\left (x\right )}{a b}-\frac {\mathrm {sign}\left (x\right )}{b \left (b x+a\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(c)*(sgn(x)/((b*x + a)*b) - sgn(x)/(a*b))

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Mupad [B]
time = 0.16, size = 22, normalized size = 0.92 \begin {gather*} -\frac {\sqrt {c\,x^2}}{b\,x\,\left (a+b\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(1/2)/(x*(a + b*x)^2),x)

[Out]

-(c*x^2)^(1/2)/(b*x*(a + b*x))

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