3.10.13 \(\int \frac {x}{\sqrt {c x^2} (a+b x)^2} \, dx\) [913]

Optimal. Leaf size=22 \[ -\frac {x}{b \sqrt {c x^2} (a+b x)} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 32} \begin {gather*} -\frac {x}{b \sqrt {c x^2} (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(Sqrt[c*x^2]*(a + b*x)^2),x]

[Out]

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

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Mathematica [A]
time = 0.00, size = 22, normalized size = 1.00 \begin {gather*} -\frac {x}{b \sqrt {c x^2} (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(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 = 2.08, size = 83, normalized size = 3.77 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\text {DirectedInfinity}\left [\frac {1}{\sqrt {c x^2}}\right ],a\text {==}0\text {\&\&}b\text {==}0\right \},\left \{\text {DirectedInfinity}\left [\frac {x^2}{\sqrt {c x^2}}\right ],a\text {==}-b x\right \},\left \{\frac {x^2}{a^2 \sqrt {c x^2}},b\text {==}0\right \}\right \},-\frac {x}{a b \sqrt {c x^2}+b^2 x \sqrt {c x^2}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Piecewise[{{DirectedInfinity[1 / Sqrt[c x ^ 2]], a == 0 && b == 0}, {DirectedInfinity[x ^ 2 / Sqrt[c x ^ 2]],
a == -b x}, {x ^ 2 / (a ^ 2 Sqrt[c x ^ 2]), b == 0}}, -x / (a b Sqrt[c x ^ 2] + b ^ 2 x Sqrt[c x ^ 2])]

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Maple [A]
time = 0.12, size = 21, normalized size = 0.95

method	result	size
gosper	\(-\frac {x}{b \left (b x +a \right ) \sqrt {c \,x^{2}}}\)	\(21\)
default	\(-\frac {x}{b \left (b x +a \right ) \sqrt {c \,x^{2}}}\)	\(21\)
risch	\(-\frac {x}{b \left (b x +a \right ) \sqrt {c \,x^{2}}}\)	\(21\)
trager	\(\frac {\left (-1+x \right ) \sqrt {c \,x^{2}}}{c \left (b x +a \right ) \left (a +b \right ) x}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 21, normalized size = 0.95 \begin {gather*} \frac {\sqrt {c x^{2}}}{a b c x + a^{2} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c*x^2)/(a*b*c*x + a^2*c)

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Fricas [A]
time = 0.29, size = 25, normalized size = 1.14 \begin {gather*} -\frac {\sqrt {c x^{2}}}{b^{2} c x^{2} + a b c x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(c*x^2)/(b^2*c*x^2 + a*b*c*x)

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Sympy [A]
time = 0.44, size = 68, normalized size = 3.09 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{\sqrt {c x^{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {\tilde {\infty } x^{2}}{\sqrt {c x^{2}}} & \text {for}\: a = - b x \\\frac {x^{2}}{a^{2} \sqrt {c x^{2}}} & \text {for}\: b = 0 \\- \frac {x}{a b \sqrt {c x^{2}} + b^{2} x \sqrt {c x^{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 26, normalized size = 1.18 \begin {gather*} \frac {\frac {\mathrm {sign}\left (x\right )}{a b}-\frac 1{b \left (b x+a\right ) \mathrm {sign}\left (x\right )}}{\sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sgn(x)/(a*b) - 1/((b*x + a)*b*sgn(x)))/sqrt(c)

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Mupad [B]
time = 0.16, size = 25, normalized size = 1.14 \begin {gather*} -\frac {\sqrt {c\,x^2}}{b\,c\,x\,\left (a+b\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/((c*x^2)^(1/2)*(a + b*x)^2),x)

[Out]

-(c*x^2)^(1/2)/(b*c*x*(a + b*x))

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