3.8.91 \(\int \frac {a+b x}{(c x^2)^{3/2}} \, dx\) [791]

Optimal. Leaf size=29 \[ -\frac {(a+b x)^2}{2 a c x \sqrt {c x^2}} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {15, 37} \begin {gather*} -\frac {(a+b x)^2}{2 a c x \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(c*x^2)^(3/2),x]

[Out]

-1/2*(a + b*x)^2/(a*c*x*Sqrt[c*x^2])

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 37

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

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 22, normalized size = 0.76 \begin {gather*} \frac {x (-a-2 b x)}{2 \left (c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/(c*x^2)^(3/2),x]

[Out]

(x*(-a - 2*b*x))/(2*(c*x^2)^(3/2))

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Maple [A]
time = 0.02, size = 17, normalized size = 0.59

method	result	size
gosper	\(-\frac {x \left (2 b x +a \right )}{2 \left (c \,x^{2}\right )^{\frac {3}{2}}}\)	\(17\)
default	\(-\frac {x \left (2 b x +a \right )}{2 \left (c \,x^{2}\right )^{\frac {3}{2}}}\)	\(17\)
risch	\(\frac {-b x -\frac {a}{2}}{c x \sqrt {c \,x^{2}}}\)	\(23\)
trager	\(\frac {\left (-1+x \right ) \left (a x +2 b x +a \right ) \sqrt {c \,x^{2}}}{2 c^{2} x^{3}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x*(2*b*x+a)/(c*x^2)^(3/2)

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Maxima [A]
time = 0.29, size = 23, normalized size = 0.79 \begin {gather*} -\frac {b}{\sqrt {c x^{2}} c} - \frac {a}{2 \, c^{\frac {3}{2}} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b/(sqrt(c*x^2)*c) - 1/2*a/(c^(3/2)*x^2)

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Fricas [A]
time = 1.16, size = 21, normalized size = 0.72 \begin {gather*} -\frac {\sqrt {c x^{2}} {\left (2 \, b x + a\right )}}{2 \, c^{2} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(c*x^2)*(2*b*x + a)/(c^2*x^3)

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Sympy [A]
time = 0.20, size = 27, normalized size = 0.93 \begin {gather*} - \frac {a x}{2 \left (c x^{2}\right )^{\frac {3}{2}}} - \frac {b x^{2}}{\left (c x^{2}\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*x/(2*(c*x**2)**(3/2)) - b*x**2/(c*x**2)**(3/2)

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Giac [A]
time = 3.15, size = 18, normalized size = 0.62 \begin {gather*} -\frac {2 \, b x + a}{2 \, c^{\frac {3}{2}} x^{2} \mathrm {sgn}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b*x + a)/(c^(3/2)*x^2*sgn(x))

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Mupad [B]
time = 0.15, size = 25, normalized size = 0.86 \begin {gather*} -\frac {2\,b\,x^3+a\,x^2}{2\,c^{3/2}\,x\,{\left (x^2\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/(c*x^2)^(3/2),x)

[Out]

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

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