3.8.94 \(\int \frac {a+b x}{x^3 (c x^2)^{3/2}} \, dx\) [794]

Optimal. Leaf size=41 \[ -\frac {a}{5 c x^4 \sqrt {c x^2}}-\frac {b}{4 c x^3 \sqrt {c x^2}} \]

[Out]

-1/5*a/c/x^4/(c*x^2)^(1/2)-1/4*b/c/x^3/(c*x^2)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 45} \begin {gather*} -\frac {a}{5 c x^4 \sqrt {c x^2}}-\frac {b}{4 c x^3 \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(x^3*(c*x^2)^(3/2)),x]

[Out]

-1/5*a/(c*x^4*Sqrt[c*x^2]) - b/(4*c*x^3*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 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {a+b x}{x^3 \left (c x^2\right )^{3/2}} \, dx &=\frac {x \int \frac {a+b x}{x^6} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a}{x^6}+\frac {b}{x^5}\right ) \, dx}{c \sqrt {c x^2}}\\ &=-\frac {a}{5 c x^4 \sqrt {c x^2}}-\frac {b}{4 c x^3 \sqrt {c x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 22, normalized size = 0.54 \begin {gather*} \frac {c (-4 a-5 b x)}{20 \left (c x^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(-4*a - 5*b*x))/(20*(c*x^2)^(5/2))

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Maple [A]
time = 0.03, size = 21, normalized size = 0.51

method	result	size
gosper	\(-\frac {5 b x +4 a}{20 x^{2} \left (c \,x^{2}\right )^{\frac {3}{2}}}\)	\(21\)
default	\(-\frac {5 b x +4 a}{20 x^{2} \left (c \,x^{2}\right )^{\frac {3}{2}}}\)	\(21\)
risch	\(\frac {-\frac {b x}{4}-\frac {a}{5}}{c \,x^{4} \sqrt {c \,x^{2}}}\)	\(23\)
trager	\(\frac {\left (-1+x \right ) \left (4 a \,x^{4}+5 b \,x^{4}+4 a \,x^{3}+5 b \,x^{3}+4 a \,x^{2}+5 x^{2} b +4 a x +5 b x +4 a \right ) \sqrt {c \,x^{2}}}{20 c^{2} x^{6}}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*(5*b*x+4*a)/x^2/(c*x^2)^(3/2)

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Maxima [A]
time = 0.28, size = 19, normalized size = 0.46 \begin {gather*} -\frac {b}{4 \, c^{\frac {3}{2}} x^{4}} - \frac {a}{5 \, c^{\frac {3}{2}} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*b/(c^(3/2)*x^4) - 1/5*a/(c^(3/2)*x^5)

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Fricas [A]
time = 1.20, size = 23, normalized size = 0.56 \begin {gather*} -\frac {\sqrt {c x^{2}} {\left (5 \, b x + 4 \, a\right )}}{20 \, c^{2} x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*sqrt(c*x^2)*(5*b*x + 4*a)/(c^2*x^6)

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Sympy [A]
time = 0.28, size = 29, normalized size = 0.71 \begin {gather*} - \frac {a}{5 x^{2} \left (c x^{2}\right )^{\frac {3}{2}}} - \frac {b}{4 x \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)/x**3/(c*x**2)**(3/2),x)

[Out]

-a/(5*x**2*(c*x**2)**(3/2)) - b/(4*x*(c*x**2)**(3/2))

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Giac [A]
time = 2.49, size = 20, normalized size = 0.49 \begin {gather*} -\frac {5 \, b x + 4 \, a}{20 \, c^{\frac {3}{2}} x^{5} \mathrm {sgn}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20*(5*b*x + 4*a)/(c^(3/2)*x^5*sgn(x))

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Mupad [B]
time = 0.15, size = 26, normalized size = 0.63 \begin {gather*} -\frac {4\,a\,\sqrt {x^2}+5\,b\,x\,\sqrt {x^2}}{20\,c^{3/2}\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/(x^3*(c*x^2)^(3/2)),x)

[Out]

-(4*a*(x^2)^(1/2) + 5*b*x*(x^2)^(1/2))/(20*c^(3/2)*x^6)

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