3.800 \(\int \frac{a+b x}{x (c x^2)^{5/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0081763, antiderivative size = 41, normalized size of antiderivative = 1., 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, 43} \[ -\frac{a}{5 c^2 x^4 \sqrt{c x^2}}-\frac{b}{4 c^2 x^3 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(x*(c*x^2)^(5/2)),x]

[Out]

-a/(5*c^2*x^4*Sqrt[c*x^2]) - b/(4*c^2*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 43

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

Mathematica [A]  time = 0.0079968, size = 27, normalized size = 0.66 \[ -\frac{\sqrt{c x^2} (4 a+5 b x)}{20 c^3 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/(x*(c*x^2)^(5/2)),x]

[Out]

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

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Maple [A]  time = 0.004, size = 18, normalized size = 0.4 \begin{align*} -{\frac{5\,bx+4\,a}{20} \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/x/(c*x^2)^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.12213, size = 26, normalized size = 0.63 \begin{align*} -\frac{b}{4 \, c^{\frac{5}{2}} x^{4}} - \frac{a}{5 \, c^{\frac{5}{2}} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x/(c*x^2)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.51195, size = 58, normalized size = 1.41 \begin{align*} -\frac{\sqrt{c x^{2}}{\left (5 \, b x + 4 \, a\right )}}{20 \, c^{3} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x/(c*x^2)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.994106, size = 32, normalized size = 0.78 \begin{align*} - \frac{a}{5 c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}} - \frac{b x}{4 c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x/(c*x**2)**(5/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b x + a}{\left (c x^{2}\right )^{\frac{5}{2}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x/(c*x^2)^(5/2),x, algorithm="giac")

[Out]

integrate((b*x + a)/((c*x^2)^(5/2)*x), x)