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

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

[Out]

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

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Rubi [A]  time = 0.0082408, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {15, 43} \[ -\frac{a}{3 c^2 x^2 \sqrt{c x^2}}-\frac{b}{2 c^2 x \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*x))/(c*x^2)^(5/2),x]

[Out]

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

Mathematica [A]  time = 0.0037681, size = 24, normalized size = 0.59 \[ \frac{x^2 (-2 a-3 b x)}{6 \left (c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(-2*a - 3*b*x))/(6*(c*x^2)^(5/2))

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Maple [A]  time = 0.001, size = 21, normalized size = 0.5 \begin{align*} -{\frac{{x}^{2} \left ( 3\,bx+2\,a \right ) }{6} \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.069, size = 31, normalized size = 0.76 \begin{align*} -\frac{a}{3 \, \left (c x^{2}\right )^{\frac{3}{2}} c} - \frac{b}{2 \, c^{\frac{5}{2}} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.59097, size = 57, normalized size = 1.39 \begin{align*} -\frac{\sqrt{c x^{2}}{\left (3 \, b x + 2 \, a\right )}}{6 \, c^{3} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.852636, size = 37, normalized size = 0.9 \begin{align*} - \frac{a x^{2}}{3 c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}} - \frac{b x^{3}}{2 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(x*(b*x+a)/(c*x**2)**(5/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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