3.795 \(\int \frac{a+b x}{x^4 (c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=41 \[ -\frac{a}{6 c x^5 \sqrt{c x^2}}-\frac{b}{5 c x^4 \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0074406, 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} \[ -\frac{a}{6 c x^5 \sqrt{c x^2}}-\frac{b}{5 c x^4 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(x^4*(c*x^2)^(3/2)),x]

[Out]

-a/(6*c*x^5*Sqrt[c*x^2]) - b/(5*c*x^4*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^4 \left (c x^2\right )^{3/2}} \, dx &=\frac{x \int \frac{a+b x}{x^7} \, dx}{c \sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{a}{x^7}+\frac{b}{x^6}\right ) \, dx}{c \sqrt{c x^2}}\\ &=-\frac{a}{6 c x^5 \sqrt{c x^2}}-\frac{b}{5 c x^4 \sqrt{c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0072439, size = 24, normalized size = 0.59 \[ \frac{-5 a-6 b x}{30 x^3 \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/(x^4*(c*x^2)^(3/2)),x]

[Out]

(-5*a - 6*b*x)/(30*x^3*(c*x^2)^(3/2))

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Maple [A]  time = 0.003, size = 21, normalized size = 0.5 \begin{align*} -{\frac{6\,bx+5\,a}{30\,{x}^{3}} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/x^4/(c*x^2)^(3/2),x)

[Out]

-1/30*(6*b*x+5*a)/x^3/(c*x^2)^(3/2)

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Maxima [A]  time = 1.06289, size = 26, normalized size = 0.63 \begin{align*} -\frac{b}{5 \, c^{\frac{3}{2}} x^{5}} - \frac{a}{6 \, c^{\frac{3}{2}} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^4/(c*x^2)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.49702, size = 58, normalized size = 1.41 \begin{align*} -\frac{\sqrt{c x^{2}}{\left (6 \, b x + 5 \, a\right )}}{30 \, c^{2} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^4/(c*x^2)^(3/2),x, algorithm="fricas")

[Out]

-1/30*sqrt(c*x^2)*(6*b*x + 5*a)/(c^2*x^7)

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Sympy [A]  time = 1.00212, size = 37, normalized size = 0.9 \begin{align*} - \frac{a}{6 c^{\frac{3}{2}} x^{3} \left (x^{2}\right )^{\frac{3}{2}}} - \frac{b}{5 c^{\frac{3}{2}} x^{2} \left (x^{2}\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x**4/(c*x**2)**(3/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^4/(c*x^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x