3.803 \(\int \frac{a+b x}{x^4 (c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=41 \[ -\frac{a}{8 c^2 x^7 \sqrt{c x^2}}-\frac{b}{7 c^2 x^6 \sqrt{c x^2}} \]

[Out]

-a/(8*c^2*x^7*Sqrt[c*x^2]) - b/(7*c^2*x^6*Sqrt[c*x^2])

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Rubi [A]  time = 0.0083773, 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} \[ -\frac{a}{8 c^2 x^7 \sqrt{c x^2}}-\frac{b}{7 c^2 x^6 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(x^4*(c*x^2)^(5/2)),x]

[Out]

-a/(8*c^2*x^7*Sqrt[c*x^2]) - b/(7*c^2*x^6*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 )^{5/2}} \, dx &=\frac{x \int \frac{a+b x}{x^9} \, dx}{c^2 \sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{a}{x^9}+\frac{b}{x^8}\right ) \, dx}{c^2 \sqrt{c x^2}}\\ &=-\frac{a}{8 c^2 x^7 \sqrt{c x^2}}-\frac{b}{7 c^2 x^6 \sqrt{c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0066191, size = 24, normalized size = 0.59 \[ \frac{-7 a-8 b x}{56 x^3 \left (c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/(x^4*(c*x^2)^(5/2)),x]

[Out]

(-7*a - 8*b*x)/(56*x^3*(c*x^2)^(5/2))

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Maple [A]  time = 0.003, size = 21, normalized size = 0.5 \begin{align*} -{\frac{8\,bx+7\,a}{56\,{x}^{3}} \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^4/(c*x^2)^(5/2),x)

[Out]

-1/56*(8*b*x+7*a)/x^3/(c*x^2)^(5/2)

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Maxima [A]  time = 1.0884, size = 26, normalized size = 0.63 \begin{align*} -\frac{b}{7 \, c^{\frac{5}{2}} x^{7}} - \frac{a}{8 \, c^{\frac{5}{2}} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^4/(c*x^2)^(5/2),x, algorithm="maxima")

[Out]

-1/7*b/(c^(5/2)*x^7) - 1/8*a/(c^(5/2)*x^8)

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Fricas [A]  time = 1.48948, size = 58, normalized size = 1.41 \begin{align*} -\frac{\sqrt{c x^{2}}{\left (8 \, b x + 7 \, a\right )}}{56 \, c^{3} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^4/(c*x^2)^(5/2),x, algorithm="fricas")

[Out]

-1/56*sqrt(c*x^2)*(8*b*x + 7*a)/(c^3*x^9)

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Sympy [A]  time = 1.7776, size = 37, normalized size = 0.9 \begin{align*} - \frac{a}{8 c^{\frac{5}{2}} x^{3} \left (x^{2}\right )^{\frac{5}{2}}} - \frac{b}{7 c^{\frac{5}{2}} x^{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**4/(c*x**2)**(5/2),x)

[Out]

-a/(8*c**(5/2)*x**3*(x**2)**(5/2)) - b/(7*c**(5/2)*x**2*(x**2)**(5/2))

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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)^(5/2),x, algorithm="giac")

[Out]

sage0*x