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

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

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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 = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {15, 43} \begin {gather*} -\frac {a}{4 c^2 x^3 \sqrt {c x^2}}-\frac {b}{3 c^2 x^2 \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(c*x^2)^(5/2),x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 27, normalized size = 0.66 \begin {gather*} -\frac {\sqrt {c x^2} (3 a+4 b x)}{12 c^3 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/(c*x^2)^(5/2),x]

[Out]

-1/12*(Sqrt[c*x^2]*(3*a + 4*b*x))/(c^3*x^5)

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IntegrateAlgebraic [A]  time = 0.02, size = 22, normalized size = 0.54 \begin {gather*} -\frac {x (3 a+4 b x)}{12 \left (c x^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*(x*(3*a + 4*b*x))/(c*x^2)^(5/2)

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fricas [A]  time = 1.00, size = 23, normalized size = 0.56 \begin {gather*} -\frac {\sqrt {c x^{2}} {\left (4 \, b x + 3 \, a\right )}}{12 \, c^{3} x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*sqrt(c*x^2)*(4*b*x + 3*a)/(c^3*x^5)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [A]  time = 0.00, size = 19, normalized size = 0.46 \begin {gather*} -\frac {\left (4 b x +3 a \right ) x}{12 \left (c \,x^{2}\right )^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*x*(4*b*x+3*a)/(c*x^2)^(5/2)

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maxima [A]  time = 1.37, size = 23, normalized size = 0.56 \begin {gather*} -\frac {b}{3 \, \left (c x^{2}\right )^{\frac {3}{2}} c} - \frac {a}{4 \, c^{\frac {5}{2}} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.16, size = 26, normalized size = 0.63 \begin {gather*} -\frac {3\,a\,\sqrt {x^2}+4\,b\,x\,\sqrt {x^2}}{12\,c^{5/2}\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)/(c*x^2)^(5/2),x)

[Out]

-(3*a*(x^2)^(1/2) + 4*b*x*(x^2)^(1/2))/(12*c^(5/2)*x^5)

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sympy [A]  time = 0.92, size = 36, normalized size = 0.88 \begin {gather*} - \frac {a x}{4 c^{\frac {5}{2}} \left (x^{2}\right )^{\frac {5}{2}}} - \frac {b x^{2}}{3 c^{\frac {5}{2}} \left (x^{2}\right )^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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