3.572 \(\int \frac{d+e x}{(a+c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=28 \[ -\frac{a e-c d x}{a c \sqrt{a+c x^2}} \]

[Out]

-((a*e - c*d*x)/(a*c*Sqrt[a + c*x^2]))

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Rubi [A]  time = 0.0070311, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {637} \[ -\frac{a e-c d x}{a c \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(a + c*x^2)^(3/2),x]

[Out]

-((a*e - c*d*x)/(a*c*Sqrt[a + c*x^2]))

Rule 637

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-(a*e) + c*d*x)/(a*c*Sqrt[a + c*x^2]),
x] /; FreeQ[{a, c, d, e}, x]

Rubi steps

\begin{align*} \int \frac{d+e x}{\left (a+c x^2\right )^{3/2}} \, dx &=-\frac{a e-c d x}{a c \sqrt{a+c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0137637, size = 27, normalized size = 0.96 \[ \frac{c d x-a e}{a c \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)/(a + c*x^2)^(3/2),x]

[Out]

(-(a*e) + c*d*x)/(a*c*Sqrt[a + c*x^2])

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Maple [A]  time = 0.042, size = 27, normalized size = 1. \begin{align*} -{\frac{-cdx+ae}{ac}{\frac{1}{\sqrt{c{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/(c*x^2+a)^(3/2),x)

[Out]

-(-c*d*x+a*e)/(c*x^2+a)^(1/2)/a/c

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Maxima [A]  time = 1.12443, size = 42, normalized size = 1.5 \begin{align*} \frac{d x}{\sqrt{c x^{2} + a} a} - \frac{e}{\sqrt{c x^{2} + a} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*x/(sqrt(c*x^2 + a)*a) - e/(sqrt(c*x^2 + a)*c)

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Fricas [A]  time = 1.82502, size = 69, normalized size = 2.46 \begin{align*} \frac{{\left (c d x - a e\right )} \sqrt{c x^{2} + a}}{a c^{2} x^{2} + a^{2} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*d*x - a*e)*sqrt(c*x^2 + a)/(a*c^2*x^2 + a^2*c)

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Sympy [A]  time = 3.98878, size = 46, normalized size = 1.64 \begin{align*} e \left (\begin{cases} - \frac{1}{c \sqrt{a + c x^{2}}} & \text{for}\: c \neq 0 \\\frac{x^{2}}{2 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + \frac{d x}{a^{\frac{3}{2}} \sqrt{1 + \frac{c x^{2}}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x**2+a)**(3/2),x)

[Out]

e*Piecewise((-1/(c*sqrt(a + c*x**2)), Ne(c, 0)), (x**2/(2*a**(3/2)), True)) + d*x/(a**(3/2)*sqrt(1 + c*x**2/a)
)

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Giac [A]  time = 1.34222, size = 32, normalized size = 1.14 \begin{align*} \frac{\frac{d x}{a} - \frac{e}{c}}{\sqrt{c x^{2} + a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x/a - e/c)/sqrt(c*x^2 + a)