3.6.72 \(\int \frac {d+e x}{(a+c x^2)^{3/2}} \, dx\) [572]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, 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 = {651} \begin {gather*} -\frac {a e-c d x}{a c \sqrt {a+c x^2}} \end {gather*}

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 651

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*}

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Mathematica [A]
time = 0.21, size = 27, normalized size = 0.96 \begin {gather*} \frac {-a e+c d x}{a c \sqrt {a+c x^2}} \end {gather*}

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 = 0.39, size = 32, normalized size = 1.14

method	result	size
gosper	\(-\frac {-c d x +a e}{\sqrt {c \,x^{2}+a}\, a c}\)	\(27\)
trager	\(-\frac {-c d x +a e}{\sqrt {c \,x^{2}+a}\, a c}\)	\(27\)
default	\(-\frac {e}{c \sqrt {c \,x^{2}+a}}+\frac {d x}{a \sqrt {c \,x^{2}+a}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 32, normalized size = 1.14 \begin {gather*} \frac {d x}{\sqrt {c x^{2} + a} a} - \frac {e}{\sqrt {c x^{2} + a} c} \end {gather*}

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 = 1.79, size = 36, normalized size = 1.29 \begin {gather*} \frac {{\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c^{2} x^{2} + a^{2} c} \end {gather*}

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 = 2.57, size = 46, normalized size = 1.64 \begin {gather*} 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 {gather*}

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 = 2.51, size = 24, normalized size = 0.86 \begin {gather*} \frac {\frac {d x}{a} - \frac {e}{c}}{\sqrt {c x^{2} + a}} \end {gather*}

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)

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Mupad [B]
time = 0.37, size = 24, normalized size = 0.86 \begin {gather*} -\frac {\frac {e}{c}-\frac {d\,x}{a}}{\sqrt {c\,x^2+a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(a + c*x^2)^(3/2),x)

[Out]

-(e/c - (d*x)/a)/(a + c*x^2)^(1/2)

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