3.3.92 \(\int \frac {x (d+e x)}{(a+c x^2)^2} \, dx\) [292]

Optimal. Leaf size=50 \[ -\frac {d+e x}{2 c \left (a+c x^2\right )}+\frac {e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 \sqrt {a} c^{3/2}} \]

[Out]

1/2*(-e*x-d)/c/(c*x^2+a)+1/2*e*arctan(x*c^(1/2)/a^(1/2))/c^(3/2)/a^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {792, 211} \begin {gather*} \frac {e \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 \sqrt {a} c^{3/2}}-\frac {d+e x}{2 c \left (a+c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x*(d + e*x))/(a + c*x^2)^2,x]

[Out]

-1/2*(d + e*x)/(c*(a + c*x^2)) + (e*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*Sqrt[a]*c^(3/2))

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 792

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a*(e*f + d*g) - (
c*d*f - a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(2*a*c*(p + 1)),
Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {x (d+e x)}{\left (a+c x^2\right )^2} \, dx &=-\frac {d+e x}{2 c \left (a+c x^2\right )}+\frac {e \int \frac {1}{a+c x^2} \, dx}{2 c}\\ &=-\frac {d+e x}{2 c \left (a+c x^2\right )}+\frac {e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 \sqrt {a} c^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 53, normalized size = 1.06 \begin {gather*} \frac {-d-e x}{2 c \left (a+c x^2\right )}+\frac {e \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 \sqrt {a} c^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-d - e*x)/(2*c*(a + c*x^2)) + (e*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*Sqrt[a]*c^(3/2))

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Maple [A]
time = 0.56, size = 46, normalized size = 0.92

method	result	size
default	\(\frac {-\frac {e x}{2 c}-\frac {d}{2 c}}{c \,x^{2}+a}+\frac {e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 c \sqrt {a c}}\)	\(46\)
risch	\(\frac {-\frac {e x}{2 c}-\frac {d}{2 c}}{c \,x^{2}+a}-\frac {e \ln \left (c x +\sqrt {-a c}\right )}{4 \sqrt {-a c}\, c}+\frac {e \ln \left (-c x +\sqrt {-a c}\right )}{4 \sqrt {-a c}\, c}\)	\(73\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 43, normalized size = 0.86 \begin {gather*} \frac {\arctan \left (\frac {c x}{\sqrt {a c}}\right ) e}{2 \, \sqrt {a c} c} - \frac {x e + d}{2 \, {\left (c^{2} x^{2} + a c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 3.56, size = 137, normalized size = 2.74 \begin {gather*} \left [-\frac {2 \, a c x e + {\left (c x^{2} + a\right )} \sqrt {-a c} e \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) + 2 \, a c d}{4 \, {\left (a c^{3} x^{2} + a^{2} c^{2}\right )}}, -\frac {a c x e - {\left (c x^{2} + a\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) e + a c d}{2 \, {\left (a c^{3} x^{2} + a^{2} c^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(2*a*c*x*e + (c*x^2 + a)*sqrt(-a*c)*e*log((c*x^2 - 2*sqrt(-a*c)*x - a)/(c*x^2 + a)) + 2*a*c*d)/(a*c^3*x^
2 + a^2*c^2), -1/2*(a*c*x*e - (c*x^2 + a)*sqrt(a*c)*arctan(sqrt(a*c)*x/a)*e + a*c*d)/(a*c^3*x^2 + a^2*c^2)]

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Sympy [A]
time = 0.25, size = 85, normalized size = 1.70 \begin {gather*} e \left (- \frac {\sqrt {- \frac {1}{a c^{3}}} \log {\left (- a c \sqrt {- \frac {1}{a c^{3}}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a c^{3}}} \log {\left (a c \sqrt {- \frac {1}{a c^{3}}} + x \right )}}{4}\right ) + \frac {- d - e x}{2 a c + 2 c^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e*(-sqrt(-1/(a*c**3))*log(-a*c*sqrt(-1/(a*c**3)) + x)/4 + sqrt(-1/(a*c**3))*log(a*c*sqrt(-1/(a*c**3)) + x)/4)
+ (-d - e*x)/(2*a*c + 2*c**2*x**2)

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Giac [A]
time = 0.97, size = 42, normalized size = 0.84 \begin {gather*} \frac {\arctan \left (\frac {c x}{\sqrt {a c}}\right ) e}{2 \, \sqrt {a c} c} - \frac {x e + d}{2 \, {\left (c x^{2} + a\right )} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.05, size = 44, normalized size = 0.88 \begin {gather*} \frac {e\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{2\,\sqrt {a}\,c^{3/2}}-\frac {\frac {d}{2\,c}+\frac {e\,x}{2\,c}}{c\,x^2+a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(d + e*x))/(a + c*x^2)^2,x)

[Out]

(e*atan((c^(1/2)*x)/a^(1/2)))/(2*a^(1/2)*c^(3/2)) - (d/(2*c) + (e*x)/(2*c))/(a + c*x^2)

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