3.290 \(\int \frac{x^3 (d+e x)}{\left (a+c x^2\right )^2} \, dx\)

Optimal. Leaf size=78 \[ -\frac{3 \sqrt{a} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 c^{5/2}}+\frac{d \log \left (a+c x^2\right )}{2 c^2}-\frac{x^2 (d+e x)}{2 c \left (a+c x^2\right )}+\frac{3 e x}{2 c^2} \]

[Out]

(3*e*x)/(2*c^2) - (x^2*(d + e*x))/(2*c*(a + c*x^2)) - (3*Sqrt[a]*e*ArcTan[(Sqrt[
c]*x)/Sqrt[a]])/(2*c^(5/2)) + (d*Log[a + c*x^2])/(2*c^2)

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Rubi [A]  time = 0.127813, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{3 \sqrt{a} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 c^{5/2}}+\frac{d \log \left (a+c x^2\right )}{2 c^2}-\frac{x^2 (d+e x)}{2 c \left (a+c x^2\right )}+\frac{3 e x}{2 c^2} \]

Antiderivative was successfully verified.

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

[Out]

(3*e*x)/(2*c^2) - (x^2*(d + e*x))/(2*c*(a + c*x^2)) - (3*Sqrt[a]*e*ArcTan[(Sqrt[
c]*x)/Sqrt[a]])/(2*c^(5/2)) + (d*Log[a + c*x^2])/(2*c^2)

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Rubi in Sympy [A]  time = 27.2228, size = 75, normalized size = 0.96 \[ - \frac{3 \sqrt{a} e \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 c^{\frac{5}{2}}} - \frac{x^{2} \left (2 d + 2 e x\right )}{4 c \left (a + c x^{2}\right )} + \frac{d \log{\left (a + c x^{2} \right )}}{2 c^{2}} + \frac{3 e x}{2 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(e*x+d)/(c*x**2+a)**2,x)

[Out]

-3*sqrt(a)*e*atan(sqrt(c)*x/sqrt(a))/(2*c**(5/2)) - x**2*(2*d + 2*e*x)/(4*c*(a +
c*x**2)) + d*log(a + c*x**2)/(2*c**2) + 3*e*x/(2*c**2)

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Mathematica [A]  time = 0.093652, size = 75, normalized size = 0.96 \[ -\frac{3 \sqrt{a} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 c^{5/2}}+\frac{a d+a e x}{2 c^2 \left (a+c x^2\right )}+\frac{d \log \left (a+c x^2\right )}{2 c^2}+\frac{e x}{c^2} \]

Antiderivative was successfully verified.

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

[Out]

(e*x)/c^2 + (a*d + a*e*x)/(2*c^2*(a + c*x^2)) - (3*Sqrt[a]*e*ArcTan[(Sqrt[c]*x)/
Sqrt[a]])/(2*c^(5/2)) + (d*Log[a + c*x^2])/(2*c^2)

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Maple [A]  time = 0.01, size = 76, normalized size = 1. \[{\frac{ex}{{c}^{2}}}+{\frac{axe}{2\,{c}^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{ad}{2\,{c}^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{d\ln \left ( c{x}^{2}+a \right ) }{2\,{c}^{2}}}-{\frac{3\,ae}{2\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(e*x+d)/(c*x^2+a)^2,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.279654, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, c e x^{3} + 6 \, a e x + 3 \,{\left (c e x^{2} + a e\right )} \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{2} - 2 \, c x \sqrt{-\frac{a}{c}} - a}{c x^{2} + a}\right ) + 2 \, a d + 2 \,{\left (c d x^{2} + a d\right )} \log \left (c x^{2} + a\right )}{4 \,{\left (c^{3} x^{2} + a c^{2}\right )}}, \frac{2 \, c e x^{3} + 3 \, a e x - 3 \,{\left (c e x^{2} + a e\right )} \sqrt{\frac{a}{c}} \arctan \left (\frac{x}{\sqrt{\frac{a}{c}}}\right ) + a d +{\left (c d x^{2} + a d\right )} \log \left (c x^{2} + a\right )}{2 \,{\left (c^{3} x^{2} + a c^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(4*c*e*x^3 + 6*a*e*x + 3*(c*e*x^2 + a*e)*sqrt(-a/c)*log((c*x^2 - 2*c*x*sqrt
(-a/c) - a)/(c*x^2 + a)) + 2*a*d + 2*(c*d*x^2 + a*d)*log(c*x^2 + a))/(c^3*x^2 +
a*c^2), 1/2*(2*c*e*x^3 + 3*a*e*x - 3*(c*e*x^2 + a*e)*sqrt(a/c)*arctan(x/sqrt(a/c
)) + a*d + (c*d*x^2 + a*d)*log(c*x^2 + a))/(c^3*x^2 + a*c^2)]

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Sympy [A]  time = 2.90978, size = 162, normalized size = 2.08 \[ \left (\frac{d}{2 c^{2}} - \frac{3 e \sqrt{- a c^{5}}}{4 c^{5}}\right ) \log{\left (x + \frac{- 4 c^{2} \left (\frac{d}{2 c^{2}} - \frac{3 e \sqrt{- a c^{5}}}{4 c^{5}}\right ) + 2 d}{3 e} \right )} + \left (\frac{d}{2 c^{2}} + \frac{3 e \sqrt{- a c^{5}}}{4 c^{5}}\right ) \log{\left (x + \frac{- 4 c^{2} \left (\frac{d}{2 c^{2}} + \frac{3 e \sqrt{- a c^{5}}}{4 c^{5}}\right ) + 2 d}{3 e} \right )} + \frac{a d + a e x}{2 a c^{2} + 2 c^{3} x^{2}} + \frac{e x}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d/(2*c**2) - 3*e*sqrt(-a*c**5)/(4*c**5))*log(x + (-4*c**2*(d/(2*c**2) - 3*e*sqr
t(-a*c**5)/(4*c**5)) + 2*d)/(3*e)) + (d/(2*c**2) + 3*e*sqrt(-a*c**5)/(4*c**5))*l
og(x + (-4*c**2*(d/(2*c**2) + 3*e*sqrt(-a*c**5)/(4*c**5)) + 2*d)/(3*e)) + (a*d +
a*e*x)/(2*a*c**2 + 2*c**3*x**2) + e*x/c**2

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GIAC/XCAS [A]  time = 0.270452, size = 90, normalized size = 1.15 \[ -\frac{3 \, a \arctan \left (\frac{c x}{\sqrt{a c}}\right ) e}{2 \, \sqrt{a c} c^{2}} + \frac{x e}{c^{2}} + \frac{d{\rm ln}\left (c x^{2} + a\right )}{2 \, c^{2}} + \frac{a x e + a d}{2 \,{\left (c x^{2} + a\right )} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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