3.289 \(\int \frac{x^4 (d+e x)}{\left (a+c x^2\right )^2} \, dx\)

Optimal. Leaf size=85 \[ -\frac{3 \sqrt{a} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 c^{5/2}}-\frac{a e \log \left (a+c x^2\right )}{c^3}-\frac{x^3 (d+e x)}{2 c \left (a+c x^2\right )}+\frac{3 d x}{2 c^2}+\frac{e x^2}{c^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.184867, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{3 \sqrt{a} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 c^{5/2}}-\frac{a e \log \left (a+c x^2\right )}{c^3}-\frac{x^3 (d+e x)}{2 c \left (a+c x^2\right )}+\frac{3 d x}{2 c^2}+\frac{e x^2}{c^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(d + e*x))/(a + c*x^2)^2,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{3 \sqrt{a} d \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 c^{\frac{5}{2}}} - \frac{a e \log{\left (a + c x^{2} \right )}}{c^{3}} - \frac{x^{3} \left (2 d + 2 e x\right )}{4 c \left (a + c x^{2}\right )} + \frac{3 d x}{2 c^{2}} + \frac{2 e \int x\, dx}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(e*x+d)/(c*x**2+a)**2,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.116992, size = 77, normalized size = 0.91 \[ \frac{\frac{a (c d x-a e)}{a+c x^2}-3 \sqrt{a} \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )-2 a e \log \left (a+c x^2\right )+2 c d x+c e x^2}{2 c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(d + e*x))/(a + c*x^2)^2,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 88, normalized size = 1. \[{\frac{e{x}^{2}}{2\,{c}^{2}}}+{\frac{dx}{{c}^{2}}}+{\frac{axd}{2\,{c}^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{{a}^{2}e}{2\,{c}^{3} \left ( c{x}^{2}+a \right ) }}-{\frac{ae\ln \left ( c{x}^{2}+a \right ) }{{c}^{3}}}-{\frac{3\,ad}{2\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(e*x+d)/(c*x^2+a)^2,x)

[Out]

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

_______________________________________________________________________________________

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^4/(c*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.280443, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, c^{2} e x^{4} + 4 \, c^{2} d x^{3} + 2 \, a c e x^{2} + 6 \, a c d x - 2 \, a^{2} e + 3 \,{\left (c^{2} d x^{2} + a c d\right )} \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{2} - 2 \, c x \sqrt{-\frac{a}{c}} - a}{c x^{2} + a}\right ) - 4 \,{\left (a c e x^{2} + a^{2} e\right )} \log \left (c x^{2} + a\right )}{4 \,{\left (c^{4} x^{2} + a c^{3}\right )}}, \frac{c^{2} e x^{4} + 2 \, c^{2} d x^{3} + a c e x^{2} + 3 \, a c d x - a^{2} e - 3 \,{\left (c^{2} d x^{2} + a c d\right )} \sqrt{\frac{a}{c}} \arctan \left (\frac{x}{\sqrt{\frac{a}{c}}}\right ) - 2 \,{\left (a c e x^{2} + a^{2} e\right )} \log \left (c x^{2} + a\right )}{2 \,{\left (c^{4} x^{2} + a c^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*x^4/(c*x^2 + a)^2,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 3.08785, size = 189, normalized size = 2.22 \[ \left (- \frac{a e}{c^{3}} - \frac{3 d \sqrt{- a c^{7}}}{4 c^{6}}\right ) \log{\left (x + \frac{- 4 a e - 4 c^{3} \left (- \frac{a e}{c^{3}} - \frac{3 d \sqrt{- a c^{7}}}{4 c^{6}}\right )}{3 c d} \right )} + \left (- \frac{a e}{c^{3}} + \frac{3 d \sqrt{- a c^{7}}}{4 c^{6}}\right ) \log{\left (x + \frac{- 4 a e - 4 c^{3} \left (- \frac{a e}{c^{3}} + \frac{3 d \sqrt{- a c^{7}}}{4 c^{6}}\right )}{3 c d} \right )} + \frac{- a^{2} e + a c d x}{2 a c^{3} + 2 c^{4} x^{2}} + \frac{d x}{c^{2}} + \frac{e x^{2}}{2 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(e*x+d)/(c*x**2+a)**2,x)

[Out]

(-a*e/c**3 - 3*d*sqrt(-a*c**7)/(4*c**6))*log(x + (-4*a*e - 4*c**3*(-a*e/c**3 - 3
*d*sqrt(-a*c**7)/(4*c**6)))/(3*c*d)) + (-a*e/c**3 + 3*d*sqrt(-a*c**7)/(4*c**6))*
log(x + (-4*a*e - 4*c**3*(-a*e/c**3 + 3*d*sqrt(-a*c**7)/(4*c**6)))/(3*c*d)) + (-
a**2*e + a*c*d*x)/(2*a*c**3 + 2*c**4*x**2) + d*x/c**2 + e*x**2/(2*c**2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.273762, size = 117, normalized size = 1.38 \[ -\frac{3 \, a d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} c^{2}} - \frac{a e{\rm ln}\left (c x^{2} + a\right )}{c^{3}} + \frac{c^{2} x^{2} e + 2 \, c^{2} d x}{2 \, c^{4}} + \frac{a c d x - a^{2} e}{2 \,{\left (c x^{2} + a\right )} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*x^4/(c*x^2 + a)^2,x, algorithm="giac")

[Out]

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