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

Optimal. Leaf size=75 \[ \frac{3 d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{c}}+\frac{3 d x}{8 a^2 \left (a+c x^2\right )}-\frac{a e-c d x}{4 a c \left (a+c x^2\right )^2} \]

[Out]

-(a*e - c*d*x)/(4*a*c*(a + c*x^2)^2) + (3*d*x)/(8*a^2*(a + c*x^2)) + (3*d*ArcTan
[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*Sqrt[c])

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Rubi [A]  time = 0.0646081, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{3 d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{c}}+\frac{3 d x}{8 a^2 \left (a+c x^2\right )}-\frac{a e-c d x}{4 a c \left (a+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)/(a + c*x^2)^3,x]

[Out]

-(a*e - c*d*x)/(4*a*c*(a + c*x^2)^2) + (3*d*x)/(8*a^2*(a + c*x^2)) + (3*d*ArcTan
[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*Sqrt[c])

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Rubi in Sympy [A]  time = 9.00467, size = 66, normalized size = 0.88 \[ - \frac{a e - c d x}{4 a c \left (a + c x^{2}\right )^{2}} + \frac{3 d x}{8 a^{2} \left (a + c x^{2}\right )} + \frac{3 d \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a*e - c*d*x)/(4*a*c*(a + c*x**2)**2) + 3*d*x/(8*a**2*(a + c*x**2)) + 3*d*atan(
sqrt(c)*x/sqrt(a))/(8*a**(5/2)*sqrt(c))

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Mathematica [A]  time = 0.0926997, size = 71, normalized size = 0.95 \[ \frac{\frac{\sqrt{a} \left (-2 a^2 e+5 a c d x+3 c^2 d x^3\right )}{\left (a+c x^2\right )^2}+3 \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[a]*(-2*a^2*e + 5*a*c*d*x + 3*c^2*d*x^3))/(a + c*x^2)^2 + 3*Sqrt[c]*d*ArcT
an[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c)

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Maple [A]  time = 0.005, size = 65, normalized size = 0.9 \[{\frac{2\,cdx-2\,ae}{8\,ac \left ( c{x}^{2}+a \right ) ^{2}}}+{\frac{3\,dx}{8\,{a}^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{3\,d}{8\,{a}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(c*x^2+a)^3,x)

[Out]

1/8*(2*c*d*x-2*a*e)/a/c/(c*x^2+a)^2+3/8*d*x/a^2/(c*x^2+a)+3/8*d/a^2/(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)/(c*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.213286, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (c^{3} d x^{4} + 2 \, a c^{2} d x^{2} + a^{2} c d\right )} \log \left (\frac{2 \, a c x +{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) + 2 \,{\left (3 \, c^{2} d x^{3} + 5 \, a c d x - 2 \, a^{2} e\right )} \sqrt{-a c}}{16 \,{\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt{-a c}}, \frac{3 \,{\left (c^{3} d x^{4} + 2 \, a c^{2} d x^{2} + a^{2} c d\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) +{\left (3 \, c^{2} d x^{3} + 5 \, a c d x - 2 \, a^{2} e\right )} \sqrt{a c}}{8 \,{\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(3*(c^3*d*x^4 + 2*a*c^2*d*x^2 + a^2*c*d)*log((2*a*c*x + (c*x^2 - a)*sqrt(-
a*c))/(c*x^2 + a)) + 2*(3*c^2*d*x^3 + 5*a*c*d*x - 2*a^2*e)*sqrt(-a*c))/((a^2*c^3
*x^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt(-a*c)), 1/8*(3*(c^3*d*x^4 + 2*a*c^2*d*x^2 + a
^2*c*d)*arctan(sqrt(a*c)*x/a) + (3*c^2*d*x^3 + 5*a*c*d*x - 2*a^2*e)*sqrt(a*c))/(
(a^2*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt(a*c))]

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Sympy [A]  time = 2.34197, size = 124, normalized size = 1.65 \[ d \left (- \frac{3 \sqrt{- \frac{1}{a^{5} c}} \log{\left (- a^{3} \sqrt{- \frac{1}{a^{5} c}} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{a^{5} c}} \log{\left (a^{3} \sqrt{- \frac{1}{a^{5} c}} + x \right )}}{16}\right ) + \frac{- 2 a^{2} e + 5 a c d x + 3 c^{2} d x^{3}}{8 a^{4} c + 16 a^{3} c^{2} x^{2} + 8 a^{2} c^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*(-3*sqrt(-1/(a**5*c))*log(-a**3*sqrt(-1/(a**5*c)) + x)/16 + 3*sqrt(-1/(a**5*c)
)*log(a**3*sqrt(-1/(a**5*c)) + x)/16) + (-2*a**2*e + 5*a*c*d*x + 3*c**2*d*x**3)/
(8*a**4*c + 16*a**3*c**2*x**2 + 8*a**2*c**3*x**4)

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GIAC/XCAS [A]  time = 0.211324, size = 82, normalized size = 1.09 \[ \frac{3 \, d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \, \sqrt{a c} a^{2}} + \frac{3 \, c^{2} d x^{3} + 5 \, a c d x - 2 \, a^{2} e}{8 \,{\left (c x^{2} + a\right )}^{2} a^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/8*d*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^2) + 1/8*(3*c^2*d*x^3 + 5*a*c*d*x - 2*a
^2*e)/((c*x^2 + a)^2*a^2*c)