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

Optimal. Leaf size=93 \[ \frac{5 d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} \sqrt{c}}+\frac{5 d x}{16 a^3 \left (a+c x^2\right )}+\frac{5 d x}{24 a^2 \left (a+c x^2\right )^2}-\frac{a e-c d x}{6 a c \left (a+c x^2\right )^3} \]

[Out]

-(a*e - c*d*x)/(6*a*c*(a + c*x^2)^3) + (5*d*x)/(24*a^2*(a + c*x^2)^2) + (5*d*x)/
(16*a^3*(a + c*x^2)) + (5*d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(16*a^(7/2)*Sqrt[c])

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Rubi [A]  time = 0.0912396, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{5 d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} \sqrt{c}}+\frac{5 d x}{16 a^3 \left (a+c x^2\right )}+\frac{5 d x}{24 a^2 \left (a+c x^2\right )^2}-\frac{a e-c d x}{6 a c \left (a+c x^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)/(a + c*x^2)^4,x]

[Out]

-(a*e - c*d*x)/(6*a*c*(a + c*x^2)^3) + (5*d*x)/(24*a^2*(a + c*x^2)^2) + (5*d*x)/
(16*a^3*(a + c*x^2)) + (5*d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(16*a^(7/2)*Sqrt[c])

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Rubi in Sympy [A]  time = 11.2073, size = 85, normalized size = 0.91 \[ - \frac{a e - c d x}{6 a c \left (a + c x^{2}\right )^{3}} + \frac{5 d x}{24 a^{2} \left (a + c x^{2}\right )^{2}} + \frac{5 d x}{16 a^{3} \left (a + c x^{2}\right )} + \frac{5 d \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 a^{\frac{7}{2}} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a*e - c*d*x)/(6*a*c*(a + c*x**2)**3) + 5*d*x/(24*a**2*(a + c*x**2)**2) + 5*d*x
/(16*a**3*(a + c*x**2)) + 5*d*atan(sqrt(c)*x/sqrt(a))/(16*a**(7/2)*sqrt(c))

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Mathematica [A]  time = 0.0883188, size = 83, normalized size = 0.89 \[ \frac{\frac{\sqrt{a} \left (-8 a^3 e+33 a^2 c d x+40 a c^2 d x^3+15 c^3 d x^5\right )}{\left (a+c x^2\right )^3}+15 \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{48 a^{7/2} c} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[a]*(-8*a^3*e + 33*a^2*c*d*x + 40*a*c^2*d*x^3 + 15*c^3*d*x^5))/(a + c*x^2)
^3 + 15*Sqrt[c]*d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(48*a^(7/2)*c)

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Maple [A]  time = 0.006, size = 81, normalized size = 0.9 \[{\frac{2\,cdx-2\,ae}{12\,ac \left ( c{x}^{2}+a \right ) ^{3}}}+{\frac{5\,dx}{24\,{a}^{2} \left ( c{x}^{2}+a \right ) ^{2}}}+{\frac{5\,dx}{16\,{a}^{3} \left ( c{x}^{2}+a \right ) }}+{\frac{5\,d}{16\,{a}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(c*x^2+a)^4,x)

[Out]

1/12*(2*c*d*x-2*a*e)/a/c/(c*x^2+a)^3+5/24*d*x/a^2/(c*x^2+a)^2+5/16*d*x/a^3/(c*x^
2+a)+5/16*d/a^3/(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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227257, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (c^{4} d x^{6} + 3 \, a c^{3} d x^{4} + 3 \, a^{2} c^{2} d x^{2} + a^{3} c d\right )} \log \left (\frac{2 \, a c x +{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) + 2 \,{\left (15 \, c^{3} d x^{5} + 40 \, a c^{2} d x^{3} + 33 \, a^{2} c d x - 8 \, a^{3} e\right )} \sqrt{-a c}}{96 \,{\left (a^{3} c^{4} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{5} c^{2} x^{2} + a^{6} c\right )} \sqrt{-a c}}, \frac{15 \,{\left (c^{4} d x^{6} + 3 \, a c^{3} d x^{4} + 3 \, a^{2} c^{2} d x^{2} + a^{3} c d\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) +{\left (15 \, c^{3} d x^{5} + 40 \, a c^{2} d x^{3} + 33 \, a^{2} c d x - 8 \, a^{3} e\right )} \sqrt{a c}}{48 \,{\left (a^{3} c^{4} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{5} c^{2} x^{2} + a^{6} c\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(15*(c^4*d*x^6 + 3*a*c^3*d*x^4 + 3*a^2*c^2*d*x^2 + a^3*c*d)*log((2*a*c*x +
(c*x^2 - a)*sqrt(-a*c))/(c*x^2 + a)) + 2*(15*c^3*d*x^5 + 40*a*c^2*d*x^3 + 33*a^
2*c*d*x - 8*a^3*e)*sqrt(-a*c))/((a^3*c^4*x^6 + 3*a^4*c^3*x^4 + 3*a^5*c^2*x^2 + a
^6*c)*sqrt(-a*c)), 1/48*(15*(c^4*d*x^6 + 3*a*c^3*d*x^4 + 3*a^2*c^2*d*x^2 + a^3*c
*d)*arctan(sqrt(a*c)*x/a) + (15*c^3*d*x^5 + 40*a*c^2*d*x^3 + 33*a^2*c*d*x - 8*a^
3*e)*sqrt(a*c))/((a^3*c^4*x^6 + 3*a^4*c^3*x^4 + 3*a^5*c^2*x^2 + a^6*c)*sqrt(a*c)
)]

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Sympy [A]  time = 3.03054, size = 150, normalized size = 1.61 \[ d \left (- \frac{5 \sqrt{- \frac{1}{a^{7} c}} \log{\left (- a^{4} \sqrt{- \frac{1}{a^{7} c}} + x \right )}}{32} + \frac{5 \sqrt{- \frac{1}{a^{7} c}} \log{\left (a^{4} \sqrt{- \frac{1}{a^{7} c}} + x \right )}}{32}\right ) + \frac{- 8 a^{3} e + 33 a^{2} c d x + 40 a c^{2} d x^{3} + 15 c^{3} d x^{5}}{48 a^{6} c + 144 a^{5} c^{2} x^{2} + 144 a^{4} c^{3} x^{4} + 48 a^{3} c^{4} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*(-5*sqrt(-1/(a**7*c))*log(-a**4*sqrt(-1/(a**7*c)) + x)/32 + 5*sqrt(-1/(a**7*c)
)*log(a**4*sqrt(-1/(a**7*c)) + x)/32) + (-8*a**3*e + 33*a**2*c*d*x + 40*a*c**2*d
*x**3 + 15*c**3*d*x**5)/(48*a**6*c + 144*a**5*c**2*x**2 + 144*a**4*c**3*x**4 + 4
8*a**3*c**4*x**6)

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GIAC/XCAS [A]  time = 0.213405, size = 99, normalized size = 1.06 \[ \frac{5 \, d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{16 \, \sqrt{a c} a^{3}} + \frac{15 \, c^{3} d x^{5} + 40 \, a c^{2} d x^{3} + 33 \, a^{2} c d x - 8 \, a^{3} e}{48 \,{\left (c x^{2} + a\right )}^{3} a^{3} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/16*d*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^3) + 1/48*(15*c^3*d*x^5 + 40*a*c^2*d*x
^3 + 33*a^2*c*d*x - 8*a^3*e)/((c*x^2 + a)^3*a^3*c)