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

Optimal. Leaf size=98 \[ \frac{3 d \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{3/2}}-\frac{3 d (d+e x) (a e-c d x)}{8 a^2 c \left (a+c x^2\right )}+\frac{x (d+e x)^3}{4 a \left (a+c x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.112806, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{3 d \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{3/2}}-\frac{3 d (d+e x) (a e-c d x)}{8 a^2 c \left (a+c x^2\right )}+\frac{x (d+e x)^3}{4 a \left (a+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.7352, size = 88, normalized size = 0.9 \[ \frac{x \left (d + e x\right )^{3}}{4 a \left (a + c x^{2}\right )^{2}} - \frac{3 d \left (d + e x\right ) \left (a e - c d x\right )}{8 a^{2} c \left (a + c x^{2}\right )} + \frac{3 d \left (a e^{2} + c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.204953, size = 127, normalized size = 1.3 \[ \frac{\frac{\sqrt{a} \left (-2 a^3 e^3-a^2 c e \left (6 d^2+3 d e x+4 e^2 x^2\right )+a c^2 d x \left (5 d^2+3 e^2 x^2\right )+3 c^3 d^3 x^3\right )}{\left (a+c x^2\right )^2}+3 \sqrt{c} d \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 133, normalized size = 1.4 \[{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}} \left ({\frac{3\,d \left ( a{e}^{2}+c{d}^{2} \right ){x}^{3}}{8\,{a}^{2}}}-{\frac{{e}^{3}{x}^{2}}{2\,c}}-{\frac{d \left ( 3\,a{e}^{2}-5\,c{d}^{2} \right ) x}{8\,ac}}-{\frac{e \left ( a{e}^{2}+3\,c{d}^{2} \right ) }{4\,{c}^{2}}} \right ) }+{\frac{3\,d{e}^{2}}{8\,ac}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,{d}^{3}}{8\,{a}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3/(c*x^2+a)^3,x)

[Out]

(3/8*d*(a*e^2+c*d^2)/a^2*x^3-1/2*e^3*x^2/c-1/8*d*(3*a*e^2-5*c*d^2)/a/c*x-1/4*e*(
a*e^2+3*c*d^2)/c^2)/(c*x^2+a)^2+3/8*d/a/c/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*e^
2+3/8*d^3/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)^3/(c*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220967, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (a^{2} c^{2} d^{3} + a^{3} c d e^{2} +{\left (c^{4} d^{3} + a c^{3} d e^{2}\right )} x^{4} + 2 \,{\left (a c^{3} d^{3} + a^{2} c^{2} d e^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, a c x +{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) - 2 \,{\left (4 \, a^{2} c e^{3} x^{2} + 6 \, a^{2} c d^{2} e + 2 \, a^{3} e^{3} - 3 \,{\left (c^{3} d^{3} + a c^{2} d e^{2}\right )} x^{3} -{\left (5 \, a c^{2} d^{3} - 3 \, a^{2} c d e^{2}\right )} x\right )} \sqrt{-a c}}{16 \,{\left (a^{2} c^{4} x^{4} + 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )} \sqrt{-a c}}, \frac{3 \,{\left (a^{2} c^{2} d^{3} + a^{3} c d e^{2} +{\left (c^{4} d^{3} + a c^{3} d e^{2}\right )} x^{4} + 2 \,{\left (a c^{3} d^{3} + a^{2} c^{2} d e^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (4 \, a^{2} c e^{3} x^{2} + 6 \, a^{2} c d^{2} e + 2 \, a^{3} e^{3} - 3 \,{\left (c^{3} d^{3} + a c^{2} d e^{2}\right )} x^{3} -{\left (5 \, a c^{2} d^{3} - 3 \, a^{2} c d e^{2}\right )} x\right )} \sqrt{a c}}{8 \,{\left (a^{2} c^{4} x^{4} + 2 \, a^{3} c^{3} x^{2} + a^{4} c^{2}\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(3*(a^2*c^2*d^3 + a^3*c*d*e^2 + (c^4*d^3 + a*c^3*d*e^2)*x^4 + 2*(a*c^3*d^3
+ a^2*c^2*d*e^2)*x^2)*log((2*a*c*x + (c*x^2 - a)*sqrt(-a*c))/(c*x^2 + a)) - 2*(
4*a^2*c*e^3*x^2 + 6*a^2*c*d^2*e + 2*a^3*e^3 - 3*(c^3*d^3 + a*c^2*d*e^2)*x^3 - (5
*a*c^2*d^3 - 3*a^2*c*d*e^2)*x)*sqrt(-a*c))/((a^2*c^4*x^4 + 2*a^3*c^3*x^2 + a^4*c
^2)*sqrt(-a*c)), 1/8*(3*(a^2*c^2*d^3 + a^3*c*d*e^2 + (c^4*d^3 + a*c^3*d*e^2)*x^4
+ 2*(a*c^3*d^3 + a^2*c^2*d*e^2)*x^2)*arctan(sqrt(a*c)*x/a) - (4*a^2*c*e^3*x^2 +
6*a^2*c*d^2*e + 2*a^3*e^3 - 3*(c^3*d^3 + a*c^2*d*e^2)*x^3 - (5*a*c^2*d^3 - 3*a^
2*c*d*e^2)*x)*sqrt(a*c))/((a^2*c^4*x^4 + 2*a^3*c^3*x^2 + a^4*c^2)*sqrt(a*c))]

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Sympy [A]  time = 5.60343, size = 272, normalized size = 2.78 \[ - \frac{3 d \sqrt{- \frac{1}{a^{5} c^{3}}} \left (a e^{2} + c d^{2}\right ) \log{\left (- \frac{3 a^{3} c d \sqrt{- \frac{1}{a^{5} c^{3}}} \left (a e^{2} + c d^{2}\right )}{3 a d e^{2} + 3 c d^{3}} + x \right )}}{16} + \frac{3 d \sqrt{- \frac{1}{a^{5} c^{3}}} \left (a e^{2} + c d^{2}\right ) \log{\left (\frac{3 a^{3} c d \sqrt{- \frac{1}{a^{5} c^{3}}} \left (a e^{2} + c d^{2}\right )}{3 a d e^{2} + 3 c d^{3}} + x \right )}}{16} + \frac{- 2 a^{3} e^{3} - 6 a^{2} c d^{2} e - 4 a^{2} c e^{3} x^{2} + x^{3} \left (3 a c^{2} d e^{2} + 3 c^{3} d^{3}\right ) + x \left (- 3 a^{2} c d e^{2} + 5 a c^{2} d^{3}\right )}{8 a^{4} c^{2} + 16 a^{3} c^{3} x^{2} + 8 a^{2} c^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*d*sqrt(-1/(a**5*c**3))*(a*e**2 + c*d**2)*log(-3*a**3*c*d*sqrt(-1/(a**5*c**3))
*(a*e**2 + c*d**2)/(3*a*d*e**2 + 3*c*d**3) + x)/16 + 3*d*sqrt(-1/(a**5*c**3))*(a
*e**2 + c*d**2)*log(3*a**3*c*d*sqrt(-1/(a**5*c**3))*(a*e**2 + c*d**2)/(3*a*d*e**
2 + 3*c*d**3) + x)/16 + (-2*a**3*e**3 - 6*a**2*c*d**2*e - 4*a**2*c*e**3*x**2 + x
**3*(3*a*c**2*d*e**2 + 3*c**3*d**3) + x*(-3*a**2*c*d*e**2 + 5*a*c**2*d**3))/(8*a
**4*c**2 + 16*a**3*c**3*x**2 + 8*a**2*c**4*x**4)

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GIAC/XCAS [A]  time = 0.211412, size = 167, normalized size = 1.7 \[ \frac{3 \,{\left (c d^{3} + a d e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \, \sqrt{a c} a^{2} c} + \frac{3 \, c^{3} d^{3} x^{3} + 3 \, a c^{2} d x^{3} e^{2} + 5 \, a c^{2} d^{3} x - 4 \, a^{2} c x^{2} e^{3} - 3 \, a^{2} c d x e^{2} - 6 \, a^{2} c d^{2} e - 2 \, a^{3} e^{3}}{8 \,{\left (c x^{2} + a\right )}^{2} a^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/8*(c*d^3 + a*d*e^2)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^2*c) + 1/8*(3*c^3*d^3*x
^3 + 3*a*c^2*d*x^3*e^2 + 5*a*c^2*d^3*x - 4*a^2*c*x^2*e^3 - 3*a^2*c*d*x*e^2 - 6*a
^2*c*d^2*e - 2*a^3*e^3)/((c*x^2 + a)^2*a^2*c^2)