3.509 \(\int \frac{(d+e x)^3}{\left (a+c x^2\right )^4} \, dx\)

Optimal. Leaf size=156 \[ \frac{d \left (3 a e^2+5 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} c^{3/2}}-\frac{4 a e \left (a e^2+5 c d^2\right )-c d x \left (15 c d^2-a e^2\right )}{48 a^3 c^2 \left (a+c x^2\right )}-\frac{(d+e x)^2 (2 a e-5 c d x)}{24 a^2 c \left (a+c x^2\right )^2}+\frac{x (d+e x)^3}{6 a \left (a+c x^2\right )^3} \]

[Out]

(x*(d + e*x)^3)/(6*a*(a + c*x^2)^3) - ((2*a*e - 5*c*d*x)*(d + e*x)^2)/(24*a^2*c*
(a + c*x^2)^2) - (4*a*e*(5*c*d^2 + a*e^2) - c*d*(15*c*d^2 - a*e^2)*x)/(48*a^3*c^
2*(a + c*x^2)) + (d*(5*c*d^2 + 3*a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(16*a^(7/2)
*c^(3/2))

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Rubi [A]  time = 0.334743, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{d \left (3 a e^2+5 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} c^{3/2}}-\frac{4 a e \left (a e^2+5 c d^2\right )-c d x \left (15 c d^2-a e^2\right )}{48 a^3 c^2 \left (a+c x^2\right )}-\frac{(d+e x)^2 (2 a e-5 c d x)}{24 a^2 c \left (a+c x^2\right )^2}+\frac{x (d+e x)^3}{6 a \left (a+c x^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3/(a + c*x^2)^4,x]

[Out]

(x*(d + e*x)^3)/(6*a*(a + c*x^2)^3) - ((2*a*e - 5*c*d*x)*(d + e*x)^2)/(24*a^2*c*
(a + c*x^2)^2) - (4*a*e*(5*c*d^2 + a*e^2) - c*d*(15*c*d^2 - a*e^2)*x)/(48*a^3*c^
2*(a + c*x^2)) + (d*(5*c*d^2 + 3*a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(16*a^(7/2)
*c^(3/2))

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Rubi in Sympy [A]  time = 38.8401, size = 139, normalized size = 0.89 \[ \frac{x \left (d + e x\right )^{3}}{6 a \left (a + c x^{2}\right )^{3}} - \frac{\left (d + e x\right )^{2} \left (2 a e - 5 c d x\right )}{24 a^{2} c \left (a + c x^{2}\right )^{2}} - \frac{\left (a e - c d x\right ) \left (4 a e^{2} + 15 c d^{2} + 5 c d e x\right )}{48 a^{3} c^{2} \left (a + c x^{2}\right )} + \frac{d \left (3 a e^{2} + 5 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 a^{\frac{7}{2}} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3/(c*x**2+a)**4,x)

[Out]

x*(d + e*x)**3/(6*a*(a + c*x**2)**3) - (d + e*x)**2*(2*a*e - 5*c*d*x)/(24*a**2*c
*(a + c*x**2)**2) - (a*e - c*d*x)*(4*a*e**2 + 15*c*d**2 + 5*c*d*e*x)/(48*a**3*c*
*2*(a + c*x**2)) + d*(3*a*e**2 + 5*c*d**2)*atan(sqrt(c)*x/sqrt(a))/(16*a**(7/2)*
c**(3/2))

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Mathematica [A]  time = 0.316386, size = 155, normalized size = 0.99 \[ \frac{\frac{\sqrt{a} \left (-4 a^4 e^3-3 a^3 c e \left (8 d^2+3 d e x+4 e^2 x^2\right )+3 a^2 c^2 d x \left (11 d^2+8 e^2 x^2\right )+a c^3 d x^3 \left (40 d^2+9 e^2 x^2\right )+15 c^4 d^3 x^5\right )}{\left (a+c x^2\right )^3}+3 \sqrt{c} d \left (3 a e^2+5 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{48 a^{7/2} c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3/(a + c*x^2)^4,x]

[Out]

((Sqrt[a]*(-4*a^4*e^3 + 15*c^4*d^3*x^5 - 3*a^3*c*e*(8*d^2 + 3*d*e*x + 4*e^2*x^2)
+ 3*a^2*c^2*d*x*(11*d^2 + 8*e^2*x^2) + a*c^3*d*x^3*(40*d^2 + 9*e^2*x^2)))/(a +
c*x^2)^3 + 3*Sqrt[c]*d*(5*c*d^2 + 3*a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(48*a^(7
/2)*c^2)

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Maple [A]  time = 0.012, size = 158, normalized size = 1. \[{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{3}} \left ({\frac{d \left ( 3\,a{e}^{2}+5\,c{d}^{2} \right ) c{x}^{5}}{16\,{a}^{3}}}+{\frac{d \left ( 3\,a{e}^{2}+5\,c{d}^{2} \right ){x}^{3}}{6\,{a}^{2}}}-{\frac{{e}^{3}{x}^{2}}{4\,c}}-{\frac{d \left ( 3\,a{e}^{2}-11\,c{d}^{2} \right ) x}{16\,ac}}-{\frac{e \left ( a{e}^{2}+6\,c{d}^{2} \right ) }{12\,{c}^{2}}} \right ) }+{\frac{3\,d{e}^{2}}{16\,{a}^{2}c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{5\,{d}^{3}}{16\,{a}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3/(c*x^2+a)^4,x)

[Out]

(1/16*d*(3*a*e^2+5*c*d^2)/a^3*c*x^5+1/6/a^2*d*(3*a*e^2+5*c*d^2)*x^3-1/4*e^3*x^2/
c-1/16*d*(3*a*e^2-11*c*d^2)/a/c*x-1/12*e*(a*e^2+6*c*d^2)/c^2)/(c*x^2+a)^3+3/16*d
/a^2/c/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*e^2+5/16*d^3/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)^3/(c*x^2 + a)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238529, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (5 \, a^{3} c^{2} d^{3} + 3 \, a^{4} c d e^{2} +{\left (5 \, c^{5} d^{3} + 3 \, a c^{4} d e^{2}\right )} x^{6} + 3 \,{\left (5 \, a c^{4} d^{3} + 3 \, a^{2} c^{3} d e^{2}\right )} x^{4} + 3 \,{\left (5 \, a^{2} c^{3} d^{3} + 3 \, a^{3} c^{2} d e^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, a c x +{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) - 2 \,{\left (12 \, a^{3} c e^{3} x^{2} + 24 \, a^{3} c d^{2} e + 4 \, a^{4} e^{3} - 3 \,{\left (5 \, c^{4} d^{3} + 3 \, a c^{3} d e^{2}\right )} x^{5} - 8 \,{\left (5 \, a c^{3} d^{3} + 3 \, a^{2} c^{2} d e^{2}\right )} x^{3} - 3 \,{\left (11 \, a^{2} c^{2} d^{3} - 3 \, a^{3} c d e^{2}\right )} x\right )} \sqrt{-a c}}{96 \,{\left (a^{3} c^{5} x^{6} + 3 \, a^{4} c^{4} x^{4} + 3 \, a^{5} c^{3} x^{2} + a^{6} c^{2}\right )} \sqrt{-a c}}, \frac{3 \,{\left (5 \, a^{3} c^{2} d^{3} + 3 \, a^{4} c d e^{2} +{\left (5 \, c^{5} d^{3} + 3 \, a c^{4} d e^{2}\right )} x^{6} + 3 \,{\left (5 \, a c^{4} d^{3} + 3 \, a^{2} c^{3} d e^{2}\right )} x^{4} + 3 \,{\left (5 \, a^{2} c^{3} d^{3} + 3 \, a^{3} c^{2} d e^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (12 \, a^{3} c e^{3} x^{2} + 24 \, a^{3} c d^{2} e + 4 \, a^{4} e^{3} - 3 \,{\left (5 \, c^{4} d^{3} + 3 \, a c^{3} d e^{2}\right )} x^{5} - 8 \,{\left (5 \, a c^{3} d^{3} + 3 \, a^{2} c^{2} d e^{2}\right )} x^{3} - 3 \,{\left (11 \, a^{2} c^{2} d^{3} - 3 \, a^{3} c d e^{2}\right )} x\right )} \sqrt{a c}}{48 \,{\left (a^{3} c^{5} x^{6} + 3 \, a^{4} c^{4} x^{4} + 3 \, a^{5} c^{3} x^{2} + a^{6} c^{2}\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*x^2 + a)^4,x, algorithm="fricas")

[Out]

[1/96*(3*(5*a^3*c^2*d^3 + 3*a^4*c*d*e^2 + (5*c^5*d^3 + 3*a*c^4*d*e^2)*x^6 + 3*(5
*a*c^4*d^3 + 3*a^2*c^3*d*e^2)*x^4 + 3*(5*a^2*c^3*d^3 + 3*a^3*c^2*d*e^2)*x^2)*log
((2*a*c*x + (c*x^2 - a)*sqrt(-a*c))/(c*x^2 + a)) - 2*(12*a^3*c*e^3*x^2 + 24*a^3*
c*d^2*e + 4*a^4*e^3 - 3*(5*c^4*d^3 + 3*a*c^3*d*e^2)*x^5 - 8*(5*a*c^3*d^3 + 3*a^2
*c^2*d*e^2)*x^3 - 3*(11*a^2*c^2*d^3 - 3*a^3*c*d*e^2)*x)*sqrt(-a*c))/((a^3*c^5*x^
6 + 3*a^4*c^4*x^4 + 3*a^5*c^3*x^2 + a^6*c^2)*sqrt(-a*c)), 1/48*(3*(5*a^3*c^2*d^3
+ 3*a^4*c*d*e^2 + (5*c^5*d^3 + 3*a*c^4*d*e^2)*x^6 + 3*(5*a*c^4*d^3 + 3*a^2*c^3*
d*e^2)*x^4 + 3*(5*a^2*c^3*d^3 + 3*a^3*c^2*d*e^2)*x^2)*arctan(sqrt(a*c)*x/a) - (1
2*a^3*c*e^3*x^2 + 24*a^3*c*d^2*e + 4*a^4*e^3 - 3*(5*c^4*d^3 + 3*a*c^3*d*e^2)*x^5
- 8*(5*a*c^3*d^3 + 3*a^2*c^2*d*e^2)*x^3 - 3*(11*a^2*c^2*d^3 - 3*a^3*c*d*e^2)*x)
*sqrt(a*c))/((a^3*c^5*x^6 + 3*a^4*c^4*x^4 + 3*a^5*c^3*x^2 + a^6*c^2)*sqrt(a*c))]

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Sympy [A]  time = 7.78768, size = 320, normalized size = 2.05 \[ - \frac{d \sqrt{- \frac{1}{a^{7} c^{3}}} \left (3 a e^{2} + 5 c d^{2}\right ) \log{\left (- \frac{a^{4} c d \sqrt{- \frac{1}{a^{7} c^{3}}} \left (3 a e^{2} + 5 c d^{2}\right )}{3 a d e^{2} + 5 c d^{3}} + x \right )}}{32} + \frac{d \sqrt{- \frac{1}{a^{7} c^{3}}} \left (3 a e^{2} + 5 c d^{2}\right ) \log{\left (\frac{a^{4} c d \sqrt{- \frac{1}{a^{7} c^{3}}} \left (3 a e^{2} + 5 c d^{2}\right )}{3 a d e^{2} + 5 c d^{3}} + x \right )}}{32} + \frac{- 4 a^{4} e^{3} - 24 a^{3} c d^{2} e - 12 a^{3} c e^{3} x^{2} + x^{5} \left (9 a c^{3} d e^{2} + 15 c^{4} d^{3}\right ) + x^{3} \left (24 a^{2} c^{2} d e^{2} + 40 a c^{3} d^{3}\right ) + x \left (- 9 a^{3} c d e^{2} + 33 a^{2} c^{2} d^{3}\right )}{48 a^{6} c^{2} + 144 a^{5} c^{3} x^{2} + 144 a^{4} c^{4} x^{4} + 48 a^{3} c^{5} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3/(c*x**2+a)**4,x)

[Out]

-d*sqrt(-1/(a**7*c**3))*(3*a*e**2 + 5*c*d**2)*log(-a**4*c*d*sqrt(-1/(a**7*c**3))
*(3*a*e**2 + 5*c*d**2)/(3*a*d*e**2 + 5*c*d**3) + x)/32 + d*sqrt(-1/(a**7*c**3))*
(3*a*e**2 + 5*c*d**2)*log(a**4*c*d*sqrt(-1/(a**7*c**3))*(3*a*e**2 + 5*c*d**2)/(3
*a*d*e**2 + 5*c*d**3) + x)/32 + (-4*a**4*e**3 - 24*a**3*c*d**2*e - 12*a**3*c*e**
3*x**2 + x**5*(9*a*c**3*d*e**2 + 15*c**4*d**3) + x**3*(24*a**2*c**2*d*e**2 + 40*
a*c**3*d**3) + x*(-9*a**3*c*d*e**2 + 33*a**2*c**2*d**3))/(48*a**6*c**2 + 144*a**
5*c**3*x**2 + 144*a**4*c**4*x**4 + 48*a**3*c**5*x**6)

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GIAC/XCAS [A]  time = 0.212662, size = 208, normalized size = 1.33 \[ \frac{{\left (5 \, c d^{3} + 3 \, a d e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{16 \, \sqrt{a c} a^{3} c} + \frac{15 \, c^{4} d^{3} x^{5} + 9 \, a c^{3} d x^{5} e^{2} + 40 \, a c^{3} d^{3} x^{3} + 24 \, a^{2} c^{2} d x^{3} e^{2} + 33 \, a^{2} c^{2} d^{3} x - 12 \, a^{3} c x^{2} e^{3} - 9 \, a^{3} c d x e^{2} - 24 \, a^{3} c d^{2} e - 4 \, a^{4} e^{3}}{48 \,{\left (c x^{2} + a\right )}^{3} a^{3} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3/(c*x^2 + a)^4,x, algorithm="giac")

[Out]

1/16*(5*c*d^3 + 3*a*d*e^2)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^3*c) + 1/48*(15*c^
4*d^3*x^5 + 9*a*c^3*d*x^5*e^2 + 40*a*c^3*d^3*x^3 + 24*a^2*c^2*d*x^3*e^2 + 33*a^2
*c^2*d^3*x - 12*a^3*c*x^2*e^3 - 9*a^3*c*d*x*e^2 - 24*a^3*c*d^2*e - 4*a^4*e^3)/((
c*x^2 + a)^3*a^3*c^2)