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

Optimal. Leaf size=145 \[ \frac{\left (a e^2+5 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} c^{3/2}}+\frac{x \left (a e^2+5 c d^2\right )}{16 a^3 c \left (a+c x^2\right )}-\frac{4 a d e-x \left (a e^2+5 c d^2\right )}{24 a^2 c \left (a+c x^2\right )^2}-\frac{(d+e x) (a e-c d x)}{6 a c \left (a+c x^2\right )^3} \]

[Out]

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

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Rubi [A]  time = 0.175987, antiderivative size = 145, 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 \[ \frac{\left (a e^2+5 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} c^{3/2}}+\frac{x \left (a e^2+5 c d^2\right )}{16 a^3 c \left (a+c x^2\right )}-\frac{4 a d e-x \left (a e^2+5 c d^2\right )}{24 a^2 c \left (a+c x^2\right )^2}-\frac{(d+e x) (a e-c d x)}{6 a c \left (a+c x^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

-((a*e - c*d*x)*(d + e*x))/(6*a*c*(a + c*x^2)^3) - (4*a*d*e - (5*c*d^2 + a*e^2)*
x)/(24*a^2*c*(a + c*x^2)^2) + ((5*c*d^2 + a*e^2)*x)/(16*a^3*c*(a + c*x^2)) + ((5
*c*d^2 + 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 = 24.4126, size = 126, normalized size = 0.87 \[ - \frac{\left (d + e x\right ) \left (a e - c d x\right )}{6 a c \left (a + c x^{2}\right )^{3}} + \frac{- 4 a d e + x \left (a e^{2} + 5 c d^{2}\right )}{24 a^{2} c \left (a + c x^{2}\right )^{2}} + \frac{x \left (a e^{2} + 5 c d^{2}\right )}{16 a^{3} c \left (a + c x^{2}\right )} + \frac{\left (a e^{2} + 5 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 a^{\frac{7}{2}} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(d + e*x)*(a*e - c*d*x)/(6*a*c*(a + c*x**2)**3) + (-4*a*d*e + x*(a*e**2 + 5*c*d
**2))/(24*a**2*c*(a + c*x**2)**2) + x*(a*e**2 + 5*c*d**2)/(16*a**3*c*(a + c*x**2
)) + (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 = 0.186942, size = 127, normalized size = 0.88 \[ \frac{\left (a e^2+5 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} c^{3/2}}+\frac{-a^3 e (16 d+3 e x)+a^2 c x \left (33 d^2+8 e^2 x^2\right )+a c^2 x^3 \left (40 d^2+3 e^2 x^2\right )+15 c^3 d^2 x^5}{48 a^3 c \left (a+c x^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(15*c^3*d^2*x^5 - a^3*e*(16*d + 3*e*x) + a*c^2*x^3*(40*d^2 + 3*e^2*x^2) + a^2*c*
x*(33*d^2 + 8*e^2*x^2))/(48*a^3*c*(a + c*x^2)^3) + ((5*c*d^2 + a*e^2)*ArcTan[(Sq
rt[c]*x)/Sqrt[a]])/(16*a^(7/2)*c^(3/2))

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Maple [A]  time = 0.012, size = 129, normalized size = 0.9 \[{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{3}} \left ({\frac{ \left ( a{e}^{2}+5\,c{d}^{2} \right ) c{x}^{5}}{16\,{a}^{3}}}+{\frac{ \left ( a{e}^{2}+5\,c{d}^{2} \right ){x}^{3}}{6\,{a}^{2}}}-{\frac{ \left ( a{e}^{2}-11\,c{d}^{2} \right ) x}{16\,ac}}-{\frac{de}{3\,c}} \right ) }+{\frac{{e}^{2}}{16\,{a}^{2}c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{5\,{d}^{2}}{16\,{a}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2/(c*x^2+a)^4,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235704, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left ({\left (5 \, c^{4} d^{2} + a c^{3} e^{2}\right )} x^{6} + 5 \, a^{3} c d^{2} + a^{4} e^{2} + 3 \,{\left (5 \, a c^{3} d^{2} + a^{2} c^{2} e^{2}\right )} x^{4} + 3 \,{\left (5 \, a^{2} c^{2} d^{2} + a^{3} c e^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, a c x +{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) + 2 \,{\left (3 \,{\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} x^{5} - 16 \, a^{3} d e + 8 \,{\left (5 \, a c^{2} d^{2} + a^{2} c e^{2}\right )} x^{3} + 3 \,{\left (11 \, a^{2} c d^{2} - a^{3} e^{2}\right )} x\right )} \sqrt{-a c}}{96 \,{\left (a^{3} c^{4} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{5} c^{2} x^{2} + a^{6} c\right )} \sqrt{-a c}}, \frac{3 \,{\left ({\left (5 \, c^{4} d^{2} + a c^{3} e^{2}\right )} x^{6} + 5 \, a^{3} c d^{2} + a^{4} e^{2} + 3 \,{\left (5 \, a c^{3} d^{2} + a^{2} c^{2} e^{2}\right )} x^{4} + 3 \,{\left (5 \, a^{2} c^{2} d^{2} + a^{3} c e^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) +{\left (3 \,{\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} x^{5} - 16 \, a^{3} d e + 8 \,{\left (5 \, a c^{2} d^{2} + a^{2} c e^{2}\right )} x^{3} + 3 \,{\left (11 \, a^{2} c d^{2} - a^{3} e^{2}\right )} x\right )} \sqrt{a c}}{48 \,{\left (a^{3} c^{4} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{5} c^{2} x^{2} + a^{6} c\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(3*((5*c^4*d^2 + a*c^3*e^2)*x^6 + 5*a^3*c*d^2 + a^4*e^2 + 3*(5*a*c^3*d^2 +
a^2*c^2*e^2)*x^4 + 3*(5*a^2*c^2*d^2 + a^3*c*e^2)*x^2)*log((2*a*c*x + (c*x^2 - a
)*sqrt(-a*c))/(c*x^2 + a)) + 2*(3*(5*c^3*d^2 + a*c^2*e^2)*x^5 - 16*a^3*d*e + 8*(
5*a*c^2*d^2 + a^2*c*e^2)*x^3 + 3*(11*a^2*c*d^2 - a^3*e^2)*x)*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*(3*((5*c^4*d^2
+ a*c^3*e^2)*x^6 + 5*a^3*c*d^2 + a^4*e^2 + 3*(5*a*c^3*d^2 + a^2*c^2*e^2)*x^4 +
3*(5*a^2*c^2*d^2 + a^3*c*e^2)*x^2)*arctan(sqrt(a*c)*x/a) + (3*(5*c^3*d^2 + a*c^2
*e^2)*x^5 - 16*a^3*d*e + 8*(5*a*c^2*d^2 + a^2*c*e^2)*x^3 + 3*(11*a^2*c*d^2 - a^3
*e^2)*x)*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 = 4.83507, size = 214, normalized size = 1.48 \[ - \frac{\sqrt{- \frac{1}{a^{7} c^{3}}} \left (a e^{2} + 5 c d^{2}\right ) \log{\left (- a^{4} c \sqrt{- \frac{1}{a^{7} c^{3}}} + x \right )}}{32} + \frac{\sqrt{- \frac{1}{a^{7} c^{3}}} \left (a e^{2} + 5 c d^{2}\right ) \log{\left (a^{4} c \sqrt{- \frac{1}{a^{7} c^{3}}} + x \right )}}{32} + \frac{- 16 a^{3} d e + x^{5} \left (3 a c^{2} e^{2} + 15 c^{3} d^{2}\right ) + x^{3} \left (8 a^{2} c e^{2} + 40 a c^{2} d^{2}\right ) + x \left (- 3 a^{3} e^{2} + 33 a^{2} c d^{2}\right )}{48 a^{6} c + 144 a^{5} c^{2} x^{2} + 144 a^{4} c^{3} x^{4} + 48 a^{3} c^{4} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-1/(a**7*c**3))*(a*e**2 + 5*c*d**2)*log(-a**4*c*sqrt(-1/(a**7*c**3)) + x)/
32 + sqrt(-1/(a**7*c**3))*(a*e**2 + 5*c*d**2)*log(a**4*c*sqrt(-1/(a**7*c**3)) +
x)/32 + (-16*a**3*d*e + x**5*(3*a*c**2*e**2 + 15*c**3*d**2) + x**3*(8*a**2*c*e**
2 + 40*a*c**2*d**2) + x*(-3*a**3*e**2 + 33*a**2*c*d**2))/(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)

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GIAC/XCAS [A]  time = 0.210876, size = 166, normalized size = 1.14 \[ \frac{{\left (5 \, c d^{2} + a e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{16 \, \sqrt{a c} a^{3} c} + \frac{15 \, c^{3} d^{2} x^{5} + 3 \, a c^{2} x^{5} e^{2} + 40 \, a c^{2} d^{2} x^{3} + 8 \, a^{2} c x^{3} e^{2} + 33 \, a^{2} c d^{2} x - 3 \, a^{3} x e^{2} - 16 \, a^{3} d e}{48 \,{\left (c x^{2} + a\right )}^{3} a^{3} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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