3.500 \(\int \frac{1}{(d+e x)^2 \left (a+c x^2\right )^2} \, dx\)

Optimal. Leaf size=205 \[ \frac{\sqrt{c} \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \left (a e^2+c d^2\right )^3}+\frac{a e+c d x}{2 a \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )}+\frac{e \left (c d^2-3 a e^2\right )}{2 a (d+e x) \left (a e^2+c d^2\right )^2}-\frac{2 c d e^3 \log \left (a+c x^2\right )}{\left (a e^2+c d^2\right )^3}+\frac{4 c d e^3 \log (d+e x)}{\left (a e^2+c d^2\right )^3} \]

[Out]

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

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Rubi [A]  time = 0.45866, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{\sqrt{c} \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \left (a e^2+c d^2\right )^3}+\frac{a e+c d x}{2 a \left (a+c x^2\right ) (d+e x) \left (a e^2+c d^2\right )}+\frac{e \left (c d^2-3 a e^2\right )}{2 a (d+e x) \left (a e^2+c d^2\right )^2}-\frac{2 c d e^3 \log \left (a+c x^2\right )}{\left (a e^2+c d^2\right )^3}+\frac{4 c d e^3 \log (d+e x)}{\left (a e^2+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^2*(a + c*x^2)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 95.3098, size = 187, normalized size = 0.91 \[ - \frac{2 c d e^{3} \log{\left (a + c x^{2} \right )}}{\left (a e^{2} + c d^{2}\right )^{3}} + \frac{4 c d e^{3} \log{\left (d + e x \right )}}{\left (a e^{2} + c d^{2}\right )^{3}} - \frac{e \left (3 a e^{2} - c d^{2}\right )}{2 a \left (d + e x\right ) \left (a e^{2} + c d^{2}\right )^{2}} + \frac{a e + c d x}{2 a \left (a + c x^{2}\right ) \left (d + e x\right ) \left (a e^{2} + c d^{2}\right )} - \frac{\sqrt{c} \left (3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**2/(c*x**2+a)**2,x)

[Out]

-2*c*d*e**3*log(a + c*x**2)/(a*e**2 + c*d**2)**3 + 4*c*d*e**3*log(d + e*x)/(a*e*
*2 + c*d**2)**3 - e*(3*a*e**2 - c*d**2)/(2*a*(d + e*x)*(a*e**2 + c*d**2)**2) + (
a*e + c*d*x)/(2*a*(a + c*x**2)*(d + e*x)*(a*e**2 + c*d**2)) - sqrt(c)*(3*a**2*e*
*4 - 6*a*c*d**2*e**2 - c**2*d**4)*atan(sqrt(c)*x/sqrt(a))/(2*a**(3/2)*(a*e**2 +
c*d**2)**3)

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Mathematica [A]  time = 0.448504, size = 162, normalized size = 0.79 \[ \frac{\frac{\sqrt{c} \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{3/2}}+\frac{c \left (a e^2+c d^2\right ) \left (a e (2 d-e x)+c d^2 x\right )}{a \left (a+c x^2\right )}-\frac{2 e^3 \left (a e^2+c d^2\right )}{d+e x}-4 c d e^3 \log \left (a+c x^2\right )+8 c d e^3 \log (d+e x)}{2 \left (a e^2+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^2*(a + c*x^2)^2),x]

[Out]

((-2*e^3*(c*d^2 + a*e^2))/(d + e*x) + (c*(c*d^2 + a*e^2)*(c*d^2*x + a*e*(2*d - e
*x)))/(a*(a + c*x^2)) + (Sqrt[c]*(c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4)*ArcTan[(S
qrt[c]*x)/Sqrt[a]])/a^(3/2) + 8*c*d*e^3*Log[d + e*x] - 4*c*d*e^3*Log[a + c*x^2])
/(2*(c*d^2 + a*e^2)^3)

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Maple [A]  time = 0.022, size = 316, normalized size = 1.5 \[ -{\frac{{e}^{3}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( ex+d \right ) }}+4\,{\frac{d{e}^{3}c\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}}-{\frac{acx{e}^{4}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) }}+{\frac{{c}^{3}x{d}^{4}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) a}}+{\frac{acd{e}^{3}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) }}+{\frac{{c}^{2}{d}^{3}e}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3} \left ( c{x}^{2}+a \right ) }}-2\,{\frac{d{e}^{3}c\ln \left ( a \left ( c{x}^{2}+a \right ) \right ) }{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}}-{\frac{3\,{e}^{4}ac}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+3\,{\frac{{d}^{2}{e}^{2}{c}^{2}}{ \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+{\frac{{d}^{4}{c}^{3}}{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{3}a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^2/(c*x^2+a)^2,x)

[Out]

-e^3/(a*e^2+c*d^2)^2/(e*x+d)+4*c*d*e^3*ln(e*x+d)/(a*e^2+c*d^2)^3-1/2*c/(a*e^2+c*
d^2)^3/(c*x^2+a)*a*x*e^4+1/2*c^3/(a*e^2+c*d^2)^3/(c*x^2+a)/a*x*d^4+c/(a*e^2+c*d^
2)^3/(c*x^2+a)*a*d*e^3+c^2/(a*e^2+c*d^2)^3/(c*x^2+a)*d^3*e-2*c/(a*e^2+c*d^2)^3*d
*e^3*ln(a*(c*x^2+a))-3/2*c/(a*e^2+c*d^2)^3*a/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))
*e^4+3*c^2/(a*e^2+c*d^2)^3/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*d^2*e^2+1/2*c^3/(
a*e^2+c*d^2)^3/a/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*d^4

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.33537, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)^2*(e*x + d)^2),x, algorithm="fricas")

[Out]

[1/4*(4*a*c^2*d^4*e - 4*a^3*e^5 + 2*(c^3*d^4*e - 2*a*c^2*d^2*e^3 - 3*a^2*c*e^5)*
x^2 - (a*c^2*d^5 + 6*a^2*c*d^3*e^2 - 3*a^3*d*e^4 + (c^3*d^4*e + 6*a*c^2*d^2*e^3
- 3*a^2*c*e^5)*x^3 + (c^3*d^5 + 6*a*c^2*d^3*e^2 - 3*a^2*c*d*e^4)*x^2 + (a*c^2*d^
4*e + 6*a^2*c*d^2*e^3 - 3*a^3*e^5)*x)*sqrt(-c/a)*log((c*x^2 - 2*a*x*sqrt(-c/a) -
a)/(c*x^2 + a)) + 2*(c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*x - 8*(a*c^2*d*e^
4*x^3 + a*c^2*d^2*e^3*x^2 + a^2*c*d*e^4*x + a^2*c*d^2*e^3)*log(c*x^2 + a) + 16*(
a*c^2*d*e^4*x^3 + a*c^2*d^2*e^3*x^2 + a^2*c*d*e^4*x + a^2*c*d^2*e^3)*log(e*x + d
))/(a^2*c^3*d^7 + 3*a^3*c^2*d^5*e^2 + 3*a^4*c*d^3*e^4 + a^5*d*e^6 + (a*c^4*d^6*e
+ 3*a^2*c^3*d^4*e^3 + 3*a^3*c^2*d^2*e^5 + a^4*c*e^7)*x^3 + (a*c^4*d^7 + 3*a^2*c
^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 + a^4*c*d*e^6)*x^2 + (a^2*c^3*d^6*e + 3*a^3*c^2*d
^4*e^3 + 3*a^4*c*d^2*e^5 + a^5*e^7)*x), 1/2*(2*a*c^2*d^4*e - 2*a^3*e^5 + (c^3*d^
4*e - 2*a*c^2*d^2*e^3 - 3*a^2*c*e^5)*x^2 + (a*c^2*d^5 + 6*a^2*c*d^3*e^2 - 3*a^3*
d*e^4 + (c^3*d^4*e + 6*a*c^2*d^2*e^3 - 3*a^2*c*e^5)*x^3 + (c^3*d^5 + 6*a*c^2*d^3
*e^2 - 3*a^2*c*d*e^4)*x^2 + (a*c^2*d^4*e + 6*a^2*c*d^2*e^3 - 3*a^3*e^5)*x)*sqrt(
c/a)*arctan(c*x/(a*sqrt(c/a))) + (c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*x - 4
*(a*c^2*d*e^4*x^3 + a*c^2*d^2*e^3*x^2 + a^2*c*d*e^4*x + a^2*c*d^2*e^3)*log(c*x^2
+ a) + 8*(a*c^2*d*e^4*x^3 + a*c^2*d^2*e^3*x^2 + a^2*c*d*e^4*x + a^2*c*d^2*e^3)*
log(e*x + d))/(a^2*c^3*d^7 + 3*a^3*c^2*d^5*e^2 + 3*a^4*c*d^3*e^4 + a^5*d*e^6 + (
a*c^4*d^6*e + 3*a^2*c^3*d^4*e^3 + 3*a^3*c^2*d^2*e^5 + a^4*c*e^7)*x^3 + (a*c^4*d^
7 + 3*a^2*c^3*d^5*e^2 + 3*a^3*c^2*d^3*e^4 + a^4*c*d*e^6)*x^2 + (a^2*c^3*d^6*e +
3*a^3*c^2*d^4*e^3 + 3*a^4*c*d^2*e^5 + a^5*e^7)*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**2/(c*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218176, size = 521, normalized size = 2.54 \[ -\frac{2 \, c d e^{3}{\rm ln}\left (c - \frac{2 \, c d}{x e + d} + \frac{c d^{2}}{{\left (x e + d\right )}^{2}} + \frac{a e^{2}}{{\left (x e + d\right )}^{2}}\right )}{c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}} + \frac{{\left (c^{3} d^{4} e^{2} + 6 \, a c^{2} d^{2} e^{4} - 3 \, a^{2} c e^{6}\right )} \arctan \left (\frac{{\left (c d - \frac{c d^{2}}{x e + d} - \frac{a e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt{a c}}\right ) e^{\left (-2\right )}}{2 \,{\left (a c^{3} d^{6} + 3 \, a^{2} c^{2} d^{4} e^{2} + 3 \, a^{3} c d^{2} e^{4} + a^{4} e^{6}\right )} \sqrt{a c}} - \frac{e^{7}}{{\left (c^{2} d^{4} e^{4} + 2 \, a c d^{2} e^{6} + a^{2} e^{8}\right )}{\left (x e + d\right )}} + \frac{\frac{c^{3} d^{3} e - 3 \, a c^{2} d e^{3}}{c d^{2} + a e^{2}} - \frac{{\left (c^{3} d^{4} e^{2} - 6 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} e^{\left (-1\right )}}{{\left (c d^{2} + a e^{2}\right )}{\left (x e + d\right )}}}{2 \,{\left (c d^{2} + a e^{2}\right )}^{2} a{\left (c - \frac{2 \, c d}{x e + d} + \frac{c d^{2}}{{\left (x e + d\right )}^{2}} + \frac{a e^{2}}{{\left (x e + d\right )}^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)^2*(e*x + d)^2),x, algorithm="giac")

[Out]

-2*c*d*e^3*ln(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + a*e^2/(x*e + d)^2)/(c^3*
d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6) + 1/2*(c^3*d^4*e^2 + 6*a*c^2*
d^2*e^4 - 3*a^2*c*e^6)*arctan((c*d - c*d^2/(x*e + d) - a*e^2/(x*e + d))*e^(-1)/s
qrt(a*c))*e^(-2)/((a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)*sq
rt(a*c)) - e^7/((c^2*d^4*e^4 + 2*a*c*d^2*e^6 + a^2*e^8)*(x*e + d)) + 1/2*((c^3*d
^3*e - 3*a*c^2*d*e^3)/(c*d^2 + a*e^2) - (c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e
^6)*e^(-1)/((c*d^2 + a*e^2)*(x*e + d)))/((c*d^2 + a*e^2)^2*a*(c - 2*c*d/(x*e + d
) + c*d^2/(x*e + d)^2 + a*e^2/(x*e + d)^2))