3.76 \(\int \frac{1}{\left (a+\frac{c}{x^2}+\frac{b}{x}\right ) x^3 (d+e x)^2} \, dx\)

Optimal. Leaf size=248 \[ -\frac{\left (a^2 d^2-a e (2 b d+c e)+b^2 e^2\right ) \log \left (a x^2+b x+c\right )}{2 c \left (a d^2-e (b d-c e)\right )^2}+\frac{\left (a^2 d (b d+4 c e)-a b e (2 b d+3 c e)+b^3 e^2\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{c \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac{e^2}{d (d+e x) \left (a d^2-b d e+c e^2\right )}-\frac{e^2 \log (d+e x) \left (3 a d^2-e (2 b d-c e)\right )}{d^2 \left (a d^2-e (b d-c e)\right )^2}+\frac{\log (x)}{c d^2} \]

[Out]

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

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Rubi [A]  time = 0.848912, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{\left (a^2 d^2-a e (2 b d+c e)+b^2 e^2\right ) \log \left (a x^2+b x+c\right )}{2 c \left (a d^2-e (b d-c e)\right )^2}+\frac{\left (a^2 d (b d+4 c e)-a b e (2 b d+3 c e)+b^3 e^2\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{c \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac{e^2}{d (d+e x) \left (a d^2-e (b d-c e)\right )}-\frac{e^2 \log (d+e x) \left (3 a d^2-e (2 b d-c e)\right )}{d^2 \left (a d^2-e (b d-c e)\right )^2}+\frac{\log (x)}{c d^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + c/x^2 + b/x)*x^3*(d + e*x)^2),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+c/x**2+b/x)/x**3/(e*x+d)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.476868, size = 246, normalized size = 0.99 \[ \frac{\left (-a^2 d^2+a e (2 b d+c e)-b^2 e^2\right ) \log (x (a x+b)+c)}{2 c \left (a d^2+e (c e-b d)\right )^2}-\frac{\left (a^2 d (b d+4 c e)-a b e (2 b d+3 c e)+b^3 e^2\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{c \sqrt{4 a c-b^2} \left (a d^2+e (c e-b d)\right )^2}+\frac{e^2}{d (d+e x) \left (a d^2+e (c e-b d)\right )}-\frac{e^2 \log (d+e x) \left (3 a d^2+e (c e-2 b d)\right )}{\left (a d^3+d e (c e-b d)\right )^2}+\frac{\log (x)}{c d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + c/x^2 + b/x)*x^3*(d + e*x)^2),x]

[Out]

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

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Maple [B]  time = 0.019, size = 589, normalized size = 2.4 \[{\frac{\ln \left ( x \right ) }{c{d}^{2}}}+{\frac{{e}^{2}}{d \left ( a{d}^{2}-bde+{e}^{2}c \right ) \left ( ex+d \right ) }}-3\,{\frac{{e}^{2}\ln \left ( ex+d \right ) a}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}+2\,{\frac{{e}^{3}\ln \left ( ex+d \right ) b}{d \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}-{\frac{{e}^{4}\ln \left ( ex+d \right ) c}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}{d}^{2}}}-{\frac{{a}^{2}\ln \left ( a{x}^{2}+bx+c \right ){d}^{2}}{2\, \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}c}}+{\frac{a\ln \left ( a{x}^{2}+bx+c \right ) bde}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}c}}+{\frac{a\ln \left ( a{x}^{2}+bx+c \right ){e}^{2}}{2\, \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ){b}^{2}{e}^{2}}{2\, \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}c}}-{\frac{{a}^{2}b{d}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}c}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-4\,{\frac{{a}^{2}de}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{a{b}^{2}de}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+3\,{\frac{ab{e}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{3}{e}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}c}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+c/x^2+b/x)/x^3/(e*x+d)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [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*(a + b/x + c/x^2)*x^3),x, algorithm="fricas")

[Out]

Timed out

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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/(a+c/x**2+b/x)/x**3/(e*x+d)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.297199, size = 528, normalized size = 2.13 \[ -\frac{{\left (a^{2} b d^{2} e^{2} - 2 \, a b^{2} d e^{3} + 4 \, a^{2} c d e^{3} + b^{3} e^{4} - 3 \, a b c e^{4}\right )} \arctan \left (\frac{{\left (2 \, a d - \frac{2 \, a d^{2}}{x e + d} - b e + \frac{2 \, b d e}{x e + d} - \frac{2 \, c e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt{-b^{2} + 4 \, a c}}\right ) e^{\left (-2\right )}}{{\left (a^{2} c d^{4} - 2 \, a b c d^{3} e + b^{2} c d^{2} e^{2} + 2 \, a c^{2} d^{2} e^{2} - 2 \, b c^{2} d e^{3} + c^{3} e^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{{\left (a^{2} d^{2} - 2 \, a b d e + b^{2} e^{2} - a c e^{2}\right )}{\rm ln}\left (a - \frac{2 \, a d}{x e + d} + \frac{a d^{2}}{{\left (x e + d\right )}^{2}} + \frac{b e}{x e + d} - \frac{b d e}{{\left (x e + d\right )}^{2}} + \frac{c e^{2}}{{\left (x e + d\right )}^{2}}\right )}{2 \,{\left (a^{2} c d^{4} - 2 \, a b c d^{3} e + b^{2} c d^{2} e^{2} + 2 \, a c^{2} d^{2} e^{2} - 2 \, b c^{2} d e^{3} + c^{3} e^{4}\right )}} + \frac{e^{5}}{{\left (a d^{3} e^{3} - b d^{2} e^{4} + c d e^{5}\right )}{\left (x e + d\right )}} + \frac{{\rm ln}\left ({\left | -\frac{d}{x e + d} + 1 \right |}\right )}{c d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((e*x + d)^2*(a + b/x + c/x^2)*x^3),x, algorithm="giac")

[Out]

-(a^2*b*d^2*e^2 - 2*a*b^2*d*e^3 + 4*a^2*c*d*e^3 + b^3*e^4 - 3*a*b*c*e^4)*arctan(
(2*a*d - 2*a*d^2/(x*e + d) - b*e + 2*b*d*e/(x*e + d) - 2*c*e^2/(x*e + d))*e^(-1)
/sqrt(-b^2 + 4*a*c))*e^(-2)/((a^2*c*d^4 - 2*a*b*c*d^3*e + b^2*c*d^2*e^2 + 2*a*c^
2*d^2*e^2 - 2*b*c^2*d*e^3 + c^3*e^4)*sqrt(-b^2 + 4*a*c)) - 1/2*(a^2*d^2 - 2*a*b*
d*e + b^2*e^2 - a*c*e^2)*ln(a - 2*a*d/(x*e + d) + a*d^2/(x*e + d)^2 + b*e/(x*e +
d) - b*d*e/(x*e + d)^2 + c*e^2/(x*e + d)^2)/(a^2*c*d^4 - 2*a*b*c*d^3*e + b^2*c*
d^2*e^2 + 2*a*c^2*d^2*e^2 - 2*b*c^2*d*e^3 + c^3*e^4) + e^5/((a*d^3*e^3 - b*d^2*e
^4 + c*d*e^5)*(x*e + d)) + ln(abs(-d/(x*e + d) + 1))/(c*d^2)