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

Optimal. Leaf size=246 \[ \frac{\left (-c \left (a d^2-c e^2\right )+b^2 d^2-2 b c d e\right ) \log \left (a x^2+b x+c\right )}{2 a \left (a d^2-e (b d-c e)\right )^2}+\frac{\left (-b c \left (3 a d^2-c e^2\right )+4 a c^2 d e+b^3 d^2-2 b^2 c d e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac{d^2 \log (d+e x) \left (a d^2-e (2 b d-3 c e)\right )}{e^2 \left (a d^2-e (b d-c e)\right )^2}+\frac{d^3}{e^2 (d+e x) \left (a d^2-e (b d-c e)\right )} \]

[Out]

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

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Rubi [A]  time = 0.833699, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ \frac{\left (-c \left (a d^2-c e^2\right )+b^2 d^2-2 b c d e\right ) \log \left (a x^2+b x+c\right )}{2 a \left (a d^2-e (b d-c e)\right )^2}+\frac{\left (-b c \left (3 a d^2-c e^2\right )+4 a c^2 d e+b^3 d^2-2 b^2 c d e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{a \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}+\frac{d^2 \log (d+e x) \left (a d^2-e (2 b d-3 c e)\right )}{e^2 \left (a d^2-e (b d-c e)\right )^2}+\frac{d^3}{e^2 (d+e x) \left (a d^2-e (b d-c e)\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [A]  time = 0.406977, size = 207, normalized size = 0.84 \[ \frac{\frac{\left (c \left (c e^2-a d^2\right )+b^2 d^2-2 b c d e\right ) \log (x (a x+b)+c)}{a}-\frac{2 \left (b c \left (c e^2-3 a d^2\right )+4 a c^2 d e+b^3 d^2-2 b^2 c d e\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{a \sqrt{4 a c-b^2}}+\frac{2 \log (d+e x) \left (a d^4+d^2 e (3 c e-2 b d)\right )}{e^2}+\frac{2 d^3 \left (a d^2+e (c e-b d)\right )}{e^2 (d+e x)}}{2 \left (a d^2+e (c e-b d)\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.014, size = 580, normalized size = 2.4 \[{\frac{{d}^{4}\ln \left ( ex+d \right ) a}{{e}^{2} \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}-2\,{\frac{{d}^{3}\ln \left ( ex+d \right ) b}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}e}}+3\,{\frac{{d}^{2}\ln \left ( ex+d \right ) c}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}+{\frac{{d}^{3}}{{e}^{2} \left ( a{d}^{2}-bde+{e}^{2}c \right ) \left ( ex+d \right ) }}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ) c{d}^{2}}{2\, \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ){b}^{2}{d}^{2}}{2\, \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}a}}-{\frac{\ln \left ( a{x}^{2}+bx+c \right ) bcde}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}a}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ){c}^{2}{e}^{2}}{2\, \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}a}}+3\,{\frac{bc{d}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\frac{{c}^{2}de}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{3}{d}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}a}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+2\,{\frac{{b}^{2}dec}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}\sqrt{4\,ac-{b}^{2}}a}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{b{c}^{2}{e}^{2}}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) ^{2}a}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 17.6323, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.28371, size = 556, normalized size = 2.26 \[ \frac{1}{2} \,{\left (\frac{2 \, d^{3} e^{2}}{{\left (a d^{2} e^{3} - b d e^{4} + c e^{5}\right )}{\left (x e + d\right )}} + \frac{2 \,{\left (b^{3} d^{2} e^{3} - 3 \, a b c d^{2} e^{3} - 2 \, b^{2} c d e^{4} + 4 \, a c^{2} d e^{4} + b c^{2} e^{5}\right )} \arctan \left (-\frac{{\left (2 \, a d - \frac{2 \, a d^{2}}{x e + d} - b e + \frac{2 \, b d e}{x e + d} - \frac{2 \, c e^{2}}{x e + d}\right )} e^{\left (-1\right )}}{\sqrt{-b^{2} + 4 \, a c}}\right ) e^{\left (-2\right )}}{{\left (a^{3} d^{4} - 2 \, a^{2} b d^{3} e + a b^{2} d^{2} e^{2} + 2 \, a^{2} c d^{2} e^{2} - 2 \, a b c d e^{3} + a c^{2} e^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\left (b^{2} d^{2} e - a c d^{2} e - 2 \, b c d e^{2} + c^{2} e^{3}\right )}{\rm ln}\left (-a + \frac{2 \, a d}{x e + d} - \frac{a d^{2}}{{\left (x e + d\right )}^{2}} - \frac{b e}{x e + d} + \frac{b d e}{{\left (x e + d\right )}^{2}} - \frac{c e^{2}}{{\left (x e + d\right )}^{2}}\right )}{a^{3} d^{4} - 2 \, a^{2} b d^{3} e + a b^{2} d^{2} e^{2} + 2 \, a^{2} c d^{2} e^{2} - 2 \, a b c d e^{3} + a c^{2} e^{4}} - \frac{2 \, e^{\left (-1\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right )}{a}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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