3.69 \(\int \frac{1}{\left (a+\frac{c}{x^2}+\frac{b}{x}\right ) x^5 (d+e x)} \, dx\)

Optimal. Leaf size=252 \[ \frac{\left (a^2 c d-a b (b d+2 c e)+b^3 e\right ) \log \left (a x^2+b x+c\right )}{2 c^3 \left (a d^2-e (b d-c e)\right )}-\frac{\left (a^2 c (3 b d+2 c e)-a b^2 (b d+4 c e)+b^4 e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{\log (x) \left (-c \left (a d^2-c e^2\right )+b^2 d^2+b c d e\right )}{c^3 d^3}-\frac{e^4 \log (d+e x)}{d^3 \left (a d^2-e (b d-c e)\right )}+\frac{b d+c e}{c^2 d^2 x}-\frac{1}{2 c d x^2} \]

[Out]

-1/(2*c*d*x^2) + (b*d + c*e)/(c^2*d^2*x) - ((b^4*e + a^2*c*(3*b*d + 2*c*e) - a*b
^2*(b*d + 4*c*e))*ArcTanh[(b + 2*a*x)/Sqrt[b^2 - 4*a*c]])/(c^3*Sqrt[b^2 - 4*a*c]
*(a*d^2 - e*(b*d - c*e))) + ((b^2*d^2 + b*c*d*e - c*(a*d^2 - c*e^2))*Log[x])/(c^
3*d^3) - (e^4*Log[d + e*x])/(d^3*(a*d^2 - e*(b*d - c*e))) + ((a^2*c*d + b^3*e -
a*b*(b*d + 2*c*e))*Log[c + b*x + a*x^2])/(2*c^3*(a*d^2 - e*(b*d - c*e)))

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Rubi [A]  time = 0.912672, antiderivative size = 252, 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 \[ \frac{\left (a^2 c d-a b (b d+2 c e)+b^3 e\right ) \log \left (a x^2+b x+c\right )}{2 c^3 \left (a d^2-e (b d-c e)\right )}-\frac{\left (a^2 c (3 b d+2 c e)-a b^2 (b d+4 c e)+b^4 e\right ) \tanh ^{-1}\left (\frac{2 a x+b}{\sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac{\log (x) \left (-c \left (a d^2-c e^2\right )+b^2 d^2+b c d e\right )}{c^3 d^3}-\frac{e^4 \log (d+e x)}{d^3 \left (a d^2-e (b d-c e)\right )}+\frac{b d+c e}{c^2 d^2 x}-\frac{1}{2 c d x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + c/x^2 + b/x)*x^5*(d + e*x)),x]

[Out]

-1/(2*c*d*x^2) + (b*d + c*e)/(c^2*d^2*x) - ((b^4*e + a^2*c*(3*b*d + 2*c*e) - a*b
^2*(b*d + 4*c*e))*ArcTanh[(b + 2*a*x)/Sqrt[b^2 - 4*a*c]])/(c^3*Sqrt[b^2 - 4*a*c]
*(a*d^2 - e*(b*d - c*e))) + ((b^2*d^2 + b*c*d*e - c*(a*d^2 - c*e^2))*Log[x])/(c^
3*d^3) - (e^4*Log[d + e*x])/(d^3*(a*d^2 - e*(b*d - c*e))) + ((a^2*c*d + b^3*e -
a*b*(b*d + 2*c*e))*Log[c + b*x + a*x^2])/(2*c^3*(a*d^2 - e*(b*d - c*e)))

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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**5/(e*x+d),x)

[Out]

Timed out

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Mathematica [A]  time = 0.376587, size = 252, normalized size = 1. \[ \frac{\left (a^2 c d-a b (b d+2 c e)+b^3 e\right ) \log (x (a x+b)+c)}{2 c^3 \left (a d^2+e (c e-b d)\right )}-\frac{\left (a^2 c (3 b d+2 c e)-a b^2 (b d+4 c e)+b^4 e\right ) \tan ^{-1}\left (\frac{2 a x+b}{\sqrt{4 a c-b^2}}\right )}{c^3 \sqrt{4 a c-b^2} \left (e (b d-c e)-a d^2\right )}+\frac{\log (x) \left (c \left (c e^2-a d^2\right )+b^2 d^2+b c d e\right )}{c^3 d^3}-\frac{e^4 \log (d+e x)}{a d^5+d^3 e (c e-b d)}+\frac{b d+c e}{c^2 d^2 x}-\frac{1}{2 c d x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + c/x^2 + b/x)*x^5*(d + e*x)),x]

[Out]

-1/(2*c*d*x^2) + (b*d + c*e)/(c^2*d^2*x) - ((b^4*e + a^2*c*(3*b*d + 2*c*e) - a*b
^2*(b*d + 4*c*e))*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]])/(c^3*Sqrt[-b^2 + 4*a*c
]*(-(a*d^2) + e*(b*d - c*e))) + ((b^2*d^2 + b*c*d*e + c*(-(a*d^2) + c*e^2))*Log[
x])/(c^3*d^3) - (e^4*Log[d + e*x])/(a*d^5 + d^3*e*(-(b*d) + c*e)) + ((a^2*c*d +
b^3*e - a*b*(b*d + 2*c*e))*Log[c + x*(b + a*x)])/(2*c^3*(a*d^2 + e*(-(b*d) + c*e
)))

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Maple [B]  time = 0.018, size = 562, normalized size = 2.2 \[ -{\frac{1}{2\,cd{x}^{2}}}+{\frac{b}{{c}^{2}xd}}+{\frac{e}{cx{d}^{2}}}-{\frac{\ln \left ( x \right ) a}{{c}^{2}d}}+{\frac{\ln \left ( x \right ){b}^{2}}{d{c}^{3}}}+{\frac{\ln \left ( x \right ) be}{{c}^{2}{d}^{2}}}+{\frac{\ln \left ( x \right ){e}^{2}}{c{d}^{3}}}-{\frac{{e}^{4}\ln \left ( ex+d \right ) }{{d}^{3} \left ( a{d}^{2}-bde+{e}^{2}c \right ) }}+{\frac{{a}^{2}\ln \left ( a{x}^{2}+bx+c \right ) d}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){c}^{2}}}-{\frac{a\ln \left ( a{x}^{2}+bx+c \right ){b}^{2}d}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){c}^{3}}}-{\frac{a\ln \left ( a{x}^{2}+bx+c \right ) be}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){c}^{2}}}+{\frac{\ln \left ( a{x}^{2}+bx+c \right ){b}^{3}e}{ \left ( 2\,a{d}^{2}-2\,bde+2\,{e}^{2}c \right ){c}^{3}}}+3\,{\frac{{a}^{2}bd}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{{a}^{2}e}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ) c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{a{b}^{3}d}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){c}^{3}}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-4\,{\frac{a{b}^{2}e}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,ax+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{4}e}{ \left ( a{d}^{2}-bde+{e}^{2}c \right ){c}^{3}}\arctan \left ({(2\,ax+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+c/x^2+b/x)/x^5/(e*x+d),x)

[Out]

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

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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)*(a + b/x + c/x^2)*x^5),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)*(a + b/x + c/x^2)*x^5),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**5/(e*x+d),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.300104, size = 377, normalized size = 1.5 \[ -\frac{{\left (a b^{2} d - a^{2} c d - b^{3} e + 2 \, a b c e\right )}{\rm ln}\left (a x^{2} + b x + c\right )}{2 \,{\left (a c^{3} d^{2} - b c^{3} d e + c^{4} e^{2}\right )}} - \frac{e^{5}{\rm ln}\left ({\left | x e + d \right |}\right )}{a d^{5} e - b d^{4} e^{2} + c d^{3} e^{3}} - \frac{{\left (a b^{3} d - 3 \, a^{2} b c d - b^{4} e + 4 \, a b^{2} c e - 2 \, a^{2} c^{2} e\right )} \arctan \left (\frac{2 \, a x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (a c^{3} d^{2} - b c^{3} d e + c^{4} e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\left (b^{2} d^{2} - a c d^{2} + b c d e + c^{2} e^{2}\right )}{\rm ln}\left ({\left | x \right |}\right )}{c^{3} d^{3}} - \frac{c^{2} d^{2} - 2 \,{\left (b c d^{2} + c^{2} d e\right )} x}{2 \, c^{3} d^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((e*x + d)*(a + b/x + c/x^2)*x^5),x, algorithm="giac")

[Out]

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