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

Optimal. Leaf size=257 \[ -\frac{\left (5 \sqrt [3]{b} d-2 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{2/3}}+\frac{\left (5 \sqrt [3]{b} d-2 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{2/3}}-\frac{\left (2 \sqrt [3]{a} e+5 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3} b^{2/3}}+\frac{x \left (5 a d+4 a e x-9 b c x^2\right )}{18 a^3 \left (a+b x^3\right )}-\frac{c \log \left (a+b x^3\right )}{3 a^3}+\frac{c \log (x)}{a^3}+\frac{x \left (a d+a e x-b c x^2\right )}{6 a^2 \left (a+b x^3\right )^2} \]

[Out]

(x*(a*d + a*e*x - b*c*x^2))/(6*a^2*(a + b*x^3)^2) + (x*(5*a*d + 4*a*e*x - 9*b*c*
x^2))/(18*a^3*(a + b*x^3)) - ((5*b^(1/3)*d + 2*a^(1/3)*e)*ArcTan[(a^(1/3) - 2*b^
(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(8/3)*b^(2/3)) + (c*Log[x])/a^3 + ((5*
b^(1/3)*d - 2*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(8/3)*b^(2/3)) - ((5*b^
(1/3)*d - 2*a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(8/
3)*b^(2/3)) - (c*Log[a + b*x^3])/(3*a^3)

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Rubi [A]  time = 0.816306, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391 \[ -\frac{\left (5 \sqrt [3]{b} d-2 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{2/3}}+\frac{\left (5 \sqrt [3]{b} d-2 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{2/3}}-\frac{\left (2 \sqrt [3]{a} e+5 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3} b^{2/3}}+\frac{x \left (5 a d+4 a e x-9 b c x^2\right )}{18 a^3 \left (a+b x^3\right )}-\frac{c \log \left (a+b x^3\right )}{3 a^3}+\frac{c \log (x)}{a^3}+\frac{x \left (a d+a e x-b c x^2\right )}{6 a^2 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a*d + a*e*x - b*c*x^2))/(6*a^2*(a + b*x^3)^2) + (x*(5*a*d + 4*a*e*x - 9*b*c*
x^2))/(18*a^3*(a + b*x^3)) - ((5*b^(1/3)*d + 2*a^(1/3)*e)*ArcTan[(a^(1/3) - 2*b^
(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(8/3)*b^(2/3)) + (c*Log[x])/a^3 + ((5*
b^(1/3)*d - 2*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(8/3)*b^(2/3)) - ((5*b^
(1/3)*d - 2*a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(8/
3)*b^(2/3)) - (c*Log[a + b*x^3])/(3*a^3)

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Rubi in Sympy [A]  time = 73.9252, size = 204, normalized size = 0.79 \[ \frac{x \left (\frac{c}{x} + d + e x\right )}{6 a \left (a + b x^{3}\right )^{2}} + \frac{x \left (5 d + 4 e x\right )}{18 a^{2} \left (a + b x^{3}\right )} - \frac{\left (2 \sqrt [3]{a} e - 5 \sqrt [3]{b} d\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{8}{3}} b^{\frac{2}{3}}} + \frac{\left (2 \sqrt [3]{a} e - 5 \sqrt [3]{b} d\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{8}{3}} b^{\frac{2}{3}}} - \frac{\sqrt{3} \left (2 \sqrt [3]{a} e + 5 \sqrt [3]{b} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{8}{3}} b^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(c/x + d + e*x)/(6*a*(a + b*x**3)**2) + x*(5*d + 4*e*x)/(18*a**2*(a + b*x**3))
- (2*a**(1/3)*e - 5*b**(1/3)*d)*log(a**(1/3) + b**(1/3)*x)/(27*a**(8/3)*b**(2/3
)) + (2*a**(1/3)*e - 5*b**(1/3)*d)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)
*x**2)/(54*a**(8/3)*b**(2/3)) - sqrt(3)*(2*a**(1/3)*e + 5*b**(1/3)*d)*atan(sqrt(
3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(27*a**(8/3)*b**(2/3))

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Mathematica [A]  time = 0.357454, size = 229, normalized size = 0.89 \[ \frac{\frac{\left (2 a^{2/3} e-5 \sqrt [3]{a} \sqrt [3]{b} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+\frac{2 \left (5 \sqrt [3]{a} \sqrt [3]{b} d-2 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac{9 a^2 (c+x (d+e x))}{\left (a+b x^3\right )^2}-\frac{2 \sqrt{3} \sqrt [3]{a} \left (2 \sqrt [3]{a} e+5 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{2/3}}+\frac{3 a (6 c+x (5 d+4 e x))}{a+b x^3}-18 c \log \left (a+b x^3\right )+54 c \log (x)}{54 a^3} \]

Antiderivative was successfully verified.

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

[Out]

((9*a^2*(c + x*(d + e*x)))/(a + b*x^3)^2 + (3*a*(6*c + x*(5*d + 4*e*x)))/(a + b*
x^3) - (2*Sqrt[3]*a^(1/3)*(5*b^(1/3)*d + 2*a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/
a^(1/3))/Sqrt[3]])/b^(2/3) + 54*c*Log[x] + (2*(5*a^(1/3)*b^(1/3)*d - 2*a^(2/3)*e
)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + ((-5*a^(1/3)*b^(1/3)*d + 2*a^(2/3)*e)*Log[
a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3) - 18*c*Log[a + b*x^3])/(54*a
^3)

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Maple [A]  time = 0.019, size = 331, normalized size = 1.3 \[{\frac{c\ln \left ( x \right ) }{{a}^{3}}}+{\frac{2\,b{x}^{5}e}{9\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{5\,bd{x}^{4}}{18\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{c{x}^{3}b}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{7\,e{x}^{2}}{18\,a \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{4\,dx}{9\,a \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{c}{2\,a \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{5\,d}{27\,{a}^{2}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,d}{54\,{a}^{2}b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,d\sqrt{3}}{27\,{a}^{2}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{2\,e}{27\,{a}^{2}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{e}{27\,{a}^{2}b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{2\,\sqrt{3}e}{27\,{a}^{2}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{c\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*ln(x)/a^3+2/9/a^2/(b*x^3+a)^2*x^5*b*e+5/18/a^2/(b*x^3+a)^2*x^4*b*d+1/3/a^2/(b*
x^3+a)^2*x^3*c*b+7/18/a/(b*x^3+a)^2*x^2*e+4/9/a/(b*x^3+a)^2*x*d+1/2/a/(b*x^3+a)^
2*c+5/27/a^2/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*d-5/54/a^2/b/(a/b)^(2/3)*ln(x^2-x*(
a/b)^(1/3)+(a/b)^(2/3))*d+5/27/a^2/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(
a/b)^(1/3)*x-1))*d-2/27/a^2/b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*e+1/27/a^2/b/(a/b)^(
1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*e+2/27/a^2/b*3^(1/2)/(a/b)^(1/3)*arctan(1
/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*e-1/3*c*ln(b*x^3+a)/a^3

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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^2 + d*x + c)/((b*x^3 + a)^3*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218052, size = 359, normalized size = 1.4 \[ -\frac{c{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac{c{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} + \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} b d - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{3} b^{2}} + \frac{4 \, a b x^{5} e + 5 \, a b d x^{4} + 6 \, a b c x^{3} + 7 \, a^{2} x^{2} e + 8 \, a^{2} d x + 9 \, a^{2} c}{18 \,{\left (b x^{3} + a\right )}^{2} a^{3}} - \frac{{\left (2 \, a^{4} b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e + 5 \, a^{4} b d\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{7} b} + \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} d + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{4} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*c*ln(abs(b*x^3 + a))/a^3 + c*ln(abs(x))/a^3 + 1/27*sqrt(3)*(5*(-a*b^2)^(1/3
)*b*d - 2*(-a*b^2)^(2/3)*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3)
)/(a^3*b^2) + 1/18*(4*a*b*x^5*e + 5*a*b*d*x^4 + 6*a*b*c*x^3 + 7*a^2*x^2*e + 8*a^
2*d*x + 9*a^2*c)/((b*x^3 + a)^2*a^3) - 1/27*(2*a^4*b*(-a/b)^(1/3)*e + 5*a^4*b*d)
*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^7*b) + 1/54*(5*(-a*b^2)^(1/3)*a*b^3*d
+ 2*(-a*b^2)^(2/3)*a*b^2*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b^4)