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

Optimal. Leaf size=267 \[ -\frac{\left (5 a^{2/3} e+14 b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{10/3} \sqrt [3]{b}}+\frac{\left (5 a^{2/3} e+14 b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt [3]{b}}+\frac{\left (14 b^{2/3} c-5 a^{2/3} e\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{10/3} \sqrt [3]{b}}+\frac{x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}-\frac{d \log \left (a+b x^3\right )}{3 a^3}-\frac{c}{a^3 x}+\frac{d \log (x)}{a^3}+\frac{x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2} \]

[Out]

-(c/(a^3*x)) + (x*(a*e - b*c*x - b*d*x^2))/(6*a^2*(a + b*x^3)^2) + (x*(5*a*e - 1
0*b*c*x - 9*b*d*x^2))/(18*a^3*(a + b*x^3)) + ((14*b^(2/3)*c - 5*a^(2/3)*e)*ArcTa
n[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(10/3)*b^(1/3)) + (d*
Log[x])/a^3 + ((14*b^(2/3)*c + 5*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(10/
3)*b^(1/3)) - ((14*b^(2/3)*c + 5*a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^
(2/3)*x^2])/(54*a^(10/3)*b^(1/3)) - (d*Log[a + b*x^3])/(3*a^3)

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Rubi [A]  time = 0.908881, antiderivative size = 267, 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 \[ -\frac{\left (5 a^{2/3} e+14 b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{10/3} \sqrt [3]{b}}+\frac{\left (5 a^{2/3} e+14 b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} \sqrt [3]{b}}+\frac{\left (14 b^{2/3} c-5 a^{2/3} e\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{10/3} \sqrt [3]{b}}+\frac{x \left (5 a e-10 b c x-9 b d x^2\right )}{18 a^3 \left (a+b x^3\right )}-\frac{d \log \left (a+b x^3\right )}{3 a^3}-\frac{c}{a^3 x}+\frac{d \log (x)}{a^3}+\frac{x \left (a e-b c x-b d x^2\right )}{6 a^2 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(c/(a^3*x)) + (x*(a*e - b*c*x - b*d*x^2))/(6*a^2*(a + b*x^3)^2) + (x*(5*a*e - 1
0*b*c*x - 9*b*d*x^2))/(18*a^3*(a + b*x^3)) + ((14*b^(2/3)*c - 5*a^(2/3)*e)*ArcTa
n[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(10/3)*b^(1/3)) + (d*
Log[x])/a^3 + ((14*b^(2/3)*c + 5*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(10/
3)*b^(1/3)) - ((14*b^(2/3)*c + 5*a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^
(2/3)*x^2])/(54*a^(10/3)*b^(1/3)) - (d*Log[a + b*x^3])/(3*a^3)

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Rubi in Sympy [A]  time = 57.4305, size = 160, normalized size = 0.6 \[ \frac{x \left (\frac{c}{x^{2}} + \frac{d}{x} + e\right )}{6 a \left (a + b x^{3}\right )^{2}} + \frac{5 e x}{18 a^{2} \left (a + b x^{3}\right )} + \frac{5 e \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{8}{3}} \sqrt [3]{b}} - \frac{5 e \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{8}{3}} \sqrt [3]{b}} - \frac{5 \sqrt{3} e \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{8}{3}} \sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.437054, size = 248, normalized size = 0.93 \[ \frac{-\frac{\left (14 a^{2/3} b^{2/3} c+5 a^{4/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}+\frac{2 \left (14 a^{2/3} b^{2/3} c+5 a^{4/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac{2 \sqrt{3} a^{2/3} \left (5 a^{2/3} e-14 b^{2/3} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{b}}+\frac{9 a^2 \left (a (d+e x)-b c x^2\right )}{\left (a+b x^3\right )^2}+\frac{3 a \left (6 a d+5 a e x-10 b c x^2\right )}{a+b x^3}-18 a d \log \left (a+b x^3\right )-\frac{54 a c}{x}+54 a d \log (x)}{54 a^4} \]

Antiderivative was successfully verified.

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

[Out]

((-54*a*c)/x + (3*a*(6*a*d + 5*a*e*x - 10*b*c*x^2))/(a + b*x^3) + (9*a^2*(-(b*c*
x^2) + a*(d + e*x)))/(a + b*x^3)^2 - (2*Sqrt[3]*a^(2/3)*(-14*b^(2/3)*c + 5*a^(2/
3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + 54*a*d*Log[x] + (2*
(14*a^(2/3)*b^(2/3)*c + 5*a^(4/3)*e)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) - ((14*a^
(2/3)*b^(2/3)*c + 5*a^(4/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b
^(1/3) - 18*a*d*Log[a + b*x^3])/(54*a^4)

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Maple [A]  time = 0.023, size = 334, normalized size = 1.3 \[{\frac{d\ln \left ( x \right ) }{{a}^{3}}}-{\frac{c}{{a}^{3}x}}-{\frac{5\,{b}^{2}{x}^{5}c}{9\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{5\,b{x}^{4}e}{18\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{b{x}^{3}d}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{13\,b{x}^{2}c}{18\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{4\,ex}{9\,a \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{d}{2\,a \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{5\,e}{27\,{a}^{2}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,e}{54\,{a}^{2}b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,\sqrt{3}e}{27\,{a}^{2}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{14\,c}{27\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{7\,c}{27\,{a}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{14\,c\sqrt{3}}{27\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{d\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*ln(x)/a^3-c/a^3/x-5/9/a^3/(b*x^3+a)^2*b^2*x^5*c+5/18/a^2/(b*x^3+a)^2*x^4*b*e+1
/3/a^2/(b*x^3+a)^2*x^3*b*d-13/18/a^2/(b*x^3+a)^2*x^2*b*c+4/9/a/(b*x^3+a)^2*x*e+1
/2/a/(b*x^3+a)^2*d+5/27/a^2/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*e-5/54/a^2/b/(a/b)^(
2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*e+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))*e+14/27/a^3/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*c-7/27
/a^3/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))*c-14/27/a^3*3^(1/2)/(a/b)^(1/
3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*c-1/3*d*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^2),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^2),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**2/(b*x**3+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220449, size = 377, normalized size = 1.41 \[ -\frac{d{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac{d{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} + \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a e - 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} c\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{4} b} + \frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} e + 14 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{2} c\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{4} b^{3}} - \frac{28 \, b^{2} c x^{6} - 5 \, a b x^{5} e - 6 \, a b d x^{4} + 49 \, a b c x^{3} - 8 \, a^{2} x^{2} e - 9 \, a^{2} d x + 18 \, a^{2} c}{18 \,{\left (b x^{3} + a\right )}^{2} a^{3} x} + \frac{{\left (14 \, a^{3} b^{2} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 5 \, a^{4} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{7} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*d*ln(abs(b*x^3 + a))/a^3 + d*ln(abs(x))/a^3 + 1/54*(5*(-a*b^2)^(1/3)*a*e -
14*(-a*b^2)^(2/3)*c)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b) + 1/27*sqrt
(3)*(5*(-a*b^2)^(1/3)*a*b^2*e + 14*(-a*b^2)^(2/3)*b^2*c)*arctan(1/3*sqrt(3)*(2*x
+ (-a/b)^(1/3))/(-a/b)^(1/3))/(a^4*b^3) - 1/18*(28*b^2*c*x^6 - 5*a*b*x^5*e - 6*
a*b*d*x^4 + 49*a*b*c*x^3 - 8*a^2*x^2*e - 9*a^2*d*x + 18*a^2*c)/((b*x^3 + a)^2*a^
3*x) + 1/27*(14*a^3*b^2*c*(-a/b)^(1/3) - 5*a^4*b*e)*(-a/b)^(1/3)*ln(abs(x - (-a/
b)^(1/3)))/(a^7*b)