3.345 \(\int \frac{c+d x+e x^2}{x^4 \left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=298 \[ \frac{\sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{11/3}}-\frac{2 \sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3}}+\frac{2 \sqrt [3]{b} \left (7 \sqrt [3]{a} e+10 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{11/3}}+\frac{b c \log \left (a+b x^3\right )}{a^4}-\frac{3 b c \log (x)}{a^4}-\frac{x \left (-\frac{15 b^2 c x^2}{a}+11 b d+10 b e x\right )}{18 a^3 \left (a+b x^3\right )}-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x}-\frac{x \left (-\frac{b^2 c x^2}{a}+b d+b e x\right )}{6 a^2 \left (a+b x^3\right )^2} \]

[Out]

-c/(3*a^3*x^3) - d/(2*a^3*x^2) - e/(a^3*x) - (x*(b*d + b*e*x - (b^2*c*x^2)/a))/(
6*a^2*(a + b*x^3)^2) - (x*(11*b*d + 10*b*e*x - (15*b^2*c*x^2)/a))/(18*a^3*(a + b
*x^3)) + (2*b^(1/3)*(10*b^(1/3)*d + 7*a^(1/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/
(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(11/3)) - (3*b*c*Log[x])/a^4 - (2*b^(1/3)*(10*b
^(1/3)*d - 7*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(11/3)) + (b^(1/3)*(10*b
^(1/3)*d - 7*a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(27*a^(1
1/3)) + (b*c*Log[a + b*x^3])/a^4

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Rubi [A]  time = 1.11201, antiderivative size = 298, 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 \[ \frac{\sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{11/3}}-\frac{2 \sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3}}+\frac{2 \sqrt [3]{b} \left (7 \sqrt [3]{a} e+10 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{11/3}}+\frac{b c \log \left (a+b x^3\right )}{a^4}-\frac{3 b c \log (x)}{a^4}-\frac{x \left (-\frac{15 b^2 c x^2}{a}+11 b d+10 b e x\right )}{18 a^3 \left (a+b x^3\right )}-\frac{c}{3 a^3 x^3}-\frac{d}{2 a^3 x^2}-\frac{e}{a^3 x}-\frac{x \left (-\frac{b^2 c x^2}{a}+b d+b e x\right )}{6 a^2 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x + e*x^2)/(x^4*(a + b*x^3)^3),x]

[Out]

-c/(3*a^3*x^3) - d/(2*a^3*x^2) - e/(a^3*x) - (x*(b*d + b*e*x - (b^2*c*x^2)/a))/(
6*a^2*(a + b*x^3)^2) - (x*(11*b*d + 10*b*e*x - (15*b^2*c*x^2)/a))/(18*a^3*(a + b
*x^3)) + (2*b^(1/3)*(10*b^(1/3)*d + 7*a^(1/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/
(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(11/3)) - (3*b*c*Log[x])/a^4 - (2*b^(1/3)*(10*b
^(1/3)*d - 7*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(11/3)) + (b^(1/3)*(10*b
^(1/3)*d - 7*a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(27*a^(1
1/3)) + (b*c*Log[a + b*x^3])/a^4

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Rubi in Sympy [A]  time = 14.1222, size = 27, normalized size = 0.09 \[ \frac{x \left (\frac{c}{x^{4}} + \frac{d}{x^{3}} + \frac{e}{x^{2}}\right )}{6 a \left (a + b x^{3}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x**2+d*x+c)/x**4/(b*x**3+a)**3,x)

[Out]

x*(c/x**4 + d/x**3 + e/x**2)/(6*a*(a + b*x**3)**2)

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Mathematica [A]  time = 0.698342, size = 255, normalized size = 0.86 \[ -\frac{-2 \sqrt [3]{b} \left (10 \sqrt [3]{a} \sqrt [3]{b} d-7 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+4 \sqrt [3]{b} \left (10 \sqrt [3]{a} \sqrt [3]{b} d-7 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\frac{9 a^2 b (c+x (d+e x))}{\left (a+b x^3\right )^2}+\frac{3 a b (12 c+x (11 d+10 e x))}{a+b x^3}-54 b c \log \left (a+b x^3\right )-4 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \left (7 \sqrt [3]{a} e+10 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+\frac{18 a c}{x^3}+\frac{27 a d}{x^2}+\frac{54 a e}{x}+162 b c \log (x)}{54 a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x + e*x^2)/(x^4*(a + b*x^3)^3),x]

[Out]

-((18*a*c)/x^3 + (27*a*d)/x^2 + (54*a*e)/x + (9*a^2*b*(c + x*(d + e*x)))/(a + b*
x^3)^2 + (3*a*b*(12*c + x*(11*d + 10*e*x)))/(a + b*x^3) - 4*Sqrt[3]*a^(1/3)*b^(1
/3)*(10*b^(1/3)*d + 7*a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 1
62*b*c*Log[x] + 4*b^(1/3)*(10*a^(1/3)*b^(1/3)*d - 7*a^(2/3)*e)*Log[a^(1/3) + b^(
1/3)*x] - 2*b^(1/3)*(10*a^(1/3)*b^(1/3)*d - 7*a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b
^(1/3)*x + b^(2/3)*x^2] - 54*b*c*Log[a + b*x^3])/(54*a^4)

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Maple [A]  time = 0.025, size = 351, normalized size = 1.2 \[ -{\frac{d}{2\,{a}^{3}{x}^{2}}}-{\frac{e}{{a}^{3}x}}-{\frac{c}{3\,{a}^{3}{x}^{3}}}-3\,{\frac{bc\ln \left ( x \right ) }{{a}^{4}}}-{\frac{5\,{x}^{5}e{b}^{2}}{9\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{11\,{x}^{4}{b}^{2}d}{18\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{2\,{b}^{2}{x}^{3}c}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{13\,be{x}^{2}}{18\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{7\,bxd}{9\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{5\,bc}{6\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{20\,d}{27\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{10\,d}{27\,{a}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{20\,d\sqrt{3}}{27\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{14\,e}{27\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{7\,e}{27\,{a}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{14\,e\sqrt{3}}{27\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{bc\ln \left ( b{x}^{3}+a \right ) }{{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^2+d*x+c)/x^4/(b*x^3+a)^3,x)

[Out]

-1/2*d/a^3/x^2-e/a^3/x-1/3*c/a^3/x^3-3*b*c*ln(x)/a^4-5/9/a^3/(b*x^3+a)^2*x^5*e*b
^2-11/18/a^3/(b*x^3+a)^2*x^4*b^2*d-2/3/a^3*b^2/(b*x^3+a)^2*x^3*c-13/18/a^2/(b*x^
3+a)^2*x^2*b*e-7/9/a^2/(b*x^3+a)^2*x*b*d-5/6/a^2*b/(b*x^3+a)^2*c-20/27/a^3/(a/b)
^(2/3)*ln(x+(a/b)^(1/3))*d+10/27/a^3/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3
))*d-20/27/a^3/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*d+14/
27/a^3*e/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-7/27/a^3*e/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/
3)+(a/b)^(2/3))-14/27/a^3*e*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3
)*x-1))+b*c*ln(b*x^3+a)/a^4

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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^4),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^4),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**4/(b*x**3+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222242, size = 421, normalized size = 1.41 \[ \frac{b c{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{a^{4}} - \frac{3 \, b c{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} - \frac{{\left (10 \, \left (-a b^{2}\right )^{\frac{1}{3}} b d + 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{27 \, a^{4} b} - \frac{2 \, \sqrt{3}{\left (10 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} d - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{5} b^{3}} + \frac{2 \,{\left (7 \, a^{5} b^{2} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e + 10 \, a^{5} b^{2} d\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{9} b} - \frac{28 \, a b^{2} x^{8} e + 20 \, a b^{2} d x^{7} + 18 \, a b^{2} c x^{6} + 49 \, a^{2} b x^{5} e + 32 \, a^{2} b d x^{4} + 27 \, a^{2} b c x^{3} + 18 \, a^{3} x^{2} e + 9 \, a^{3} d x + 6 \, a^{3} c}{18 \,{\left (b x^{3} + a\right )}^{2} a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^2 + d*x + c)/((b*x^3 + a)^3*x^4),x, algorithm="giac")

[Out]

b*c*ln(abs(b*x^3 + a))/a^4 - 3*b*c*ln(abs(x))/a^4 - 1/27*(10*(-a*b^2)^(1/3)*b*d
+ 7*(-a*b^2)^(2/3)*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b) - 2/27*sqr
t(3)*(10*(-a*b^2)^(1/3)*a*b^3*d - 7*(-a*b^2)^(2/3)*a*b^2*e)*arctan(1/3*sqrt(3)*(
2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^5*b^3) + 2/27*(7*a^5*b^2*(-a/b)^(1/3)*e + 1
0*a^5*b^2*d)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^9*b) - 1/18*(28*a*b^2*x^8
*e + 20*a*b^2*d*x^7 + 18*a*b^2*c*x^6 + 49*a^2*b*x^5*e + 32*a^2*b*d*x^4 + 27*a^2*
b*c*x^3 + 18*a^3*x^2*e + 9*a^3*d*x + 6*a^3*c)/((b*x^3 + a)^2*a^4*x^3)