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

Optimal. Leaf size=276 \[ \frac{\sqrt [3]{b} \left (10 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{11/3}}-\frac{2 \sqrt [3]{b} \left (10 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3}}+\frac{2 \sqrt [3]{b} \left (7 \sqrt [3]{a} d+10 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{11/3}}-\frac{x \left (11 b c+10 b d x+9 b e x^2\right )}{18 a^3 \left (a+b x^3\right )}-\frac{e \log \left (a+b x^3\right )}{3 a^3}-\frac{c}{2 a^3 x^2}-\frac{d}{a^3 x}+\frac{e \log (x)}{a^3}-\frac{x \left (b c+b d x+b e x^2\right )}{6 a^2 \left (a+b x^3\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.976423, antiderivative size = 276, 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 \[ \frac{\sqrt [3]{b} \left (10 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{11/3}}-\frac{2 \sqrt [3]{b} \left (10 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3}}+\frac{2 \sqrt [3]{b} \left (7 \sqrt [3]{a} d+10 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{11/3}}-\frac{x \left (11 b c+10 b d x+9 b e x^2\right )}{18 a^3 \left (a+b x^3\right )}-\frac{e \log \left (a+b x^3\right )}{3 a^3}-\frac{c}{2 a^3 x^2}-\frac{d}{a^3 x}+\frac{e \log (x)}{a^3}-\frac{x \left (b c+b d x+b e x^2\right )}{6 a^2 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.184, size = 26, normalized size = 0.09 \[ \frac{x \left (\frac{c}{x^{3}} + \frac{d}{x^{2}} + \frac{e}{x}\right )}{6 a \left (a + b x^{3}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.413425, size = 253, normalized size = 0.92 \[ \frac{2 \sqrt [3]{b} \left (10 \sqrt [3]{a} \sqrt [3]{b} c-7 a^{2/3} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+4 \sqrt [3]{b} \left (7 a^{2/3} d-10 \sqrt [3]{a} \sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\frac{9 a^2 (a e-b x (c+d x))}{\left (a+b x^3\right )^2}+\frac{3 a (6 a e-b x (11 c+10 d x))}{a+b x^3}+4 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \left (7 \sqrt [3]{a} d+10 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )-18 a e \log \left (a+b x^3\right )-\frac{27 a c}{x^2}-\frac{54 a d}{x}+54 a e \log (x)}{54 a^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.025, size = 337, normalized size = 1.2 \[ -{\frac{d}{{a}^{3}x}}+{\frac{e\ln \left ( x \right ) }{{a}^{3}}}-{\frac{c}{2\,{a}^{3}{x}^{2}}}-{\frac{5\,{x}^{5}{b}^{2}d}{9\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{11\,{x}^{4}{b}^{2}c}{18\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{be{x}^{3}}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{13\,b{x}^{2}d}{18\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{7\,bcx}{9\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{e}{2\,a \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{20\,c}{27\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{10\,c}{27\,{a}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{20\,c\sqrt{3}}{27\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{14\,d}{27\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{7\,d}{27\,{a}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{14\,d\sqrt{3}}{27\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{e\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218231, size = 390, normalized size = 1.41 \[ -\frac{e{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac{e{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} - \frac{{\left (10 \, \left (-a b^{2}\right )^{\frac{1}{3}} b c + 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} d\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{27 \, a^{4} b} - \frac{28 \, b^{2} d x^{7} + 20 \, b^{2} c x^{6} - 6 \, a b x^{5} e + 49 \, a b d x^{4} + 32 \, a b c x^{3} - 9 \, a^{2} x^{2} e + 18 \, a^{2} d x + 9 \, a^{2} c}{18 \,{\left (b x^{4} + a x\right )}^{2} a^{3}} - \frac{2 \, \sqrt{3}{\left (10 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{5} b^{3}} + \frac{2 \,{\left (7 \, a^{3} b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 10 \, a^{3} b^{2} c\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{7} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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