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

Optimal. Leaf size=291 \[ -\frac{\left (20 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}}+\frac{2 \left (20 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac{2 \left (7 \sqrt [3]{a} e+20 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{11/3} b^{2/3}}+\frac{x \left (40 a d+28 a e x-99 b c x^2\right )}{162 a^4 \left (a+b x^3\right )}-\frac{c \log \left (a+b x^3\right )}{3 a^4}+\frac{c \log (x)}{a^4}+\frac{x \left (8 a d+7 a e x-15 b c x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac{x \left (a d+a e x-b c x^2\right )}{9 a^2 \left (a+b x^3\right )^3} \]

[Out]

(x*(a*d + a*e*x - b*c*x^2))/(9*a^2*(a + b*x^3)^3) + (x*(8*a*d + 7*a*e*x - 15*b*c
*x^2))/(54*a^3*(a + b*x^3)^2) + (x*(40*a*d + 28*a*e*x - 99*b*c*x^2))/(162*a^4*(a
+ b*x^3)) - (2*(20*b^(1/3)*d + 7*a^(1/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqr
t[3]*a^(1/3))])/(81*Sqrt[3]*a^(11/3)*b^(2/3)) + (c*Log[x])/a^4 + (2*(20*b^(1/3)*
d - 7*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(243*a^(11/3)*b^(2/3)) - ((20*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])/(243*a^(11/3)*
b^(2/3)) - (c*Log[a + b*x^3])/(3*a^4)

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Rubi [A]  time = 1.0053, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391 \[ -\frac{\left (20 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{11/3} b^{2/3}}+\frac{2 \left (20 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{11/3} b^{2/3}}-\frac{2 \left (7 \sqrt [3]{a} e+20 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{11/3} b^{2/3}}+\frac{x \left (40 a d+28 a e x-99 b c x^2\right )}{162 a^4 \left (a+b x^3\right )}-\frac{c \log \left (a+b x^3\right )}{3 a^4}+\frac{c \log (x)}{a^4}+\frac{x \left (8 a d+7 a e x-15 b c x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac{x \left (a d+a e x-b c x^2\right )}{9 a^2 \left (a+b x^3\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a*d + a*e*x - b*c*x^2))/(9*a^2*(a + b*x^3)^3) + (x*(8*a*d + 7*a*e*x - 15*b*c
*x^2))/(54*a^3*(a + b*x^3)^2) + (x*(40*a*d + 28*a*e*x - 99*b*c*x^2))/(162*a^4*(a
+ b*x^3)) - (2*(20*b^(1/3)*d + 7*a^(1/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqr
t[3]*a^(1/3))])/(81*Sqrt[3]*a^(11/3)*b^(2/3)) + (c*Log[x])/a^4 + (2*(20*b^(1/3)*
d - 7*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(243*a^(11/3)*b^(2/3)) - ((20*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])/(243*a^(11/3)*
b^(2/3)) - (c*Log[a + b*x^3])/(3*a^4)

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Rubi in Sympy [A]  time = 86.48, size = 231, normalized size = 0.79 \[ \frac{x \left (\frac{c}{x} + d + e x\right )}{9 a \left (a + b x^{3}\right )^{3}} + \frac{x \left (8 d + 7 e x\right )}{54 a^{2} \left (a + b x^{3}\right )^{2}} + \frac{x \left (40 d + 28 e x\right )}{162 a^{3} \left (a + b x^{3}\right )} - \frac{2 \left (7 \sqrt [3]{a} e - 20 \sqrt [3]{b} d\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{243 a^{\frac{11}{3}} b^{\frac{2}{3}}} + \frac{\left (7 \sqrt [3]{a} e - 20 \sqrt [3]{b} d\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{243 a^{\frac{11}{3}} b^{\frac{2}{3}}} - \frac{2 \sqrt{3} \left (7 \sqrt [3]{a} e + 20 \sqrt [3]{b} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{243 a^{\frac{11}{3}} b^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(c/x + d + e*x)/(9*a*(a + b*x**3)**3) + x*(8*d + 7*e*x)/(54*a**2*(a + b*x**3)*
*2) + x*(40*d + 28*e*x)/(162*a**3*(a + b*x**3)) - 2*(7*a**(1/3)*e - 20*b**(1/3)*
d)*log(a**(1/3) + b**(1/3)*x)/(243*a**(11/3)*b**(2/3)) + (7*a**(1/3)*e - 20*b**(
1/3)*d)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(243*a**(11/3)*b**(2
/3)) - 2*sqrt(3)*(7*a**(1/3)*e + 20*b**(1/3)*d)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**
(1/3)*x/3)/a**(1/3))/(243*a**(11/3)*b**(2/3))

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Mathematica [A]  time = 0.450075, size = 259, normalized size = 0.89 \[ \frac{\frac{2 \left (7 a^{2/3} e-20 \sqrt [3]{a} \sqrt [3]{b} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+\frac{4 \left (20 \sqrt [3]{a} \sqrt [3]{b} d-7 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac{54 a^3 (c+x (d+e x))}{\left (a+b x^3\right )^3}+\frac{9 a^2 (9 c+x (8 d+7 e x))}{\left (a+b x^3\right )^2}-\frac{4 \sqrt{3} \sqrt [3]{a} \left (7 \sqrt [3]{a} e+20 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{2/3}}+\frac{6 a (27 c+2 x (10 d+7 e x))}{a+b x^3}-162 c \log \left (a+b x^3\right )+486 c \log (x)}{486 a^4} \]

Antiderivative was successfully verified.

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

[Out]

((54*a^3*(c + x*(d + e*x)))/(a + b*x^3)^3 + (9*a^2*(9*c + x*(8*d + 7*e*x)))/(a +
b*x^3)^2 + (6*a*(27*c + 2*x*(10*d + 7*e*x)))/(a + b*x^3) - (4*Sqrt[3]*a^(1/3)*(
20*b^(1/3)*d + 7*a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3)
+ 486*c*Log[x] + (4*(20*a^(1/3)*b^(1/3)*d - 7*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*
x])/b^(2/3) + (2*(-20*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])/b^(2/3) - 162*c*Log[a + b*x^3])/(486*a^4)

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Maple [A]  time = 0.023, size = 394, normalized size = 1.4 \[{\frac{c\ln \left ( x \right ) }{{a}^{4}}}+{\frac{14\,{x}^{8}{b}^{2}e}{81\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{20\,{x}^{7}{b}^{2}d}{81\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{{b}^{2}c{x}^{6}}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{77\,b{x}^{5}e}{162\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{52\,bd{x}^{4}}{81\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{5\,b{x}^{3}c}{6\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{67\,e{x}^{2}}{162\,a \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{41\,dx}{81\,a \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{11\,c}{18\,a \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{40\,d}{243\,{a}^{3}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{20\,d}{243\,{a}^{3}b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{40\,d\sqrt{3}}{243\,{a}^{3}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{14\,e}{243\,{a}^{3}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{7\,e}{243\,{a}^{3}b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{14\,e\sqrt{3}}{243\,{a}^{3}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{c\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*ln(x)/a^4+14/81/a^3/(b*x^3+a)^3*x^8*b^2*e+20/81/a^3/(b*x^3+a)^3*x^7*b^2*d+1/3/
a^3/(b*x^3+a)^3*x^6*b^2*c+77/162/a^2/(b*x^3+a)^3*x^5*b*e+52/81/a^2/(b*x^3+a)^3*x
^4*b*d+5/6/a^2/(b*x^3+a)^3*x^3*b*c+67/162/a/(b*x^3+a)^3*x^2*e+41/81/a/(b*x^3+a)^
3*x*d+11/18/a/(b*x^3+a)^3*c+40/243/a^3*d/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-20/243/
a^3*d/b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+40/243/a^3*d/b/(a/b)^(2/3)
*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-14/243/a^3*e/b/(a/b)^(1/3)*ln(x
+(a/b)^(1/3))+7/243/a^3*e/b/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+14/243
/a^3*e*3^(1/2)/b/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/3*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)^4*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)^4*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)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219856, size = 409, normalized size = 1.41 \[ -\frac{c{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{4}} + \frac{c{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} + \frac{2 \, \sqrt{3}{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} b d - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{243 \, a^{4} b^{2}} + \frac{28 \, a b^{2} x^{8} e + 40 \, a b^{2} d x^{7} + 54 \, a b^{2} c x^{6} + 77 \, a^{2} b x^{5} e + 104 \, a^{2} b d x^{4} + 135 \, a^{2} b c x^{3} + 67 \, a^{3} x^{2} e + 82 \, a^{3} d x + 99 \, a^{3} c}{162 \,{\left (b x^{3} + a\right )}^{3} a^{4}} + \frac{{\left (20 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} d + 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{243 \, a^{5} b^{4}} - \frac{2 \,{\left (7 \, a^{5} b \left (-\frac{a}{b}\right )^{\frac{1}{3}} e + 20 \, a^{5} b d\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{243 \, a^{9} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*c*ln(abs(b*x^3 + a))/a^4 + c*ln(abs(x))/a^4 + 2/243*sqrt(3)*(20*(-a*b^2)^(1
/3)*b*d - 7*(-a*b^2)^(2/3)*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/
3))/(a^4*b^2) + 1/162*(28*a*b^2*x^8*e + 40*a*b^2*d*x^7 + 54*a*b^2*c*x^6 + 77*a^2
*b*x^5*e + 104*a^2*b*d*x^4 + 135*a^2*b*c*x^3 + 67*a^3*x^2*e + 82*a^3*d*x + 99*a^
3*c)/((b*x^3 + a)^3*a^4) + 1/243*(20*(-a*b^2)^(1/3)*a*b^3*d + 7*(-a*b^2)^(2/3)*a
*b^2*e)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^5*b^4) - 2/243*(7*a^5*b*(-a/b
)^(1/3)*e + 20*a^5*b*d)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^9*b)