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

Optimal. Leaf size=301 \[ -\frac{10 \left (2 a^{2/3} e+7 b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{13/3} \sqrt [3]{b}}+\frac{20 \left (2 a^{2/3} e+7 b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{13/3} \sqrt [3]{b}}+\frac{20 \left (7 b^{2/3} c-2 a^{2/3} e\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{13/3} \sqrt [3]{b}}+\frac{x \left (40 a e-118 b c x-99 b d x^2\right )}{162 a^4 \left (a+b x^3\right )}-\frac{d \log \left (a+b x^3\right )}{3 a^4}-\frac{c}{a^4 x}+\frac{d \log (x)}{a^4}+\frac{x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac{x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3} \]

[Out]

-(c/(a^4*x)) + (x*(a*e - b*c*x - b*d*x^2))/(9*a^2*(a + b*x^3)^3) + (x*(8*a*e - 1
6*b*c*x - 15*b*d*x^2))/(54*a^3*(a + b*x^3)^2) + (x*(40*a*e - 118*b*c*x - 99*b*d*
x^2))/(162*a^4*(a + b*x^3)) + (20*(7*b^(2/3)*c - 2*a^(2/3)*e)*ArcTan[(a^(1/3) -
2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(81*Sqrt[3]*a^(13/3)*b^(1/3)) + (d*Log[x])/a^4
+ (20*(7*b^(2/3)*c + 2*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(243*a^(13/3)*b^(1/3
)) - (10*(7*b^(2/3)*c + 2*a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x
^2])/(243*a^(13/3)*b^(1/3)) - (d*Log[a + b*x^3])/(3*a^4)

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Rubi [A]  time = 1.14169, antiderivative size = 301, 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 \[ -\frac{10 \left (2 a^{2/3} e+7 b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{13/3} \sqrt [3]{b}}+\frac{20 \left (2 a^{2/3} e+7 b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{13/3} \sqrt [3]{b}}+\frac{20 \left (7 b^{2/3} c-2 a^{2/3} e\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{13/3} \sqrt [3]{b}}+\frac{x \left (40 a e-118 b c x-99 b d x^2\right )}{162 a^4 \left (a+b x^3\right )}-\frac{d \log \left (a+b x^3\right )}{3 a^4}-\frac{c}{a^4 x}+\frac{d \log (x)}{a^4}+\frac{x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac{x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

-(c/(a^4*x)) + (x*(a*e - b*c*x - b*d*x^2))/(9*a^2*(a + b*x^3)^3) + (x*(8*a*e - 1
6*b*c*x - 15*b*d*x^2))/(54*a^3*(a + b*x^3)^2) + (x*(40*a*e - 118*b*c*x - 99*b*d*
x^2))/(162*a^4*(a + b*x^3)) + (20*(7*b^(2/3)*c - 2*a^(2/3)*e)*ArcTan[(a^(1/3) -
2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(81*Sqrt[3]*a^(13/3)*b^(1/3)) + (d*Log[x])/a^4
+ (20*(7*b^(2/3)*c + 2*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(243*a^(13/3)*b^(1/3
)) - (10*(7*b^(2/3)*c + 2*a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x
^2])/(243*a^(13/3)*b^(1/3)) - (d*Log[a + b*x^3])/(3*a^4)

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Rubi in Sympy [A]  time = 63.2244, size = 178, normalized size = 0.59 \[ \frac{x \left (\frac{c}{x^{2}} + \frac{d}{x} + e\right )}{9 a \left (a + b x^{3}\right )^{3}} + \frac{4 e x}{27 a^{2} \left (a + b x^{3}\right )^{2}} + \frac{20 e x}{81 a^{3} \left (a + b x^{3}\right )} + \frac{40 e \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{243 a^{\frac{11}{3}} \sqrt [3]{b}} - \frac{20 e \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{243 a^{\frac{11}{3}} \sqrt [3]{b}} - \frac{40 \sqrt{3} e \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{243 a^{\frac{11}{3}} \sqrt [3]{b}} \]

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)**4,x)

[Out]

x*(c/x**2 + d/x + e)/(9*a*(a + b*x**3)**3) + 4*e*x/(27*a**2*(a + b*x**3)**2) + 2
0*e*x/(81*a**3*(a + b*x**3)) + 40*e*log(a**(1/3) + b**(1/3)*x)/(243*a**(11/3)*b*
*(1/3)) - 20*e*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(243*a**(11/3
)*b**(1/3)) - 40*sqrt(3)*e*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/
(243*a**(11/3)*b**(1/3))

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Mathematica [A]  time = 0.553533, size = 279, normalized size = 0.93 \[ \frac{-\frac{20 \left (7 a^{2/3} b^{2/3} c+2 a^{4/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}+\frac{40 \left (7 a^{2/3} b^{2/3} c+2 a^{4/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac{40 \sqrt{3} a^{2/3} \left (2 a^{2/3} e-7 b^{2/3} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{b}}+\frac{54 a^3 \left (a (d+e x)-b c x^2\right )}{\left (a+b x^3\right )^3}+\frac{9 a^2 \left (9 a d+8 a e x-16 b c x^2\right )}{\left (a+b x^3\right )^2}+\frac{6 a \left (27 a d+20 a e x-59 b c x^2\right )}{a+b x^3}-162 a d \log \left (a+b x^3\right )-\frac{486 a c}{x}+486 a d \log (x)}{486 a^5} \]

Antiderivative was successfully verified.

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

[Out]

((-486*a*c)/x + (9*a^2*(9*a*d + 8*a*e*x - 16*b*c*x^2))/(a + b*x^3)^2 + (6*a*(27*
a*d + 20*a*e*x - 59*b*c*x^2))/(a + b*x^3) + (54*a^3*(-(b*c*x^2) + a*(d + e*x)))/
(a + b*x^3)^3 - (40*Sqrt[3]*a^(2/3)*(-7*b^(2/3)*c + 2*a^(2/3)*e)*ArcTan[(1 - (2*
b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + 486*a*d*Log[x] + (40*(7*a^(2/3)*b^(2/3)*
c + 2*a^(4/3)*e)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) - (20*(7*a^(2/3)*b^(2/3)*c +
2*a^(4/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3) - 162*a*d*L
og[a + b*x^3])/(486*a^5)

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Maple [A]  time = 0.027, size = 397, normalized size = 1.3 \[ -{\frac{c}{{a}^{4}x}}+{\frac{d\ln \left ( x \right ) }{{a}^{4}}}-{\frac{59\,{x}^{8}{b}^{3}c}{81\,{a}^{4} \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{20\,{x}^{7}{b}^{2}e}{81\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{{b}^{2}d{x}^{6}}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{3}}}-{\frac{142\,{x}^{5}{b}^{2}c}{81\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{52\,b{x}^{4}e}{81\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{5\,b{x}^{3}d}{6\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{3}}}-{\frac{92\,b{x}^{2}c}{81\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{41\,ex}{81\,a \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{11\,d}{18\,a \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{40\,e}{243\,{a}^{3}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{20\,e}{243\,{a}^{3}b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{40\,e\sqrt{3}}{243\,{a}^{3}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{140\,c}{243\,{a}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{70\,c}{243\,{a}^{4}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{140\,c\sqrt{3}}{243\,{a}^{4}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{d\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c/a^4/x+d*ln(x)/a^4-59/81/a^4/(b*x^3+a)^3*x^8*b^3*c+20/81/a^3/(b*x^3+a)^3*x^7*b
^2*e+1/3/a^3/(b*x^3+a)^3*x^6*b^2*d-142/81/a^3/(b*x^3+a)^3*x^5*b^2*c+52/81/a^2/(b
*x^3+a)^3*x^4*b*e+5/6/a^2/(b*x^3+a)^3*x^3*b*d-92/81/a^2/(b*x^3+a)^3*x^2*b*c+41/8
1/a/(b*x^3+a)^3*x*e+11/18/a/(b*x^3+a)^3*d+40/243/a^3*e/b/(a/b)^(2/3)*ln(x+(a/b)^
(1/3))-20/243/a^3*e/b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+40/243/a^3*e
/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+140/243/a^4*c/(a/
b)^(1/3)*ln(x+(a/b)^(1/3))-70/243/a^4*c/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(
2/3))-140/243/a^4*c*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-
1/3*d*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^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)^4*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)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219696, size = 427, normalized size = 1.42 \[ -\frac{d{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{4}} + \frac{d{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} + \frac{10 \,{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a e - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} c\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{243 \, a^{5} b} + \frac{20 \, \sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} e + 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} b^{2} c\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{243 \, a^{5} b^{3}} - \frac{280 \, b^{3} c x^{9} - 40 \, a b^{2} x^{8} e - 54 \, a b^{2} d x^{7} + 770 \, a b^{2} c x^{6} - 104 \, a^{2} b x^{5} e - 135 \, a^{2} b d x^{4} + 670 \, a^{2} b c x^{3} - 82 \, a^{3} x^{2} e - 99 \, a^{3} d x + 162 \, a^{3} c}{162 \,{\left (b x^{3} + a\right )}^{3} a^{4} x} + \frac{20 \,{\left (7 \, a^{4} b^{2} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 2 \, a^{5} b e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{243 \, a^{9} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*d*ln(abs(b*x^3 + a))/a^4 + d*ln(abs(x))/a^4 + 10/243*(2*(-a*b^2)^(1/3)*a*e
- 7*(-a*b^2)^(2/3)*c)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^5*b) + 20/243*s
qrt(3)*(2*(-a*b^2)^(1/3)*a*b^2*e + 7*(-a*b^2)^(2/3)*b^2*c)*arctan(1/3*sqrt(3)*(2
*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^5*b^3) - 1/162*(280*b^3*c*x^9 - 40*a*b^2*x^8
*e - 54*a*b^2*d*x^7 + 770*a*b^2*c*x^6 - 104*a^2*b*x^5*e - 135*a^2*b*d*x^4 + 670*
a^2*b*c*x^3 - 82*a^3*x^2*e - 99*a^3*d*x + 162*a^3*c)/((b*x^3 + a)^3*a^4*x) + 20/
243*(7*a^4*b^2*c*(-a/b)^(1/3) - 2*a^5*b*e)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3))
)/(a^9*b)