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

Optimal. Leaf size=310 \[ \frac{10 \sqrt [3]{b} \left (11 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{14/3}}-\frac{20 \sqrt [3]{b} \left (11 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{14/3}}+\frac{20 \sqrt [3]{b} \left (7 \sqrt [3]{a} d+11 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{14/3}}-\frac{x \left (139 b c+118 b d x+99 b e x^2\right )}{162 a^4 \left (a+b x^3\right )}-\frac{e \log \left (a+b x^3\right )}{3 a^4}-\frac{c}{2 a^4 x^2}-\frac{d}{a^4 x}+\frac{e \log (x)}{a^4}-\frac{x \left (17 b c+16 b d x+15 b e x^2\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac{x \left (b c+b d x+b e x^2\right )}{9 a^2 \left (a+b x^3\right )^3} \]

[Out]

-c/(2*a^4*x^2) - d/(a^4*x) - (x*(b*c + b*d*x + b*e*x^2))/(9*a^2*(a + b*x^3)^3) -
(x*(17*b*c + 16*b*d*x + 15*b*e*x^2))/(54*a^3*(a + b*x^3)^2) - (x*(139*b*c + 118
*b*d*x + 99*b*e*x^2))/(162*a^4*(a + b*x^3)) + (20*b^(1/3)*(11*b^(1/3)*c + 7*a^(1
/3)*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(81*Sqrt[3]*a^(14/3))
+ (e*Log[x])/a^4 - (20*b^(1/3)*(11*b^(1/3)*c - 7*a^(1/3)*d)*Log[a^(1/3) + b^(1/3
)*x])/(243*a^(14/3)) + (10*b^(1/3)*(11*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])/(243*a^(14/3)) - (e*Log[a + b*x^3])/(3*a^4)

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Rubi [A]  time = 1.23059, antiderivative size = 310, 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 \[ \frac{10 \sqrt [3]{b} \left (11 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{14/3}}-\frac{20 \sqrt [3]{b} \left (11 \sqrt [3]{b} c-7 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{14/3}}+\frac{20 \sqrt [3]{b} \left (7 \sqrt [3]{a} d+11 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{81 \sqrt{3} a^{14/3}}-\frac{x \left (139 b c+118 b d x+99 b e x^2\right )}{162 a^4 \left (a+b x^3\right )}-\frac{e \log \left (a+b x^3\right )}{3 a^4}-\frac{c}{2 a^4 x^2}-\frac{d}{a^4 x}+\frac{e \log (x)}{a^4}-\frac{x \left (17 b c+16 b d x+15 b e x^2\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac{x \left (b c+b d x+b e x^2\right )}{9 a^2 \left (a+b x^3\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

-c/(2*a^4*x^2) - d/(a^4*x) - (x*(b*c + b*d*x + b*e*x^2))/(9*a^2*(a + b*x^3)^3) -
(x*(17*b*c + 16*b*d*x + 15*b*e*x^2))/(54*a^3*(a + b*x^3)^2) - (x*(139*b*c + 118
*b*d*x + 99*b*e*x^2))/(162*a^4*(a + b*x^3)) + (20*b^(1/3)*(11*b^(1/3)*c + 7*a^(1
/3)*d)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(81*Sqrt[3]*a^(14/3))
+ (e*Log[x])/a^4 - (20*b^(1/3)*(11*b^(1/3)*c - 7*a^(1/3)*d)*Log[a^(1/3) + b^(1/3
)*x])/(243*a^(14/3)) + (10*b^(1/3)*(11*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])/(243*a^(14/3)) - (e*Log[a + b*x^3])/(3*a^4)

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Rubi in Sympy [A]  time = 13.9475, size = 26, normalized size = 0.08 \[ \frac{x \left (\frac{c}{x^{3}} + \frac{d}{x^{2}} + \frac{e}{x}\right )}{9 a \left (a + b x^{3}\right )^{3}} \]

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)**4,x)

[Out]

x*(c/x**3 + d/x**2 + e/x)/(9*a*(a + b*x**3)**3)

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Mathematica [A]  time = 0.608032, size = 284, normalized size = 0.92 \[ \frac{20 \sqrt [3]{b} \left (11 \sqrt [3]{a} \sqrt [3]{b} c-7 a^{2/3} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+40 \sqrt [3]{b} \left (7 a^{2/3} d-11 \sqrt [3]{a} \sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\frac{54 a^3 (a e-b x (c+d x))}{\left (a+b x^3\right )^3}+\frac{9 a^2 (9 a e-b x (17 c+16 d x))}{\left (a+b x^3\right )^2}+\frac{3 a (54 a e-b x (139 c+118 d x))}{a+b x^3}+40 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \left (7 \sqrt [3]{a} d+11 \sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )-162 a e \log \left (a+b x^3\right )-\frac{243 a c}{x^2}-\frac{486 a d}{x}+486 a e \log (x)}{486 a^5} \]

Antiderivative was successfully verified.

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

[Out]

((-243*a*c)/x^2 - (486*a*d)/x + (54*a^3*(a*e - b*x*(c + d*x)))/(a + b*x^3)^3 + (
9*a^2*(9*a*e - b*x*(17*c + 16*d*x)))/(a + b*x^3)^2 + (3*a*(54*a*e - b*x*(139*c +
118*d*x)))/(a + b*x^3) + 40*Sqrt[3]*a^(1/3)*b^(1/3)*(11*b^(1/3)*c + 7*a^(1/3)*d
)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 486*a*e*Log[x] + 40*b^(1/3)*(-11
*a^(1/3)*b^(1/3)*c + 7*a^(2/3)*d)*Log[a^(1/3) + b^(1/3)*x] + 20*b^(1/3)*(11*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] - 16
2*a*e*Log[a + b*x^3])/(486*a^5)

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Maple [A]  time = 0.028, size = 400, normalized size = 1.3 \[ -{\frac{c}{2\,{a}^{4}{x}^{2}}}-{\frac{d}{{a}^{4}x}}+{\frac{e\ln \left ( x \right ) }{{a}^{4}}}-{\frac{59\,{b}^{3}d{x}^{8}}{81\,{a}^{4} \left ( b{x}^{3}+a \right ) ^{3}}}-{\frac{139\,{b}^{3}c{x}^{7}}{162\,{a}^{4} \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{{b}^{2}e{x}^{6}}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{3}}}-{\frac{142\,{b}^{2}d{x}^{5}}{81\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{3}}}-{\frac{329\,{b}^{2}c{x}^{4}}{162\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{5\,be{x}^{3}}{6\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{3}}}-{\frac{92\,b{x}^{2}d}{81\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{3}}}-{\frac{104\,bcx}{81\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{3}}}+{\frac{11\,e}{18\,a \left ( b{x}^{3}+a \right ) ^{3}}}-{\frac{220\,c}{243\,{a}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{110\,c}{243\,{a}^{4}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{220\,c\sqrt{3}}{243\,{a}^{4}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{140\,d}{243\,{a}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{70\,d}{243\,{a}^{4}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{140\,d\sqrt{3}}{243\,{a}^{4}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{e\ln \left ( b{x}^{3}+a \right ) }{3\,{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*c/a^4/x^2-d/a^4/x+e*ln(x)/a^4-59/81*b^3/a^4/(b*x^3+a)^3*d*x^8-139/162*b^3/a
^4/(b*x^3+a)^3*c*x^7+1/3*b^2/a^3/(b*x^3+a)^3*e*x^6-142/81*b^2/a^3/(b*x^3+a)^3*d*
x^5-329/162*b^2/a^3/(b*x^3+a)^3*c*x^4+5/6*b/a^2/(b*x^3+a)^3*e*x^3-92/81*b/a^2/(b
*x^3+a)^3*x^2*d-104/81*b/a^2/(b*x^3+a)^3*c*x+11/18/a/(b*x^3+a)^3*e-220/243/a^4*c
/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+110/243/a^4*c/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a
/b)^(2/3))-220/243/a^4*c/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x
-1))+140/243/a^4*d/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-70/243/a^4*d/(a/b)^(1/3)*ln(x^2
-x*(a/b)^(1/3)+(a/b)^(2/3))-140/243/a^4*d*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)
*(2/(a/b)^(1/3)*x-1))-1/3*e*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^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)^4*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)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21822, size = 441, normalized size = 1.42 \[ -\frac{e{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{4}} + \frac{e{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} - \frac{10 \,{\left (11 \, \left (-a b^{2}\right )^{\frac{1}{3}} b c + 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} d\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{243 \, a^{5} b} - \frac{20 \, \sqrt{3}{\left (11 \, \left (-a b^{2}\right )^{\frac{1}{3}} a b^{3} c - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} a b^{2} d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{243 \, a^{6} b^{3}} + \frac{20 \,{\left (7 \, a^{4} b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + 11 \, a^{4} b^{2} c\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{243 \, a^{9} b} - \frac{280 \, b^{3} d x^{10} + 220 \, b^{3} c x^{9} - 54 \, a b^{2} x^{8} e + 770 \, a b^{2} d x^{7} + 572 \, a b^{2} c x^{6} - 135 \, a^{2} b x^{5} e + 670 \, a^{2} b d x^{4} + 451 \, a^{2} b c x^{3} - 99 \, a^{3} x^{2} e + 162 \, a^{3} d x + 81 \, a^{3} c}{162 \,{\left (b x^{3} + a\right )}^{3} a^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*e*ln(abs(b*x^3 + a))/a^4 + e*ln(abs(x))/a^4 - 10/243*(11*(-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^5*b) - 20/243*
sqrt(3)*(11*(-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^6*b^3) + 20/243*(7*a^4*b^2*d*(-a/b)^(1/3
) + 11*a^4*b^2*c)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a^9*b) - 1/162*(280*b^
3*d*x^10 + 220*b^3*c*x^9 - 54*a*b^2*x^8*e + 770*a*b^2*d*x^7 + 572*a*b^2*c*x^6 -
135*a^2*b*x^5*e + 670*a^2*b*d*x^4 + 451*a^2*b*c*x^3 - 99*a^3*x^2*e + 162*a^3*d*x
+ 81*a^3*c)/((b*x^3 + a)^3*a^4*x^2)