3.8 \(\int \frac{a+b x^3+c x^6}{\left (d+e x^3\right )^3} \, dx\)

Optimal. Leaf size=242 \[ -\frac{x \left (7 c d^2-e (5 a e+b d)\right )}{18 d^2 e^2 \left (d+e x^3\right )}+\frac{x \left (a e^2-b d e+c d^2\right )}{6 d e^2 \left (d+e x^3\right )^2}-\frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (e (5 a e+b d)+2 c d^2\right )}{54 d^{8/3} e^{7/3}}+\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (e (5 a e+b d)+2 c d^2\right )}{27 d^{8/3} e^{7/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right ) \left (e (5 a e+b d)+2 c d^2\right )}{9 \sqrt{3} d^{8/3} e^{7/3}} \]

[Out]

((c*d^2 - b*d*e + a*e^2)*x)/(6*d*e^2*(d + e*x^3)^2) - ((7*c*d^2 - e*(b*d + 5*a*e
))*x)/(18*d^2*e^2*(d + e*x^3)) - ((2*c*d^2 + e*(b*d + 5*a*e))*ArcTan[(d^(1/3) -
2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(9*Sqrt[3]*d^(8/3)*e^(7/3)) + ((2*c*d^2 + e*(b*
d + 5*a*e))*Log[d^(1/3) + e^(1/3)*x])/(27*d^(8/3)*e^(7/3)) - ((2*c*d^2 + e*(b*d
+ 5*a*e))*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(54*d^(8/3)*e^(7/3))

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Rubi [A]  time = 0.524643, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{x \left (7 c d^2-e (5 a e+b d)\right )}{18 d^2 e^2 \left (d+e x^3\right )}+\frac{x \left (a e^2-b d e+c d^2\right )}{6 d e^2 \left (d+e x^3\right )^2}-\frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (e (5 a e+b d)+2 c d^2\right )}{54 d^{8/3} e^{7/3}}+\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (e (5 a e+b d)+2 c d^2\right )}{27 d^{8/3} e^{7/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right ) \left (e (5 a e+b d)+2 c d^2\right )}{9 \sqrt{3} d^{8/3} e^{7/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3 + c*x^6)/(d + e*x^3)^3,x]

[Out]

((c*d^2 - b*d*e + a*e^2)*x)/(6*d*e^2*(d + e*x^3)^2) - ((7*c*d^2 - e*(b*d + 5*a*e
))*x)/(18*d^2*e^2*(d + e*x^3)) - ((2*c*d^2 + e*(b*d + 5*a*e))*ArcTan[(d^(1/3) -
2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(9*Sqrt[3]*d^(8/3)*e^(7/3)) + ((2*c*d^2 + e*(b*
d + 5*a*e))*Log[d^(1/3) + e^(1/3)*x])/(27*d^(8/3)*e^(7/3)) - ((2*c*d^2 + e*(b*d
+ 5*a*e))*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(54*d^(8/3)*e^(7/3))

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Rubi in Sympy [A]  time = 66.6659, size = 233, normalized size = 0.96 \[ \frac{x \left (a e^{2} - b d e + c d^{2}\right )}{6 d e^{2} \left (d + e x^{3}\right )^{2}} + \frac{x \left (5 a e^{2} + b d e - 7 c d^{2}\right )}{18 d^{2} e^{2} \left (d + e x^{3}\right )} + \frac{\left (5 a e^{2} + b d e + 2 c d^{2}\right ) \log{\left (\sqrt [3]{d} + \sqrt [3]{e} x \right )}}{27 d^{\frac{8}{3}} e^{\frac{7}{3}}} - \frac{\left (5 a e^{2} + b d e + 2 c d^{2}\right ) \log{\left (d^{\frac{2}{3}} - \sqrt [3]{d} \sqrt [3]{e} x + e^{\frac{2}{3}} x^{2} \right )}}{54 d^{\frac{8}{3}} e^{\frac{7}{3}}} - \frac{\sqrt{3} \left (5 a e^{2} + b d e + 2 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{d}}{3} - \frac{2 \sqrt [3]{e} x}{3}\right )}{\sqrt [3]{d}} \right )}}{27 d^{\frac{8}{3}} e^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**6+b*x**3+a)/(e*x**3+d)**3,x)

[Out]

x*(a*e**2 - b*d*e + c*d**2)/(6*d*e**2*(d + e*x**3)**2) + x*(5*a*e**2 + b*d*e - 7
*c*d**2)/(18*d**2*e**2*(d + e*x**3)) + (5*a*e**2 + b*d*e + 2*c*d**2)*log(d**(1/3
) + e**(1/3)*x)/(27*d**(8/3)*e**(7/3)) - (5*a*e**2 + b*d*e + 2*c*d**2)*log(d**(2
/3) - d**(1/3)*e**(1/3)*x + e**(2/3)*x**2)/(54*d**(8/3)*e**(7/3)) - sqrt(3)*(5*a
*e**2 + b*d*e + 2*c*d**2)*atan(sqrt(3)*(d**(1/3)/3 - 2*e**(1/3)*x/3)/d**(1/3))/(
27*d**(8/3)*e**(7/3))

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Mathematica [A]  time = 0.490439, size = 209, normalized size = 0.86 \[ \frac{2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (e (5 a e+b d)+2 c d^2\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt{3}}\right ) \left (e (5 a e+b d)+2 c d^2\right )-\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (e (5 a e+b d)+2 c d^2\right )-\frac{3 d^{2/3} \sqrt [3]{e} x \left (c d^2 \left (4 d+7 e x^3\right )-e \left (a e \left (8 d+5 e x^3\right )+b d \left (e x^3-2 d\right )\right )\right )}{\left (d+e x^3\right )^2}}{54 d^{8/3} e^{7/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3 + c*x^6)/(d + e*x^3)^3,x]

[Out]

((-3*d^(2/3)*e^(1/3)*x*(c*d^2*(4*d + 7*e*x^3) - e*(b*d*(-2*d + e*x^3) + a*e*(8*d
+ 5*e*x^3))))/(d + e*x^3)^2 - 2*Sqrt[3]*(2*c*d^2 + e*(b*d + 5*a*e))*ArcTan[(1 -
(2*e^(1/3)*x)/d^(1/3))/Sqrt[3]] + 2*(2*c*d^2 + e*(b*d + 5*a*e))*Log[d^(1/3) + e
^(1/3)*x] - (2*c*d^2 + e*(b*d + 5*a*e))*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3
)*x^2])/(54*d^(8/3)*e^(7/3))

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Maple [A]  time = 0.015, size = 362, normalized size = 1.5 \[{\frac{1}{ \left ( e{x}^{3}+d \right ) ^{2}} \left ({\frac{ \left ( 5\,a{e}^{2}+bde-7\,c{d}^{2} \right ){x}^{4}}{18\,{d}^{2}e}}+{\frac{ \left ( 4\,a{e}^{2}-bde-2\,c{d}^{2} \right ) x}{9\,{e}^{2}d}} \right ) }+{\frac{5\,a}{27\,{d}^{2}e}\ln \left ( x+\sqrt [3]{{\frac{d}{e}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{b}{27\,{e}^{2}d}\ln \left ( x+\sqrt [3]{{\frac{d}{e}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,c}{27\,{e}^{3}}\ln \left ( x+\sqrt [3]{{\frac{d}{e}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,a}{54\,{d}^{2}e}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{d}{e}}}+ \left ({\frac{d}{e}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}-{\frac{b}{54\,{e}^{2}d}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{d}{e}}}+ \left ({\frac{d}{e}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}-{\frac{c}{27\,{e}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{d}{e}}}+ \left ({\frac{d}{e}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,\sqrt{3}a}{27\,{d}^{2}e}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}-1 \right ) } \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}b}{27\,{e}^{2}d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}-1 \right ) } \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,\sqrt{3}c}{27\,{e}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}-1 \right ) } \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^6+b*x^3+a)/(e*x^3+d)^3,x)

[Out]

(1/18*(5*a*e^2+b*d*e-7*c*d^2)/d^2/e*x^4+1/9*(4*a*e^2-b*d*e-2*c*d^2)/e^2/d*x)/(e*
x^3+d)^2+5/27/d^2/e/(d/e)^(2/3)*ln(x+(d/e)^(1/3))*a+1/27/d/e^2/(d/e)^(2/3)*ln(x+
(d/e)^(1/3))*b+2/27/e^3/(d/e)^(2/3)*ln(x+(d/e)^(1/3))*c-5/54/d^2/e/(d/e)^(2/3)*l
n(x^2-x*(d/e)^(1/3)+(d/e)^(2/3))*a-1/54/d/e^2/(d/e)^(2/3)*ln(x^2-x*(d/e)^(1/3)+(
d/e)^(2/3))*b-1/27/e^3/(d/e)^(2/3)*ln(x^2-x*(d/e)^(1/3)+(d/e)^(2/3))*c+5/27/d^2/
e/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*a+1/27/d/e^2/(d/e)
^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*b+2/27/e^3/(d/e)^(2/3)*3^
(1/2)*arctan(1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*c

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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((c*x^6 + b*x^3 + a)/(e*x^3 + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.26324, size = 552, normalized size = 2.28 \[ -\frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (2 \, c d^{2} e^{2} + b d e^{3} + 5 \, a e^{4}\right )} x^{6} + 2 \, c d^{4} + b d^{3} e + 5 \, a d^{2} e^{2} + 2 \,{\left (2 \, c d^{3} e + b d^{2} e^{2} + 5 \, a d e^{3}\right )} x^{3}\right )} \log \left (\left (d^{2} e\right )^{\frac{2}{3}} x^{2} - \left (d^{2} e\right )^{\frac{1}{3}} d x + d^{2}\right ) - 2 \, \sqrt{3}{\left ({\left (2 \, c d^{2} e^{2} + b d e^{3} + 5 \, a e^{4}\right )} x^{6} + 2 \, c d^{4} + b d^{3} e + 5 \, a d^{2} e^{2} + 2 \,{\left (2 \, c d^{3} e + b d^{2} e^{2} + 5 \, a d e^{3}\right )} x^{3}\right )} \log \left (\left (d^{2} e\right )^{\frac{1}{3}} x + d\right ) - 6 \,{\left ({\left (2 \, c d^{2} e^{2} + b d e^{3} + 5 \, a e^{4}\right )} x^{6} + 2 \, c d^{4} + b d^{3} e + 5 \, a d^{2} e^{2} + 2 \,{\left (2 \, c d^{3} e + b d^{2} e^{2} + 5 \, a d e^{3}\right )} x^{3}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (d^{2} e\right )^{\frac{1}{3}} x - \sqrt{3} d}{3 \, d}\right ) + 3 \, \sqrt{3}{\left ({\left (7 \, c d^{2} e - b d e^{2} - 5 \, a e^{3}\right )} x^{4} + 2 \,{\left (2 \, c d^{3} + b d^{2} e - 4 \, a d e^{2}\right )} x\right )} \left (d^{2} e\right )^{\frac{1}{3}}\right )}}{162 \,{\left (d^{2} e^{4} x^{6} + 2 \, d^{3} e^{3} x^{3} + d^{4} e^{2}\right )} \left (d^{2} e\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^3,x, algorithm="fricas")

[Out]

-1/162*sqrt(3)*(sqrt(3)*((2*c*d^2*e^2 + b*d*e^3 + 5*a*e^4)*x^6 + 2*c*d^4 + b*d^3
*e + 5*a*d^2*e^2 + 2*(2*c*d^3*e + b*d^2*e^2 + 5*a*d*e^3)*x^3)*log((d^2*e)^(2/3)*
x^2 - (d^2*e)^(1/3)*d*x + d^2) - 2*sqrt(3)*((2*c*d^2*e^2 + b*d*e^3 + 5*a*e^4)*x^
6 + 2*c*d^4 + b*d^3*e + 5*a*d^2*e^2 + 2*(2*c*d^3*e + b*d^2*e^2 + 5*a*d*e^3)*x^3)
*log((d^2*e)^(1/3)*x + d) - 6*((2*c*d^2*e^2 + b*d*e^3 + 5*a*e^4)*x^6 + 2*c*d^4 +
b*d^3*e + 5*a*d^2*e^2 + 2*(2*c*d^3*e + b*d^2*e^2 + 5*a*d*e^3)*x^3)*arctan(1/3*(
2*sqrt(3)*(d^2*e)^(1/3)*x - sqrt(3)*d)/d) + 3*sqrt(3)*((7*c*d^2*e - b*d*e^2 - 5*
a*e^3)*x^4 + 2*(2*c*d^3 + b*d^2*e - 4*a*d*e^2)*x)*(d^2*e)^(1/3))/((d^2*e^4*x^6 +
2*d^3*e^3*x^3 + d^4*e^2)*(d^2*e)^(1/3))

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Sympy [A]  time = 22.2383, size = 246, normalized size = 1.02 \[ \frac{x^{4} \left (5 a e^{3} + b d e^{2} - 7 c d^{2} e\right ) + x \left (8 a d e^{2} - 2 b d^{2} e - 4 c d^{3}\right )}{18 d^{4} e^{2} + 36 d^{3} e^{3} x^{3} + 18 d^{2} e^{4} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} d^{8} e^{7} - 125 a^{3} e^{6} - 75 a^{2} b d e^{5} - 150 a^{2} c d^{2} e^{4} - 15 a b^{2} d^{2} e^{4} - 60 a b c d^{3} e^{3} - 60 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 b^{2} c d^{4} e^{2} - 12 b c^{2} d^{5} e - 8 c^{3} d^{6}, \left ( t \mapsto t \log{\left (\frac{27 t d^{3} e^{2}}{5 a e^{2} + b d e + 2 c d^{2}} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**6+b*x**3+a)/(e*x**3+d)**3,x)

[Out]

(x**4*(5*a*e**3 + b*d*e**2 - 7*c*d**2*e) + x*(8*a*d*e**2 - 2*b*d**2*e - 4*c*d**3
))/(18*d**4*e**2 + 36*d**3*e**3*x**3 + 18*d**2*e**4*x**6) + RootSum(19683*_t**3*
d**8*e**7 - 125*a**3*e**6 - 75*a**2*b*d*e**5 - 150*a**2*c*d**2*e**4 - 15*a*b**2*
d**2*e**4 - 60*a*b*c*d**3*e**3 - 60*a*c**2*d**4*e**2 - b**3*d**3*e**3 - 6*b**2*c
*d**4*e**2 - 12*b*c**2*d**5*e - 8*c**3*d**6, Lambda(_t, _t*log(27*_t*d**3*e**2/(
5*a*e**2 + b*d*e + 2*c*d**2) + x)))

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GIAC/XCAS [A]  time = 0.280327, size = 340, normalized size = 1.4 \[ \frac{\sqrt{3}{\left (2 \, \left (-d e^{2}\right )^{\frac{1}{3}} c d^{2} + \left (-d e^{2}\right )^{\frac{1}{3}} b d e + 5 \, \left (-d e^{2}\right )^{\frac{1}{3}} a e^{2}\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}}}\right ) e^{\left (-3\right )}}{27 \, d^{3}} - \frac{{\left (2 \, c d^{2} + b d e + 5 \, a e^{2}\right )} \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} e^{\left (-2\right )}{\rm ln}\left ({\left | x - \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} \right |}\right )}{27 \, d^{3}} + \frac{{\left (2 \, \left (-d e^{2}\right )^{\frac{1}{3}} c d^{2} + \left (-d e^{2}\right )^{\frac{1}{3}} b d e + 5 \, \left (-d e^{2}\right )^{\frac{1}{3}} a e^{2}\right )} e^{\left (-3\right )}{\rm ln}\left (x^{2} + \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} x + \left (-d e^{\left (-1\right )}\right )^{\frac{2}{3}}\right )}{54 \, d^{3}} - \frac{{\left (7 \, c d^{2} x^{4} e - b d x^{4} e^{2} - 5 \, a x^{4} e^{3} + 4 \, c d^{3} x + 2 \, b d^{2} x e - 8 \, a d x e^{2}\right )} e^{\left (-2\right )}}{18 \,{\left (x^{3} e + d\right )}^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^3,x, algorithm="giac")

[Out]

1/27*sqrt(3)*(2*(-d*e^2)^(1/3)*c*d^2 + (-d*e^2)^(1/3)*b*d*e + 5*(-d*e^2)^(1/3)*a
*e^2)*arctan(1/3*sqrt(3)*(2*x + (-d*e^(-1))^(1/3))/(-d*e^(-1))^(1/3))*e^(-3)/d^3
- 1/27*(2*c*d^2 + b*d*e + 5*a*e^2)*(-d*e^(-1))^(1/3)*e^(-2)*ln(abs(x - (-d*e^(-
1))^(1/3)))/d^3 + 1/54*(2*(-d*e^2)^(1/3)*c*d^2 + (-d*e^2)^(1/3)*b*d*e + 5*(-d*e^
2)^(1/3)*a*e^2)*e^(-3)*ln(x^2 + (-d*e^(-1))^(1/3)*x + (-d*e^(-1))^(2/3))/d^3 - 1
/18*(7*c*d^2*x^4*e - b*d*x^4*e^2 - 5*a*x^4*e^3 + 4*c*d^3*x + 2*b*d^2*x*e - 8*a*d
*x*e^2)*e^(-2)/((x^3*e + d)^2*d^2)