3.7 \(\int \frac{a+b x^3+c x^6}{\left (d+e x^3\right )^2} \, dx\)

Optimal. Leaf size=213 \[ \frac{x \left (a e^2-b d e+c d^2\right )}{3 d e^2 \left (d+e x^3\right )}+\frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (4 c d^2-e (2 a e+b d)\right )}{18 d^{5/3} e^{7/3}}-\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (4 c d^2-e (2 a e+b d)\right )}{9 d^{5/3} e^{7/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right ) \left (4 c d^2-e (2 a e+b d)\right )}{3 \sqrt{3} d^{5/3} e^{7/3}}+\frac{c x}{e^2} \]

[Out]

(c*x)/e^2 + ((c*d^2 - b*d*e + a*e^2)*x)/(3*d*e^2*(d + e*x^3)) + ((4*c*d^2 - e*(b
*d + 2*a*e))*ArcTan[(d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(3*Sqrt[3]*d^(5/
3)*e^(7/3)) - ((4*c*d^2 - e*(b*d + 2*a*e))*Log[d^(1/3) + e^(1/3)*x])/(9*d^(5/3)*
e^(7/3)) + ((4*c*d^2 - e*(b*d + 2*a*e))*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3
)*x^2])/(18*d^(5/3)*e^(7/3))

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Rubi [A]  time = 0.448877, antiderivative size = 213, 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 \[ \frac{x \left (a e^2-b d e+c d^2\right )}{3 d e^2 \left (d+e x^3\right )}+\frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (4 c d^2-e (2 a e+b d)\right )}{18 d^{5/3} e^{7/3}}-\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (4 c d^2-e (2 a e+b d)\right )}{9 d^{5/3} e^{7/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right ) \left (4 c d^2-e (2 a e+b d)\right )}{3 \sqrt{3} d^{5/3} e^{7/3}}+\frac{c x}{e^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3 + c*x^6)/(d + e*x^3)^2,x]

[Out]

(c*x)/e^2 + ((c*d^2 - b*d*e + a*e^2)*x)/(3*d*e^2*(d + e*x^3)) + ((4*c*d^2 - e*(b
*d + 2*a*e))*ArcTan[(d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(3*Sqrt[3]*d^(5/
3)*e^(7/3)) - ((4*c*d^2 - e*(b*d + 2*a*e))*Log[d^(1/3) + e^(1/3)*x])/(9*d^(5/3)*
e^(7/3)) + ((4*c*d^2 - e*(b*d + 2*a*e))*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3
)*x^2])/(18*d^(5/3)*e^(7/3))

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Rubi in Sympy [A]  time = 64.4256, size = 202, normalized size = 0.95 \[ \frac{c x}{e^{2}} + \frac{x \left (a e^{2} - b d e + c d^{2}\right )}{3 d e^{2} \left (d + e x^{3}\right )} + \frac{\left (2 a e^{2} + b d e - 4 c d^{2}\right ) \log{\left (\sqrt [3]{d} + \sqrt [3]{e} x \right )}}{9 d^{\frac{5}{3}} e^{\frac{7}{3}}} - \frac{\left (2 a e^{2} + b d e - 4 c d^{2}\right ) \log{\left (d^{\frac{2}{3}} - \sqrt [3]{d} \sqrt [3]{e} x + e^{\frac{2}{3}} x^{2} \right )}}{18 d^{\frac{5}{3}} e^{\frac{7}{3}}} - \frac{\sqrt{3} \left (2 a e^{2} + b d e - 4 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{d}}{3} - \frac{2 \sqrt [3]{e} x}{3}\right )}{\sqrt [3]{d}} \right )}}{9 d^{\frac{5}{3}} e^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**6+b*x**3+a)/(e*x**3+d)**2,x)

[Out]

c*x/e**2 + x*(a*e**2 - b*d*e + c*d**2)/(3*d*e**2*(d + e*x**3)) + (2*a*e**2 + b*d
*e - 4*c*d**2)*log(d**(1/3) + e**(1/3)*x)/(9*d**(5/3)*e**(7/3)) - (2*a*e**2 + b*
d*e - 4*c*d**2)*log(d**(2/3) - d**(1/3)*e**(1/3)*x + e**(2/3)*x**2)/(18*d**(5/3)
*e**(7/3)) - sqrt(3)*(2*a*e**2 + b*d*e - 4*c*d**2)*atan(sqrt(3)*(d**(1/3)/3 - 2*
e**(1/3)*x/3)/d**(1/3))/(9*d**(5/3)*e**(7/3))

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Mathematica [A]  time = 0.492418, size = 199, normalized size = 0.93 \[ \frac{\frac{6 \sqrt [3]{e} x \left (e (a e-b d)+c d^2\right )}{d \left (d+e x^3\right )}+\frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (4 c d^2-e (2 a e+b d)\right )}{d^{5/3}}-\frac{2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (4 c d^2-e (2 a e+b d)\right )}{d^{5/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt{3}}\right ) \left (4 c d^2-e (2 a e+b d)\right )}{d^{5/3}}+18 c \sqrt [3]{e} x}{18 e^{7/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3 + c*x^6)/(d + e*x^3)^2,x]

[Out]

(18*c*e^(1/3)*x + (6*e^(1/3)*(c*d^2 + e*(-(b*d) + a*e))*x)/(d*(d + e*x^3)) + (2*
Sqrt[3]*(4*c*d^2 - e*(b*d + 2*a*e))*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]])
/d^(5/3) - (2*(4*c*d^2 - e*(b*d + 2*a*e))*Log[d^(1/3) + e^(1/3)*x])/d^(5/3) + ((
4*c*d^2 - e*(b*d + 2*a*e))*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/d^(5/
3))/(18*e^(7/3))

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Maple [A]  time = 0.014, size = 345, normalized size = 1.6 \[{\frac{cx}{{e}^{2}}}+{\frac{ax}{3\,d \left ( e{x}^{3}+d \right ) }}-{\frac{bx}{3\,e \left ( e{x}^{3}+d \right ) }}+{\frac{cdx}{3\,{e}^{2} \left ( e{x}^{3}+d \right ) }}+{\frac{2\,a}{9\,de}\ln \left ( x+\sqrt [3]{{\frac{d}{e}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{b}{9\,{e}^{2}}\ln \left ( x+\sqrt [3]{{\frac{d}{e}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}-{\frac{4\,cd}{9\,{e}^{3}}\ln \left ( x+\sqrt [3]{{\frac{d}{e}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}-{\frac{a}{9\,de}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{d}{e}}}+ \left ({\frac{d}{e}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}-{\frac{b}{18\,{e}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{d}{e}}}+ \left ({\frac{d}{e}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,cd}{9\,{e}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{d}{e}}}+ \left ({\frac{d}{e}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,\sqrt{3}a}{9\,de}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}-1 \right ) } \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}b}{9\,{e}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}-1 \right ) } \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}-{\frac{4\,d\sqrt{3}c}{9\,{e}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}-1 \right ) } \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^6+b*x^3+a)/(e*x^3+d)^2,x)

[Out]

c*x/e^2+1/3/d*x/(e*x^3+d)*a-1/3/e*x/(e*x^3+d)*b+1/3/e^2*d*x/(e*x^3+d)*c+2/9/e/d/
(d/e)^(2/3)*ln(x+(d/e)^(1/3))*a+1/9/e^2/(d/e)^(2/3)*ln(x+(d/e)^(1/3))*b-4/9/e^3*
d/(d/e)^(2/3)*ln(x+(d/e)^(1/3))*c-1/9/e/d/(d/e)^(2/3)*ln(x^2-x*(d/e)^(1/3)+(d/e)
^(2/3))*a-1/18/e^2/(d/e)^(2/3)*ln(x^2-x*(d/e)^(1/3)+(d/e)^(2/3))*b+2/9/e^3*d/(d/
e)^(2/3)*ln(x^2-x*(d/e)^(1/3)+(d/e)^(2/3))*c+2/9/e/d/(d/e)^(2/3)*3^(1/2)*arctan(
1/3*3^(1/2)*(2/(d/e)^(1/3)*x-1))*a+1/9/e^2/(d/e)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2
)*(2/(d/e)^(1/3)*x-1))*b-4/9/e^3*d/(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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.291672, size = 385, normalized size = 1.81 \[ \frac{\sqrt{3}{\left (\sqrt{3}{\left (4 \, c d^{3} - b d^{2} e - 2 \, a d e^{2} +{\left (4 \, c d^{2} e - b d e^{2} - 2 \, a e^{3}\right )} x^{3}\right )} \log \left (\left (d^{2} e\right )^{\frac{2}{3}} x^{2} - \left (d^{2} e\right )^{\frac{1}{3}} d x + d^{2}\right ) - 2 \, \sqrt{3}{\left (4 \, c d^{3} - b d^{2} e - 2 \, a d e^{2} +{\left (4 \, c d^{2} e - b d e^{2} - 2 \, a e^{3}\right )} x^{3}\right )} \log \left (\left (d^{2} e\right )^{\frac{1}{3}} x + d\right ) - 6 \,{\left (4 \, c d^{3} - b d^{2} e - 2 \, a d e^{2} +{\left (4 \, c d^{2} e - b d e^{2} - 2 \, a e^{3}\right )} x^{3}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (d^{2} e\right )^{\frac{1}{3}} x - \sqrt{3} d}{3 \, d}\right ) + 6 \, \sqrt{3}{\left (3 \, c d e x^{4} +{\left (4 \, c d^{2} - b d e + a e^{2}\right )} x\right )} \left (d^{2} e\right )^{\frac{1}{3}}\right )}}{54 \,{\left (d e^{3} x^{3} + d^{2} e^{2}\right )} \left (d^{2} e\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^2,x, algorithm="fricas")

[Out]

1/54*sqrt(3)*(sqrt(3)*(4*c*d^3 - b*d^2*e - 2*a*d*e^2 + (4*c*d^2*e - b*d*e^2 - 2*
a*e^3)*x^3)*log((d^2*e)^(2/3)*x^2 - (d^2*e)^(1/3)*d*x + d^2) - 2*sqrt(3)*(4*c*d^
3 - b*d^2*e - 2*a*d*e^2 + (4*c*d^2*e - b*d*e^2 - 2*a*e^3)*x^3)*log((d^2*e)^(1/3)
*x + d) - 6*(4*c*d^3 - b*d^2*e - 2*a*d*e^2 + (4*c*d^2*e - b*d*e^2 - 2*a*e^3)*x^3
)*arctan(1/3*(2*sqrt(3)*(d^2*e)^(1/3)*x - sqrt(3)*d)/d) + 6*sqrt(3)*(3*c*d*e*x^4
+ (4*c*d^2 - b*d*e + a*e^2)*x)*(d^2*e)^(1/3))/((d*e^3*x^3 + d^2*e^2)*(d^2*e)^(1
/3))

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Sympy [A]  time = 6.96547, size = 206, normalized size = 0.97 \[ \frac{c x}{e^{2}} + \frac{x \left (a e^{2} - b d e + c d^{2}\right )}{3 d^{2} e^{2} + 3 d e^{3} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} d^{5} e^{7} - 8 a^{3} e^{6} - 12 a^{2} b d e^{5} + 48 a^{2} c d^{2} e^{4} - 6 a b^{2} d^{2} e^{4} + 48 a b c d^{3} e^{3} - 96 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} + 12 b^{2} c d^{4} e^{2} - 48 b c^{2} d^{5} e + 64 c^{3} d^{6}, \left ( t \mapsto t \log{\left (\frac{9 t d^{2} e^{2}}{2 a e^{2} + b d e - 4 c d^{2}} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**6+b*x**3+a)/(e*x**3+d)**2,x)

[Out]

c*x/e**2 + x*(a*e**2 - b*d*e + c*d**2)/(3*d**2*e**2 + 3*d*e**3*x**3) + RootSum(7
29*_t**3*d**5*e**7 - 8*a**3*e**6 - 12*a**2*b*d*e**5 + 48*a**2*c*d**2*e**4 - 6*a*
b**2*d**2*e**4 + 48*a*b*c*d**3*e**3 - 96*a*c**2*d**4*e**2 - b**3*d**3*e**3 + 12*
b**2*c*d**4*e**2 - 48*b*c**2*d**5*e + 64*c**3*d**6, Lambda(_t, _t*log(9*_t*d**2*
e**2/(2*a*e**2 + b*d*e - 4*c*d**2) + x)))

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GIAC/XCAS [A]  time = 0.279325, size = 306, normalized size = 1.44 \[ c x e^{\left (-2\right )} - \frac{\sqrt{3}{\left (4 \, \left (-d e^{2}\right )^{\frac{1}{3}} c d^{2} - \left (-d e^{2}\right )^{\frac{1}{3}} b d e - 2 \, \left (-d e^{2}\right )^{\frac{1}{3}} a e^{2}\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}}}\right ) e^{\left (-3\right )}}{9 \, d^{2}} + \frac{{\left (4 \, c d^{2} - b d e - 2 \, a e^{2}\right )} \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} e^{\left (-2\right )}{\rm ln}\left ({\left | x - \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} \right |}\right )}{9 \, d^{2}} - \frac{{\left (4 \, \left (-d e^{2}\right )^{\frac{1}{3}} c d^{2} - \left (-d e^{2}\right )^{\frac{1}{3}} b d e - 2 \, \left (-d e^{2}\right )^{\frac{1}{3}} a e^{2}\right )} e^{\left (-3\right )}{\rm ln}\left (x^{2} + \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} x + \left (-d e^{\left (-1\right )}\right )^{\frac{2}{3}}\right )}{18 \, d^{2}} + \frac{{\left (c d^{2} x - b d x e + a x e^{2}\right )} e^{\left (-2\right )}}{3 \,{\left (x^{3} e + d\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)/(e*x^3 + d)^2,x, algorithm="giac")

[Out]

c*x*e^(-2) - 1/9*sqrt(3)*(4*(-d*e^2)^(1/3)*c*d^2 - (-d*e^2)^(1/3)*b*d*e - 2*(-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^2 + 1/9*(4*c*d^2 - b*d*e - 2*a*e^2)*(-d*e^(-1))^(1/3)*e^(-2)*ln(abs(x
- (-d*e^(-1))^(1/3)))/d^2 - 1/18*(4*(-d*e^2)^(1/3)*c*d^2 - (-d*e^2)^(1/3)*b*d*e
- 2*(-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^2 + 1/3*(c*d^2*x - b*d*x*e + a*x*e^2)*e^(-2)/((x^3*e + d)*d)