3.9 \(\int \frac{x^8 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx\)

Optimal. Leaf size=132 \[ -\frac{\left (a c e+b^2 (-e)+b c d\right ) \log \left (a+b x^3+c x^6\right )}{6 c^3}-\frac{\left (3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 c^3 \sqrt{b^2-4 a c}}+\frac{x^3 (c d-b e)}{3 c^2}+\frac{e x^6}{6 c} \]

[Out]

((c*d - b*e)*x^3)/(3*c^2) + (e*x^6)/(6*c) - ((b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*
b*c*e)*ArcTanh[(b + 2*c*x^3)/Sqrt[b^2 - 4*a*c]])/(3*c^3*Sqrt[b^2 - 4*a*c]) - ((b
*c*d - b^2*e + a*c*e)*Log[a + b*x^3 + c*x^6])/(6*c^3)

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Rubi [A]  time = 0.441345, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{\left (a c e+b^2 (-e)+b c d\right ) \log \left (a+b x^3+c x^6\right )}{6 c^3}-\frac{\left (3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 c^3 \sqrt{b^2-4 a c}}+\frac{x^3 (c d-b e)}{3 c^2}+\frac{e x^6}{6 c} \]

Antiderivative was successfully verified.

[In]  Int[(x^8*(d + e*x^3))/(a + b*x^3 + c*x^6),x]

[Out]

((c*d - b*e)*x^3)/(3*c^2) + (e*x^6)/(6*c) - ((b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*
b*c*e)*ArcTanh[(b + 2*c*x^3)/Sqrt[b^2 - 4*a*c]])/(3*c^3*Sqrt[b^2 - 4*a*c]) - ((b
*c*d - b^2*e + a*c*e)*Log[a + b*x^3 + c*x^6])/(6*c^3)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \left (\frac{b e}{3} - \frac{c d}{3}\right ) \int ^{x^{3}} \frac{1}{c^{2}}\, dx + \frac{e \int ^{x^{3}} x\, dx}{3 c} + \frac{\left (- a c e + b^{2} e - b c d\right ) \log{\left (a + b x^{3} + c x^{6} \right )}}{6 c^{3}} + \frac{\left (- 3 a b c e + 2 a c^{2} d + b^{3} e - b^{2} c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{\sqrt{- 4 a c + b^{2}}} \right )}}{3 c^{3} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8*(e*x**3+d)/(c*x**6+b*x**3+a),x)

[Out]

-(b*e/3 - c*d/3)*Integral(c**(-2), (x, x**3)) + e*Integral(x, (x, x**3))/(3*c) +
(-a*c*e + b**2*e - b*c*d)*log(a + b*x**3 + c*x**6)/(6*c**3) + (-3*a*b*c*e + 2*a
*c**2*d + b**3*e - b**2*c*d)*atanh((b + 2*c*x**3)/sqrt(-4*a*c + b**2))/(3*c**3*s
qrt(-4*a*c + b**2))

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Mathematica [A]  time = 0.108351, size = 126, normalized size = 0.95 \[ \frac{\left (-a c e+b^2 e-b c d\right ) \log \left (a+b x^3+c x^6\right )+\frac{2 \left (3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d\right ) \tan ^{-1}\left (\frac{b+2 c x^3}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+2 c x^3 (c d-b e)+c^2 e x^6}{6 c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^8*(d + e*x^3))/(a + b*x^3 + c*x^6),x]

[Out]

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

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Maple [B]  time = 0.008, size = 260, normalized size = 2. \[{\frac{e{x}^{6}}{6\,c}}-{\frac{be{x}^{3}}{3\,{c}^{2}}}+{\frac{d{x}^{3}}{3\,c}}-{\frac{\ln \left ( c{x}^{6}+b{x}^{3}+a \right ) ae}{6\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{6}+b{x}^{3}+a \right ){b}^{2}e}{6\,{c}^{3}}}-{\frac{\ln \left ( c{x}^{6}+b{x}^{3}+a \right ) bd}{6\,{c}^{2}}}+{\frac{abe}{{c}^{2}}\arctan \left ({(2\,c{x}^{3}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{2\,ad}{3\,c}\arctan \left ({(2\,c{x}^{3}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{{b}^{3}e}{3\,{c}^{3}}\arctan \left ({(2\,c{x}^{3}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}d}{3\,{c}^{2}}\arctan \left ({(2\,c{x}^{3}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(e*x^3+d)/(c*x^6+b*x^3+a),x)

[Out]

1/6*e*x^6/c-1/3/c^2*x^3*b*e+1/3/c*d*x^3-1/6/c^2*ln(c*x^6+b*x^3+a)*a*e+1/6/c^3*ln
(c*x^6+b*x^3+a)*b^2*e-1/6/c^2*ln(c*x^6+b*x^3+a)*b*d+1/c^2/(4*a*c-b^2)^(1/2)*arct
an((2*c*x^3+b)/(4*a*c-b^2)^(1/2))*a*b*e-2/3/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x^3+
b)/(4*a*c-b^2)^(1/2))*a*d-1/3/c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x^3+b)/(4*a*c-b^
2)^(1/2))*b^3*e+1/3/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x^3+b)/(4*a*c-b^2)^(1/2))*
b^2*d

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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^3 + d)*x^8/(c*x^6 + b*x^3 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.612669, size = 1, normalized size = 0.01 \[ \left [\frac{{\left ({\left (b^{2} c - 2 \, a c^{2}\right )} d -{\left (b^{3} - 3 \, a b c\right )} e\right )} \log \left (-\frac{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{3} + b^{3} - 4 \, a b c -{\left (2 \, c^{2} x^{6} + 2 \, b c x^{3} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{6} + b x^{3} + a}\right ) +{\left (c^{2} e x^{6} + 2 \,{\left (c^{2} d - b c e\right )} x^{3} -{\left (b c d -{\left (b^{2} - a c\right )} e\right )} \log \left (c x^{6} + b x^{3} + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{6 \, \sqrt{b^{2} - 4 \, a c} c^{3}}, \frac{2 \,{\left ({\left (b^{2} c - 2 \, a c^{2}\right )} d -{\left (b^{3} - 3 \, a b c\right )} e\right )} \arctan \left (-\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) +{\left (c^{2} e x^{6} + 2 \,{\left (c^{2} d - b c e\right )} x^{3} -{\left (b c d -{\left (b^{2} - a c\right )} e\right )} \log \left (c x^{6} + b x^{3} + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{6 \, \sqrt{-b^{2} + 4 \, a c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^3 + d)*x^8/(c*x^6 + b*x^3 + a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 46.1755, size = 619, normalized size = 4.69 \[ \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{6 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c e - b^{2} e + b c d}{6 c^{3}}\right ) \log{\left (x^{3} + \frac{2 a^{2} c e - a b^{2} e + a b c d + 12 a c^{3} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{6 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c e - b^{2} e + b c d}{6 c^{3}}\right ) - 3 b^{2} c^{2} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{6 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c e - b^{2} e + b c d}{6 c^{3}}\right )}{3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d} \right )} + \left (\frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{6 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c e - b^{2} e + b c d}{6 c^{3}}\right ) \log{\left (x^{3} + \frac{2 a^{2} c e - a b^{2} e + a b c d + 12 a c^{3} \left (\frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{6 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c e - b^{2} e + b c d}{6 c^{3}}\right ) - 3 b^{2} c^{2} \left (\frac{\sqrt{- 4 a c + b^{2}} \left (3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d\right )}{6 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c e - b^{2} e + b c d}{6 c^{3}}\right )}{3 a b c e - 2 a c^{2} d - b^{3} e + b^{2} c d} \right )} + \frac{e x^{6}}{6 c} - \frac{x^{3} \left (b e - c d\right )}{3 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8*(e*x**3+d)/(c*x**6+b*x**3+a),x)

[Out]

(-sqrt(-4*a*c + b**2)*(3*a*b*c*e - 2*a*c**2*d - b**3*e + b**2*c*d)/(6*c**3*(4*a*
c - b**2)) - (a*c*e - b**2*e + b*c*d)/(6*c**3))*log(x**3 + (2*a**2*c*e - a*b**2*
e + a*b*c*d + 12*a*c**3*(-sqrt(-4*a*c + b**2)*(3*a*b*c*e - 2*a*c**2*d - b**3*e +
b**2*c*d)/(6*c**3*(4*a*c - b**2)) - (a*c*e - b**2*e + b*c*d)/(6*c**3)) - 3*b**2
*c**2*(-sqrt(-4*a*c + b**2)*(3*a*b*c*e - 2*a*c**2*d - b**3*e + b**2*c*d)/(6*c**3
*(4*a*c - b**2)) - (a*c*e - b**2*e + b*c*d)/(6*c**3)))/(3*a*b*c*e - 2*a*c**2*d -
b**3*e + b**2*c*d)) + (sqrt(-4*a*c + b**2)*(3*a*b*c*e - 2*a*c**2*d - b**3*e + b
**2*c*d)/(6*c**3*(4*a*c - b**2)) - (a*c*e - b**2*e + b*c*d)/(6*c**3))*log(x**3 +
(2*a**2*c*e - a*b**2*e + a*b*c*d + 12*a*c**3*(sqrt(-4*a*c + b**2)*(3*a*b*c*e -
2*a*c**2*d - b**3*e + b**2*c*d)/(6*c**3*(4*a*c - b**2)) - (a*c*e - b**2*e + b*c*
d)/(6*c**3)) - 3*b**2*c**2*(sqrt(-4*a*c + b**2)*(3*a*b*c*e - 2*a*c**2*d - b**3*e
+ b**2*c*d)/(6*c**3*(4*a*c - b**2)) - (a*c*e - b**2*e + b*c*d)/(6*c**3)))/(3*a*
b*c*e - 2*a*c**2*d - b**3*e + b**2*c*d)) + e*x**6/(6*c) - x**3*(b*e - c*d)/(3*c*
*2)

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GIAC/XCAS [A]  time = 0.269936, size = 177, normalized size = 1.34 \[ \frac{c x^{6} e + 2 \, c d x^{3} - 2 \, b x^{3} e}{6 \, c^{2}} - \frac{{\left (b c d - b^{2} e + a c e\right )}{\rm ln}\left (c x^{6} + b x^{3} + a\right )}{6 \, c^{3}} + \frac{{\left (b^{2} c d - 2 \, a c^{2} d - b^{3} e + 3 \, a b c e\right )} \arctan \left (\frac{2 \, c x^{3} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{3 \, \sqrt{-b^{2} + 4 \, a c} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^3 + d)*x^8/(c*x^6 + b*x^3 + a),x, algorithm="giac")

[Out]

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