3.10 \(\int \frac{x^5 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx\)

Optimal. Leaf size=97 \[ \frac{\left (2 a c e+b^2 (-e)+b c d\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 c^2 \sqrt{b^2-4 a c}}+\frac{(c d-b e) \log \left (a+b x^3+c x^6\right )}{6 c^2}+\frac{e x^3}{3 c} \]

[Out]

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

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Rubi [A]  time = 0.255665, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{\left (2 a c e+b^2 (-e)+b c d\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 c^2 \sqrt{b^2-4 a c}}+\frac{(c d-b e) \log \left (a+b x^3+c x^6\right )}{6 c^2}+\frac{e x^3}{3 c} \]

Antiderivative was successfully verified.

[In]  Int[(x^5*(d + e*x^3))/(a + b*x^3 + c*x^6),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(e*x**3+d)/(c*x**6+b*x**3+a),x)

[Out]

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

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Mathematica [A]  time = 0.11327, size = 93, normalized size = 0.96 \[ \frac{\frac{2 \left (-2 a c e+b^2 e-b c d\right ) \tan ^{-1}\left (\frac{b+2 c x^3}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+(c d-b e) \log \left (a+b x^3+c x^6\right )+2 c e x^3}{6 c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^5*(d + e*x^3))/(a + b*x^3 + c*x^6),x]

[Out]

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

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Maple [A]  time = 0.005, size = 175, normalized size = 1.8 \[{\frac{e{x}^{3}}{3\,c}}-{\frac{\ln \left ( c{x}^{6}+b{x}^{3}+a \right ) be}{6\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{6}+b{x}^{3}+a \right ) d}{6\,c}}-{\frac{2\,ae}{3\,c}\arctan \left ({(2\,c{x}^{3}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}e}{3\,{c}^{2}}\arctan \left ({(2\,c{x}^{3}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{bd}{3\,c}\arctan \left ({(2\,c{x}^{3}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(e*x^3+d)/(c*x^6+b*x^3+a),x)

[Out]

1/3/c*x^3*e-1/6/c^2*ln(c*x^6+b*x^3+a)*b*e+1/6/c*ln(c*x^6+b*x^3+a)*d-2/3/c/(4*a*c
-b^2)^(1/2)*arctan((2*c*x^3+b)/(4*a*c-b^2)^(1/2))*a*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*e-1/3/c/(4*a*c-b^2)^(1/2)*arctan((2*c*
x^3+b)/(4*a*c-b^2)^(1/2))*b*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^5/(c*x^6 + b*x^3 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.353454, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (b c d -{\left (b^{2} - 2 \, a c\right )} e\right )} \log \left (\frac{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{3} + b^{3} - 4 \, a b c +{\left (2 \, c^{2} x^{6} + 2 \, b c x^{3} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{6} + b x^{3} + a}\right ) +{\left (2 \, c e x^{3} +{\left (c d - b e\right )} \log \left (c x^{6} + b x^{3} + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{6 \, \sqrt{b^{2} - 4 \, a c} c^{2}}, -\frac{2 \,{\left (b c d -{\left (b^{2} - 2 \, a c\right )} e\right )} \arctan \left (-\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) -{\left (2 \, c e x^{3} +{\left (c d - b e\right )} \log \left (c x^{6} + b x^{3} + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{6 \, \sqrt{-b^{2} + 4 \, a c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^3 + d)*x^5/(c*x^6 + b*x^3 + a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 29.424, size = 434, normalized size = 4.47 \[ \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \left (4 a c - b^{2}\right )} - \frac{b e - c d}{6 c^{2}}\right ) \log{\left (x^{3} + \frac{- a b e - 12 a c^{2} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \left (4 a c - b^{2}\right )} - \frac{b e - c d}{6 c^{2}}\right ) + 2 a c d + 3 b^{2} c \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \left (4 a c - b^{2}\right )} - \frac{b e - c d}{6 c^{2}}\right )}{2 a c e - b^{2} e + b c d} \right )} + \left (\frac{\sqrt{- 4 a c + b^{2}} \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \left (4 a c - b^{2}\right )} - \frac{b e - c d}{6 c^{2}}\right ) \log{\left (x^{3} + \frac{- a b e - 12 a c^{2} \left (\frac{\sqrt{- 4 a c + b^{2}} \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \left (4 a c - b^{2}\right )} - \frac{b e - c d}{6 c^{2}}\right ) + 2 a c d + 3 b^{2} c \left (\frac{\sqrt{- 4 a c + b^{2}} \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \left (4 a c - b^{2}\right )} - \frac{b e - c d}{6 c^{2}}\right )}{2 a c e - b^{2} e + b c d} \right )} + \frac{e x^{3}}{3 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(e*x**3+d)/(c*x**6+b*x**3+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.271663, size = 128, normalized size = 1.32 \[ \frac{x^{3} e}{3 \, c} + \frac{{\left (c d - b e\right )}{\rm ln}\left (c x^{6} + b x^{3} + a\right )}{6 \, c^{2}} - \frac{{\left (b c d - b^{2} e + 2 \, a c e\right )} \arctan \left (\frac{2 \, c x^{3} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{3 \, \sqrt{-b^{2} + 4 \, a c} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^3 + d)*x^5/(c*x^6 + b*x^3 + a),x, algorithm="giac")

[Out]

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