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

Optimal. Leaf size=193 \[ \frac{\sqrt [3]{a} \left (d-\frac{\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3}}-\frac{\sqrt [3]{a} \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{5/3}}+\frac{\sqrt [3]{a} \left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3}}+\frac{c \log \left (a+b x^3\right )}{3 b}+\frac{d x}{b}+\frac{e x^2}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.49715, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348 \[ \frac{\sqrt [3]{a} \left (d-\frac{\sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3}}-\frac{\sqrt [3]{a} \left (\sqrt [3]{b} d-\sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{5/3}}+\frac{\sqrt [3]{a} \left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3}}+\frac{c \log \left (a+b x^3\right )}{3 b}+\frac{d x}{b}+\frac{e x^2}{2 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\sqrt [3]{a} \left (\sqrt [3]{a} e - \sqrt [3]{b} d\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 b^{\frac{5}{3}}} - \frac{\sqrt [3]{a} \left (\sqrt [3]{a} e - \sqrt [3]{b} d\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 b^{\frac{5}{3}}} + \frac{\sqrt{3} \sqrt [3]{a} \left (\sqrt [3]{a} e + \sqrt [3]{b} d\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 b^{\frac{5}{3}}} + \frac{c \log{\left (a + b x^{3} \right )}}{3 b} + \frac{e \int x\, dx}{b} + \frac{\int d\, dx}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(1/3)*(a**(1/3)*e - b**(1/3)*d)*log(a**(1/3) + b**(1/3)*x)/(3*b**(5/3)) - a**
(1/3)*(a**(1/3)*e - b**(1/3)*d)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x*
*2)/(6*b**(5/3)) + sqrt(3)*a**(1/3)*(a**(1/3)*e + b**(1/3)*d)*atan(sqrt(3)*(a**(
1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(3*b**(5/3)) + c*log(a + b*x**3)/(3*b) + e*In
tegral(x, x)/b + Integral(d, x)/b

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Mathematica [A]  time = 0.165772, size = 184, normalized size = 0.95 \[ \frac{-\left (a^{2/3} e-\sqrt [3]{a} \sqrt [3]{b} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \left (a^{2/3} e-\sqrt [3]{a} \sqrt [3]{b} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 b^{2/3} c \log \left (a+b x^3\right )+2 \sqrt{3} \sqrt [3]{a} \left (\sqrt [3]{a} e+\sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )+6 b^{2/3} d x+3 b^{2/3} e x^2}{6 b^{5/3}} \]

Antiderivative was successfully verified.

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

[Out]

(6*b^(2/3)*d*x + 3*b^(2/3)*e*x^2 + 2*Sqrt[3]*a^(1/3)*(b^(1/3)*d + a^(1/3)*e)*Arc
Tan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 2*(-(a^(1/3)*b^(1/3)*d) + a^(2/3)*e)*
Log[a^(1/3) + b^(1/3)*x] - (-(a^(1/3)*b^(1/3)*d) + a^(2/3)*e)*Log[a^(2/3) - a^(1
/3)*b^(1/3)*x + b^(2/3)*x^2] + 2*b^(2/3)*c*Log[a + b*x^3])/(6*b^(5/3))

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Maple [A]  time = 0.006, size = 221, normalized size = 1.2 \[{\frac{e{x}^{2}}{2\,b}}+{\frac{dx}{b}}-{\frac{ad}{3\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{ad}{6\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{a\sqrt{3}d}{3\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{ae}{3\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{ae}{6\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{a\sqrt{3}e}{3\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{c\ln \left ( b{x}^{3}+a \right ) }{3\,b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)*x^2/(b*x^3 + a),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)*x^2/(b*x^3 + a),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 2.82039, size = 150, normalized size = 0.78 \[ \operatorname{RootSum}{\left (27 t^{3} b^{5} - 27 t^{2} b^{4} c + t \left (9 a b^{2} d e + 9 b^{3} c^{2}\right ) - a^{2} e^{3} - 3 a b c d e + a b d^{3} - b^{2} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{9 t^{2} b^{3} e - 6 t b^{2} c e - 3 t b^{2} d^{2} + 2 a d e^{2} + b c^{2} e + b c d^{2}}{a e^{3} + b d^{3}} \right )} \right )\right )} + \frac{d x}{b} + \frac{e x^{2}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(27*_t**3*b**5 - 27*_t**2*b**4*c + _t*(9*a*b**2*d*e + 9*b**3*c**2) - a**2
*e**3 - 3*a*b*c*d*e + a*b*d**3 - b**2*c**3, Lambda(_t, _t*log(x + (9*_t**2*b**3*
e - 6*_t*b**2*c*e - 3*_t*b**2*d**2 + 2*a*d*e**2 + b*c**2*e + b*c*d**2)/(a*e**3 +
b*d**3)))) + d*x/b + e*x**2/(2*b)

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GIAC/XCAS [A]  time = 0.215265, size = 285, normalized size = 1.48 \[ \frac{c{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b} + \frac{b x^{2} e + 2 \, b d x}{2 \, b^{2}} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac{2}{3}} a b e\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{4}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} d + \left (-a b^{2}\right )^{\frac{2}{3}} a b e\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{4}} + \frac{{\left (a b^{4} \left (-\frac{a}{b}\right )^{\frac{1}{3}} e + a b^{4} d\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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