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

Optimal. Leaf size=183 \[ \frac{\left (a^{2/3} e+b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{4/3}}-\frac{\left (a^{2/3} e+b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{4/3}}-\frac{\left (b^{2/3} c-a^{2/3} e\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} b^{4/3}}+\frac{d \log \left (a+b x^3\right )}{3 b}+\frac{e x}{b} \]

[Out]

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

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Rubi [A]  time = 0.469453, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381 \[ \frac{\left (a^{2/3} e+b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{4/3}}-\frac{\left (a^{2/3} e+b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{4/3}}-\frac{\left (b^{2/3} c-a^{2/3} e\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} b^{4/3}}+\frac{d \log \left (a+b x^3\right )}{3 b}+\frac{e x}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*log(a + b*x**3)/(3*b) + Integral(e, x)/b + sqrt(3)*(a**(2/3)*e - b**(2/3)*c)*a
tan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(3*a**(1/3)*b**(4/3)) - (a**
(2/3)*e + b**(2/3)*c)*log(a**(1/3) + b**(1/3)*x)/(3*a**(1/3)*b**(4/3)) + (a**(2/
3)*e + b**(2/3)*c)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(6*a**(1/
3)*b**(4/3))

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Mathematica [A]  time = 0.126427, size = 200, normalized size = 1.09 \[ -\frac{\left (a^{4/3} \left (-\sqrt [3]{b}\right ) e-a^{2/3} b c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a b^{5/3}}+\frac{\left (a^{4/3} \left (-\sqrt [3]{b}\right ) e-a^{2/3} b c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a b^{5/3}}+\frac{\left (a^{2/3} b c-a^{4/3} \sqrt [3]{b} e\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a b^{5/3}}+\frac{d \log \left (a+b x^3\right )}{3 b}+\frac{e x}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 209, normalized size = 1.1 \[{\frac{ex}{b}}-{\frac{ae}{3\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{ae}{6\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{a\sqrt{3}e}{3\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{c}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{c}{6\,b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{c\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{d\ln \left ( b{x}^{3}+a \right ) }{3\,b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 2.95915, size = 160, normalized size = 0.87 \[ \operatorname{RootSum}{\left (27 t^{3} a b^{4} - 27 t^{2} a b^{3} d + t \left (- 9 a b^{2} c e + 9 a b^{2} d^{2}\right ) + a^{2} e^{3} + 3 a b c d e - a b d^{3} + b^{2} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{- 9 t^{2} a b^{3} c - 3 t a^{2} b e^{2} + 6 t a b^{2} c d + a^{2} d e^{2} + 2 a b c^{2} e - a b c d^{2}}{a^{2} e^{3} - b^{2} c^{3}} \right )} \right )\right )} + \frac{e x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.216157, size = 257, normalized size = 1.4 \[ \frac{x e}{b} + \frac{d{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a e + \left (-a b^{2}\right )^{\frac{2}{3}} c\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{2}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a e - \left (-a b^{2}\right )^{\frac{2}{3}} c\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{2}} - \frac{{\left (b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a b^{2} e\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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