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

Optimal. Leaf size=205 \[ \frac{\sqrt [3]{a} \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3}}-\frac{\sqrt [3]{a} \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{5/3}}+\frac{\sqrt [3]{a} \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3}}-\frac{a e \log \left (a+b x^3\right )}{3 b^2}+\frac{c x}{b}+\frac{d x^2}{2 b}+\frac{e x^3}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.529306, antiderivative size = 205, 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 \[ \frac{\sqrt [3]{a} \left (c-\frac{\sqrt [3]{a} d}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{4/3}}-\frac{\sqrt [3]{a} \left (\sqrt [3]{b} c-\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{5/3}}+\frac{\sqrt [3]{a} \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} b^{5/3}}-\frac{a e \log \left (a+b x^3\right )}{3 b^2}+\frac{c x}{b}+\frac{d x^2}{2 b}+\frac{e x^3}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.226753, size = 191, normalized size = 0.93 \[ \frac{\sqrt [3]{b} \left (\sqrt [3]{a} \sqrt [3]{b} c-a^{2/3} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 \sqrt [3]{b} \left (a^{2/3} d-\sqrt [3]{a} \sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 \sqrt{3} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )-2 a e \log \left (a+b x^3\right )+6 b c x+3 b d x^2+2 b e x^3}{6 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 231, normalized size = 1.1 \[{\frac{e{x}^{3}}{3\,b}}+{\frac{d{x}^{2}}{2\,b}}+{\frac{cx}{b}}-{\frac{ac}{3\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{ac}{6\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{a\sqrt{3}c}{3\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{ad}{3\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{ad}{6\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{a\sqrt{3}d}{3\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{ae\ln \left ( b{x}^{3}+a \right ) }{3\,{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 2.88853, size = 178, normalized size = 0.87 \[ \operatorname{RootSum}{\left (27 t^{3} b^{6} + 27 t^{2} a b^{4} e + t \left (9 a^{2} b^{2} e^{2} + 9 a b^{3} c d\right ) + a^{3} e^{3} + 3 a^{2} b c d e - a^{2} b d^{3} + a b^{2} c^{3}, \left ( t \mapsto t \log{\left (x + \frac{9 t^{2} b^{4} d + 6 t a b^{2} d e - 3 t b^{3} c^{2} + a^{2} d e^{2} - a b c^{2} e + 2 a b c d^{2}}{a b d^{3} + b^{2} c^{3}} \right )} \right )\right )} + \frac{c x}{b} + \frac{d x^{2}}{2 b} + \frac{e x^{3}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.215968, size = 302, normalized size = 1.47 \[ -\frac{a e{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} + \frac{2 \, b^{2} x^{3} e + 3 \, b^{2} d x^{2} + 6 \, b^{2} c x}{6 \, b^{3}} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} c - \left (-a b^{2}\right )^{\frac{2}{3}} a b d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a b^{4}} - \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} a b^{2} c + \left (-a b^{2}\right )^{\frac{2}{3}} a b d\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{4}} + \frac{{\left (a b^{6} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a b^{6} c\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*a*e*ln(abs(b*x^3 + a))/b^2 + 1/6*(2*b^2*x^3*e + 3*b^2*d*x^2 + 6*b^2*c*x)/b^
3 - 1/3*sqrt(3)*((-a*b^2)^(1/3)*a*b^2*c - (-a*b^2)^(2/3)*a*b*d)*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*c + (-
a*b^2)^(2/3)*a*b*d)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a*b^4) + 1/3*(a*b^6
*d*(-a/b)^(1/3) + a*b^6*c)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/(a*b^7)