3.18 \(\int \frac{d+e x^3}{x^2 \left (a+b x^3+c x^6\right )} \, dx\)

Optimal. Leaf size=653 \[ -\frac{\sqrt [3]{c} \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} a \sqrt [3]{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt [3]{c} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} a \sqrt [3]{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt [3]{c} \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} a \sqrt [3]{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt [3]{c} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} a \sqrt [3]{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt [3]{c} \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} a \sqrt [3]{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt [3]{c} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} a \sqrt [3]{\sqrt{b^2-4 a c}+b}}-\frac{d}{a x} \]

[Out]

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

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Rubi [A]  time = 2.60559, antiderivative size = 653, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32 \[ -\frac{\sqrt [3]{c} \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} a \sqrt [3]{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt [3]{c} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} a \sqrt [3]{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt [3]{c} \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} a \sqrt [3]{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt [3]{c} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} a \sqrt [3]{\sqrt{b^2-4 a c}+b}}+\frac{\sqrt [3]{c} \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} a \sqrt [3]{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt [3]{c} \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} a \sqrt [3]{\sqrt{b^2-4 a c}+b}}-\frac{d}{a x} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x^3)/(x^2*(a + b*x^3 + c*x^6)),x]

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x**3+d)/x**2/(c*x**6+b*x**3+a),x)

[Out]

Timed out

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Mathematica [C]  time = 0.0747336, size = 85, normalized size = 0.13 \[ -\frac{\text{RootSum}\left [\text{$\#$1}^6 c+\text{$\#$1}^3 b+a\&,\frac{\text{$\#$1}^3 c d \log (x-\text{$\#$1})-a e \log (x-\text{$\#$1})+b d \log (x-\text{$\#$1})}{2 \text{$\#$1}^4 c+\text{$\#$1} b}\&\right ]}{3 a}-\frac{d}{a x} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x^3)/(x^2*(a + b*x^3 + c*x^6)),x]

[Out]

-(d/(a*x)) - RootSum[a + b*#1^3 + c*#1^6 & , (b*d*Log[x - #1] - a*e*Log[x - #1]
+ c*d*Log[x - #1]*#1^3)/(b*#1 + 2*c*#1^4) & ]/(3*a)

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Maple [C]  time = 0.01, size = 70, normalized size = 0.1 \[ -{\frac{d}{ax}}-{\frac{1}{3\,a}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}c+{{\it \_Z}}^{3}b+a \right ) }{\frac{ \left ( cd{{\it \_R}}^{4}+ \left ( -ae+bd \right ){\it \_R} \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}c+{{\it \_R}}^{2}b}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x^3+d)/x^2/(c*x^6+b*x^3+a),x)

[Out]

-d/a/x-1/3/a*sum((c*d*_R^4+(-a*e+b*d)*_R)/(2*_R^5*c+_R^2*b)*ln(x-_R),_R=RootOf(_
Z^6*c+_Z^3*b+a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{\int \frac{c d x^{4} +{\left (b d - a e\right )} x}{c x^{6} + b x^{3} + a}\,{d x}}{a} - \frac{d}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^3 + d)/((c*x^6 + b*x^3 + a)*x^2),x, algorithm="maxima")

[Out]

-integrate((c*d*x^4 + (b*d - a*e)*x)/(c*x^6 + b*x^3 + a), x)/a - d/(a*x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^3 + d)/((c*x^6 + b*x^3 + a)*x^2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x**3+d)/x**2/(c*x**6+b*x**3+a),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{e x^{3} + d}{{\left (c x^{6} + b x^{3} + a\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x^3 + d)/((c*x^6 + b*x^3 + a)*x^2),x, algorithm="giac")

[Out]

integrate((e*x^3 + d)/((c*x^6 + b*x^3 + a)*x^2), x)