3.5 \(\int \frac{a+b x^3}{c+d x^3} \, dx\)

Optimal. Leaf size=144 \[ \frac{(b c-a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{4/3}}-\frac{(b c-a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{4/3}}+\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} d^{4/3}}+\frac{b x}{d} \]

[Out]

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

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Rubi [A]  time = 0.211318, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412 \[ \frac{(b c-a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{2/3} d^{4/3}}-\frac{(b c-a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} d^{4/3}}+\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{\sqrt{3} c^{2/3} d^{4/3}}+\frac{b x}{d} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)/(c + d*x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 32.6833, size = 134, normalized size = 0.93 \[ \frac{b x}{d} + \frac{\left (a d - b c\right ) \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{3 c^{\frac{2}{3}} d^{\frac{4}{3}}} - \frac{\left (a d - b c\right ) \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{6 c^{\frac{2}{3}} d^{\frac{4}{3}}} - \frac{\sqrt{3} \left (a d - b c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{3 c^{\frac{2}{3}} d^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x/d + (a*d - b*c)*log(c**(1/3) + d**(1/3)*x)/(3*c**(2/3)*d**(4/3)) - (a*d - b*
c)*log(c**(2/3) - c**(1/3)*d**(1/3)*x + d**(2/3)*x**2)/(6*c**(2/3)*d**(4/3)) - s
qrt(3)*(a*d - b*c)*atan(sqrt(3)*(c**(1/3)/3 - 2*d**(1/3)*x/3)/c**(1/3))/(3*c**(2
/3)*d**(4/3))

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Mathematica [A]  time = 0.140246, size = 128, normalized size = 0.89 \[ \frac{(b c-a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )-2 (b c-a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )+2 \sqrt{3} (b c-a d) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )+6 b c^{2/3} \sqrt [3]{d} x}{6 c^{2/3} d^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 195, normalized size = 1.4 \[{\frac{bx}{d}}+{\frac{a}{3\,d}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{bc}{3\,{d}^{2}}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{a}{6\,d}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{bc}{6\,{d}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}a}{3\,d}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{\sqrt{3}bc}{3\,{d}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)/(d*x^3+c),x)

[Out]

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

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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((b*x^3 + a)/(d*x^3 + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.215331, size = 176, normalized size = 1.22 \[ \frac{\sqrt{3}{\left (6 \, \sqrt{3} \left (c^{2} d\right )^{\frac{1}{3}} b x + \sqrt{3}{\left (b c - a d\right )} \log \left (\left (c^{2} d\right )^{\frac{2}{3}} x^{2} - \left (c^{2} d\right )^{\frac{1}{3}} c x + c^{2}\right ) - 2 \, \sqrt{3}{\left (b c - a d\right )} \log \left (\left (c^{2} d\right )^{\frac{1}{3}} x + c\right ) - 6 \,{\left (b c - a d\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (c^{2} d\right )^{\frac{1}{3}} x - \sqrt{3} c}{3 \, c}\right )\right )}}{18 \, \left (c^{2} d\right )^{\frac{1}{3}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*sqrt(3)*(6*sqrt(3)*(c^2*d)^(1/3)*b*x + sqrt(3)*(b*c - a*d)*log((c^2*d)^(2/3
)*x^2 - (c^2*d)^(1/3)*c*x + c^2) - 2*sqrt(3)*(b*c - a*d)*log((c^2*d)^(1/3)*x + c
) - 6*(b*c - a*d)*arctan(1/3*(2*sqrt(3)*(c^2*d)^(1/3)*x - sqrt(3)*c)/c))/((c^2*d
)^(1/3)*d)

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Sympy [A]  time = 1.93706, size = 71, normalized size = 0.49 \[ \frac{b x}{d} + \operatorname{RootSum}{\left (27 t^{3} c^{2} d^{4} - a^{3} d^{3} + 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + b^{3} c^{3}, \left ( t \mapsto t \log{\left (\frac{3 t c d}{a d - b c} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.21749, size = 217, normalized size = 1.51 \[ \frac{b x}{d} + \frac{{\left (b c - a d\right )} \left (-\frac{c}{d}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{3 \, c d} - \frac{\sqrt{3}{\left (\left (-c d^{2}\right )^{\frac{1}{3}} b c - \left (-c d^{2}\right )^{\frac{1}{3}} a d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{3 \, c d^{2}} - \frac{{\left (\left (-c d^{2}\right )^{\frac{1}{3}} b c - \left (-c d^{2}\right )^{\frac{1}{3}} a d\right )}{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{6 \, c d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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