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

Optimal. Leaf size=169 \[ -\frac{(2 a d+b c) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{18 c^{5/3} d^{4/3}}+\frac{(2 a d+b c) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} d^{4/3}}-\frac{(2 a d+b c) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{3 \sqrt{3} c^{5/3} d^{4/3}}-\frac{x (b c-a d)}{3 c d \left (c+d x^3\right )} \]

[Out]

-((b*c - a*d)*x)/(3*c*d*(c + d*x^3)) - ((b*c + 2*a*d)*ArcTan[(c^(1/3) - 2*d^(1/3
)*x)/(Sqrt[3]*c^(1/3))])/(3*Sqrt[3]*c^(5/3)*d^(4/3)) + ((b*c + 2*a*d)*Log[c^(1/3
) + d^(1/3)*x])/(9*c^(5/3)*d^(4/3)) - ((b*c + 2*a*d)*Log[c^(2/3) - c^(1/3)*d^(1/
3)*x + d^(2/3)*x^2])/(18*c^(5/3)*d^(4/3))

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Rubi [A]  time = 0.195134, antiderivative size = 169, 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 \[ -\frac{(2 a d+b c) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{18 c^{5/3} d^{4/3}}+\frac{(2 a d+b c) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} d^{4/3}}-\frac{(2 a d+b c) \tan ^{-1}\left (\frac{\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt{3} \sqrt [3]{c}}\right )}{3 \sqrt{3} c^{5/3} d^{4/3}}-\frac{x (b c-a d)}{3 c d \left (c+d x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - a*d)*x)/(3*c*d*(c + d*x^3)) - ((b*c + 2*a*d)*ArcTan[(c^(1/3) - 2*d^(1/3
)*x)/(Sqrt[3]*c^(1/3))])/(3*Sqrt[3]*c^(5/3)*d^(4/3)) + ((b*c + 2*a*d)*Log[c^(1/3
) + d^(1/3)*x])/(9*c^(5/3)*d^(4/3)) - ((b*c + 2*a*d)*Log[c^(2/3) - c^(1/3)*d^(1/
3)*x + d^(2/3)*x^2])/(18*c^(5/3)*d^(4/3))

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Rubi in Sympy [A]  time = 34.2323, size = 155, normalized size = 0.92 \[ \frac{x \left (a d - b c\right )}{3 c d \left (c + d x^{3}\right )} + \frac{\left (2 a d + b c\right ) \log{\left (\sqrt [3]{c} + \sqrt [3]{d} x \right )}}{9 c^{\frac{5}{3}} d^{\frac{4}{3}}} - \frac{\left (2 a d + b c\right ) \log{\left (c^{\frac{2}{3}} - \sqrt [3]{c} \sqrt [3]{d} x + d^{\frac{2}{3}} x^{2} \right )}}{18 c^{\frac{5}{3}} d^{\frac{4}{3}}} - \frac{\sqrt{3} \left (2 a d + b c\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{c}}{3} - \frac{2 \sqrt [3]{d} x}{3}\right )}{\sqrt [3]{c}} \right )}}{9 c^{\frac{5}{3}} d^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a*d - b*c)/(3*c*d*(c + d*x**3)) + (2*a*d + b*c)*log(c**(1/3) + d**(1/3)*x)/(9
*c**(5/3)*d**(4/3)) - (2*a*d + b*c)*log(c**(2/3) - c**(1/3)*d**(1/3)*x + d**(2/3
)*x**2)/(18*c**(5/3)*d**(4/3)) - sqrt(3)*(2*a*d + b*c)*atan(sqrt(3)*(c**(1/3)/3
- 2*d**(1/3)*x/3)/c**(1/3))/(9*c**(5/3)*d**(4/3))

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Mathematica [A]  time = 0.1699, size = 145, normalized size = 0.86 \[ \frac{-(2 a d+b c) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )-\frac{6 c^{2/3} \sqrt [3]{d} x (b c-a d)}{c+d x^3}+2 (2 a d+b c) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )-2 \sqrt{3} (2 a d+b c) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt{3}}\right )}{18 c^{5/3} d^{4/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-6*c^(2/3)*d^(1/3)*(b*c - a*d)*x)/(c + d*x^3) - 2*Sqrt[3]*(b*c + 2*a*d)*ArcTan
[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]] + 2*(b*c + 2*a*d)*Log[c^(1/3) + d^(1/3)*x]
- (b*c + 2*a*d)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(18*c^(5/3)*d^(
4/3))

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Maple [A]  time = 0.011, size = 221, normalized size = 1.3 \[{\frac{ \left ( ad-bc \right ) x}{3\,cd \left ( d{x}^{3}+c \right ) }}+{\frac{2\,a}{9\,cd}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{b}{9\,{d}^{2}}\ln \left ( x+\sqrt [3]{{\frac{c}{d}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{a}{9\,cd}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}-{\frac{b}{18\,{d}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{c}{d}}}+ \left ({\frac{c}{d}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,\sqrt{3}a}{9\,cd}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}b}{9\,{d}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{c}{d}}}}}}-1 \right ) } \right ) \left ({\frac{c}{d}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)/(d*x^3+c)^2,x)

[Out]

1/3*(a*d-b*c)/c/d*x/(d*x^3+c)+2/9/c/d/(c/d)^(2/3)*ln(x+(c/d)^(1/3))*a+1/9/d^2/(c
/d)^(2/3)*ln(x+(c/d)^(1/3))*b-1/9/c/d/(c/d)^(2/3)*ln(x^2-x*(c/d)^(1/3)+(c/d)^(2/
3))*a-1/18/d^2/(c/d)^(2/3)*ln(x^2-x*(c/d)^(1/3)+(c/d)^(2/3))*b+2/9/c/d/(c/d)^(2/
3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1))*a+1/9/d^2/(c/d)^(2/3)*3^(1/2)
*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1))*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((b*x^3 + a)/(d*x^3 + c)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.214528, size = 275, normalized size = 1.63 \[ -\frac{\sqrt{3}{\left (6 \, \sqrt{3} \left (c^{2} d\right )^{\frac{1}{3}}{\left (b c - a d\right )} x + \sqrt{3}{\left ({\left (b c d + 2 \, a d^{2}\right )} x^{3} + b c^{2} + 2 \, a c d\right )} \log \left (\left (c^{2} d\right )^{\frac{2}{3}} x^{2} - \left (c^{2} d\right )^{\frac{1}{3}} c x + c^{2}\right ) - 2 \, \sqrt{3}{\left ({\left (b c d + 2 \, a d^{2}\right )} x^{3} + b c^{2} + 2 \, a c d\right )} \log \left (\left (c^{2} d\right )^{\frac{1}{3}} x + c\right ) - 6 \,{\left ({\left (b c d + 2 \, a d^{2}\right )} x^{3} + b c^{2} + 2 \, a c d\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (c^{2} d\right )^{\frac{1}{3}} x - \sqrt{3} c}{3 \, c}\right )\right )}}{54 \,{\left (c d^{2} x^{3} + c^{2} d\right )} \left (c^{2} d\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/54*sqrt(3)*(6*sqrt(3)*(c^2*d)^(1/3)*(b*c - a*d)*x + sqrt(3)*((b*c*d + 2*a*d^2
)*x^3 + b*c^2 + 2*a*c*d)*log((c^2*d)^(2/3)*x^2 - (c^2*d)^(1/3)*c*x + c^2) - 2*sq
rt(3)*((b*c*d + 2*a*d^2)*x^3 + b*c^2 + 2*a*c*d)*log((c^2*d)^(1/3)*x + c) - 6*((b
*c*d + 2*a*d^2)*x^3 + b*c^2 + 2*a*c*d)*arctan(1/3*(2*sqrt(3)*(c^2*d)^(1/3)*x - s
qrt(3)*c)/c))/((c*d^2*x^3 + c^2*d)*(c^2*d)^(1/3))

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Sympy [A]  time = 2.61964, size = 97, normalized size = 0.57 \[ \frac{x \left (a d - b c\right )}{3 c^{2} d + 3 c d^{2} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} c^{5} d^{4} - 8 a^{3} d^{3} - 12 a^{2} b c d^{2} - 6 a b^{2} c^{2} d - b^{3} c^{3}, \left ( t \mapsto t \log{\left (\frac{9 t c^{2} d}{2 a d + b c} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(a*d - b*c)/(3*c**2*d + 3*c*d**2*x**3) + RootSum(729*_t**3*c**5*d**4 - 8*a**3*
d**3 - 12*a**2*b*c*d**2 - 6*a*b**2*c**2*d - b**3*c**3, Lambda(_t, _t*log(9*_t*c*
*2*d/(2*a*d + b*c) + x)))

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GIAC/XCAS [A]  time = 0.217759, size = 246, normalized size = 1.46 \[ -\frac{{\left (b c + 2 \, a d\right )} \left (-\frac{c}{d}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{c}{d}\right )^{\frac{1}{3}} \right |}\right )}{9 \, c^{2} d} + \frac{\sqrt{3}{\left (\left (-c d^{2}\right )^{\frac{1}{3}} b c + 2 \, \left (-c d^{2}\right )^{\frac{1}{3}} a d\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{c}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{c}{d}\right )^{\frac{1}{3}}}\right )}{9 \, c^{2} d^{2}} - \frac{b c x - a d x}{3 \,{\left (d x^{3} + c\right )} c d} + \frac{{\left (\left (-c d^{2}\right )^{\frac{1}{3}} b c + 2 \, \left (-c d^{2}\right )^{\frac{1}{3}} a d\right )}{\rm ln}\left (x^{2} + x \left (-\frac{c}{d}\right )^{\frac{1}{3}} + \left (-\frac{c}{d}\right )^{\frac{2}{3}}\right )}{18 \, c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*(b*c + 2*a*d)*(-c/d)^(1/3)*ln(abs(x - (-c/d)^(1/3)))/(c^2*d) + 1/9*sqrt(3)*
((-c*d^2)^(1/3)*b*c + 2*(-c*d^2)^(1/3)*a*d)*arctan(1/3*sqrt(3)*(2*x + (-c/d)^(1/
3))/(-c/d)^(1/3))/(c^2*d^2) - 1/3*(b*c*x - a*d*x)/((d*x^3 + c)*c*d) + 1/18*((-c*
d^2)^(1/3)*b*c + 2*(-c*d^2)^(1/3)*a*d)*ln(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/(
c^2*d^2)