3.82 \(\int \frac{A+B x^3}{x^2 \left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=196 \[ -\frac{(4 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{7/3} b^{2/3}}+\frac{(4 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3} b^{2/3}}+\frac{(4 A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{7/3} b^{2/3}}-\frac{4 A b-a B}{3 a^2 b x}+\frac{A b-a B}{3 a b x \left (a+b x^3\right )} \]

[Out]

-(4*A*b - a*B)/(3*a^2*b*x) + (A*b - a*B)/(3*a*b*x*(a + b*x^3)) + ((4*A*b - a*B)*
ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(7/3)*b^(2/3)) +
((4*A*b - a*B)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(7/3)*b^(2/3)) - ((4*A*b - a*B)*L
og[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(7/3)*b^(2/3))

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Rubi [A]  time = 0.295535, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{(4 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{7/3} b^{2/3}}+\frac{(4 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{7/3} b^{2/3}}+\frac{(4 A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{7/3} b^{2/3}}-\frac{4 A b-a B}{3 a^2 b x}+\frac{A b-a B}{3 a b x \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(4*A*b - a*B)/(3*a^2*b*x) + (A*b - a*B)/(3*a*b*x*(a + b*x^3)) + ((4*A*b - a*B)*
ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(7/3)*b^(2/3)) +
((4*A*b - a*B)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(7/3)*b^(2/3)) - ((4*A*b - a*B)*L
og[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(7/3)*b^(2/3))

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Rubi in Sympy [A]  time = 39.9923, size = 172, normalized size = 0.88 \[ \frac{A b - B a}{3 a b x \left (a + b x^{3}\right )} - \frac{4 A b - B a}{3 a^{2} b x} + \frac{\left (4 A b - B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{7}{3}} b^{\frac{2}{3}}} - \frac{\left (4 A b - B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{7}{3}} b^{\frac{2}{3}}} + \frac{\sqrt{3} \left (4 A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{7}{3}} b^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**3+A)/x**2/(b*x**3+a)**2,x)

[Out]

(A*b - B*a)/(3*a*b*x*(a + b*x**3)) - (4*A*b - B*a)/(3*a**2*b*x) + (4*A*b - B*a)*
log(a**(1/3) + b**(1/3)*x)/(9*a**(7/3)*b**(2/3)) - (4*A*b - B*a)*log(a**(2/3) -
a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(18*a**(7/3)*b**(2/3)) + sqrt(3)*(4*A*b - B
*a)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(9*a**(7/3)*b**(2/3))

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Mathematica [A]  time = 0.254926, size = 164, normalized size = 0.84 \[ \frac{\frac{(a B-4 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+\frac{2 (4 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac{2 \sqrt{3} (4 A b-a B) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{2/3}}+\frac{6 \sqrt [3]{a} x^2 (a B-A b)}{a+b x^3}-\frac{18 \sqrt [3]{a} A}{x}}{18 a^{7/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-18*a^(1/3)*A)/x + (6*a^(1/3)*(-(A*b) + a*B)*x^2)/(a + b*x^3) + (2*Sqrt[3]*(4*
A*b - a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + (2*(4*A*b - a*
B)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) + ((-4*A*b + a*B)*Log[a^(2/3) - a^(1/3)*b^(
1/3)*x + b^(2/3)*x^2])/b^(2/3))/(18*a^(7/3))

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Maple [A]  time = 0.016, size = 241, normalized size = 1.2 \[ -{\frac{A}{x{a}^{2}}}-{\frac{A{x}^{2}b}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{{x}^{2}B}{3\,a \left ( b{x}^{3}+a \right ) }}+{\frac{4\,A}{9\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{2\,A}{9\,{a}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{4\,A\sqrt{3}}{9\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{B}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B}{18\,ab}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B\sqrt{3}}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)/x^2/(b*x^3+a)^2,x)

[Out]

-A/a^2/x-1/3/a^2*x^2/(b*x^3+a)*A*b+1/3/a*x^2/(b*x^3+a)*B+4/9/a^2*A/(a/b)^(1/3)*l
n(x+(a/b)^(1/3))-2/9/a^2*A/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-4/9/a^2
*A*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/9/a*B/b/(a/b)^(
1/3)*ln(x+(a/b)^(1/3))+1/18/a*B/b/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+
1/9/a*B*3^(1/2)/b/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))

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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)/((b*x^3 + a)^2*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236828, size = 305, normalized size = 1.56 \[ \frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (B a b - 4 \, A b^{2}\right )} x^{4} +{\left (B a^{2} - 4 \, A a b\right )} x\right )} \log \left (\left (a b^{2}\right )^{\frac{1}{3}} b x^{2} + a b - \left (a b^{2}\right )^{\frac{2}{3}} x\right ) - 2 \, \sqrt{3}{\left ({\left (B a b - 4 \, A b^{2}\right )} x^{4} +{\left (B a^{2} - 4 \, A a b\right )} x\right )} \log \left (a b + \left (a b^{2}\right )^{\frac{2}{3}} x\right ) + 6 \,{\left ({\left (B a b - 4 \, A b^{2}\right )} x^{4} +{\left (B a^{2} - 4 \, A a b\right )} x\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) + 6 \, \sqrt{3}{\left ({\left (B a - 4 \, A b\right )} x^{3} - 3 \, A a\right )} \left (a b^{2}\right )^{\frac{1}{3}}\right )}}{54 \,{\left (a^{2} b x^{4} + a^{3} x\right )} \left (a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/54*sqrt(3)*(sqrt(3)*((B*a*b - 4*A*b^2)*x^4 + (B*a^2 - 4*A*a*b)*x)*log((a*b^2)^
(1/3)*b*x^2 + a*b - (a*b^2)^(2/3)*x) - 2*sqrt(3)*((B*a*b - 4*A*b^2)*x^4 + (B*a^2
- 4*A*a*b)*x)*log(a*b + (a*b^2)^(2/3)*x) + 6*((B*a*b - 4*A*b^2)*x^4 + (B*a^2 -
4*A*a*b)*x)*arctan(-1/3*(sqrt(3)*a*b - 2*sqrt(3)*(a*b^2)^(2/3)*x)/(a*b)) + 6*sqr
t(3)*((B*a - 4*A*b)*x^3 - 3*A*a)*(a*b^2)^(1/3))/((a^2*b*x^4 + a^3*x)*(a*b^2)^(1/
3))

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Sympy [A]  time = 3.18261, size = 122, normalized size = 0.62 \[ \frac{- 3 A a + x^{3} \left (- 4 A b + B a\right )}{3 a^{3} x + 3 a^{2} b x^{4}} + \operatorname{RootSum}{\left (729 t^{3} a^{7} b^{2} - 64 A^{3} b^{3} + 48 A^{2} B a b^{2} - 12 A B^{2} a^{2} b + B^{3} a^{3}, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a^{5} b}{16 A^{2} b^{2} - 8 A B a b + B^{2} a^{2}} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**3+A)/x**2/(b*x**3+a)**2,x)

[Out]

(-3*A*a + x**3*(-4*A*b + B*a))/(3*a**3*x + 3*a**2*b*x**4) + RootSum(729*_t**3*a*
*7*b**2 - 64*A**3*b**3 + 48*A**2*B*a*b**2 - 12*A*B**2*a**2*b + B**3*a**3, Lambda
(_t, _t*log(81*_t**2*a**5*b/(16*A**2*b**2 - 8*A*B*a*b + B**2*a**2) + x)))

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GIAC/XCAS [A]  time = 0.216819, size = 273, normalized size = 1.39 \[ -\frac{{\left (B a \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 4 \, A b \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{3}} + \frac{B a x^{3} - 4 \, A b x^{3} - 3 \, A a}{3 \,{\left (b x^{4} + a x\right )} a^{2}} - \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} B a - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{3} b^{2}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} B a - 4 \, \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*(B*a*(-a/b)^(1/3) - 4*A*b*(-a/b)^(1/3))*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3
)))/a^3 + 1/3*(B*a*x^3 - 4*A*b*x^3 - 3*A*a)/((b*x^4 + a*x)*a^2) - 1/9*sqrt(3)*((
-a*b^2)^(2/3)*B*a - 4*(-a*b^2)^(2/3)*A*b)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3)
)/(-a/b)^(1/3))/(a^3*b^2) + 1/18*((-a*b^2)^(2/3)*B*a - 4*(-a*b^2)^(2/3)*A*b)*ln(
x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b^2)