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

Optimal. Leaf size=196 \[ \frac{(5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3} \sqrt [3]{b}}-\frac{(5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} \sqrt [3]{b}}+\frac{(5 A b-2 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{8/3} \sqrt [3]{b}}-\frac{5 A b-2 a B}{6 a^2 b x^2}+\frac{A b-a B}{3 a b x^2 \left (a+b x^3\right )} \]

[Out]

-(5*A*b - 2*a*B)/(6*a^2*b*x^2) + (A*b - a*B)/(3*a*b*x^2*(a + b*x^3)) + ((5*A*b -
2*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(8/3)*b^
(1/3)) - ((5*A*b - 2*a*B)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(8/3)*b^(1/3)) + ((5*A*
b - 2*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(8/3)*b^(1/3))

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Rubi [A]  time = 0.284863, 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 \[ \frac{(5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3} \sqrt [3]{b}}-\frac{(5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} \sqrt [3]{b}}+\frac{(5 A b-2 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{8/3} \sqrt [3]{b}}-\frac{5 A b-2 a B}{6 a^2 b x^2}+\frac{A b-a B}{3 a b x^2 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(5*A*b - 2*a*B)/(6*a^2*b*x^2) + (A*b - a*B)/(3*a*b*x^2*(a + b*x^3)) + ((5*A*b -
2*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(8/3)*b^
(1/3)) - ((5*A*b - 2*a*B)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(8/3)*b^(1/3)) + ((5*A*
b - 2*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(8/3)*b^(1/3))

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Rubi in Sympy [A]  time = 40.9844, size = 182, normalized size = 0.93 \[ \frac{A b - B a}{3 a b x^{2} \left (a + b x^{3}\right )} - \frac{5 A b - 2 B a}{6 a^{2} b x^{2}} - \frac{\left (5 A b - 2 B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{8}{3}} \sqrt [3]{b}} + \frac{\left (5 A b - 2 B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{8}{3}} \sqrt [3]{b}} + \frac{\sqrt{3} \left (5 A b - 2 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{8}{3}} \sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(A*b - B*a)/(3*a*b*x**2*(a + b*x**3)) - (5*A*b - 2*B*a)/(6*a**2*b*x**2) - (5*A*b
- 2*B*a)*log(a**(1/3) + b**(1/3)*x)/(9*a**(8/3)*b**(1/3)) + (5*A*b - 2*B*a)*log
(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(18*a**(8/3)*b**(1/3)) + sqrt(3
)*(5*A*b - 2*B*a)*atan(sqrt(3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(9*a**(8/
3)*b**(1/3))

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Mathematica [A]  time = 0.305306, size = 163, normalized size = 0.83 \[ \frac{\frac{(5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}+\frac{6 a^{2/3} x (a B-A b)}{a+b x^3}-\frac{9 a^{2/3} A}{x^2}+\frac{2 (2 a B-5 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac{2 \sqrt{3} (5 A b-2 a B) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{b}}}{18 a^{8/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-9*a^(2/3)*A)/x^2 + (6*a^(2/3)*(-(A*b) + a*B)*x)/(a + b*x^3) + (2*Sqrt[3]*(5*A
*b - 2*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + (2*(-5*A*b +
2*a*B)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) + ((5*A*b - 2*a*B)*Log[a^(2/3) - a^(1/3
)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3))/(18*a^(8/3))

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Maple [A]  time = 0.014, size = 237, normalized size = 1.2 \[ -{\frac{A}{2\,{a}^{2}{x}^{2}}}-{\frac{Axb}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{xB}{3\,a \left ( b{x}^{3}+a \right ) }}-{\frac{5\,A}{9\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,A}{18\,{a}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,A\sqrt{3}}{9\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,B}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B}{9\,ab}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,B\sqrt{3}}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*A/a^2/x^2-1/3/a^2*x/(b*x^3+a)*A*b+1/3/a*x/(b*x^3+a)*B-5/9/a^2*A/(a/b)^(2/3)
*ln(x+(a/b)^(1/3))+5/18/a^2*A/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-5/9/
a^2*A/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+2/9/a*B/b/(a/b
)^(2/3)*ln(x+(a/b)^(1/3))-1/9/a*B/b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3)
)+2/9/a*B/b/(a/b)^(2/3)*3^(1/2)*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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240346, size = 327, normalized size = 1.67 \[ \frac{\sqrt{3}{\left (\sqrt{3}{\left ({\left (2 \, B a b - 5 \, A b^{2}\right )} x^{5} +{\left (2 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \log \left (\left (-a^{2} b\right )^{\frac{2}{3}} x^{2} + \left (-a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 2 \, \sqrt{3}{\left ({\left (2 \, B a b - 5 \, A b^{2}\right )} x^{5} +{\left (2 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \log \left (\left (-a^{2} b\right )^{\frac{1}{3}} x - a\right ) + 6 \,{\left ({\left (2 \, B a b - 5 \, A b^{2}\right )} x^{5} +{\left (2 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} x + \sqrt{3} a}{3 \, a}\right ) + 3 \, \sqrt{3}{\left ({\left (2 \, B a - 5 \, A b\right )} x^{3} - 3 \, A a\right )} \left (-a^{2} b\right )^{\frac{1}{3}}\right )}}{54 \,{\left (a^{2} b x^{5} + a^{3} x^{2}\right )} \left (-a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/54*sqrt(3)*(sqrt(3)*((2*B*a*b - 5*A*b^2)*x^5 + (2*B*a^2 - 5*A*a*b)*x^2)*log((-
a^2*b)^(2/3)*x^2 + (-a^2*b)^(1/3)*a*x + a^2) - 2*sqrt(3)*((2*B*a*b - 5*A*b^2)*x^
5 + (2*B*a^2 - 5*A*a*b)*x^2)*log((-a^2*b)^(1/3)*x - a) + 6*((2*B*a*b - 5*A*b^2)*
x^5 + (2*B*a^2 - 5*A*a*b)*x^2)*arctan(1/3*(2*sqrt(3)*(-a^2*b)^(1/3)*x + sqrt(3)*
a)/a) + 3*sqrt(3)*((2*B*a - 5*A*b)*x^3 - 3*A*a)*(-a^2*b)^(1/3))/((a^2*b*x^5 + a^
3*x^2)*(-a^2*b)^(1/3))

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Sympy [A]  time = 3.63286, size = 109, normalized size = 0.56 \[ \frac{- 3 A a + x^{3} \left (- 5 A b + 2 B a\right )}{6 a^{3} x^{2} + 6 a^{2} b x^{5}} + \operatorname{RootSum}{\left (729 t^{3} a^{8} b + 125 A^{3} b^{3} - 150 A^{2} B a b^{2} + 60 A B^{2} a^{2} b - 8 B^{3} a^{3}, \left ( t \mapsto t \log{\left (\frac{9 t a^{3}}{- 5 A b + 2 B a} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-3*A*a + x**3*(-5*A*b + 2*B*a))/(6*a**3*x**2 + 6*a**2*b*x**5) + RootSum(729*_t*
*3*a**8*b + 125*A**3*b**3 - 150*A**2*B*a*b**2 + 60*A*B**2*a**2*b - 8*B**3*a**3,
Lambda(_t, _t*log(9*_t*a**3/(-5*A*b + 2*B*a) + x)))

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GIAC/XCAS [A]  time = 0.222569, size = 254, normalized size = 1.3 \[ -\frac{{\left (2 \, B a - 5 \, A b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{3}} + \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{3} b} + \frac{B a x - A b x}{3 \,{\left (b x^{3} + a\right )} a^{2}} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - 5 \, \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{3} b} - \frac{A}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*(2*B*a - 5*A*b)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/a^3 + 1/9*sqrt(3)*(2
*(-a*b^2)^(1/3)*B*a - 5*(-a*b^2)^(1/3)*A*b)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/
3))/(-a/b)^(1/3))/(a^3*b) + 1/3*(B*a*x - A*b*x)/((b*x^3 + a)*a^2) + 1/18*(2*(-a*
b^2)^(1/3)*B*a - 5*(-a*b^2)^(1/3)*A*b)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(
a^3*b) - 1/2*A/(a^2*x^2)