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

Optimal. Leaf size=149 \[ \frac{(A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} \sqrt [3]{b}}-\frac{(A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} \sqrt [3]{b}}+\frac{(A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3} \sqrt [3]{b}}-\frac{A}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.237478, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ \frac{(A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} \sqrt [3]{b}}-\frac{(A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} \sqrt [3]{b}}+\frac{(A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{5/3} \sqrt [3]{b}}-\frac{A}{2 a x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 33.2997, size = 138, normalized size = 0.93 \[ - \frac{A}{2 a x^{2}} - \frac{\left (A b - B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{3 a^{\frac{5}{3}} \sqrt [3]{b}} + \frac{\left (A b - B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{6 a^{\frac{5}{3}} \sqrt [3]{b}} + \frac{\sqrt{3} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{3 a^{\frac{5}{3}} \sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.222422, size = 135, normalized size = 0.91 \[ \frac{\frac{(A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}-\frac{3 a^{2/3} A}{x^2}+\frac{2 (a B-A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac{2 \sqrt{3} (A b-a B) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{b}}}{6 a^{5/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 195, normalized size = 1.3 \[ -{\frac{A}{3\,a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{B}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{A}{6\,a}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{B}{6\,b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{\sqrt{3}A}{3\,a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}B}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{A}{2\,a{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)/x^3/(b*x^3+a),x)

[Out]

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

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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)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.252096, size = 198, normalized size = 1.33 \[ \frac{\sqrt{3}{\left (\sqrt{3}{\left (B a - A b\right )} x^{2} \log \left (\left (-a^{2} b\right )^{\frac{2}{3}} x^{2} + \left (-a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 2 \, \sqrt{3}{\left (B a - A b\right )} x^{2} \log \left (\left (-a^{2} b\right )^{\frac{1}{3}} x - a\right ) + 6 \,{\left (B a - A b\right )} x^{2} \arctan \left (\frac{2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} x + \sqrt{3} a}{3 \, a}\right ) - 3 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} A\right )}}{18 \, \left (-a^{2} b\right )^{\frac{1}{3}} a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.32981, size = 73, normalized size = 0.49 \[ - \frac{A}{2 a x^{2}} + \operatorname{RootSum}{\left (27 t^{3} a^{5} b + A^{3} b^{3} - 3 A^{2} B a b^{2} + 3 A B^{2} a^{2} b - B^{3} a^{3}, \left ( t \mapsto t \log{\left (\frac{3 t a^{2}}{- A b + B a} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(2*a*x**2) + RootSum(27*_t**3*a**5*b + A**3*b**3 - 3*A**2*B*a*b**2 + 3*A*B**2
*a**2*b - B**3*a**3, Lambda(_t, _t*log(3*_t*a**2/(-A*b + B*a) + x)))

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GIAC/XCAS [A]  time = 0.219927, size = 217, normalized size = 1.46 \[ -\frac{{\left (B a - A b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{2}} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} B a - \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, a^{2} b} + \frac{{\left (\left (-a b^{2}\right )^{\frac{1}{3}} B a - \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b} - \frac{A}{2 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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