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

Optimal. Leaf size=171 \[ \frac{(2 a B+A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{4/3} b^{5/3}}-\frac{(2 a B+A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{5/3}}-\frac{(2 a B+A b) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{4/3} b^{5/3}}+\frac{x^2 (A b-a B)}{3 a b \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.343308, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389 \[ \frac{(2 a B+A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{4/3} b^{5/3}}-\frac{(2 a B+A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{5/3}}-\frac{(2 a B+A b) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{4/3} b^{5/3}}+\frac{x^2 (A b-a B)}{3 a b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 33.8584, size = 156, normalized size = 0.91 \[ \frac{x^{2} \left (A b - B a\right )}{3 a b \left (a + b x^{3}\right )} - \frac{\left (A b + 2 B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{4}{3}} b^{\frac{5}{3}}} + \frac{\left (A b + 2 B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{4}{3}} b^{\frac{5}{3}}} - \frac{\sqrt{3} \left (A b + 2 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{4}{3}} b^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.17991, size = 146, normalized size = 0.85 \[ \frac{(2 a B+A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac{6 \sqrt [3]{a} b^{2/3} x^2 (a B-A b)}{a+b x^3}-2 (2 a B+A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} (2 a B+A b) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{18 a^{4/3} b^{5/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-6*a^(1/3)*b^(2/3)*(-(A*b) + a*B)*x^2)/(a + b*x^3) - 2*Sqrt[3]*(A*b + 2*a*B)*A
rcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 2*(A*b + 2*a*B)*Log[a^(1/3) + b^(1/
3)*x] + (A*b + 2*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(4/3
)*b^(5/3))

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Maple [A]  time = 0.012, size = 223, normalized size = 1.3 \[{\frac{ \left ( Ab-Ba \right ){x}^{2}}{3\,ab \left ( b{x}^{3}+a \right ) }}-{\frac{A}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{2\,B}{9\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{A}{18\,ab}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B}{9\,{b}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{\sqrt{3}A}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{2\,\sqrt{3}B}{9\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(B*x^3+A)/(b*x^3+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238198, size = 293, normalized size = 1.71 \[ -\frac{\sqrt{3}{\left (6 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}}{\left (B a - A b\right )} x^{2} + \sqrt{3}{\left ({\left (2 \, B a b + A b^{2}\right )} x^{3} + 2 \, B a^{2} + A a b\right )} \log \left (\left (-a b^{2}\right )^{\frac{1}{3}} b x^{2} - a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) - 2 \, \sqrt{3}{\left ({\left (2 \, B a b + A b^{2}\right )} x^{3} + 2 \, B a^{2} + A a b\right )} \log \left (a b + \left (-a b^{2}\right )^{\frac{2}{3}} x\right ) + 6 \,{\left ({\left (2 \, B a b + A b^{2}\right )} x^{3} + 2 \, B a^{2} + A a b\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right )\right )}}{54 \,{\left (a b^{2} x^{3} + a^{2} b\right )} \left (-a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.02384, size = 117, normalized size = 0.68 \[ - \frac{x^{2} \left (- A b + B a\right )}{3 a^{2} b + 3 a b^{2} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} a^{4} b^{5} + A^{3} b^{3} + 6 A^{2} B a b^{2} + 12 A B^{2} a^{2} b + 8 B^{3} a^{3}, \left ( t \mapsto t \log{\left (\frac{81 t^{2} a^{3} b^{3}}{A^{2} b^{2} + 4 A B a b + 4 B^{2} a^{2}} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**2*(-A*b + B*a)/(3*a**2*b + 3*a*b**2*x**3) + RootSum(729*_t**3*a**4*b**5 + A*
*3*b**3 + 6*A**2*B*a*b**2 + 12*A*B**2*a**2*b + 8*B**3*a**3, Lambda(_t, _t*log(81
*_t**2*a**3*b**3/(A**2*b**2 + 4*A*B*a*b + 4*B**2*a**2) + x)))

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GIAC/XCAS [A]  time = 0.22073, size = 273, normalized size = 1.6 \[ -\frac{{\left (2 \, B a \left (-\frac{a}{b}\right )^{\frac{1}{3}} + A b \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{2} b} - \frac{B a x^{2} - A b x^{2}}{3 \,{\left (b x^{3} + a\right )} a b} - \frac{\sqrt{3}{\left (2 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a + \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{2} b^{3}} + \frac{{\left (2 \, \left (-a b^{2}\right )^{\frac{2}{3}} B a + \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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