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

Optimal. Leaf size=227 \[ -\frac{(7 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{10/3} b^{2/3}}+\frac{2 (7 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} b^{2/3}}+\frac{2 (7 A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{10/3} b^{2/3}}-\frac{2 (7 A b-a B)}{9 a^3 b x}+\frac{7 A b-a B}{18 a^2 b x \left (a+b x^3\right )}+\frac{A b-a B}{6 a b x \left (a+b x^3\right )^2} \]

[Out]

(-2*(7*A*b - a*B))/(9*a^3*b*x) + (A*b - a*B)/(6*a*b*x*(a + b*x^3)^2) + (7*A*b -
a*B)/(18*a^2*b*x*(a + b*x^3)) + (2*(7*A*b - a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/
(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(10/3)*b^(2/3)) + (2*(7*A*b - a*B)*Log[a^(1/3)
+ b^(1/3)*x])/(27*a^(10/3)*b^(2/3)) - ((7*A*b - a*B)*Log[a^(2/3) - a^(1/3)*b^(1/
3)*x + b^(2/3)*x^2])/(27*a^(10/3)*b^(2/3))

_______________________________________________________________________________________

Rubi [A]  time = 0.335766, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45 \[ -\frac{(7 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{10/3} b^{2/3}}+\frac{2 (7 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} b^{2/3}}+\frac{2 (7 A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{10/3} b^{2/3}}-\frac{2 (7 A b-a B)}{9 a^3 b x}+\frac{7 A b-a B}{18 a^2 b x \left (a+b x^3\right )}+\frac{A b-a B}{6 a b x \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(7*A*b - a*B))/(9*a^3*b*x) + (A*b - a*B)/(6*a*b*x*(a + b*x^3)^2) + (7*A*b -
a*B)/(18*a^2*b*x*(a + b*x^3)) + (2*(7*A*b - a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/
(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(10/3)*b^(2/3)) + (2*(7*A*b - a*B)*Log[a^(1/3)
+ b^(1/3)*x])/(27*a^(10/3)*b^(2/3)) - ((7*A*b - a*B)*Log[a^(2/3) - a^(1/3)*b^(1/
3)*x + b^(2/3)*x^2])/(27*a^(10/3)*b^(2/3))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 46.4679, size = 202, normalized size = 0.89 \[ \frac{A b - B a}{6 a b x \left (a + b x^{3}\right )^{2}} + \frac{7 A b - B a}{18 a^{2} b x \left (a + b x^{3}\right )} - \frac{2 \left (7 A b - B a\right )}{9 a^{3} b x} + \frac{2 \left (7 A b - B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{10}{3}} b^{\frac{2}{3}}} - \frac{\left (7 A b - B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{27 a^{\frac{10}{3}} b^{\frac{2}{3}}} + \frac{2 \sqrt{3} \left (7 A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{10}{3}} b^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(A*b - B*a)/(6*a*b*x*(a + b*x**3)**2) + (7*A*b - B*a)/(18*a**2*b*x*(a + b*x**3))
- 2*(7*A*b - B*a)/(9*a**3*b*x) + 2*(7*A*b - B*a)*log(a**(1/3) + b**(1/3)*x)/(27
*a**(10/3)*b**(2/3)) - (7*A*b - B*a)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/
3)*x**2)/(27*a**(10/3)*b**(2/3)) + 2*sqrt(3)*(7*A*b - B*a)*atan(sqrt(3)*(a**(1/3
)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(27*a**(10/3)*b**(2/3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.326996, size = 193, normalized size = 0.85 \[ \frac{\frac{2 (a B-7 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+\frac{9 a^{4/3} x^2 (a B-A b)}{\left (a+b x^3\right )^2}+\frac{4 (7 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}+\frac{4 \sqrt{3} (7 A b-a B) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{b^{2/3}}+\frac{6 \sqrt [3]{a} x^2 (2 a B-5 A b)}{a+b x^3}-\frac{54 \sqrt [3]{a} A}{x}}{54 a^{10/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-54*a^(1/3)*A)/x + (9*a^(4/3)*(-(A*b) + a*B)*x^2)/(a + b*x^3)^2 + (6*a^(1/3)*(
-5*A*b + 2*a*B)*x^2)/(a + b*x^3) + (4*Sqrt[3]*(7*A*b - a*B)*ArcTan[(1 - (2*b^(1/
3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) + (4*(7*A*b - a*B)*Log[a^(1/3) + b^(1/3)*x])/b^
(2/3) + (2*(-7*A*b + a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(2/3
))/(54*a^(10/3))

_______________________________________________________________________________________

Maple [A]  time = 0.018, size = 281, normalized size = 1.2 \[ -{\frac{A}{{a}^{3}x}}-{\frac{5\,A{x}^{5}{b}^{2}}{9\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{2\,bB{x}^{5}}{9\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{13\,A{x}^{2}b}{18\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{7\,B{x}^{2}}{18\,a \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{14\,A}{27\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{7\,A}{27\,{a}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{14\,A\sqrt{3}}{27\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{2\,B}{27\,{a}^{2}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B}{27\,{a}^{2}b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{2\,B\sqrt{3}}{27\,{a}^{2}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/a^3/x-5/9/a^3/(b*x^3+a)^2*A*x^5*b^2+2/9/a^2/(b*x^3+a)^2*B*x^5*b-13/18/a^2/(b*
x^3+a)^2*A*x^2*b+7/18/a/(b*x^3+a)^2*B*x^2+14/27/a^3*A/(a/b)^(1/3)*ln(x+(a/b)^(1/
3))-7/27/a^3*A/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-14/27/a^3*A*3^(1/2)
/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-2/27/a^2*B/b/(a/b)^(1/3)*ln
(x+(a/b)^(1/3))+1/27/a^2*B/b/(a/b)^(1/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+2/27/
a^2*B*3^(1/2)/b/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))

_______________________________________________________________________________________

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)^3*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.244597, size = 444, normalized size = 1.96 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3}{\left ({\left (B a b^{2} - 7 \, A b^{3}\right )} x^{7} + 2 \,{\left (B a^{2} b - 7 \, A a b^{2}\right )} x^{4} +{\left (B a^{3} - 7 \, A a^{2} b\right )} x\right )} \log \left (\left (a b^{2}\right )^{\frac{1}{3}} b x^{2} + a b - \left (a b^{2}\right )^{\frac{2}{3}} x\right ) - 4 \, \sqrt{3}{\left ({\left (B a b^{2} - 7 \, A b^{3}\right )} x^{7} + 2 \,{\left (B a^{2} b - 7 \, A a b^{2}\right )} x^{4} +{\left (B a^{3} - 7 \, A a^{2} b\right )} x\right )} \log \left (a b + \left (a b^{2}\right )^{\frac{2}{3}} x\right ) + 12 \,{\left ({\left (B a b^{2} - 7 \, A b^{3}\right )} x^{7} + 2 \,{\left (B a^{2} b - 7 \, A a b^{2}\right )} x^{4} +{\left (B a^{3} - 7 \, A a^{2} b\right )} x\right )} \arctan \left (-\frac{\sqrt{3} a b - 2 \, \sqrt{3} \left (a b^{2}\right )^{\frac{2}{3}} x}{3 \, a b}\right ) + 3 \, \sqrt{3}{\left (4 \,{\left (B a b - 7 \, A b^{2}\right )} x^{6} + 7 \,{\left (B a^{2} - 7 \, A a b\right )} x^{3} - 18 \, A a^{2}\right )} \left (a b^{2}\right )^{\frac{1}{3}}\right )}}{162 \,{\left (a^{3} b^{2} x^{7} + 2 \, a^{4} b x^{4} + a^{5} x\right )} \left (a b^{2}\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/162*sqrt(3)*(2*sqrt(3)*((B*a*b^2 - 7*A*b^3)*x^7 + 2*(B*a^2*b - 7*A*a*b^2)*x^4
+ (B*a^3 - 7*A*a^2*b)*x)*log((a*b^2)^(1/3)*b*x^2 + a*b - (a*b^2)^(2/3)*x) - 4*sq
rt(3)*((B*a*b^2 - 7*A*b^3)*x^7 + 2*(B*a^2*b - 7*A*a*b^2)*x^4 + (B*a^3 - 7*A*a^2*
b)*x)*log(a*b + (a*b^2)^(2/3)*x) + 12*((B*a*b^2 - 7*A*b^3)*x^7 + 2*(B*a^2*b - 7*
A*a*b^2)*x^4 + (B*a^3 - 7*A*a^2*b)*x)*arctan(-1/3*(sqrt(3)*a*b - 2*sqrt(3)*(a*b^
2)^(2/3)*x)/(a*b)) + 3*sqrt(3)*(4*(B*a*b - 7*A*b^2)*x^6 + 7*(B*a^2 - 7*A*a*b)*x^
3 - 18*A*a^2)*(a*b^2)^(1/3))/((a^3*b^2*x^7 + 2*a^4*b*x^4 + a^5*x)*(a*b^2)^(1/3))

_______________________________________________________________________________________

Sympy [A]  time = 6.30941, size = 162, normalized size = 0.71 \[ \frac{- 18 A a^{2} + x^{6} \left (- 28 A b^{2} + 4 B a b\right ) + x^{3} \left (- 49 A a b + 7 B a^{2}\right )}{18 a^{5} x + 36 a^{4} b x^{4} + 18 a^{3} b^{2} x^{7}} + \operatorname{RootSum}{\left (19683 t^{3} a^{10} b^{2} - 2744 A^{3} b^{3} + 1176 A^{2} B a b^{2} - 168 A B^{2} a^{2} b + 8 B^{3} a^{3}, \left ( t \mapsto t \log{\left (\frac{729 t^{2} a^{7} b}{196 A^{2} b^{2} - 56 A B a b + 4 B^{2} a^{2}} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-18*A*a**2 + x**6*(-28*A*b**2 + 4*B*a*b) + x**3*(-49*A*a*b + 7*B*a**2))/(18*a**
5*x + 36*a**4*b*x**4 + 18*a**3*b**2*x**7) + RootSum(19683*_t**3*a**10*b**2 - 274
4*A**3*b**3 + 1176*A**2*B*a*b**2 - 168*A*B**2*a**2*b + 8*B**3*a**3, Lambda(_t, _
t*log(729*_t**2*a**7*b/(196*A**2*b**2 - 56*A*B*a*b + 4*B**2*a**2) + x)))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.222318, size = 305, normalized size = 1.34 \[ -\frac{2 \,{\left (B a \left (-\frac{a}{b}\right )^{\frac{1}{3}} - 7 \, A b \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{4}} - \frac{A}{a^{3} x} - \frac{2 \, \sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{2}{3}} B a - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{4} b^{2}} + \frac{4 \, B a b x^{5} - 10 \, A b^{2} x^{5} + 7 \, B a^{2} x^{2} - 13 \, A a b x^{2}}{18 \,{\left (b x^{3} + a\right )}^{2} a^{3}} + \frac{{\left (\left (-a b^{2}\right )^{\frac{2}{3}} B a - 7 \, \left (-a b^{2}\right )^{\frac{2}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{27 \, a^{4} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/27*(B*a*(-a/b)^(1/3) - 7*A*b*(-a/b)^(1/3))*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/
3)))/a^4 - A/(a^3*x) - 2/27*sqrt(3)*((-a*b^2)^(2/3)*B*a - 7*(-a*b^2)^(2/3)*A*b)*
arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^4*b^2) + 1/18*(4*B*a*b*
x^5 - 10*A*b^2*x^5 + 7*B*a^2*x^2 - 13*A*a*b*x^2)/((b*x^3 + a)^2*a^3) + 1/27*((-a
*b^2)^(2/3)*B*a - 7*(-a*b^2)^(2/3)*A*b)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/
(a^4*b^2)