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

Optimal. Leaf size=227 \[ \frac{5 (4 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} \sqrt [3]{b}}-\frac{5 (4 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}+\frac{5 (4 A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{11/3} \sqrt [3]{b}}-\frac{5 (4 A b-a B)}{18 a^3 b x^2}+\frac{4 A b-a B}{9 a^2 b x^2 \left (a+b x^3\right )}+\frac{A b-a B}{6 a b x^2 \left (a+b x^3\right )^2} \]

[Out]

(-5*(4*A*b - a*B))/(18*a^3*b*x^2) + (A*b - a*B)/(6*a*b*x^2*(a + b*x^3)^2) + (4*A
*b - a*B)/(9*a^2*b*x^2*(a + b*x^3)) + (5*(4*A*b - a*B)*ArcTan[(a^(1/3) - 2*b^(1/
3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(11/3)*b^(1/3)) - (5*(4*A*b - a*B)*Log[a^
(1/3) + b^(1/3)*x])/(27*a^(11/3)*b^(1/3)) + (5*(4*A*b - a*B)*Log[a^(2/3) - a^(1/
3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(11/3)*b^(1/3))

_______________________________________________________________________________________

Rubi [A]  time = 0.362378, 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 \[ \frac{5 (4 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} \sqrt [3]{b}}-\frac{5 (4 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}+\frac{5 (4 A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{11/3} \sqrt [3]{b}}-\frac{5 (4 A b-a B)}{18 a^3 b x^2}+\frac{4 A b-a B}{9 a^2 b x^2 \left (a+b x^3\right )}+\frac{A b-a B}{6 a b x^2 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(-5*(4*A*b - a*B))/(18*a^3*b*x^2) + (A*b - a*B)/(6*a*b*x^2*(a + b*x^3)^2) + (4*A
*b - a*B)/(9*a^2*b*x^2*(a + b*x^3)) + (5*(4*A*b - a*B)*ArcTan[(a^(1/3) - 2*b^(1/
3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(11/3)*b^(1/3)) - (5*(4*A*b - a*B)*Log[a^
(1/3) + b^(1/3)*x])/(27*a^(11/3)*b^(1/3)) + (5*(4*A*b - a*B)*Log[a^(2/3) - a^(1/
3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(11/3)*b^(1/3))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 47.4783, size = 209, normalized size = 0.92 \[ \frac{A b - B a}{6 a b x^{2} \left (a + b x^{3}\right )^{2}} + \frac{4 A b - B a}{9 a^{2} b x^{2} \left (a + b x^{3}\right )} - \frac{5 \left (4 A b - B a\right )}{18 a^{3} b x^{2}} - \frac{5 \left (4 A b - B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{11}{3}} \sqrt [3]{b}} + \frac{5 \left (4 A b - B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{54 a^{\frac{11}{3}} \sqrt [3]{b}} + \frac{5 \sqrt{3} \left (4 A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{11}{3}} \sqrt [3]{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(A*b - B*a)/(6*a*b*x**2*(a + b*x**3)**2) + (4*A*b - B*a)/(9*a**2*b*x**2*(a + b*x
**3)) - 5*(4*A*b - B*a)/(18*a**3*b*x**2) - 5*(4*A*b - B*a)*log(a**(1/3) + b**(1/
3)*x)/(27*a**(11/3)*b**(1/3)) + 5*(4*A*b - B*a)*log(a**(2/3) - a**(1/3)*b**(1/3)
*x + b**(2/3)*x**2)/(54*a**(11/3)*b**(1/3)) + 5*sqrt(3)*(4*A*b - B*a)*atan(sqrt(
3)*(a**(1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(27*a**(11/3)*b**(1/3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.350573, size = 189, normalized size = 0.83 \[ \frac{\frac{5 (4 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}+\frac{9 a^{5/3} x (a B-A b)}{\left (a+b x^3\right )^2}+\frac{3 a^{2/3} x (5 a B-11 A b)}{a+b x^3}-\frac{27 a^{2/3} A}{x^2}+\frac{10 (a B-4 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac{10 \sqrt{3} (4 A b-a B) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt [3]{b}}}{54 a^{11/3}} \]

Antiderivative was successfully verified.

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

[Out]

((-27*a^(2/3)*A)/x^2 + (9*a^(5/3)*(-(A*b) + a*B)*x)/(a + b*x^3)^2 + (3*a^(2/3)*(
-11*A*b + 5*a*B)*x)/(a + b*x^3) + (10*Sqrt[3]*(4*A*b - a*B)*ArcTan[(1 - (2*b^(1/
3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + (10*(-4*A*b + a*B)*Log[a^(1/3) + b^(1/3)*x])/
b^(1/3) + (5*(4*A*b - a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/
3))/(54*a^(11/3))

_______________________________________________________________________________________

Maple [A]  time = 0.019, size = 277, normalized size = 1.2 \[ -{\frac{A}{2\,{x}^{2}{a}^{3}}}-{\frac{11\,A{x}^{4}{b}^{2}}{18\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{5\,bB{x}^{4}}{18\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{7\,Axb}{9\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{4\,Bx}{9\,a \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{20\,A}{27\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{10\,A}{27\,{a}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{20\,A\sqrt{3}}{27\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,B}{27\,{a}^{2}b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,B}{54\,{a}^{2}b}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,B\sqrt{3}}{27\,{a}^{2}b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*A/a^3/x^2-11/18/a^3/(b*x^3+a)^2*A*x^4*b^2+5/18/a^2/(b*x^3+a)^2*B*x^4*b-7/9/
a^2/(b*x^3+a)^2*A*x*b+4/9/a/(b*x^3+a)^2*B*x-20/27/a^3*A/(a/b)^(2/3)*ln(x+(a/b)^(
1/3))+10/27/a^3*A/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-20/27/a^3*A/(a/b
)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+5/27/a^2*B/b/(a/b)^(2/3)
*ln(x+(a/b)^(1/3))-5/54/a^2*B/b/(a/b)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+5/
27/a^2*B/b/(a/b)^(2/3)*3^(1/2)*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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.240154, size = 456, normalized size = 2.01 \[ \frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left ({\left (B a b^{2} - 4 \, A b^{3}\right )} x^{8} + 2 \,{\left (B a^{2} b - 4 \, A a b^{2}\right )} x^{5} +{\left (B a^{3} - 4 \, A a^{2} b\right )} x^{2}\right )} \log \left (\left (-a^{2} b\right )^{\frac{2}{3}} x^{2} + \left (-a^{2} b\right )^{\frac{1}{3}} a x + a^{2}\right ) - 10 \, \sqrt{3}{\left ({\left (B a b^{2} - 4 \, A b^{3}\right )} x^{8} + 2 \,{\left (B a^{2} b - 4 \, A a b^{2}\right )} x^{5} +{\left (B a^{3} - 4 \, A a^{2} b\right )} x^{2}\right )} \log \left (\left (-a^{2} b\right )^{\frac{1}{3}} x - a\right ) + 30 \,{\left ({\left (B a b^{2} - 4 \, A b^{3}\right )} x^{8} + 2 \,{\left (B a^{2} b - 4 \, A a b^{2}\right )} x^{5} +{\left (B a^{3} - 4 \, A a^{2} b\right )} x^{2}\right )} \arctan \left (\frac{2 \, \sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} x + \sqrt{3} a}{3 \, a}\right ) + 3 \, \sqrt{3}{\left (5 \,{\left (B a b - 4 \, A b^{2}\right )} x^{6} + 8 \,{\left (B a^{2} - 4 \, A a b\right )} x^{3} - 9 \, A a^{2}\right )} \left (-a^{2} b\right )^{\frac{1}{3}}\right )}}{162 \,{\left (a^{3} b^{2} x^{8} + 2 \, a^{4} b x^{5} + a^{5} x^{2}\right )} \left (-a^{2} b\right )^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/162*sqrt(3)*(5*sqrt(3)*((B*a*b^2 - 4*A*b^3)*x^8 + 2*(B*a^2*b - 4*A*a*b^2)*x^5
+ (B*a^3 - 4*A*a^2*b)*x^2)*log((-a^2*b)^(2/3)*x^2 + (-a^2*b)^(1/3)*a*x + a^2) -
10*sqrt(3)*((B*a*b^2 - 4*A*b^3)*x^8 + 2*(B*a^2*b - 4*A*a*b^2)*x^5 + (B*a^3 - 4*A
*a^2*b)*x^2)*log((-a^2*b)^(1/3)*x - a) + 30*((B*a*b^2 - 4*A*b^3)*x^8 + 2*(B*a^2*
b - 4*A*a*b^2)*x^5 + (B*a^3 - 4*A*a^2*b)*x^2)*arctan(1/3*(2*sqrt(3)*(-a^2*b)^(1/
3)*x + sqrt(3)*a)/a) + 3*sqrt(3)*(5*(B*a*b - 4*A*b^2)*x^6 + 8*(B*a^2 - 4*A*a*b)*
x^3 - 9*A*a^2)*(-a^2*b)^(1/3))/((a^3*b^2*x^8 + 2*a^4*b*x^5 + a^5*x^2)*(-a^2*b)^(
1/3))

_______________________________________________________________________________________

Sympy [A]  time = 8.22282, size = 143, normalized size = 0.63 \[ \frac{- 9 A a^{2} + x^{6} \left (- 20 A b^{2} + 5 B a b\right ) + x^{3} \left (- 32 A a b + 8 B a^{2}\right )}{18 a^{5} x^{2} + 36 a^{4} b x^{5} + 18 a^{3} b^{2} x^{8}} + \operatorname{RootSum}{\left (19683 t^{3} a^{11} b + 8000 A^{3} b^{3} - 6000 A^{2} B a b^{2} + 1500 A B^{2} a^{2} b - 125 B^{3} a^{3}, \left ( t \mapsto t \log{\left (\frac{27 t a^{4}}{- 20 A b + 5 B a} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-9*A*a**2 + x**6*(-20*A*b**2 + 5*B*a*b) + x**3*(-32*A*a*b + 8*B*a**2))/(18*a**5
*x**2 + 36*a**4*b*x**5 + 18*a**3*b**2*x**8) + RootSum(19683*_t**3*a**11*b + 8000
*A**3*b**3 - 6000*A**2*B*a*b**2 + 1500*A*B**2*a**2*b - 125*B**3*a**3, Lambda(_t,
_t*log(27*_t*a**4/(-20*A*b + 5*B*a) + x)))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.219063, size = 282, normalized size = 1.24 \[ -\frac{5 \,{\left (B a - 4 \, A b\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{4}} + \frac{5 \, \sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} B a - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{4} b} + \frac{5 \,{\left (\left (-a b^{2}\right )^{\frac{1}{3}} B a - 4 \, \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{4} b} + \frac{5 \, B a b x^{6} - 20 \, A b^{2} x^{6} + 8 \, B a^{2} x^{3} - 32 \, A a b x^{3} - 9 \, A a^{2}}{18 \,{\left (b x^{4} + a x\right )}^{2} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/27*(B*a - 4*A*b)*(-a/b)^(1/3)*ln(abs(x - (-a/b)^(1/3)))/a^4 + 5/27*sqrt(3)*((
-a*b^2)^(1/3)*B*a - 4*(-a*b^2)^(1/3)*A*b)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3)
)/(-a/b)^(1/3))/(a^4*b) + 5/54*((-a*b^2)^(1/3)*B*a - 4*(-a*b^2)^(1/3)*A*b)*ln(x^
2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b) + 1/18*(5*B*a*b*x^6 - 20*A*b^2*x^6 +
8*B*a^2*x^3 - 32*A*a*b*x^3 - 9*A*a^2)/((b*x^4 + a*x)^2*a^3)