3.106 \(\int \frac{A+B x^3}{x^6 \left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=246 \[ -\frac{2 b^{2/3} (11 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{14/3}}+\frac{4 b^{2/3} (11 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{14/3}}-\frac{4 b^{2/3} (11 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{14/3}}+\frac{2 (11 A b-5 a B)}{9 a^4 x^2}-\frac{4 (11 A b-5 a B)}{45 a^3 b x^5}+\frac{11 A b-5 a B}{18 a^2 b x^5 \left (a+b x^3\right )}+\frac{A b-a B}{6 a b x^5 \left (a+b x^3\right )^2} \]

[Out]

(-4*(11*A*b - 5*a*B))/(45*a^3*b*x^5) + (2*(11*A*b - 5*a*B))/(9*a^4*x^2) + (A*b -
a*B)/(6*a*b*x^5*(a + b*x^3)^2) + (11*A*b - 5*a*B)/(18*a^2*b*x^5*(a + b*x^3)) -
(4*b^(2/3)*(11*A*b - 5*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(
9*Sqrt[3]*a^(14/3)) + (4*b^(2/3)*(11*A*b - 5*a*B)*Log[a^(1/3) + b^(1/3)*x])/(27*
a^(14/3)) - (2*b^(2/3)*(11*A*b - 5*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3
)*x^2])/(27*a^(14/3))

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Rubi [A]  time = 0.392378, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45 \[ -\frac{2 b^{2/3} (11 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{14/3}}+\frac{4 b^{2/3} (11 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{14/3}}-\frac{4 b^{2/3} (11 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{14/3}}+\frac{2 (11 A b-5 a B)}{9 a^4 x^2}-\frac{4 (11 A b-5 a B)}{45 a^3 b x^5}+\frac{11 A b-5 a B}{18 a^2 b x^5 \left (a+b x^3\right )}+\frac{A b-a B}{6 a b x^5 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^3)/(x^6*(a + b*x^3)^3),x]

[Out]

(-4*(11*A*b - 5*a*B))/(45*a^3*b*x^5) + (2*(11*A*b - 5*a*B))/(9*a^4*x^2) + (A*b -
a*B)/(6*a*b*x^5*(a + b*x^3)^2) + (11*A*b - 5*a*B)/(18*a^2*b*x^5*(a + b*x^3)) -
(4*b^(2/3)*(11*A*b - 5*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(
9*Sqrt[3]*a^(14/3)) + (4*b^(2/3)*(11*A*b - 5*a*B)*Log[a^(1/3) + b^(1/3)*x])/(27*
a^(14/3)) - (2*b^(2/3)*(11*A*b - 5*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3
)*x^2])/(27*a^(14/3))

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Rubi in Sympy [A]  time = 54.2734, size = 235, normalized size = 0.96 \[ \frac{A b - B a}{6 a b x^{5} \left (a + b x^{3}\right )^{2}} + \frac{11 A b - 5 B a}{18 a^{2} b x^{5} \left (a + b x^{3}\right )} - \frac{4 \left (11 A b - 5 B a\right )}{45 a^{3} b x^{5}} + \frac{2 \left (11 A b - 5 B a\right )}{9 a^{4} x^{2}} + \frac{4 b^{\frac{2}{3}} \left (11 A b - 5 B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{27 a^{\frac{14}{3}}} - \frac{2 b^{\frac{2}{3}} \left (11 A b - 5 B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{27 a^{\frac{14}{3}}} - \frac{4 \sqrt{3} b^{\frac{2}{3}} \left (11 A b - 5 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{27 a^{\frac{14}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**3+A)/x**6/(b*x**3+a)**3,x)

[Out]

(A*b - B*a)/(6*a*b*x**5*(a + b*x**3)**2) + (11*A*b - 5*B*a)/(18*a**2*b*x**5*(a +
b*x**3)) - 4*(11*A*b - 5*B*a)/(45*a**3*b*x**5) + 2*(11*A*b - 5*B*a)/(9*a**4*x**
2) + 4*b**(2/3)*(11*A*b - 5*B*a)*log(a**(1/3) + b**(1/3)*x)/(27*a**(14/3)) - 2*b
**(2/3)*(11*A*b - 5*B*a)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**(2/3)*x**2)/(27
*a**(14/3)) - 4*sqrt(3)*b**(2/3)*(11*A*b - 5*B*a)*atan(sqrt(3)*(a**(1/3)/3 - 2*b
**(1/3)*x/3)/a**(1/3))/(27*a**(14/3))

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Mathematica [A]  time = 0.358666, size = 210, normalized size = 0.85 \[ \frac{20 b^{2/3} (5 a B-11 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac{45 a^{5/3} b x (a B-A b)}{\left (a+b x^3\right )^2}-\frac{15 a^{2/3} b x (11 a B-17 A b)}{a+b x^3}-\frac{135 a^{2/3} (a B-3 A b)}{x^2}-\frac{54 a^{5/3} A}{x^5}+40 b^{2/3} (11 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-40 \sqrt{3} b^{2/3} (11 A b-5 a B) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{270 a^{14/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^3)/(x^6*(a + b*x^3)^3),x]

[Out]

((-54*a^(5/3)*A)/x^5 - (135*a^(2/3)*(-3*A*b + a*B))/x^2 - (45*a^(5/3)*b*(-(A*b)
+ a*B)*x)/(a + b*x^3)^2 - (15*a^(2/3)*b*(-17*A*b + 11*a*B)*x)/(a + b*x^3) - 40*S
qrt[3]*b^(2/3)*(11*A*b - 5*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 40
*b^(2/3)*(11*A*b - 5*a*B)*Log[a^(1/3) + b^(1/3)*x] + 20*b^(2/3)*(-11*A*b + 5*a*B
)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(270*a^(14/3))

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Maple [A]  time = 0.023, size = 295, normalized size = 1.2 \[ -{\frac{A}{5\,{a}^{3}{x}^{5}}}+{\frac{3\,Ab}{2\,{a}^{4}{x}^{2}}}-{\frac{B}{2\,{x}^{2}{a}^{3}}}+{\frac{17\,{b}^{3}A{x}^{4}}{18\,{a}^{4} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{11\,{b}^{2}B{x}^{4}}{18\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{10\,Ax{b}^{2}}{9\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}}-{\frac{7\,bBx}{9\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}}+{\frac{44\,Ab}{27\,{a}^{4}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{22\,Ab}{27\,{a}^{4}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{44\,Ab\sqrt{3}}{27\,{a}^{4}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{20\,B}{27\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{10\,B}{27\,{a}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{20\,B\sqrt{3}}{27\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)/x^6/(b*x^3+a)^3,x)

[Out]

-1/5*A/a^3/x^5+3/2/a^4/x^2*A*b-1/2/a^3/x^2*B+17/18/a^4*b^3/(b*x^3+a)^2*A*x^4-11/
18/a^3*b^2/(b*x^3+a)^2*B*x^4+10/9/a^3*b^2/(b*x^3+a)^2*A*x-7/9/a^2*b/(b*x^3+a)^2*
B*x+44/27/a^4*b*A/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-22/27/a^4*b*A/(a/b)^(2/3)*ln(x^2
-x*(a/b)^(1/3)+(a/b)^(2/3))+44/27/a^4*b*A/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)
*(2/(a/b)^(1/3)*x-1))-20/27/a^3*B/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+10/27/a^3*B/(a/b
)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-20/27/a^3*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)^3*x^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237807, size = 547, normalized size = 2.22 \[ \frac{\sqrt{3}{\left (20 \, \sqrt{3}{\left ({\left (5 \, B a b^{2} - 11 \, A b^{3}\right )} x^{11} + 2 \,{\left (5 \, B a^{2} b - 11 \, A a b^{2}\right )} x^{8} +{\left (5 \, B a^{3} - 11 \, A a^{2} b\right )} x^{5}\right )} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{2} - a b x \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 40 \, \sqrt{3}{\left ({\left (5 \, B a b^{2} - 11 \, A b^{3}\right )} x^{11} + 2 \,{\left (5 \, B a^{2} b - 11 \, A a b^{2}\right )} x^{8} +{\left (5 \, B a^{3} - 11 \, A a^{2} b\right )} x^{5}\right )} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 120 \,{\left ({\left (5 \, B a b^{2} - 11 \, A b^{3}\right )} x^{11} + 2 \,{\left (5 \, B a^{2} b - 11 \, A a b^{2}\right )} x^{8} +{\left (5 \, B a^{3} - 11 \, A a^{2} b\right )} x^{5}\right )} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}{3 \, a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) - 3 \, \sqrt{3}{\left (20 \,{\left (5 \, B a b^{2} - 11 \, A b^{3}\right )} x^{9} + 32 \,{\left (5 \, B a^{2} b - 11 \, A a b^{2}\right )} x^{6} + 18 \, A a^{3} + 9 \,{\left (5 \, B a^{3} - 11 \, A a^{2} b\right )} x^{3}\right )}\right )}}{810 \,{\left (a^{4} b^{2} x^{11} + 2 \, a^{5} b x^{8} + a^{6} x^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)^3*x^6),x, algorithm="fricas")

[Out]

1/810*sqrt(3)*(20*sqrt(3)*((5*B*a*b^2 - 11*A*b^3)*x^11 + 2*(5*B*a^2*b - 11*A*a*b
^2)*x^8 + (5*B*a^3 - 11*A*a^2*b)*x^5)*(b^2/a^2)^(1/3)*log(b^2*x^2 - a*b*x*(b^2/a
^2)^(1/3) + a^2*(b^2/a^2)^(2/3)) - 40*sqrt(3)*((5*B*a*b^2 - 11*A*b^3)*x^11 + 2*(
5*B*a^2*b - 11*A*a*b^2)*x^8 + (5*B*a^3 - 11*A*a^2*b)*x^5)*(b^2/a^2)^(1/3)*log(b*
x + a*(b^2/a^2)^(1/3)) + 120*((5*B*a*b^2 - 11*A*b^3)*x^11 + 2*(5*B*a^2*b - 11*A*
a*b^2)*x^8 + (5*B*a^3 - 11*A*a^2*b)*x^5)*(b^2/a^2)^(1/3)*arctan(-1/3*(2*sqrt(3)*
b*x - sqrt(3)*a*(b^2/a^2)^(1/3))/(a*(b^2/a^2)^(1/3))) - 3*sqrt(3)*(20*(5*B*a*b^2
- 11*A*b^3)*x^9 + 32*(5*B*a^2*b - 11*A*a*b^2)*x^6 + 18*A*a^3 + 9*(5*B*a^3 - 11*
A*a^2*b)*x^3))/(a^4*b^2*x^11 + 2*a^5*b*x^8 + a^6*x^5)

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Sympy [A]  time = 27.74, size = 173, normalized size = 0.7 \[ \operatorname{RootSum}{\left (19683 t^{3} a^{14} - 85184 A^{3} b^{5} + 116160 A^{2} B a b^{4} - 52800 A B^{2} a^{2} b^{3} + 8000 B^{3} a^{3} b^{2}, \left ( t \mapsto t \log{\left (- \frac{27 t a^{5}}{- 44 A b^{2} + 20 B a b} + x \right )} \right )\right )} - \frac{18 A a^{3} + x^{9} \left (- 220 A b^{3} + 100 B a b^{2}\right ) + x^{6} \left (- 352 A a b^{2} + 160 B a^{2} b\right ) + x^{3} \left (- 99 A a^{2} b + 45 B a^{3}\right )}{90 a^{6} x^{5} + 180 a^{5} b x^{8} + 90 a^{4} b^{2} x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**3+A)/x**6/(b*x**3+a)**3,x)

[Out]

RootSum(19683*_t**3*a**14 - 85184*A**3*b**5 + 116160*A**2*B*a*b**4 - 52800*A*B**
2*a**2*b**3 + 8000*B**3*a**3*b**2, Lambda(_t, _t*log(-27*_t*a**5/(-44*A*b**2 + 2
0*B*a*b) + x))) - (18*A*a**3 + x**9*(-220*A*b**3 + 100*B*a*b**2) + x**6*(-352*A*
a*b**2 + 160*B*a**2*b) + x**3*(-99*A*a**2*b + 45*B*a**3))/(90*a**6*x**5 + 180*a*
*5*b*x**8 + 90*a**4*b**2*x**11)

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GIAC/XCAS [A]  time = 0.222518, size = 309, normalized size = 1.26 \[ -\frac{4 \, \sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - 11 \, \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{5}} + \frac{4 \,{\left (5 \, B a b - 11 \, A b^{2}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{5}} - \frac{2 \,{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - 11 \, \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{27 \, a^{5}} - \frac{11 \, B a b^{2} x^{4} - 17 \, A b^{3} x^{4} + 14 \, B a^{2} b x - 20 \, A a b^{2} x}{18 \,{\left (b x^{3} + a\right )}^{2} a^{4}} - \frac{5 \, B a x^{3} - 15 \, A b x^{3} + 2 \, A a}{10 \, a^{4} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)^3*x^6),x, algorithm="giac")

[Out]

-4/27*sqrt(3)*(5*(-a*b^2)^(1/3)*B*a - 11*(-a*b^2)^(1/3)*A*b)*arctan(1/3*sqrt(3)*
(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/a^5 + 4/27*(5*B*a*b - 11*A*b^2)*(-a/b)^(1/3)*
ln(abs(x - (-a/b)^(1/3)))/a^5 - 2/27*(5*(-a*b^2)^(1/3)*B*a - 11*(-a*b^2)^(1/3)*A
*b)*ln(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/a^5 - 1/18*(11*B*a*b^2*x^4 - 17*A*b^
3*x^4 + 14*B*a^2*b*x - 20*A*a*b^2*x)/((b*x^3 + a)^2*a^4) - 1/10*(5*B*a*x^3 - 15*
A*b*x^3 + 2*A*a)/(a^4*x^5)