3.86 \(\int \frac{A+B x^3}{x^6 \left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=215 \[ -\frac{b^{2/3} (8 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{11/3}}+\frac{b^{2/3} (8 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{11/3}}-\frac{b^{2/3} (8 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{11/3}}+\frac{8 A b-5 a B}{6 a^3 x^2}-\frac{8 A b-5 a B}{15 a^2 b x^5}+\frac{A b-a B}{3 a b x^5 \left (a+b x^3\right )} \]

[Out]

-(8*A*b - 5*a*B)/(15*a^2*b*x^5) + (8*A*b - 5*a*B)/(6*a^3*x^2) + (A*b - a*B)/(3*a
*b*x^5*(a + b*x^3)) - (b^(2/3)*(8*A*b - 5*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(S
qrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(11/3)) + (b^(2/3)*(8*A*b - 5*a*B)*Log[a^(1/3) +
b^(1/3)*x])/(9*a^(11/3)) - (b^(2/3)*(8*A*b - 5*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3
)*x + b^(2/3)*x^2])/(18*a^(11/3))

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Rubi [A]  time = 0.349572, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{b^{2/3} (8 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{11/3}}+\frac{b^{2/3} (8 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{11/3}}-\frac{b^{2/3} (8 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{11/3}}+\frac{8 A b-5 a B}{6 a^3 x^2}-\frac{8 A b-5 a B}{15 a^2 b x^5}+\frac{A b-a B}{3 a b x^5 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^3)/(x^6*(a + b*x^3)^2),x]

[Out]

-(8*A*b - 5*a*B)/(15*a^2*b*x^5) + (8*A*b - 5*a*B)/(6*a^3*x^2) + (A*b - a*B)/(3*a
*b*x^5*(a + b*x^3)) - (b^(2/3)*(8*A*b - 5*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(S
qrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(11/3)) + (b^(2/3)*(8*A*b - 5*a*B)*Log[a^(1/3) +
b^(1/3)*x])/(9*a^(11/3)) - (b^(2/3)*(8*A*b - 5*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3
)*x + b^(2/3)*x^2])/(18*a^(11/3))

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Rubi in Sympy [A]  time = 47.1074, size = 201, normalized size = 0.93 \[ \frac{A b - B a}{3 a b x^{5} \left (a + b x^{3}\right )} - \frac{8 A b - 5 B a}{15 a^{2} b x^{5}} + \frac{8 A b - 5 B a}{6 a^{3} x^{2}} + \frac{b^{\frac{2}{3}} \left (8 A b - 5 B a\right ) \log{\left (\sqrt [3]{a} + \sqrt [3]{b} x \right )}}{9 a^{\frac{11}{3}}} - \frac{b^{\frac{2}{3}} \left (8 A b - 5 B a\right ) \log{\left (a^{\frac{2}{3}} - \sqrt [3]{a} \sqrt [3]{b} x + b^{\frac{2}{3}} x^{2} \right )}}{18 a^{\frac{11}{3}}} - \frac{\sqrt{3} b^{\frac{2}{3}} \left (8 A b - 5 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} - \frac{2 \sqrt [3]{b} x}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{11}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**3+A)/x**6/(b*x**3+a)**2,x)

[Out]

(A*b - B*a)/(3*a*b*x**5*(a + b*x**3)) - (8*A*b - 5*B*a)/(15*a**2*b*x**5) + (8*A*
b - 5*B*a)/(6*a**3*x**2) + b**(2/3)*(8*A*b - 5*B*a)*log(a**(1/3) + b**(1/3)*x)/(
9*a**(11/3)) - b**(2/3)*(8*A*b - 5*B*a)*log(a**(2/3) - a**(1/3)*b**(1/3)*x + b**
(2/3)*x**2)/(18*a**(11/3)) - sqrt(3)*b**(2/3)*(8*A*b - 5*B*a)*atan(sqrt(3)*(a**(
1/3)/3 - 2*b**(1/3)*x/3)/a**(1/3))/(9*a**(11/3))

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Mathematica [A]  time = 0.321587, size = 183, normalized size = 0.85 \[ \frac{5 b^{2/3} (5 a B-8 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac{30 a^{2/3} b x (a B-A b)}{a+b x^3}-\frac{45 a^{2/3} (a B-2 A b)}{x^2}-\frac{18 a^{5/3} A}{x^5}+10 b^{2/3} (8 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-10 \sqrt{3} b^{2/3} (8 A b-5 a B) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{90 a^{11/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^3)/(x^6*(a + b*x^3)^2),x]

[Out]

((-18*a^(5/3)*A)/x^5 - (45*a^(2/3)*(-2*A*b + a*B))/x^2 - (30*a^(2/3)*b*(-(A*b) +
a*B)*x)/(a + b*x^3) - 10*Sqrt[3]*b^(2/3)*(8*A*b - 5*a*B)*ArcTan[(1 - (2*b^(1/3)
*x)/a^(1/3))/Sqrt[3]] + 10*b^(2/3)*(8*A*b - 5*a*B)*Log[a^(1/3) + b^(1/3)*x] + 5*
b^(2/3)*(-8*A*b + 5*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(90*a^(
11/3))

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Maple [A]  time = 0.018, size = 252, normalized size = 1.2 \[ -{\frac{A}{5\,{a}^{2}{x}^{5}}}+{\frac{Ab}{{x}^{2}{a}^{3}}}-{\frac{B}{2\,{a}^{2}{x}^{2}}}+{\frac{Ax{b}^{2}}{3\,{a}^{3} \left ( b{x}^{3}+a \right ) }}-{\frac{bBx}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }}+{\frac{8\,Ab}{9\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{4\,Ab}{9\,{a}^{3}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{8\,Ab\sqrt{3}}{9\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,B}{9\,{a}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{5\,B}{18\,{a}^{2}}\ln \left ({x}^{2}-x\sqrt [3]{{\frac{a}{b}}}+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{5\,B\sqrt{3}}{9\,{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)/x^6/(b*x^3+a)^2,x)

[Out]

-1/5*A/a^2/x^5+1/a^3/x^2*A*b-1/2/a^2/x^2*B+1/3*b^2/a^3*x/(b*x^3+a)*A-1/3*b/a^2*x
/(b*x^3+a)*B+8/9*b/a^3*A/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-4/9*b/a^3*A/(a/b)^(2/3)*l
n(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))+8/9*b/a^3*A/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1
/2)*(2/(a/b)^(1/3)*x-1))-5/9/a^2*B/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+5/18/a^2*B/(a/b
)^(2/3)*ln(x^2-x*(a/b)^(1/3)+(a/b)^(2/3))-5/9/a^2*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)^2*x^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236258, size = 402, normalized size = 1.87 \[ \frac{\sqrt{3}{\left (5 \, \sqrt{3}{\left ({\left (5 \, B a b - 8 \, A b^{2}\right )} x^{8} +{\left (5 \, B a^{2} - 8 \, A a b\right )} x^{5}\right )} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{2} - a b x \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 10 \, \sqrt{3}{\left ({\left (5 \, B a b - 8 \, A b^{2}\right )} x^{8} +{\left (5 \, B a^{2} - 8 \, A a b\right )} x^{5}\right )} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) + 30 \,{\left ({\left (5 \, B a b - 8 \, A b^{2}\right )} x^{8} +{\left (5 \, B a^{2} - 8 \, A a b\right )} x^{5}\right )} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3} b x - \sqrt{3} a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}{3 \, a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}}\right ) - 3 \, \sqrt{3}{\left (5 \,{\left (5 \, B a b - 8 \, A b^{2}\right )} x^{6} + 3 \,{\left (5 \, B a^{2} - 8 \, A a b\right )} x^{3} + 6 \, A a^{2}\right )}\right )}}{270 \,{\left (a^{3} b x^{8} + a^{4} x^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)^2*x^6),x, algorithm="fricas")

[Out]

1/270*sqrt(3)*(5*sqrt(3)*((5*B*a*b - 8*A*b^2)*x^8 + (5*B*a^2 - 8*A*a*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)) - 10*sqr
t(3)*((5*B*a*b - 8*A*b^2)*x^8 + (5*B*a^2 - 8*A*a*b)*x^5)*(b^2/a^2)^(1/3)*log(b*x
+ a*(b^2/a^2)^(1/3)) + 30*((5*B*a*b - 8*A*b^2)*x^8 + (5*B*a^2 - 8*A*a*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)*(5*(5*B*a*b - 8*A*b^2)*x^6 + 3*(5*B*a^2 - 8*A*a*b)*x^3 +
6*A*a^2))/(a^3*b*x^8 + a^4*x^5)

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Sympy [A]  time = 6.7474, size = 138, normalized size = 0.64 \[ \operatorname{RootSum}{\left (729 t^{3} a^{11} - 512 A^{3} b^{5} + 960 A^{2} B a b^{4} - 600 A B^{2} a^{2} b^{3} + 125 B^{3} a^{3} b^{2}, \left ( t \mapsto t \log{\left (- \frac{9 t a^{4}}{- 8 A b^{2} + 5 B a b} + x \right )} \right )\right )} - \frac{6 A a^{2} + x^{6} \left (- 40 A b^{2} + 25 B a b\right ) + x^{3} \left (- 24 A a b + 15 B a^{2}\right )}{30 a^{4} x^{5} + 30 a^{3} b x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**3+A)/x**6/(b*x**3+a)**2,x)

[Out]

RootSum(729*_t**3*a**11 - 512*A**3*b**5 + 960*A**2*B*a*b**4 - 600*A*B**2*a**2*b*
*3 + 125*B**3*a**3*b**2, Lambda(_t, _t*log(-9*_t*a**4/(-8*A*b**2 + 5*B*a*b) + x)
)) - (6*A*a**2 + x**6*(-40*A*b**2 + 25*B*a*b) + x**3*(-24*A*a*b + 15*B*a**2))/(3
0*a**4*x**5 + 30*a**3*b*x**8)

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GIAC/XCAS [A]  time = 0.219244, size = 278, normalized size = 1.29 \[ -\frac{\sqrt{3}{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - 8 \, \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{9 \, a^{4}} + \frac{{\left (5 \, B a b - 8 \, A b^{2}\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{9 \, a^{4}} - \frac{{\left (5 \, \left (-a b^{2}\right )^{\frac{1}{3}} B a - 8 \, \left (-a b^{2}\right )^{\frac{1}{3}} A b\right )}{\rm ln}\left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{18 \, a^{4}} - \frac{B a b x - A b^{2} x}{3 \,{\left (b x^{3} + a\right )} a^{3}} - \frac{5 \, B a x^{3} - 10 \, A b x^{3} + 2 \, A a}{10 \, a^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)^2*x^6),x, algorithm="giac")

[Out]

-1/9*sqrt(3)*(5*(-a*b^2)^(1/3)*B*a - 8*(-a*b^2)^(1/3)*A*b)*arctan(1/3*sqrt(3)*(2
*x + (-a/b)^(1/3))/(-a/b)^(1/3))/a^4 + 1/9*(5*B*a*b - 8*A*b^2)*(-a/b)^(1/3)*ln(a
bs(x - (-a/b)^(1/3)))/a^4 - 1/18*(5*(-a*b^2)^(1/3)*B*a - 8*(-a*b^2)^(1/3)*A*b)*l
n(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/a^4 - 1/3*(B*a*b*x - A*b^2*x)/((b*x^3 + a
)*a^3) - 1/10*(5*B*a*x^3 - 10*A*b*x^3 + 2*A*a)/(a^3*x^5)