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

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

________________________________________________________________________________________

Rubi [A]  time = 0.13, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {457, 325, 200, 31, 634, 617, 204, 628} \begin {gather*} -\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 )} \end {gather*}

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)/(Sqrt[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))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 457

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d
)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*b*e*n*(p + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b
*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] &&
LeQ[-1, m, -(n*(p + 1))]))

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {A+B x^3}{x^6 \left (a+b x^3\right )^2} \, dx &=\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}+\frac {(8 A b-5 a B) \int \frac {1}{x^6 \left (a+b x^3\right )} \, dx}{3 a b}\\ &=-\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 )}-\frac {(8 A b-5 a B) \int \frac {1}{x^3 \left (a+b x^3\right )} \, dx}{3 a^2}\\ &=-\frac {8 A b-5 a B}{15 a^2 b x^5}+\frac {8 A b-5 a B}{6 a^3 x^2}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}+\frac {(b (8 A b-5 a B)) \int \frac {1}{a+b x^3} \, dx}{3 a^3}\\ &=-\frac {8 A b-5 a B}{15 a^2 b x^5}+\frac {8 A b-5 a B}{6 a^3 x^2}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}+\frac {(b (8 A b-5 a B)) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{11/3}}+\frac {(b (8 A b-5 a B)) \int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{11/3}}\\ &=-\frac {8 A b-5 a B}{15 a^2 b x^5}+\frac {8 A b-5 a B}{6 a^3 x^2}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}+\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 {\left (b^{2/3} (8 A b-5 a B)\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{11/3}}+\frac {(b (8 A b-5 a B)) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{10/3}}\\ &=-\frac {8 A b-5 a B}{15 a^2 b x^5}+\frac {8 A b-5 a B}{6 a^3 x^2}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}+\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) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{11/3}}+\frac {\left (b^{2/3} (8 A b-5 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a^{11/3}}\\ &=-\frac {8 A b-5 a B}{15 a^2 b x^5}+\frac {8 A b-5 a B}{6 a^3 x^2}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}-\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 {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) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{11/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.18, size = 183, normalized size = 0.85 \begin {gather*} \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}} \end {gather*}

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))

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x^3}{x^6 \left (a+b x^3\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.53, size = 277, normalized size = 1.29 \begin {gather*} -\frac {15 \, {\left (5 \, B a b - 8 \, A b^{2}\right )} x^{6} + 9 \, {\left (5 \, B a^{2} - 8 \, A a b\right )} x^{3} + 18 \, A a^{2} + 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}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - 5 \, {\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 \, {\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 )}{90 \, {\left (a^{3} b x^{8} + a^{4} x^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/90*(15*(5*B*a*b - 8*A*b^2)*x^6 + 9*(5*B*a^2 - 8*A*a*b)*x^3 + 18*A*a^2 + 10*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)*arctan(1/3*(2*sqrt(3)*a*x*(b^2/a^2)^(2/3) - sqrt(3)*b)/b) - 5*((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*((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)))/(a^3*b*x^8 + a^4*x^5)

________________________________________________________________________________________

giac [A]  time = 0.20, size = 206, normalized size = 0.96 \begin {gather*} -\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}} \log \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 )} \log \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^3+A)/x^6/(b*x^3+a)^2,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)*log(abs(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)*log(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)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 252, normalized size = 1.17 \begin {gather*} \frac {A \,b^{2} x}{3 \left (b \,x^{3}+a \right ) a^{3}}-\frac {B b x}{3 \left (b \,x^{3}+a \right ) a^{2}}+\frac {8 \sqrt {3}\, A b \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}+\frac {8 A b \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}-\frac {4 A b \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}-\frac {5 \sqrt {3}\, B \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}}-\frac {5 B \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}}+\frac {5 B \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}}+\frac {A b}{a^{3} x^{2}}-\frac {B}{2 a^{2} x^{2}}-\frac {A}{5 a^{2} x^{5}} \end {gather*}

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]

1/3*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)*ln(x^2-(a/b)^(1/3)*x+(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-(a/b)^(1/3)*x+(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))-1/5/a^2*A/x^5+1/a^3/x^2*A*b-1/2/a^2/x^2*B

________________________________________________________________________________________

maxima [A]  time = 1.32, size = 186, normalized size = 0.87 \begin {gather*} -\frac {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}}{30 \, {\left (a^{3} b x^{8} + a^{4} x^{5}\right )}} - \frac {\sqrt {3} {\left (5 \, B a - 8 \, 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^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (5 \, B a - 8 \, A b\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (5 \, B a - 8 \, A b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^3+A)/x^6/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

-1/30*(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) - 1/9*sqrt(3)*(5
*B*a - 8*A*b)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^3*(a/b)^(2/3)) + 1/18*(5*B*a - 8*A*b)*log
(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^3*(a/b)^(2/3)) - 1/9*(5*B*a - 8*A*b)*log(x + (a/b)^(1/3))/(a^3*(a/b)^(2
/3))

________________________________________________________________________________________

mupad [B]  time = 2.57, size = 176, normalized size = 0.82 \begin {gather*} \frac {\frac {x^3\,\left (8\,A\,b-5\,B\,a\right )}{10\,a^2}-\frac {A}{5\,a}+\frac {b\,x^6\,\left (8\,A\,b-5\,B\,a\right )}{6\,a^3}}{b\,x^8+a\,x^5}+\frac {b^{2/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (8\,A\,b-5\,B\,a\right )}{9\,a^{11/3}}-\frac {b^{2/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (8\,A\,b-5\,B\,a\right )}{9\,a^{11/3}}+\frac {b^{2/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (8\,A\,b-5\,B\,a\right )}{9\,a^{11/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x^3*(8*A*b - 5*B*a))/(10*a^2) - A/(5*a) + (b*x^6*(8*A*b - 5*B*a))/(6*a^3))/(a*x^5 + b*x^8) + (b^(2/3)*log(b^
(1/3)*x + a^(1/3))*(8*A*b - 5*B*a))/(9*a^(11/3)) - (b^(2/3)*log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((
3^(1/2)*1i)/2 + 1/2)*(8*A*b - 5*B*a))/(9*a^(11/3)) + (b^(2/3)*log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*
((3^(1/2)*1i)/2 - 1/2)*(8*A*b - 5*B*a))/(9*a^(11/3))

________________________________________________________________________________________

sympy [A]  time = 1.84, size = 138, normalized size = 0.64 \begin {gather*} \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}} \end {gather*}

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, Lambd
a(_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))/(30*a**4*x**5 + 30*a**3*b*x**8)

________________________________________________________________________________________