3.85 \(\int \frac {A+B x^3}{x^5 (a+b x^3)^2} \, dx\)

Optimal. Leaf size=215 \[ \frac {\sqrt [3]{b} (7 A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{10/3}}-\frac {\sqrt [3]{b} (7 A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3}}-\frac {\sqrt [3]{b} (7 A b-4 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{10/3}}+\frac {7 A b-4 a B}{3 a^3 x}+\frac {4 a B-7 A b}{12 a^2 b x^4}+\frac {A b-a B}{3 a b x^4 \left (a+b x^3\right )} \]

[Out]

1/12*(-7*A*b+4*B*a)/a^2/b/x^4+1/3*(7*A*b-4*B*a)/a^3/x+1/3*(A*b-B*a)/a/b/x^4/(b*x^3+a)-1/9*b^(1/3)*(7*A*b-4*B*a
)*ln(a^(1/3)+b^(1/3)*x)/a^(10/3)+1/18*b^(1/3)*(7*A*b-4*B*a)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(10/3)
-1/9*b^(1/3)*(7*A*b-4*B*a)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(10/3)*3^(1/2)

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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, 292, 31, 634, 617, 204, 628} \[ \frac {\sqrt [3]{b} (7 A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{10/3}}-\frac {7 A b-4 a B}{12 a^2 b x^4}+\frac {7 A b-4 a B}{3 a^3 x}-\frac {\sqrt [3]{b} (7 A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3}}-\frac {\sqrt [3]{b} (7 A b-4 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{10/3}}+\frac {A b-a B}{3 a b x^4 \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^3)/(x^5*(a + b*x^3)^2),x]

[Out]

-(7*A*b - 4*a*B)/(12*a^2*b*x^4) + (7*A*b - 4*a*B)/(3*a^3*x) + (A*b - a*B)/(3*a*b*x^4*(a + b*x^3)) - (b^(1/3)*(
7*A*b - 4*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(10/3)) - (b^(1/3)*(7*A*b - 4*a
*B)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(10/3)) + (b^(1/3)*(7*A*b - 4*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3
)*x^2])/(18*a^(10/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, 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 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(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 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^5 \left (a+b x^3\right )^2} \, dx &=\frac {A b-a B}{3 a b x^4 \left (a+b x^3\right )}+\frac {(7 A b-4 a B) \int \frac {1}{x^5 \left (a+b x^3\right )} \, dx}{3 a b}\\ &=-\frac {7 A b-4 a B}{12 a^2 b x^4}+\frac {A b-a B}{3 a b x^4 \left (a+b x^3\right )}-\frac {(7 A b-4 a B) \int \frac {1}{x^2 \left (a+b x^3\right )} \, dx}{3 a^2}\\ &=-\frac {7 A b-4 a B}{12 a^2 b x^4}+\frac {7 A b-4 a B}{3 a^3 x}+\frac {A b-a B}{3 a b x^4 \left (a+b x^3\right )}+\frac {(b (7 A b-4 a B)) \int \frac {x}{a+b x^3} \, dx}{3 a^3}\\ &=-\frac {7 A b-4 a B}{12 a^2 b x^4}+\frac {7 A b-4 a B}{3 a^3 x}+\frac {A b-a B}{3 a b x^4 \left (a+b x^3\right )}-\frac {\left (b^{2/3} (7 A b-4 a B)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{10/3}}+\frac {\left (b^{2/3} (7 A b-4 a B)\right ) \int \frac {\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^{10/3}}\\ &=-\frac {7 A b-4 a B}{12 a^2 b x^4}+\frac {7 A b-4 a B}{3 a^3 x}+\frac {A b-a B}{3 a b x^4 \left (a+b x^3\right )}-\frac {\sqrt [3]{b} (7 A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3}}+\frac {\left (\sqrt [3]{b} (7 A b-4 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^{10/3}}+\frac {\left (b^{2/3} (7 A b-4 a B)\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^3}\\ &=-\frac {7 A b-4 a B}{12 a^2 b x^4}+\frac {7 A b-4 a B}{3 a^3 x}+\frac {A b-a B}{3 a b x^4 \left (a+b x^3\right )}-\frac {\sqrt [3]{b} (7 A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3}}+\frac {\sqrt [3]{b} (7 A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{10/3}}+\frac {\left (\sqrt [3]{b} (7 A b-4 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^{10/3}}\\ &=-\frac {7 A b-4 a B}{12 a^2 b x^4}+\frac {7 A b-4 a B}{3 a^3 x}+\frac {A b-a B}{3 a b x^4 \left (a+b x^3\right )}-\frac {\sqrt [3]{b} (7 A b-4 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{10/3}}-\frac {\sqrt [3]{b} (7 A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3}}+\frac {\sqrt [3]{b} (7 A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{10/3}}\\ \end {align*}

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Mathematica [A]  time = 0.24, size = 185, normalized size = 0.86 \[ \frac {2 \sqrt [3]{b} (7 A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac {9 a^{4/3} A}{x^4}-\frac {12 \sqrt [3]{a} b x^2 (a B-A b)}{a+b x^3}-\frac {36 \sqrt [3]{a} (a B-2 A b)}{x}+4 \sqrt [3]{b} (4 a B-7 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-4 \sqrt {3} \sqrt [3]{b} (7 A b-4 a B) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{36 a^{10/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^3)/(x^5*(a + b*x^3)^2),x]

[Out]

((-9*a^(4/3)*A)/x^4 - (36*a^(1/3)*(-2*A*b + a*B))/x - (12*a^(1/3)*b*(-(A*b) + a*B)*x^2)/(a + b*x^3) - 4*Sqrt[3
]*b^(1/3)*(7*A*b - 4*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 4*b^(1/3)*(-7*A*b + 4*a*B)*Log[a^(1/3)
+ b^(1/3)*x] + 2*b^(1/3)*(7*A*b - 4*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(36*a^(10/3))

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fricas [A]  time = 0.88, size = 259, normalized size = 1.20 \[ -\frac {12 \, {\left (4 \, B a b - 7 \, A b^{2}\right )} x^{6} + 9 \, {\left (4 \, B a^{2} - 7 \, A a b\right )} x^{3} + 9 \, A a^{2} + 4 \, \sqrt {3} {\left ({\left (4 \, B a b - 7 \, A b^{2}\right )} x^{7} + {\left (4 \, B a^{2} - 7 \, A a b\right )} x^{4}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - 2 \, {\left ({\left (4 \, B a b - 7 \, A b^{2}\right )} x^{7} + {\left (4 \, B a^{2} - 7 \, A a b\right )} x^{4}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (-\frac {b}{a}\right )^{\frac {2}{3}} - a \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right ) + 4 \, {\left ({\left (4 \, B a b - 7 \, A b^{2}\right )} x^{7} + {\left (4 \, B a^{2} - 7 \, A a b\right )} x^{4}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{36 \, {\left (a^{3} b x^{7} + a^{4} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^3+A)/x^5/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

-1/36*(12*(4*B*a*b - 7*A*b^2)*x^6 + 9*(4*B*a^2 - 7*A*a*b)*x^3 + 9*A*a^2 + 4*sqrt(3)*((4*B*a*b - 7*A*b^2)*x^7 +
(4*B*a^2 - 7*A*a*b)*x^4)*(-b/a)^(1/3)*arctan(2/3*sqrt(3)*x*(-b/a)^(1/3) + 1/3*sqrt(3)) - 2*((4*B*a*b - 7*A*b^
2)*x^7 + (4*B*a^2 - 7*A*a*b)*x^4)*(-b/a)^(1/3)*log(b*x^2 - a*x*(-b/a)^(2/3) - a*(-b/a)^(1/3)) + 4*((4*B*a*b -
7*A*b^2)*x^7 + (4*B*a^2 - 7*A*a*b)*x^4)*(-b/a)^(1/3)*log(b*x + a*(-b/a)^(2/3)))/(a^3*b*x^7 + a^4*x^4)

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giac [A]  time = 0.22, size = 231, normalized size = 1.07 \[ \frac {{\left (4 \, B a b \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 7 \, A b^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{4}} + \frac {\sqrt {3} {\left (4 \, \left (-a b^{2}\right )^{\frac {2}{3}} B a - 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} A b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{4} b} - \frac {B a b x^{2} - A b^{2} x^{2}}{3 \, {\left (b x^{3} + a\right )} a^{3}} - \frac {{\left (4 \, \left (-a b^{2}\right )^{\frac {2}{3}} B a - 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} A b\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{4} b} - \frac {4 \, B a x^{3} - 8 \, A b x^{3} + A a}{4 \, a^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^3+A)/x^5/(b*x^3+a)^2,x, algorithm="giac")

[Out]

1/9*(4*B*a*b*(-a/b)^(1/3) - 7*A*b^2*(-a/b)^(1/3))*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^4 + 1/9*sqrt(3)*(4
*(-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) - 1/
3*(B*a*b*x^2 - A*b^2*x^2)/((b*x^3 + a)*a^3) - 1/18*(4*(-a*b^2)^(2/3)*B*a - 7*(-a*b^2)^(2/3)*A*b)*log(x^2 + x*(
-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b) - 1/4*(4*B*a*x^3 - 8*A*b*x^3 + A*a)/(a^3*x^4)

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maple [A]  time = 0.06, size = 257, normalized size = 1.20 \[ \frac {A \,b^{2} x^{2}}{3 \left (b \,x^{3}+a \right ) a^{3}}-\frac {B b \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{2}}+\frac {7 \sqrt {3}\, A b \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}-\frac {7 A b \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}+\frac {7 A b \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}-\frac {4 \sqrt {3}\, B \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}+\frac {4 B \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}-\frac {2 B \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}+\frac {2 A b}{a^{3} x}-\frac {B}{a^{2} x}-\frac {A}{4 a^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^3+A)/x^5/(b*x^3+a)^2,x)

[Out]

1/3*b^2/a^3*x^2/(b*x^3+a)*A-1/3*b/a^2*x^2/(b*x^3+a)*B-7/9*b/a^3*A/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+7/18*b/a^3*A/(
a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+7/9*b/a^3*A*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)
*x-1))+4/9/a^2*B/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-2/9/a^2*B/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-4/9/a^2
*B*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/4/a^2*A/x^4+2*A/a^3*b/x-B/a^2/x

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maxima [A]  time = 1.42, size = 186, normalized size = 0.87 \[ -\frac {4 \, {\left (4 \, B a b - 7 \, A b^{2}\right )} x^{6} + 3 \, {\left (4 \, B a^{2} - 7 \, A a b\right )} x^{3} + 3 \, A a^{2}}{12 \, {\left (a^{3} b x^{7} + a^{4} x^{4}\right )}} - \frac {\sqrt {3} {\left (4 \, B a - 7 \, A b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (4 \, B a - 7 \, A b\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (4 \, B a - 7 \, A b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^3+A)/x^5/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

-1/12*(4*(4*B*a*b - 7*A*b^2)*x^6 + 3*(4*B*a^2 - 7*A*a*b)*x^3 + 3*A*a^2)/(a^3*b*x^7 + a^4*x^4) - 1/9*sqrt(3)*(4
*B*a - 7*A*b)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^3*(a/b)^(1/3)) - 1/18*(4*B*a - 7*A*b)*log
(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^3*(a/b)^(1/3)) + 1/9*(4*B*a - 7*A*b)*log(x + (a/b)^(1/3))/(a^3*(a/b)^(1
/3))

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mupad [B]  time = 2.62, size = 209, normalized size = 0.97 \[ \frac {\frac {x^3\,\left (7\,A\,b-4\,B\,a\right )}{4\,a^2}-\frac {A}{4\,a}+\frac {b\,x^6\,\left (7\,A\,b-4\,B\,a\right )}{3\,a^3}}{b\,x^7+a\,x^4}+\frac {{\left (-b\right )}^{1/3}\,\ln \left (a^{1/3}\,{\left (-b\right )}^{8/3}+b^3\,x\right )\,\left (7\,A\,b-4\,B\,a\right )}{9\,a^{10/3}}+\frac {{\left (-b\right )}^{1/3}\,\ln \left (a^{1/3}\,{\left (-b\right )}^{8/3}-2\,b^3\,x+\sqrt {3}\,a^{1/3}\,{\left (-b\right )}^{8/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (7\,A\,b-4\,B\,a\right )}{9\,a^{10/3}}-\frac {{\left (-b\right )}^{1/3}\,\ln \left (2\,b^3\,x-a^{1/3}\,{\left (-b\right )}^{8/3}+\sqrt {3}\,a^{1/3}\,{\left (-b\right )}^{8/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (7\,A\,b-4\,B\,a\right )}{9\,a^{10/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^3)/(x^5*(a + b*x^3)^2),x)

[Out]

((x^3*(7*A*b - 4*B*a))/(4*a^2) - A/(4*a) + (b*x^6*(7*A*b - 4*B*a))/(3*a^3))/(a*x^4 + b*x^7) + ((-b)^(1/3)*log(
a^(1/3)*(-b)^(8/3) + b^3*x)*(7*A*b - 4*B*a))/(9*a^(10/3)) + ((-b)^(1/3)*log(a^(1/3)*(-b)^(8/3) - 2*b^3*x + 3^(
1/2)*a^(1/3)*(-b)^(8/3)*1i)*((3^(1/2)*1i)/2 - 1/2)*(7*A*b - 4*B*a))/(9*a^(10/3)) - ((-b)^(1/3)*log(2*b^3*x - a
^(1/3)*(-b)^(8/3) + 3^(1/2)*a^(1/3)*(-b)^(8/3)*1i)*((3^(1/2)*1i)/2 + 1/2)*(7*A*b - 4*B*a))/(9*a^(10/3))

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sympy [A]  time = 2.32, size = 153, normalized size = 0.71 \[ \operatorname {RootSum} {\left (729 t^{3} a^{10} + 343 A^{3} b^{4} - 588 A^{2} B a b^{3} + 336 A B^{2} a^{2} b^{2} - 64 B^{3} a^{3} b, \left (t \mapsto t \log {\left (\frac {81 t^{2} a^{7}}{49 A^{2} b^{3} - 56 A B a b^{2} + 16 B^{2} a^{2} b} + x \right )} \right )\right )} + \frac {- 3 A a^{2} + x^{6} \left (28 A b^{2} - 16 B a b\right ) + x^{3} \left (21 A a b - 12 B a^{2}\right )}{12 a^{4} x^{4} + 12 a^{3} b x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**3+A)/x**5/(b*x**3+a)**2,x)

[Out]

RootSum(729*_t**3*a**10 + 343*A**3*b**4 - 588*A**2*B*a*b**3 + 336*A*B**2*a**2*b**2 - 64*B**3*a**3*b, Lambda(_t
, _t*log(81*_t**2*a**7/(49*A**2*b**3 - 56*A*B*a*b**2 + 16*B**2*a**2*b) + x))) + (-3*A*a**2 + x**6*(28*A*b**2 -
16*B*a*b) + x**3*(21*A*a*b - 12*B*a**2))/(12*a**4*x**4 + 12*a**3*b*x**7)

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