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

Optimal. Leaf size=246 \[ \frac {7 \sqrt [3]{b} (5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{13/3}}-\frac {7 \sqrt [3]{b} (5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{13/3}}-\frac {7 \sqrt [3]{b} (5 A b-2 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{13/3}}+\frac {7 (5 A b-2 a B)}{9 a^4 x}-\frac {7 (5 A b-2 a B)}{36 a^3 b x^4}+\frac {5 A b-2 a B}{9 a^2 b x^4 \left (a+b x^3\right )}+\frac {A b-a B}{6 a b x^4 \left (a+b x^3\right )^2} \]

[Out]

-7/36*(5*A*b-2*B*a)/a^3/b/x^4+7/9*(5*A*b-2*B*a)/a^4/x+1/6*(A*b-B*a)/a/b/x^4/(b*x^3+a)^2+1/9*(5*A*b-2*B*a)/a^2/
b/x^4/(b*x^3+a)-7/27*b^(1/3)*(5*A*b-2*B*a)*ln(a^(1/3)+b^(1/3)*x)/a^(13/3)+7/54*b^(1/3)*(5*A*b-2*B*a)*ln(a^(2/3
)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(13/3)-7/27*b^(1/3)*(5*A*b-2*B*a)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*
3^(1/2))/a^(13/3)*3^(1/2)

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Rubi [A]  time = 0.16, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {457, 290, 325, 292, 31, 634, 617, 204, 628} \[ \frac {7 \sqrt [3]{b} (5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{13/3}}+\frac {5 A b-2 a B}{9 a^2 b x^4 \left (a+b x^3\right )}-\frac {7 (5 A b-2 a B)}{36 a^3 b x^4}+\frac {7 (5 A b-2 a B)}{9 a^4 x}-\frac {7 \sqrt [3]{b} (5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{13/3}}-\frac {7 \sqrt [3]{b} (5 A b-2 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{13/3}}+\frac {A b-a B}{6 a b x^4 \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*(5*A*b - 2*a*B))/(36*a^3*b*x^4) + (7*(5*A*b - 2*a*B))/(9*a^4*x) + (A*b - a*B)/(6*a*b*x^4*(a + b*x^3)^2) +
(5*A*b - 2*a*B)/(9*a^2*b*x^4*(a + b*x^3)) - (7*b^(1/3)*(5*A*b - 2*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]
*a^(1/3))])/(9*Sqrt[3]*a^(13/3)) - (7*b^(1/3)*(5*A*b - 2*a*B)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(13/3)) + (7*b^(
1/3)*(5*A*b - 2*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(13/3))

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 290

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

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 )^3} \, dx &=\frac {A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}+\frac {(10 A b-4 a B) \int \frac {1}{x^5 \left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac {A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}+\frac {5 A b-2 a B}{9 a^2 b x^4 \left (a+b x^3\right )}+\frac {(7 (5 A b-2 a B)) \int \frac {1}{x^5 \left (a+b x^3\right )} \, dx}{9 a^2 b}\\ &=-\frac {7 (5 A b-2 a B)}{36 a^3 b x^4}+\frac {A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}+\frac {5 A b-2 a B}{9 a^2 b x^4 \left (a+b x^3\right )}-\frac {(7 (5 A b-2 a B)) \int \frac {1}{x^2 \left (a+b x^3\right )} \, dx}{9 a^3}\\ &=-\frac {7 (5 A b-2 a B)}{36 a^3 b x^4}+\frac {7 (5 A b-2 a B)}{9 a^4 x}+\frac {A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}+\frac {5 A b-2 a B}{9 a^2 b x^4 \left (a+b x^3\right )}+\frac {(7 b (5 A b-2 a B)) \int \frac {x}{a+b x^3} \, dx}{9 a^4}\\ &=-\frac {7 (5 A b-2 a B)}{36 a^3 b x^4}+\frac {7 (5 A b-2 a B)}{9 a^4 x}+\frac {A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}+\frac {5 A b-2 a B}{9 a^2 b x^4 \left (a+b x^3\right )}-\frac {\left (7 b^{2/3} (5 A b-2 a B)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{13/3}}+\frac {\left (7 b^{2/3} (5 A b-2 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}{27 a^{13/3}}\\ &=-\frac {7 (5 A b-2 a B)}{36 a^3 b x^4}+\frac {7 (5 A b-2 a B)}{9 a^4 x}+\frac {A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}+\frac {5 A b-2 a B}{9 a^2 b x^4 \left (a+b x^3\right )}-\frac {7 \sqrt [3]{b} (5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{13/3}}+\frac {\left (7 \sqrt [3]{b} (5 A b-2 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}{54 a^{13/3}}+\frac {\left (7 b^{2/3} (5 A b-2 a B)\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^4}\\ &=-\frac {7 (5 A b-2 a B)}{36 a^3 b x^4}+\frac {7 (5 A b-2 a B)}{9 a^4 x}+\frac {A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}+\frac {5 A b-2 a B}{9 a^2 b x^4 \left (a+b x^3\right )}-\frac {7 \sqrt [3]{b} (5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{13/3}}+\frac {7 \sqrt [3]{b} (5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{13/3}}+\frac {\left (7 \sqrt [3]{b} (5 A b-2 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 )}{9 a^{13/3}}\\ &=-\frac {7 (5 A b-2 a B)}{36 a^3 b x^4}+\frac {7 (5 A b-2 a B)}{9 a^4 x}+\frac {A b-a B}{6 a b x^4 \left (a+b x^3\right )^2}+\frac {5 A b-2 a B}{9 a^2 b x^4 \left (a+b x^3\right )}-\frac {7 \sqrt [3]{b} (5 A b-2 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{13/3}}-\frac {7 \sqrt [3]{b} (5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{13/3}}+\frac {7 \sqrt [3]{b} (5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{13/3}}\\ \end {align*}

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Mathematica [A]  time = 0.27, size = 214, normalized size = 0.87 \[ \frac {14 \sqrt [3]{b} (5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac {18 a^{4/3} b x^2 (a B-A b)}{\left (a+b x^3\right )^2}-\frac {27 a^{4/3} A}{x^4}-\frac {12 \sqrt [3]{a} b x^2 (5 a B-8 A b)}{a+b x^3}-\frac {108 \sqrt [3]{a} (a B-3 A b)}{x}+28 \sqrt [3]{b} (2 a B-5 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-28 \sqrt {3} \sqrt [3]{b} (5 A b-2 a B) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{108 a^{13/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-27*a^(4/3)*A)/x^4 - (108*a^(1/3)*(-3*A*b + a*B))/x - (18*a^(4/3)*b*(-(A*b) + a*B)*x^2)/(a + b*x^3)^2 - (12*
a^(1/3)*b*(-8*A*b + 5*a*B)*x^2)/(a + b*x^3) - 28*Sqrt[3]*b^(1/3)*(5*A*b - 2*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(
1/3))/Sqrt[3]] + 28*b^(1/3)*(-5*A*b + 2*a*B)*Log[a^(1/3) + b^(1/3)*x] + 14*b^(1/3)*(5*A*b - 2*a*B)*Log[a^(2/3)
- a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(108*a^(13/3))

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fricas [A]  time = 0.98, size = 366, normalized size = 1.49 \[ -\frac {84 \, {\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{9} + 147 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{6} + 27 \, A a^{3} + 54 \, {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{3} + 28 \, \sqrt {3} {\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{10} + 2 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{7} + {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{4}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - 14 \, {\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{10} + 2 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{7} + {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{4}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (-\frac {b}{a}\right )^{\frac {2}{3}} - a \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right ) + 28 \, {\left ({\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{10} + 2 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{7} + {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{4}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{108 \, {\left (a^{4} b^{2} x^{10} + 2 \, a^{5} b x^{7} + a^{6} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/108*(84*(2*B*a*b^2 - 5*A*b^3)*x^9 + 147*(2*B*a^2*b - 5*A*a*b^2)*x^6 + 27*A*a^3 + 54*(2*B*a^3 - 5*A*a^2*b)*x
^3 + 28*sqrt(3)*((2*B*a*b^2 - 5*A*b^3)*x^10 + 2*(2*B*a^2*b - 5*A*a*b^2)*x^7 + (2*B*a^3 - 5*A*a^2*b)*x^4)*(-b/a
)^(1/3)*arctan(2/3*sqrt(3)*x*(-b/a)^(1/3) + 1/3*sqrt(3)) - 14*((2*B*a*b^2 - 5*A*b^3)*x^10 + 2*(2*B*a^2*b - 5*A
*a*b^2)*x^7 + (2*B*a^3 - 5*A*a^2*b)*x^4)*(-b/a)^(1/3)*log(b*x^2 - a*x*(-b/a)^(2/3) - a*(-b/a)^(1/3)) + 28*((2*
B*a*b^2 - 5*A*b^3)*x^10 + 2*(2*B*a^2*b - 5*A*a*b^2)*x^7 + (2*B*a^3 - 5*A*a^2*b)*x^4)*(-b/a)^(1/3)*log(b*x + a*
(-b/a)^(2/3)))/(a^4*b^2*x^10 + 2*a^5*b*x^7 + a^6*x^4)

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giac [A]  time = 0.20, size = 254, normalized size = 1.03 \[ \frac {7 \, {\left (2 \, B a b \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, A b^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{5}} + \frac {7 \, \sqrt {3} {\left (2 \, \left (-a b^{2}\right )^{\frac {2}{3}} B a - 5 \, \left (-a b^{2}\right )^{\frac {2}{3}} A b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{5} b} - \frac {7 \, {\left (2 \, \left (-a b^{2}\right )^{\frac {2}{3}} B a - 5 \, \left (-a b^{2}\right )^{\frac {2}{3}} A b\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{5} b} - \frac {10 \, B a b^{2} x^{5} - 16 \, A b^{3} x^{5} + 13 \, B a^{2} b x^{2} - 19 \, A a b^{2} x^{2}}{18 \, {\left (b x^{3} + a\right )}^{2} a^{4}} - \frac {4 \, B a x^{3} - 12 \, A b x^{3} + A a}{4 \, a^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/27*(2*B*a*b*(-a/b)^(1/3) - 5*A*b^2*(-a/b)^(1/3))*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^5 + 7/27*sqrt(3)*
(2*(-a*b^2)^(2/3)*B*a - 5*(-a*b^2)^(2/3)*A*b)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^5*b) -
7/54*(2*(-a*b^2)^(2/3)*B*a - 5*(-a*b^2)^(2/3)*A*b)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^5*b) - 1/18*(10
*B*a*b^2*x^5 - 16*A*b^3*x^5 + 13*B*a^2*b*x^2 - 19*A*a*b^2*x^2)/((b*x^3 + a)^2*a^4) - 1/4*(4*B*a*x^3 - 12*A*b*x
^3 + A*a)/(a^4*x^4)

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maple [A]  time = 0.06, size = 299, normalized size = 1.22 \[ \frac {8 A \,b^{3} x^{5}}{9 \left (b \,x^{3}+a \right )^{2} a^{4}}-\frac {5 B \,b^{2} x^{5}}{9 \left (b \,x^{3}+a \right )^{2} a^{3}}+\frac {19 A \,b^{2} x^{2}}{18 \left (b \,x^{3}+a \right )^{2} a^{3}}-\frac {13 B b \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} a^{2}}+\frac {35 \sqrt {3}\, A b \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}}-\frac {35 A b \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}}+\frac {35 A b \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}}-\frac {14 \sqrt {3}\, B \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}+\frac {14 B \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}-\frac {7 B \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}+\frac {3 A b}{a^{4} x}-\frac {B}{a^{3} x}-\frac {A}{4 a^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/9/a^4*b^3/(b*x^3+a)^2*x^5*A-5/9/a^3*b^2/(b*x^3+a)^2*x^5*B+19/18/a^3*b^2/(b*x^3+a)^2*A*x^2-13/18/a^2*b/(b*x^3
+a)^2*B*x^2-35/27/a^4*b*A/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+35/54/a^4*b*A/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(
2/3))+35/27/a^4*b*A*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+14/27/a^3*B/(a/b)^(1/3)*ln(x+(
a/b)^(1/3))-7/27/a^3*B/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-14/27/a^3*B*3^(1/2)/(a/b)^(1/3)*arctan(1/
3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/4/a^3*A/x^4+3/a^4/x*A*b-B/a^3/x

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maxima [A]  time = 1.42, size = 221, normalized size = 0.90 \[ -\frac {28 \, {\left (2 \, B a b^{2} - 5 \, A b^{3}\right )} x^{9} + 49 \, {\left (2 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{6} + 9 \, A a^{3} + 18 \, {\left (2 \, B a^{3} - 5 \, A a^{2} b\right )} x^{3}}{36 \, {\left (a^{4} b^{2} x^{10} + 2 \, a^{5} b x^{7} + a^{6} x^{4}\right )}} - \frac {7 \, \sqrt {3} {\left (2 \, B a - 5 \, A b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {7 \, {\left (2 \, B a - 5 \, A b\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {7 \, {\left (2 \, B a - 5 \, A b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36*(28*(2*B*a*b^2 - 5*A*b^3)*x^9 + 49*(2*B*a^2*b - 5*A*a*b^2)*x^6 + 9*A*a^3 + 18*(2*B*a^3 - 5*A*a^2*b)*x^3)
/(a^4*b^2*x^10 + 2*a^5*b*x^7 + a^6*x^4) - 7/27*sqrt(3)*(2*B*a - 5*A*b)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/
(a/b)^(1/3))/(a^4*(a/b)^(1/3)) - 7/54*(2*B*a - 5*A*b)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^4*(a/b)^(1/3))
+ 7/27*(2*B*a - 5*A*b)*log(x + (a/b)^(1/3))/(a^4*(a/b)^(1/3))

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mupad [B]  time = 2.64, size = 240, normalized size = 0.98 \[ \frac {\frac {x^3\,\left (5\,A\,b-2\,B\,a\right )}{2\,a^2}-\frac {A}{4\,a}+\frac {7\,b^2\,x^9\,\left (5\,A\,b-2\,B\,a\right )}{9\,a^4}+\frac {49\,b\,x^6\,\left (5\,A\,b-2\,B\,a\right )}{36\,a^3}}{a^2\,x^4+2\,a\,b\,x^7+b^2\,x^{10}}+\frac {7\,{\left (-b\right )}^{1/3}\,\ln \left (a^{1/3}\,{\left (-b\right )}^{8/3}+b^3\,x\right )\,\left (5\,A\,b-2\,B\,a\right )}{27\,a^{13/3}}+\frac {7\,{\left (-b\right )}^{1/3}\,\ln \left (a^{1/3}\,{\left (-b\right )}^{8/3}-2\,b^3\,x+\sqrt {3}\,a^{1/3}\,{\left (-b\right )}^{8/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5\,A\,b-2\,B\,a\right )}{27\,a^{13/3}}-\frac {7\,{\left (-b\right )}^{1/3}\,\ln \left (2\,b^3\,x-a^{1/3}\,{\left (-b\right )}^{8/3}+\sqrt {3}\,a^{1/3}\,{\left (-b\right )}^{8/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5\,A\,b-2\,B\,a\right )}{27\,a^{13/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x^3*(5*A*b - 2*B*a))/(2*a^2) - A/(4*a) + (7*b^2*x^9*(5*A*b - 2*B*a))/(9*a^4) + (49*b*x^6*(5*A*b - 2*B*a))/(3
6*a^3))/(a^2*x^4 + b^2*x^10 + 2*a*b*x^7) + (7*(-b)^(1/3)*log(a^(1/3)*(-b)^(8/3) + b^3*x)*(5*A*b - 2*B*a))/(27*
a^(13/3)) + (7*(-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)*(5*A*b - 2*B*a))/(27*a^(13/3)) - (7*(-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)*(5*A*b - 2*B*a))/(27*a^(13/3))

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sympy [A]  time = 1.91, size = 189, normalized size = 0.77 \[ \operatorname {RootSum} {\left (19683 t^{3} a^{13} + 42875 A^{3} b^{4} - 51450 A^{2} B a b^{3} + 20580 A B^{2} a^{2} b^{2} - 2744 B^{3} a^{3} b, \left (t \mapsto t \log {\left (\frac {729 t^{2} a^{9}}{1225 A^{2} b^{3} - 980 A B a b^{2} + 196 B^{2} a^{2} b} + x \right )} \right )\right )} + \frac {- 9 A a^{3} + x^{9} \left (140 A b^{3} - 56 B a b^{2}\right ) + x^{6} \left (245 A a b^{2} - 98 B a^{2} b\right ) + x^{3} \left (90 A a^{2} b - 36 B a^{3}\right )}{36 a^{6} x^{4} + 72 a^{5} b x^{7} + 36 a^{4} b^{2} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(19683*_t**3*a**13 + 42875*A**3*b**4 - 51450*A**2*B*a*b**3 + 20580*A*B**2*a**2*b**2 - 2744*B**3*a**3*b,
Lambda(_t, _t*log(729*_t**2*a**9/(1225*A**2*b**3 - 980*A*B*a*b**2 + 196*B**2*a**2*b) + x))) + (-9*A*a**3 + x*
*9*(140*A*b**3 - 56*B*a*b**2) + x**6*(245*A*a*b**2 - 98*B*a**2*b) + x**3*(90*A*a**2*b - 36*B*a**3))/(36*a**6*x
**4 + 72*a**5*b*x**7 + 36*a**4*b**2*x**10)

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