3.1.96 \(\int \frac {x^9 (A+B x^3)}{(a+b x^3)^3} \, dx\)

Optimal. Leaf size=244 \[ \frac {7 \sqrt [3]{a} (2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{13/3}}-\frac {7 \sqrt [3]{a} (2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{13/3}}+\frac {7 \sqrt [3]{a} (2 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} b^{13/3}}+\frac {7 x (2 A b-5 a B)}{9 b^4}-\frac {7 x^4 (2 A b-5 a B)}{36 a b^3}+\frac {x^7 (2 A b-5 a B)}{9 a b^2 \left (a+b x^3\right )}+\frac {x^{10} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

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Rubi [A]  time = 0.17, antiderivative size = 244, 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, 288, 302, 200, 31, 634, 617, 204, 628} \begin {gather*} \frac {7 \sqrt [3]{a} (2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{13/3}}+\frac {x^7 (2 A b-5 a B)}{9 a b^2 \left (a+b x^3\right )}-\frac {7 x^4 (2 A b-5 a B)}{36 a b^3}+\frac {7 x (2 A b-5 a B)}{9 b^4}-\frac {7 \sqrt [3]{a} (2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{13/3}}+\frac {7 \sqrt [3]{a} (2 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} b^{13/3}}+\frac {x^{10} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^9*(A + B*x^3))/(a + b*x^3)^3,x]

[Out]

(7*(2*A*b - 5*a*B)*x)/(9*b^4) - (7*(2*A*b - 5*a*B)*x^4)/(36*a*b^3) + ((A*b - a*B)*x^10)/(6*a*b*(a + b*x^3)^2)
+ ((2*A*b - 5*a*B)*x^7)/(9*a*b^2*(a + b*x^3)) + (7*a^(1/3)*(2*A*b - 5*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqr
t[3]*a^(1/3))])/(9*Sqrt[3]*b^(13/3)) - (7*a^(1/3)*(2*A*b - 5*a*B)*Log[a^(1/3) + b^(1/3)*x])/(27*b^(13/3)) + (7
*a^(1/3)*(2*A*b - 5*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*b^(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 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 288

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

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

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

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Mathematica [A]  time = 0.24, size = 210, normalized size = 0.86 \begin {gather*} \frac {-14 \sqrt [3]{a} (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^2 \sqrt [3]{b} x (a B-A b)}{\left (a+b x^3\right )^2}+\frac {6 a \sqrt [3]{b} x (13 A b-19 a B)}{a+b x^3}+108 \sqrt [3]{b} x (A b-3 a B)+28 \sqrt [3]{a} (5 a B-2 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-28 \sqrt {3} \sqrt [3]{a} (5 a B-2 A b) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+27 b^{4/3} B x^4}{108 b^{13/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^9*(A + B*x^3))/(a + b*x^3)^3,x]

[Out]

(108*b^(1/3)*(A*b - 3*a*B)*x + 27*b^(4/3)*B*x^4 + (18*a^2*b^(1/3)*(-(A*b) + a*B)*x)/(a + b*x^3)^2 + (6*a*b^(1/
3)*(13*A*b - 19*a*B)*x)/(a + b*x^3) - 28*Sqrt[3]*a^(1/3)*(-2*A*b + 5*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/S
qrt[3]] + 28*a^(1/3)*(-2*A*b + 5*a*B)*Log[a^(1/3) + b^(1/3)*x] - 14*a^(1/3)*(-2*A*b + 5*a*B)*Log[a^(2/3) - a^(
1/3)*b^(1/3)*x + b^(2/3)*x^2])/(108*b^(13/3))

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^9 \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(x^9*(A + B*x^3))/(a + b*x^3)^3,x]

[Out]

IntegrateAlgebraic[(x^9*(A + B*x^3))/(a + b*x^3)^3, x]

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fricas [A]  time = 0.60, size = 347, normalized size = 1.42 \begin {gather*} \frac {27 \, B b^{3} x^{10} - 54 \, {\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{7} - 147 \, {\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{4} - 28 \, \sqrt {3} {\left ({\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{6} + 5 \, B a^{3} - 2 \, A a^{2} b + 2 \, {\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{3}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (-\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) + 14 \, {\left ({\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{6} + 5 \, B a^{3} - 2 \, A a^{2} b + 2 \, {\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{3}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ) - 28 \, {\left ({\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{6} + 5 \, B a^{3} - 2 \, A a^{2} b + 2 \, {\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{3}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ) - 84 \, {\left (5 \, B a^{3} - 2 \, A a^{2} b\right )} x}{108 \, {\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.24, size = 234, normalized size = 0.96 \begin {gather*} \frac {7 \, \sqrt {3} {\left (5 \, \left (-a b^{2}\right )^{\frac {1}{3}} B a - 2 \, \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 \, b^{5}} - \frac {7 \, {\left (5 \, B a^{2} - 2 \, A a b\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 b^{4}} + \frac {7 \, {\left (5 \, \left (-a b^{2}\right )^{\frac {1}{3}} B a - 2 \, \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 )}{54 \, b^{5}} - \frac {19 \, B a^{2} b x^{4} - 13 \, A a b^{2} x^{4} + 16 \, B a^{3} x - 10 \, A a^{2} b x}{18 \, {\left (b x^{3} + a\right )}^{2} b^{4}} + \frac {B b^{9} x^{4} - 12 \, B a b^{8} x + 4 \, A b^{9} x}{4 \, b^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="giac")

[Out]

7/27*sqrt(3)*(5*(-a*b^2)^(1/3)*B*a - 2*(-a*b^2)^(1/3)*A*b)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3
))/b^5 - 7/27*(5*B*a^2 - 2*A*a*b)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^4) + 7/54*(5*(-a*b^2)^(1/3)*B*a
- 2*(-a*b^2)^(1/3)*A*b)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^5 - 1/18*(19*B*a^2*b*x^4 - 13*A*a*b^2*x^4
+ 16*B*a^3*x - 10*A*a^2*b*x)/((b*x^3 + a)^2*b^4) + 1/4*(B*b^9*x^4 - 12*B*a*b^8*x + 4*A*b^9*x)/b^12

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maple [A]  time = 0.06, size = 299, normalized size = 1.23 \begin {gather*} \frac {13 A a \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} b^{2}}-\frac {19 B \,a^{2} x^{4}}{18 \left (b \,x^{3}+a \right )^{2} b^{3}}+\frac {B \,x^{4}}{4 b^{3}}+\frac {5 A \,a^{2} x}{9 \left (b \,x^{3}+a \right )^{2} b^{3}}-\frac {8 B \,a^{3} x}{9 \left (b \,x^{3}+a \right )^{2} b^{4}}-\frac {14 \sqrt {3}\, A a \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 {2}{3}} b^{4}}-\frac {14 A a \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}+\frac {7 A a \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 {2}{3}} b^{4}}+\frac {A x}{b^{3}}+\frac {35 \sqrt {3}\, B \,a^{2} \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 {2}{3}} b^{5}}+\frac {35 B \,a^{2} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{5}}-\frac {35 B \,a^{2} \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 {2}{3}} b^{5}}-\frac {3 B a x}{b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9*(B*x^3+A)/(b*x^3+a)^3,x)

[Out]

1/4/b^3*B*x^4+1/b^3*A*x-3/b^4*B*a*x+13/18*a/b^2/(b*x^3+a)^2*A*x^4-19/18*a^2/b^3/(b*x^3+a)^2*B*x^4+5/9*a^2/b^3/
(b*x^3+a)^2*A*x-8/9*a^3/b^4/(b*x^3+a)^2*B*x-14/27*a/b^4*A/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+7/27*a/b^4*A/(a/b)^(2/
3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-14/27*a/b^4*A/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))
+35/27*a^2/b^5*B/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-35/54*a^2/b^5*B/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+3
5/27*a^2/b^5*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 [A]  time = 1.24, size = 223, normalized size = 0.91 \begin {gather*} -\frac {{\left (19 \, B a^{2} b - 13 \, A a b^{2}\right )} x^{4} + 2 \, {\left (8 \, B a^{3} - 5 \, A a^{2} b\right )} x}{18 \, {\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} + \frac {B b x^{4} - 4 \, {\left (3 \, B a - A b\right )} x}{4 \, b^{4}} + \frac {7 \, \sqrt {3} {\left (5 \, B a^{2} - 2 \, A 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 \, b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {7 \, {\left (5 \, B a^{2} - 2 \, A 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 \, b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {7 \, {\left (5 \, B a^{2} - 2 \, A a b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

-1/18*((19*B*a^2*b - 13*A*a*b^2)*x^4 + 2*(8*B*a^3 - 5*A*a^2*b)*x)/(b^6*x^6 + 2*a*b^5*x^3 + a^2*b^4) + 1/4*(B*b
*x^4 - 4*(3*B*a - A*b)*x)/b^4 + 7/27*sqrt(3)*(5*B*a^2 - 2*A*a*b)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^
(1/3))/(b^5*(a/b)^(2/3)) - 7/54*(5*B*a^2 - 2*A*a*b)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^5*(a/b)^(2/3)) +
7/27*(5*B*a^2 - 2*A*a*b)*log(x + (a/b)^(1/3))/(b^5*(a/b)^(2/3))

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mupad [B]  time = 0.32, size = 227, normalized size = 0.93 \begin {gather*} \frac {x^4\,\left (\frac {13\,A\,a\,b^2}{18}-\frac {19\,B\,a^2\,b}{18}\right )-x\,\left (\frac {8\,B\,a^3}{9}-\frac {5\,A\,a^2\,b}{9}\right )}{a^2\,b^4+2\,a\,b^5\,x^3+b^6\,x^6}+x\,\left (\frac {A}{b^3}-\frac {3\,B\,a}{b^4}\right )+\frac {B\,x^4}{4\,b^3}+\frac {7\,{\left (-a\right )}^{1/3}\,\ln \left ({\left (-a\right )}^{4/3}+a\,b^{1/3}\,x\right )\,\left (2\,A\,b-5\,B\,a\right )}{27\,b^{13/3}}-\frac {7\,{\left (-a\right )}^{1/3}\,\ln \left ({\left (-a\right )}^{4/3}-2\,a\,b^{1/3}\,x+\sqrt {3}\,{\left (-a\right )}^{4/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (2\,A\,b-5\,B\,a\right )}{27\,b^{13/3}}+\frac {7\,{\left (-a\right )}^{1/3}\,\ln \left (2\,a\,b^{1/3}\,x-{\left (-a\right )}^{4/3}+\sqrt {3}\,{\left (-a\right )}^{4/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (2\,A\,b-5\,B\,a\right )}{27\,b^{13/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^9*(A + B*x^3))/(a + b*x^3)^3,x)

[Out]

(x^4*((13*A*a*b^2)/18 - (19*B*a^2*b)/18) - x*((8*B*a^3)/9 - (5*A*a^2*b)/9))/(a^2*b^4 + b^6*x^6 + 2*a*b^5*x^3)
+ x*(A/b^3 - (3*B*a)/b^4) + (B*x^4)/(4*b^3) + (7*(-a)^(1/3)*log((-a)^(4/3) + a*b^(1/3)*x)*(2*A*b - 5*B*a))/(27
*b^(13/3)) - (7*(-a)^(1/3)*log((-a)^(4/3) + 3^(1/2)*(-a)^(4/3)*1i - 2*a*b^(1/3)*x)*((3^(1/2)*1i)/2 + 1/2)*(2*A
*b - 5*B*a))/(27*b^(13/3)) + (7*(-a)^(1/3)*log(3^(1/2)*(-a)^(4/3)*1i - (-a)^(4/3) + 2*a*b^(1/3)*x)*((3^(1/2)*1
i)/2 - 1/2)*(2*A*b - 5*B*a))/(27*b^(13/3))

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sympy [A]  time = 4.01, size = 163, normalized size = 0.67 \begin {gather*} \frac {B x^{4}}{4 b^{3}} + x \left (\frac {A}{b^{3}} - \frac {3 B a}{b^{4}}\right ) + \frac {x^{4} \left (13 A a b^{2} - 19 B a^{2} b\right ) + x \left (10 A a^{2} b - 16 B a^{3}\right )}{18 a^{2} b^{4} + 36 a b^{5} x^{3} + 18 b^{6} x^{6}} + \operatorname {RootSum} {\left (19683 t^{3} b^{13} + 2744 A^{3} a b^{3} - 20580 A^{2} B a^{2} b^{2} + 51450 A B^{2} a^{3} b - 42875 B^{3} a^{4}, \left (t \mapsto t \log {\left (\frac {27 t b^{4}}{- 14 A b + 35 B a} + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9*(B*x**3+A)/(b*x**3+a)**3,x)

[Out]

B*x**4/(4*b**3) + x*(A/b**3 - 3*B*a/b**4) + (x**4*(13*A*a*b**2 - 19*B*a**2*b) + x*(10*A*a**2*b - 16*B*a**3))/(
18*a**2*b**4 + 36*a*b**5*x**3 + 18*b**6*x**6) + RootSum(19683*_t**3*b**13 + 2744*A**3*a*b**3 - 20580*A**2*B*a*
*2*b**2 + 51450*A*B**2*a**3*b - 42875*B**3*a**4, Lambda(_t, _t*log(27*_t*b**4/(-14*A*b + 35*B*a) + x)))

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