3.71 \(\int \frac {x^9 (A+B x^3)}{(a+b x^3)^2} \, dx\)

Optimal. Leaf size=233 \[ -\frac {a^{4/3} (7 A b-10 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{13/3}}+\frac {a^{4/3} (7 A b-10 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{13/3}}-\frac {a^{4/3} (7 A b-10 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} b^{13/3}}-\frac {a x (7 A b-10 a B)}{3 b^4}+\frac {x^4 (7 A b-10 a B)}{12 b^3}-\frac {x^7 (7 A b-10 a B)}{21 a b^2}+\frac {x^{10} (A b-a B)}{3 a b \left (a+b x^3\right )} \]

[Out]

-1/3*a*(7*A*b-10*B*a)*x/b^4+1/12*(7*A*b-10*B*a)*x^4/b^3-1/21*(7*A*b-10*B*a)*x^7/a/b^2+1/3*(A*b-B*a)*x^10/a/b/(
b*x^3+a)+1/9*a^(4/3)*(7*A*b-10*B*a)*ln(a^(1/3)+b^(1/3)*x)/b^(13/3)-1/18*a^(4/3)*(7*A*b-10*B*a)*ln(a^(2/3)-a^(1
/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(13/3)-1/9*a^(4/3)*(7*A*b-10*B*a)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2
))/b^(13/3)*3^(1/2)

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Rubi [A]  time = 0.14, antiderivative size = 233, 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, 302, 200, 31, 634, 617, 204, 628} \[ -\frac {a^{4/3} (7 A b-10 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{13/3}}+\frac {a^{4/3} (7 A b-10 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{13/3}}-\frac {a^{4/3} (7 A b-10 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} b^{13/3}}-\frac {x^7 (7 A b-10 a B)}{21 a b^2}+\frac {x^4 (7 A b-10 a B)}{12 b^3}-\frac {a x (7 A b-10 a B)}{3 b^4}+\frac {x^{10} (A b-a B)}{3 a b \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]

Int[(x^9*(A + B*x^3))/(a + b*x^3)^2,x]

[Out]

-(a*(7*A*b - 10*a*B)*x)/(3*b^4) + ((7*A*b - 10*a*B)*x^4)/(12*b^3) - ((7*A*b - 10*a*B)*x^7)/(21*a*b^2) + ((A*b
- a*B)*x^10)/(3*a*b*(a + b*x^3)) - (a^(4/3)*(7*A*b - 10*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))]
)/(3*Sqrt[3]*b^(13/3)) + (a^(4/3)*(7*A*b - 10*a*B)*Log[a^(1/3) + b^(1/3)*x])/(9*b^(13/3)) - (a^(4/3)*(7*A*b -
10*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*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 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 )^2} \, dx &=\frac {(A b-a B) x^{10}}{3 a b \left (a+b x^3\right )}+\frac {(-7 A b+10 a B) \int \frac {x^9}{a+b x^3} \, dx}{3 a b}\\ &=\frac {(A b-a B) x^{10}}{3 a b \left (a+b x^3\right )}+\frac {(-7 A b+10 a B) \int \left (\frac {a^2}{b^3}-\frac {a x^3}{b^2}+\frac {x^6}{b}-\frac {a^3}{b^3 \left (a+b x^3\right )}\right ) \, dx}{3 a b}\\ &=-\frac {a (7 A b-10 a B) x}{3 b^4}+\frac {(7 A b-10 a B) x^4}{12 b^3}-\frac {(7 A b-10 a B) x^7}{21 a b^2}+\frac {(A b-a B) x^{10}}{3 a b \left (a+b x^3\right )}+\frac {\left (a^2 (7 A b-10 a B)\right ) \int \frac {1}{a+b x^3} \, dx}{3 b^4}\\ &=-\frac {a (7 A b-10 a B) x}{3 b^4}+\frac {(7 A b-10 a B) x^4}{12 b^3}-\frac {(7 A b-10 a B) x^7}{21 a b^2}+\frac {(A b-a B) x^{10}}{3 a b \left (a+b x^3\right )}+\frac {\left (a^{4/3} (7 A b-10 a B)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 b^4}+\frac {\left (a^{4/3} (7 A b-10 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}{9 b^4}\\ &=-\frac {a (7 A b-10 a B) x}{3 b^4}+\frac {(7 A b-10 a B) x^4}{12 b^3}-\frac {(7 A b-10 a B) x^7}{21 a b^2}+\frac {(A b-a B) x^{10}}{3 a b \left (a+b x^3\right )}+\frac {a^{4/3} (7 A b-10 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{13/3}}-\frac {\left (a^{4/3} (7 A b-10 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 b^{13/3}}+\frac {\left (a^{5/3} (7 A b-10 a B)\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 b^4}\\ &=-\frac {a (7 A b-10 a B) x}{3 b^4}+\frac {(7 A b-10 a B) x^4}{12 b^3}-\frac {(7 A b-10 a B) x^7}{21 a b^2}+\frac {(A b-a B) x^{10}}{3 a b \left (a+b x^3\right )}+\frac {a^{4/3} (7 A b-10 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{13/3}}-\frac {a^{4/3} (7 A b-10 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{13/3}}+\frac {\left (a^{4/3} (7 A b-10 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 b^{13/3}}\\ &=-\frac {a (7 A b-10 a B) x}{3 b^4}+\frac {(7 A b-10 a B) x^4}{12 b^3}-\frac {(7 A b-10 a B) x^7}{21 a b^2}+\frac {(A b-a B) x^{10}}{3 a b \left (a+b x^3\right )}-\frac {a^{4/3} (7 A b-10 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} b^{13/3}}+\frac {a^{4/3} (7 A b-10 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{13/3}}-\frac {a^{4/3} (7 A b-10 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{13/3}}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 203, normalized size = 0.87 \[ \frac {14 a^{4/3} (10 a B-7 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-28 a^{4/3} (10 a B-7 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+28 \sqrt {3} a^{4/3} (10 a B-7 A b) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+\frac {84 a^2 \sqrt [3]{b} x (a B-A b)}{a+b x^3}+63 b^{4/3} x^4 (A b-2 a B)+252 a \sqrt [3]{b} x (3 a B-2 A b)+36 b^{7/3} B x^7}{252 b^{13/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^9*(A + B*x^3))/(a + b*x^3)^2,x]

[Out]

(252*a*b^(1/3)*(-2*A*b + 3*a*B)*x + 63*b^(4/3)*(A*b - 2*a*B)*x^4 + 36*b^(7/3)*B*x^7 + (84*a^2*b^(1/3)*(-(A*b)
+ a*B)*x)/(a + b*x^3) + 28*Sqrt[3]*a^(4/3)*(-7*A*b + 10*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 28*
a^(4/3)*(-7*A*b + 10*a*B)*Log[a^(1/3) + b^(1/3)*x] + 14*a^(4/3)*(-7*A*b + 10*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3
)*x + b^(2/3)*x^2])/(252*b^(13/3))

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fricas [A]  time = 1.07, size = 271, normalized size = 1.16 \[ \frac {36 \, B b^{3} x^{10} - 9 \, {\left (10 \, B a b^{2} - 7 \, A b^{3}\right )} x^{7} + 63 \, {\left (10 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} - 28 \, \sqrt {3} {\left (10 \, B a^{3} - 7 \, A a^{2} b + {\left (10 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) + 14 \, {\left (10 \, B a^{3} - 7 \, A a^{2} b + {\left (10 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) - 28 \, {\left (10 \, B a^{3} - 7 \, A a^{2} b + {\left (10 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) + 84 \, {\left (10 \, B a^{3} - 7 \, A a^{2} b\right )} x}{252 \, {\left (b^{5} x^{3} + a b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(B*x^3+A)/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

1/252*(36*B*b^3*x^10 - 9*(10*B*a*b^2 - 7*A*b^3)*x^7 + 63*(10*B*a^2*b - 7*A*a*b^2)*x^4 - 28*sqrt(3)*(10*B*a^3 -
7*A*a^2*b + (10*B*a^2*b - 7*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*(10*B*a^3 - 7*A*a^2*b + (10*B*a^2*b - 7*A*a*b^2)*x^3)*(a/b)^(1/3)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3)) -
28*(10*B*a^3 - 7*A*a^2*b + (10*B*a^2*b - 7*A*a*b^2)*x^3)*(a/b)^(1/3)*log(x + (a/b)^(1/3)) + 84*(10*B*a^3 - 7*
A*a^2*b)*x)/(b^5*x^3 + a*b^4)

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giac [A]  time = 0.20, size = 244, normalized size = 1.05 \[ -\frac {\sqrt {3} {\left (10 \, \left (-a b^{2}\right )^{\frac {1}{3}} B a^{2} - 7 \, \left (-a b^{2}\right )^{\frac {1}{3}} A a b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, b^{5}} + \frac {{\left (10 \, B a^{3} - 7 \, A a^{2} b\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a b^{4}} - \frac {{\left (10 \, \left (-a b^{2}\right )^{\frac {1}{3}} B a^{2} - 7 \, \left (-a b^{2}\right )^{\frac {1}{3}} A a b\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, b^{5}} + \frac {B a^{3} x - A a^{2} b x}{3 \, {\left (b x^{3} + a\right )} b^{4}} + \frac {4 \, B b^{12} x^{7} - 14 \, B a b^{11} x^{4} + 7 \, A b^{12} x^{4} + 84 \, B a^{2} b^{10} x - 56 \, A a b^{11} x}{28 \, b^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(B*x^3+A)/(b*x^3+a)^2,x, algorithm="giac")

[Out]

-1/9*sqrt(3)*(10*(-a*b^2)^(1/3)*B*a^2 - 7*(-a*b^2)^(1/3)*A*a*b)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)
^(1/3))/b^5 + 1/9*(10*B*a^3 - 7*A*a^2*b)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^4) - 1/18*(10*(-a*b^2)^(
1/3)*B*a^2 - 7*(-a*b^2)^(1/3)*A*a*b)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^5 + 1/3*(B*a^3*x - A*a^2*b*x)/
((b*x^3 + a)*b^4) + 1/28*(4*B*b^12*x^7 - 14*B*a*b^11*x^4 + 7*A*b^12*x^4 + 84*B*a^2*b^10*x - 56*A*a*b^11*x)/b^1
4

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maple [A]  time = 0.06, size = 288, normalized size = 1.24 \[ \frac {B \,x^{7}}{7 b^{2}}+\frac {A \,x^{4}}{4 b^{2}}-\frac {B a \,x^{4}}{2 b^{3}}-\frac {A \,a^{2} x}{3 \left (b \,x^{3}+a \right ) b^{3}}+\frac {B \,a^{3} x}{3 \left (b \,x^{3}+a \right ) b^{4}}+\frac {7 \sqrt {3}\, A \,a^{2} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}+\frac {7 A \,a^{2} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}-\frac {7 A \,a^{2} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}-\frac {2 A a x}{b^{3}}-\frac {10 \sqrt {3}\, B \,a^{3} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{5}}-\frac {10 B \,a^{3} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{5}}+\frac {5 B \,a^{3} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{5}}+\frac {3 B \,a^{2} x}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9*(B*x^3+A)/(b*x^3+a)^2,x)

[Out]

1/7/b^2*B*x^7+1/4/b^2*A*x^4-1/2/b^3*B*x^4*a-2/b^3*a*A*x+3/b^4*a^2*B*x-1/3*a^2/b^3*x/(b*x^3+a)*A+1/3*a^3/b^4*x/
(b*x^3+a)*B+7/9*a^2/b^4*A/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-7/18*a^2/b^4*A/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^
(2/3))+7/9*a^2/b^4*A/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-10/9*a^3/b^5*B/(a/b)^(2/3)*ln
(x+(a/b)^(1/3))+5/9*a^3/b^5*B/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-10/9*a^3/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 = 1.23, size = 218, normalized size = 0.94 \[ \frac {{\left (B a^{3} - A a^{2} b\right )} x}{3 \, {\left (b^{5} x^{3} + a b^{4}\right )}} + \frac {4 \, B b^{2} x^{7} - 7 \, {\left (2 \, B a b - A b^{2}\right )} x^{4} + 28 \, {\left (3 \, B a^{2} - 2 \, A a b\right )} x}{28 \, b^{4}} - \frac {\sqrt {3} {\left (10 \, B a^{3} - 7 \, A a^{2} b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (10 \, B a^{3} - 7 \, A a^{2} b\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (10 \, B a^{3} - 7 \, A a^{2} b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(B*x^3+A)/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.62, size = 209, normalized size = 0.90 \[ x^4\,\left (\frac {A}{4\,b^2}-\frac {B\,a}{2\,b^3}\right )-x\,\left (\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{b^4}\right )+\frac {B\,x^7}{7\,b^2}+\frac {x\,\left (\frac {B\,a^3}{3}-\frac {A\,a^2\,b}{3}\right )}{b^5\,x^3+a\,b^4}+\frac {a^{4/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (7\,A\,b-10\,B\,a\right )}{9\,b^{13/3}}-\frac {a^{4/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (7\,A\,b-10\,B\,a\right )}{9\,b^{13/3}}+\frac {a^{4/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (7\,A\,b-10\,B\,a\right )}{9\,b^{13/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^9*(A + B*x^3))/(a + b*x^3)^2,x)

[Out]

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

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sympy [A]  time = 2.12, size = 156, normalized size = 0.67 \[ \frac {B x^{7}}{7 b^{2}} + x^{4} \left (\frac {A}{4 b^{2}} - \frac {B a}{2 b^{3}}\right ) + x \left (- \frac {2 A a}{b^{3}} + \frac {3 B a^{2}}{b^{4}}\right ) + \frac {x \left (- A a^{2} b + B a^{3}\right )}{3 a b^{4} + 3 b^{5} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} b^{13} - 343 A^{3} a^{4} b^{3} + 1470 A^{2} B a^{5} b^{2} - 2100 A B^{2} a^{6} b + 1000 B^{3} a^{7}, \left (t \mapsto t \log {\left (- \frac {9 t b^{4}}{- 7 A a b + 10 B a^{2}} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9*(B*x**3+A)/(b*x**3+a)**2,x)

[Out]

B*x**7/(7*b**2) + x**4*(A/(4*b**2) - B*a/(2*b**3)) + x*(-2*A*a/b**3 + 3*B*a**2/b**4) + x*(-A*a**2*b + B*a**3)/
(3*a*b**4 + 3*b**5*x**3) + RootSum(729*_t**3*b**13 - 343*A**3*a**4*b**3 + 1470*A**2*B*a**5*b**2 - 2100*A*B**2*
a**6*b + 1000*B**3*a**7, Lambda(_t, _t*log(-9*_t*b**4/(-7*A*a*b + 10*B*a**2) + x)))

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