3.97 \(\int \frac {x^7 (A+B x^3)}{(a+b x^3)^3} \, dx\)

Optimal. Leaf size=222 \[ \frac {5 (A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 \sqrt [3]{a} b^{11/3}}-\frac {5 (A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{11/3}}-\frac {5 (A b-4 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} \sqrt [3]{a} b^{11/3}}-\frac {5 x^2 (A b-4 a B)}{18 a b^3}+\frac {x^5 (A b-4 a B)}{9 a b^2 \left (a+b x^3\right )}+\frac {x^8 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

[Out]

-5/18*(A*b-4*B*a)*x^2/a/b^3+1/6*(A*b-B*a)*x^8/a/b/(b*x^3+a)^2+1/9*(A*b-4*B*a)*x^5/a/b^2/(b*x^3+a)-5/27*(A*b-4*
B*a)*ln(a^(1/3)+b^(1/3)*x)/a^(1/3)/b^(11/3)+5/54*(A*b-4*B*a)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(1/3)
/b^(11/3)-5/27*(A*b-4*B*a)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(1/3)/b^(11/3)*3^(1/2)

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Rubi [A]  time = 0.14, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {457, 288, 321, 292, 31, 634, 617, 204, 628} \[ \frac {5 (A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 \sqrt [3]{a} b^{11/3}}+\frac {x^5 (A b-4 a B)}{9 a b^2 \left (a+b x^3\right )}-\frac {5 x^2 (A b-4 a B)}{18 a b^3}-\frac {5 (A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{11/3}}-\frac {5 (A b-4 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} \sqrt [3]{a} b^{11/3}}+\frac {x^8 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^7*(A + B*x^3))/(a + b*x^3)^3,x]

[Out]

(-5*(A*b - 4*a*B)*x^2)/(18*a*b^3) + ((A*b - a*B)*x^8)/(6*a*b*(a + b*x^3)^2) + ((A*b - 4*a*B)*x^5)/(9*a*b^2*(a
+ b*x^3)) - (5*(A*b - 4*a*B)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(1/3)*b^(11/3)) -
(5*(A*b - 4*a*B)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(1/3)*b^(11/3)) + (5*(A*b - 4*a*B)*Log[a^(2/3) - a^(1/3)*b^(
1/3)*x + b^(2/3)*x^2])/(54*a^(1/3)*b^(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 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 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 321

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*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && 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 {x^7 \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx &=\frac {(A b-a B) x^8}{6 a b \left (a+b x^3\right )^2}+\frac {(-2 A b+8 a B) \int \frac {x^7}{\left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac {(A b-a B) x^8}{6 a b \left (a+b x^3\right )^2}+\frac {(A b-4 a B) x^5}{9 a b^2 \left (a+b x^3\right )}-\frac {(5 (A b-4 a B)) \int \frac {x^4}{a+b x^3} \, dx}{9 a b^2}\\ &=-\frac {5 (A b-4 a B) x^2}{18 a b^3}+\frac {(A b-a B) x^8}{6 a b \left (a+b x^3\right )^2}+\frac {(A b-4 a B) x^5}{9 a b^2 \left (a+b x^3\right )}+\frac {(5 (A b-4 a B)) \int \frac {x}{a+b x^3} \, dx}{9 b^3}\\ &=-\frac {5 (A b-4 a B) x^2}{18 a b^3}+\frac {(A b-a B) x^8}{6 a b \left (a+b x^3\right )^2}+\frac {(A b-4 a B) x^5}{9 a b^2 \left (a+b x^3\right )}-\frac {(5 (A b-4 a B)) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 \sqrt [3]{a} b^{10/3}}+\frac {(5 (A b-4 a B)) \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 \sqrt [3]{a} b^{10/3}}\\ &=-\frac {5 (A b-4 a B) x^2}{18 a b^3}+\frac {(A b-a B) x^8}{6 a b \left (a+b x^3\right )^2}+\frac {(A b-4 a B) x^5}{9 a b^2 \left (a+b x^3\right )}-\frac {5 (A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{11/3}}+\frac {(5 (A b-4 a B)) \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 \sqrt [3]{a} b^{11/3}}+\frac {(5 (A b-4 a B)) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 b^{10/3}}\\ &=-\frac {5 (A b-4 a B) x^2}{18 a b^3}+\frac {(A b-a B) x^8}{6 a b \left (a+b x^3\right )^2}+\frac {(A b-4 a B) x^5}{9 a b^2 \left (a+b x^3\right )}-\frac {5 (A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{11/3}}+\frac {5 (A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 \sqrt [3]{a} b^{11/3}}+\frac {(5 (A b-4 a B)) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 \sqrt [3]{a} b^{11/3}}\\ &=-\frac {5 (A b-4 a B) x^2}{18 a b^3}+\frac {(A b-a B) x^8}{6 a b \left (a+b x^3\right )^2}+\frac {(A b-4 a B) x^5}{9 a b^2 \left (a+b x^3\right )}-\frac {5 (A b-4 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} \sqrt [3]{a} b^{11/3}}-\frac {5 (A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 \sqrt [3]{a} b^{11/3}}+\frac {5 (A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 \sqrt [3]{a} b^{11/3}}\\ \end {align*}

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Mathematica [A]  time = 0.24, size = 194, normalized size = 0.87 \[ \frac {\frac {5 (A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{a}}-\frac {6 b^{2/3} x^2 (4 A b-7 a B)}{a+b x^3}+\frac {9 a b^{2/3} x^2 (A b-a B)}{\left (a+b x^3\right )^2}+\frac {10 (4 a B-A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}+\frac {10 \sqrt {3} (4 a B-A b) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}+27 b^{2/3} B x^2}{54 b^{11/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^7*(A + B*x^3))/(a + b*x^3)^3,x]

[Out]

(27*b^(2/3)*B*x^2 + (9*a*b^(2/3)*(A*b - a*B)*x^2)/(a + b*x^3)^2 - (6*b^(2/3)*(4*A*b - 7*a*B)*x^2)/(a + b*x^3)
+ (10*Sqrt[3]*(-(A*b) + 4*a*B)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(1/3) + (10*(-(A*b) + 4*a*B)*Log
[a^(1/3) + b^(1/3)*x])/a^(1/3) + (5*(A*b - 4*a*B)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(1/3))/(54
*b^(11/3))

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fricas [B]  time = 0.68, size = 792, normalized size = 3.57 \[ \left [\frac {27 \, B a b^{4} x^{8} + 24 \, {\left (4 \, B a^{2} b^{3} - A a b^{4}\right )} x^{5} + 15 \, {\left (4 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{2} - 15 \, \sqrt {\frac {1}{3}} {\left ({\left (4 \, B a^{2} b^{3} - A a b^{4}\right )} x^{6} + 4 \, B a^{4} b - A a^{3} b^{2} + 2 \, {\left (4 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{3}\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{3} - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} x}{b x^{3} + a}\right ) - 5 \, {\left ({\left (4 \, B a b^{2} - A b^{3}\right )} x^{6} + 4 \, B a^{3} - A a^{2} b + 2 \, {\left (4 \, B a^{2} b - A a b^{2}\right )} x^{3}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) + 10 \, {\left ({\left (4 \, B a b^{2} - A b^{3}\right )} x^{6} + 4 \, B a^{3} - A a^{2} b + 2 \, {\left (4 \, B a^{2} b - A a b^{2}\right )} x^{3}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{54 \, {\left (a b^{7} x^{6} + 2 \, a^{2} b^{6} x^{3} + a^{3} b^{5}\right )}}, \frac {27 \, B a b^{4} x^{8} + 24 \, {\left (4 \, B a^{2} b^{3} - A a b^{4}\right )} x^{5} + 15 \, {\left (4 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{2} - 30 \, \sqrt {\frac {1}{3}} {\left ({\left (4 \, B a^{2} b^{3} - A a b^{4}\right )} x^{6} + 4 \, B a^{4} b - A a^{3} b^{2} + 2 \, {\left (4 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{3}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x + \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) - 5 \, {\left ({\left (4 \, B a b^{2} - A b^{3}\right )} x^{6} + 4 \, B a^{3} - A a^{2} b + 2 \, {\left (4 \, B a^{2} b - A a b^{2}\right )} x^{3}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) + 10 \, {\left ({\left (4 \, B a b^{2} - A b^{3}\right )} x^{6} + 4 \, B a^{3} - A a^{2} b + 2 \, {\left (4 \, B a^{2} b - A a b^{2}\right )} x^{3}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{54 \, {\left (a b^{7} x^{6} + 2 \, a^{2} b^{6} x^{3} + a^{3} b^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

[1/54*(27*B*a*b^4*x^8 + 24*(4*B*a^2*b^3 - A*a*b^4)*x^5 + 15*(4*B*a^3*b^2 - A*a^2*b^3)*x^2 - 15*sqrt(1/3)*((4*B
*a^2*b^3 - A*a*b^4)*x^6 + 4*B*a^4*b - A*a^3*b^2 + 2*(4*B*a^3*b^2 - A*a^2*b^3)*x^3)*sqrt((-a*b^2)^(1/3)/a)*log(
(2*b^2*x^3 - a*b + 3*sqrt(1/3)*(a*b*x + 2*(-a*b^2)^(2/3)*x^2 + (-a*b^2)^(1/3)*a)*sqrt((-a*b^2)^(1/3)/a) - 3*(-
a*b^2)^(2/3)*x)/(b*x^3 + a)) - 5*((4*B*a*b^2 - A*b^3)*x^6 + 4*B*a^3 - A*a^2*b + 2*(4*B*a^2*b - A*a*b^2)*x^3)*(
-a*b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) + 10*((4*B*a*b^2 - A*b^3)*x^6 + 4*B*a^3 - A*a
^2*b + 2*(4*B*a^2*b - A*a*b^2)*x^3)*(-a*b^2)^(2/3)*log(b*x - (-a*b^2)^(1/3)))/(a*b^7*x^6 + 2*a^2*b^6*x^3 + a^3
*b^5), 1/54*(27*B*a*b^4*x^8 + 24*(4*B*a^2*b^3 - A*a*b^4)*x^5 + 15*(4*B*a^3*b^2 - A*a^2*b^3)*x^2 - 30*sqrt(1/3)
*((4*B*a^2*b^3 - A*a*b^4)*x^6 + 4*B*a^4*b - A*a^3*b^2 + 2*(4*B*a^3*b^2 - A*a^2*b^3)*x^3)*sqrt(-(-a*b^2)^(1/3)/
a)*arctan(sqrt(1/3)*(2*b*x + (-a*b^2)^(1/3))*sqrt(-(-a*b^2)^(1/3)/a)/b) - 5*((4*B*a*b^2 - A*b^3)*x^6 + 4*B*a^3
- A*a^2*b + 2*(4*B*a^2*b - A*a*b^2)*x^3)*(-a*b^2)^(2/3)*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) +
10*((4*B*a*b^2 - A*b^3)*x^6 + 4*B*a^3 - A*a^2*b + 2*(4*B*a^2*b - A*a*b^2)*x^3)*(-a*b^2)^(2/3)*log(b*x - (-a*b^
2)^(1/3)))/(a*b^7*x^6 + 2*a^2*b^6*x^3 + a^3*b^5)]

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giac [A]  time = 0.21, size = 210, normalized size = 0.95 \[ \frac {B x^{2}}{2 \, b^{3}} - \frac {5 \, \sqrt {3} {\left (4 \, B a - A b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3}} + \frac {5 \, {\left (4 \, B a - A b\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{3}} + \frac {5 \, {\left (4 \, B a \left (-\frac {a}{b}\right )^{\frac {1}{3}} - A b \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a b^{3}} + \frac {14 \, B a b x^{5} - 8 \, A b^{2} x^{5} + 11 \, B a^{2} x^{2} - 5 \, A a b x^{2}}{18 \, {\left (b x^{3} + a\right )}^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="giac")

[Out]

1/2*B*x^2/b^3 - 5/27*sqrt(3)*(4*B*a - A*b)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(1/
3)*b^3) + 5/54*(4*B*a - A*b)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(1/3)*b^3) + 5/27*(4*B*a*(-a/b
)^(1/3) - A*b*(-a/b)^(1/3))*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^3) + 1/18*(14*B*a*b*x^5 - 8*A*b^2*x^5
+ 11*B*a^2*x^2 - 5*A*a*b*x^2)/((b*x^3 + a)^2*b^3)

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maple [A]  time = 0.06, size = 275, normalized size = 1.24 \[ -\frac {4 A \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} b}+\frac {7 B a \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} b^{2}}-\frac {5 A a \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} b^{2}}+\frac {11 B \,a^{2} x^{2}}{18 \left (b \,x^{3}+a \right )^{2} b^{3}}+\frac {B \,x^{2}}{2 b^{3}}+\frac {5 \sqrt {3}\, A \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}-\frac {5 A \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}+\frac {5 A \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}-\frac {20 \sqrt {3}\, B a \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4}}+\frac {20 B a \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4}}-\frac {10 B a \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(B*x^3+A)/(b*x^3+a)^3,x)

[Out]

1/2*B/b^3*x^2-4/9/b/(b*x^3+a)^2*A*x^5+7/9/b^2/(b*x^3+a)^2*B*x^5*a-5/18/b^2/(b*x^3+a)^2*A*x^2*a+11/18/b^3/(b*x^
3+a)^2*B*x^2*a^2-5/27/b^3*A/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+5/54/b^3*A/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2
/3))+5/27/b^3*A*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+20/27/b^4*B*a/(a/b)^(1/3)*ln(x+(a/
b)^(1/3))-10/27/b^4*B*a/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-20/27/b^4*B*a*3^(1/2)/(a/b)^(1/3)*arctan
(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))

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maxima [A]  time = 1.32, size = 196, normalized size = 0.88 \[ \frac {2 \, {\left (7 \, B a b - 4 \, A b^{2}\right )} x^{5} + {\left (11 \, B a^{2} - 5 \, A a b\right )} x^{2}}{18 \, {\left (b^{5} x^{6} + 2 \, a b^{4} x^{3} + a^{2} b^{3}\right )}} + \frac {B x^{2}}{2 \, b^{3}} - \frac {5 \, \sqrt {3} {\left (4 \, B a - A b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, b^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {5 \, {\left (4 \, B a - A b\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, b^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {5 \, {\left (4 \, B a - A b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, b^{4} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(B*x^3+A)/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

1/18*(2*(7*B*a*b - 4*A*b^2)*x^5 + (11*B*a^2 - 5*A*a*b)*x^2)/(b^5*x^6 + 2*a*b^4*x^3 + a^2*b^3) + 1/2*B*x^2/b^3
- 5/27*sqrt(3)*(4*B*a - A*b)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^4*(a/b)^(1/3)) - 5/54*(4*B
*a - A*b)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^4*(a/b)^(1/3)) + 5/27*(4*B*a - A*b)*log(x + (a/b)^(1/3))/(
b^4*(a/b)^(1/3))

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mupad [B]  time = 2.56, size = 187, normalized size = 0.84 \[ \frac {x^2\,\left (\frac {11\,B\,a^2}{18}-\frac {5\,A\,a\,b}{18}\right )-x^5\,\left (\frac {4\,A\,b^2}{9}-\frac {7\,B\,a\,b}{9}\right )}{a^2\,b^3+2\,a\,b^4\,x^3+b^5\,x^6}+\frac {B\,x^2}{2\,b^3}-\frac {5\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (A\,b-4\,B\,a\right )}{27\,a^{1/3}\,b^{11/3}}-\frac {5\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (A\,b-4\,B\,a\right )}{27\,a^{1/3}\,b^{11/3}}+\frac {5\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (A\,b-4\,B\,a\right )}{27\,a^{1/3}\,b^{11/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^7*(A + B*x^3))/(a + b*x^3)^3,x)

[Out]

(x^2*((11*B*a^2)/18 - (5*A*a*b)/18) - x^5*((4*A*b^2)/9 - (7*B*a*b)/9))/(a^2*b^3 + b^5*x^6 + 2*a*b^4*x^3) + (B*
x^2)/(2*b^3) - (5*log(b^(1/3)*x + a^(1/3))*(A*b - 4*B*a))/(27*a^(1/3)*b^(11/3)) - (5*log(3^(1/2)*a^(1/3)*1i -
2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(A*b - 4*B*a))/(27*a^(1/3)*b^(11/3)) + (5*log(3^(1/2)*a^(1/3)*1i
+ 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(A*b - 4*B*a))/(27*a^(1/3)*b^(11/3))

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sympy [A]  time = 5.85, size = 162, normalized size = 0.73 \[ \frac {B x^{2}}{2 b^{3}} + \frac {x^{5} \left (- 8 A b^{2} + 14 B a b\right ) + x^{2} \left (- 5 A a b + 11 B a^{2}\right )}{18 a^{2} b^{3} + 36 a b^{4} x^{3} + 18 b^{5} x^{6}} + \operatorname {RootSum} {\left (19683 t^{3} a b^{11} + 125 A^{3} b^{3} - 1500 A^{2} B a b^{2} + 6000 A B^{2} a^{2} b - 8000 B^{3} a^{3}, \left (t \mapsto t \log {\left (\frac {729 t^{2} a b^{7}}{25 A^{2} b^{2} - 200 A B a b + 400 B^{2} a^{2}} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(B*x**3+A)/(b*x**3+a)**3,x)

[Out]

B*x**2/(2*b**3) + (x**5*(-8*A*b**2 + 14*B*a*b) + x**2*(-5*A*a*b + 11*B*a**2))/(18*a**2*b**3 + 36*a*b**4*x**3 +
18*b**5*x**6) + RootSum(19683*_t**3*a*b**11 + 125*A**3*b**3 - 1500*A**2*B*a*b**2 + 6000*A*B**2*a**2*b - 8000*
B**3*a**3, Lambda(_t, _t*log(729*_t**2*a*b**7/(25*A**2*b**2 - 200*A*B*a*b + 400*B**2*a**2) + x)))

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