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

Optimal. Leaf size=321 \[ -\frac {5 (a B+11 A b) \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{144 \sqrt {3} a^{17/6} b^{7/6}}+\frac {5 (a B+11 A b) \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{144 \sqrt {3} a^{17/6} b^{7/6}}-\frac {5 (a B+11 A b) \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{216 a^{17/6} b^{7/6}}+\frac {5 (a B+11 A b) \tan ^{-1}\left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}+\sqrt {3}\right )}{216 a^{17/6} b^{7/6}}+\frac {5 (a B+11 A b) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{108 a^{17/6} b^{7/6}}+\frac {\sqrt {x} (a B+11 A b)}{36 a^2 b \left (a+b x^3\right )}+\frac {\sqrt {x} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]

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Rubi [A]  time = 0.52, antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {457, 290, 329, 209, 634, 618, 204, 628, 205} \begin {gather*} -\frac {5 (a B+11 A b) \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{144 \sqrt {3} a^{17/6} b^{7/6}}+\frac {5 (a B+11 A b) \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{144 \sqrt {3} a^{17/6} b^{7/6}}-\frac {5 (a B+11 A b) \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{216 a^{17/6} b^{7/6}}+\frac {5 (a B+11 A b) \tan ^{-1}\left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}+\sqrt {3}\right )}{216 a^{17/6} b^{7/6}}+\frac {5 (a B+11 A b) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{108 a^{17/6} b^{7/6}}+\frac {\sqrt {x} (a B+11 A b)}{36 a^2 b \left (a+b x^3\right )}+\frac {\sqrt {x} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((A*b - a*B)*Sqrt[x])/(6*a*b*(a + b*x^3)^2) + ((11*A*b + a*B)*Sqrt[x])/(36*a^2*b*(a + b*x^3)) - (5*(11*A*b + a
*B)*ArcTan[Sqrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(216*a^(17/6)*b^(7/6)) + (5*(11*A*b + a*B)*ArcTan[Sqrt[3] +
(2*b^(1/6)*Sqrt[x])/a^(1/6)])/(216*a^(17/6)*b^(7/6)) + (5*(11*A*b + a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(
108*a^(17/6)*b^(7/6)) - (5*(11*A*b + a*B)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(144*Sqr
t[3]*a^(17/6)*b^(7/6)) + (5*(11*A*b + a*B)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(144*Sq
rt[3]*a^(17/6)*b^(7/6))

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 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 209

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/b, n]]
, k, u, v}, Simp[u = Int[(r - s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x] +
Int[(r + s*Cos[((2*k - 1)*Pi)/n]*x)/(r^2 + 2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (2*r^2*Int[1/(r^2 +
s^2*x^2), x])/(a*n) + Dist[(2*r)/(a*n), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)
/4, 0] && PosQ[a/b]

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 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && 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 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; 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}{\sqrt {x} \left (a+b x^3\right )^3} \, dx &=\frac {(A b-a B) \sqrt {x}}{6 a b \left (a+b x^3\right )^2}+\frac {(11 A b+a B) \int \frac {1}{\sqrt {x} \left (a+b x^3\right )^2} \, dx}{12 a b}\\ &=\frac {(A b-a B) \sqrt {x}}{6 a b \left (a+b x^3\right )^2}+\frac {(11 A b+a B) \sqrt {x}}{36 a^2 b \left (a+b x^3\right )}+\frac {(5 (11 A b+a B)) \int \frac {1}{\sqrt {x} \left (a+b x^3\right )} \, dx}{72 a^2 b}\\ &=\frac {(A b-a B) \sqrt {x}}{6 a b \left (a+b x^3\right )^2}+\frac {(11 A b+a B) \sqrt {x}}{36 a^2 b \left (a+b x^3\right )}+\frac {(5 (11 A b+a B)) \operatorname {Subst}\left (\int \frac {1}{a+b x^6} \, dx,x,\sqrt {x}\right )}{36 a^2 b}\\ &=\frac {(A b-a B) \sqrt {x}}{6 a b \left (a+b x^3\right )^2}+\frac {(11 A b+a B) \sqrt {x}}{36 a^2 b \left (a+b x^3\right )}+\frac {(5 (11 A b+a B)) \operatorname {Subst}\left (\int \frac {\sqrt [6]{a}-\frac {1}{2} \sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{108 a^{17/6} b}+\frac {(5 (11 A b+a B)) \operatorname {Subst}\left (\int \frac {\sqrt [6]{a}+\frac {1}{2} \sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{108 a^{17/6} b}+\frac {(5 (11 A b+a B)) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{108 a^{8/3} b}\\ &=\frac {(A b-a B) \sqrt {x}}{6 a b \left (a+b x^3\right )^2}+\frac {(11 A b+a B) \sqrt {x}}{36 a^2 b \left (a+b x^3\right )}+\frac {5 (11 A b+a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{108 a^{17/6} b^{7/6}}-\frac {(5 (11 A b+a B)) \operatorname {Subst}\left (\int \frac {-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{144 \sqrt {3} a^{17/6} b^{7/6}}+\frac {(5 (11 A b+a B)) \operatorname {Subst}\left (\int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{144 \sqrt {3} a^{17/6} b^{7/6}}+\frac {(5 (11 A b+a B)) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{432 a^{8/3} b}+\frac {(5 (11 A b+a B)) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{432 a^{8/3} b}\\ &=\frac {(A b-a B) \sqrt {x}}{6 a b \left (a+b x^3\right )^2}+\frac {(11 A b+a B) \sqrt {x}}{36 a^2 b \left (a+b x^3\right )}+\frac {5 (11 A b+a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{108 a^{17/6} b^{7/6}}-\frac {5 (11 A b+a B) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{144 \sqrt {3} a^{17/6} b^{7/6}}+\frac {5 (11 A b+a B) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{144 \sqrt {3} a^{17/6} b^{7/6}}+\frac {(5 (11 A b+a B)) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )}{216 \sqrt {3} a^{17/6} b^{7/6}}-\frac {(5 (11 A b+a B)) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )}{216 \sqrt {3} a^{17/6} b^{7/6}}\\ &=\frac {(A b-a B) \sqrt {x}}{6 a b \left (a+b x^3\right )^2}+\frac {(11 A b+a B) \sqrt {x}}{36 a^2 b \left (a+b x^3\right )}-\frac {5 (11 A b+a B) \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{216 a^{17/6} b^{7/6}}+\frac {5 (11 A b+a B) \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{216 a^{17/6} b^{7/6}}+\frac {5 (11 A b+a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{108 a^{17/6} b^{7/6}}-\frac {5 (11 A b+a B) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{144 \sqrt {3} a^{17/6} b^{7/6}}+\frac {5 (11 A b+a B) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{144 \sqrt {3} a^{17/6} b^{7/6}}\\ \end {align*}

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Mathematica [C]  time = 0.05, size = 91, normalized size = 0.28 \begin {gather*} \frac {\sqrt {x} \left (a \left (-5 a^2 B+a b \left (17 A+B x^3\right )+11 A b^2 x^3\right )+5 \left (a+b x^3\right )^2 (a B+11 A b) \, _2F_1\left (\frac {1}{6},1;\frac {7}{6};-\frac {b x^3}{a}\right )\right )}{36 a^3 b \left (a+b x^3\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*(a*(-5*a^2*B + 11*A*b^2*x^3 + a*b*(17*A + B*x^3)) + 5*(11*A*b + a*B)*(a + b*x^3)^2*Hypergeometric2F1[
1/6, 1, 7/6, -((b*x^3)/a)]))/(36*a^3*b*(a + b*x^3)^2)

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IntegrateAlgebraic [A]  time = 0.79, size = 210, normalized size = 0.65 \begin {gather*} \frac {5 (a B+11 A b) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{108 a^{17/6} b^{7/6}}-\frac {5 (a B+11 A b) \tan ^{-1}\left (\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}\right )}{216 a^{17/6} b^{7/6}}+\frac {5 (a B+11 A b) \tanh ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{a}+\sqrt [3]{b} x}\right )}{72 \sqrt {3} a^{17/6} b^{7/6}}-\frac {\sqrt {x} \left (5 a^2 B-17 a A b-a b B x^3-11 A b^2 x^3\right )}{36 a^2 b \left (a+b x^3\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/36*(Sqrt[x]*(-17*a*A*b + 5*a^2*B - 11*A*b^2*x^3 - a*b*B*x^3))/(a^2*b*(a + b*x^3)^2) + (5*(11*A*b + a*B)*Arc
Tan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(108*a^(17/6)*b^(7/6)) - (5*(11*A*b + a*B)*ArcTan[(a^(1/3) - b^(1/3)*x)/(a^(1/
6)*b^(1/6)*Sqrt[x])])/(216*a^(17/6)*b^(7/6)) + (5*(11*A*b + a*B)*ArcTanh[(Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x])/(a^
(1/3) + b^(1/3)*x)])/(72*Sqrt[3]*a^(17/6)*b^(7/6))

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fricas [B]  time = 1.00, size = 2674, normalized size = 8.33

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^3+A)/(b*x^3+a)^3/x^(1/2),x, algorithm="fricas")

[Out]

1/432*(20*sqrt(3)*(a^2*b^3*x^6 + 2*a^3*b^2*x^3 + a^4*b)*(-(B^6*a^6 + 66*A*B^5*a^5*b + 1815*A^2*B^4*a^4*b^2 + 2
6620*A^3*B^3*a^3*b^3 + 219615*A^4*B^2*a^2*b^4 + 966306*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b^7))^(1/6)*arctan
(1/3*(2*sqrt(3)*sqrt(a^6*b^2*(-(B^6*a^6 + 66*A*B^5*a^5*b + 1815*A^2*B^4*a^4*b^2 + 26620*A^3*B^3*a^3*b^3 + 2196
15*A^4*B^2*a^2*b^4 + 966306*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b^7))^(1/3) + (B^2*a^2 + 22*A*B*a*b + 121*A^2
*b^2)*x + (B*a^4*b + 11*A*a^3*b^2)*sqrt(x)*(-(B^6*a^6 + 66*A*B^5*a^5*b + 1815*A^2*B^4*a^4*b^2 + 26620*A^3*B^3*
a^3*b^3 + 219615*A^4*B^2*a^2*b^4 + 966306*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b^7))^(1/6))*a^14*b^6*(-(B^6*a^
6 + 66*A*B^5*a^5*b + 1815*A^2*B^4*a^4*b^2 + 26620*A^3*B^3*a^3*b^3 + 219615*A^4*B^2*a^2*b^4 + 966306*A^5*B*a*b^
5 + 1771561*A^6*b^6)/(a^17*b^7))^(5/6) - 2*sqrt(3)*(B*a^15*b^6 + 11*A*a^14*b^7)*sqrt(x)*(-(B^6*a^6 + 66*A*B^5*
a^5*b + 1815*A^2*B^4*a^4*b^2 + 26620*A^3*B^3*a^3*b^3 + 219615*A^4*B^2*a^2*b^4 + 966306*A^5*B*a*b^5 + 1771561*A
^6*b^6)/(a^17*b^7))^(5/6) + sqrt(3)*(B^6*a^6 + 66*A*B^5*a^5*b + 1815*A^2*B^4*a^4*b^2 + 26620*A^3*B^3*a^3*b^3 +
219615*A^4*B^2*a^2*b^4 + 966306*A^5*B*a*b^5 + 1771561*A^6*b^6))/(B^6*a^6 + 66*A*B^5*a^5*b + 1815*A^2*B^4*a^4*
b^2 + 26620*A^3*B^3*a^3*b^3 + 219615*A^4*B^2*a^2*b^4 + 966306*A^5*B*a*b^5 + 1771561*A^6*b^6)) + 20*sqrt(3)*(a^
2*b^3*x^6 + 2*a^3*b^2*x^3 + a^4*b)*(-(B^6*a^6 + 66*A*B^5*a^5*b + 1815*A^2*B^4*a^4*b^2 + 26620*A^3*B^3*a^3*b^3
+ 219615*A^4*B^2*a^2*b^4 + 966306*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b^7))^(1/6)*arctan(1/3*(2*sqrt(3)*sqrt(
a^6*b^2*(-(B^6*a^6 + 66*A*B^5*a^5*b + 1815*A^2*B^4*a^4*b^2 + 26620*A^3*B^3*a^3*b^3 + 219615*A^4*B^2*a^2*b^4 +
966306*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b^7))^(1/3) + (B^2*a^2 + 22*A*B*a*b + 121*A^2*b^2)*x - (B*a^4*b +
11*A*a^3*b^2)*sqrt(x)*(-(B^6*a^6 + 66*A*B^5*a^5*b + 1815*A^2*B^4*a^4*b^2 + 26620*A^3*B^3*a^3*b^3 + 219615*A^4*
B^2*a^2*b^4 + 966306*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b^7))^(1/6))*a^14*b^6*(-(B^6*a^6 + 66*A*B^5*a^5*b +
1815*A^2*B^4*a^4*b^2 + 26620*A^3*B^3*a^3*b^3 + 219615*A^4*B^2*a^2*b^4 + 966306*A^5*B*a*b^5 + 1771561*A^6*b^6)/
(a^17*b^7))^(5/6) - 2*sqrt(3)*(B*a^15*b^6 + 11*A*a^14*b^7)*sqrt(x)*(-(B^6*a^6 + 66*A*B^5*a^5*b + 1815*A^2*B^4*
a^4*b^2 + 26620*A^3*B^3*a^3*b^3 + 219615*A^4*B^2*a^2*b^4 + 966306*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b^7))^(
5/6) - sqrt(3)*(B^6*a^6 + 66*A*B^5*a^5*b + 1815*A^2*B^4*a^4*b^2 + 26620*A^3*B^3*a^3*b^3 + 219615*A^4*B^2*a^2*b
^4 + 966306*A^5*B*a*b^5 + 1771561*A^6*b^6))/(B^6*a^6 + 66*A*B^5*a^5*b + 1815*A^2*B^4*a^4*b^2 + 26620*A^3*B^3*a
^3*b^3 + 219615*A^4*B^2*a^2*b^4 + 966306*A^5*B*a*b^5 + 1771561*A^6*b^6)) + 5*(a^2*b^3*x^6 + 2*a^3*b^2*x^3 + a^
4*b)*(-(B^6*a^6 + 66*A*B^5*a^5*b + 1815*A^2*B^4*a^4*b^2 + 26620*A^3*B^3*a^3*b^3 + 219615*A^4*B^2*a^2*b^4 + 966
306*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b^7))^(1/6)*log(25*a^6*b^2*(-(B^6*a^6 + 66*A*B^5*a^5*b + 1815*A^2*B^4
*a^4*b^2 + 26620*A^3*B^3*a^3*b^3 + 219615*A^4*B^2*a^2*b^4 + 966306*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b^7))^
(1/3) + 25*(B^2*a^2 + 22*A*B*a*b + 121*A^2*b^2)*x + 25*(B*a^4*b + 11*A*a^3*b^2)*sqrt(x)*(-(B^6*a^6 + 66*A*B^5*
a^5*b + 1815*A^2*B^4*a^4*b^2 + 26620*A^3*B^3*a^3*b^3 + 219615*A^4*B^2*a^2*b^4 + 966306*A^5*B*a*b^5 + 1771561*A
^6*b^6)/(a^17*b^7))^(1/6)) - 5*(a^2*b^3*x^6 + 2*a^3*b^2*x^3 + a^4*b)*(-(B^6*a^6 + 66*A*B^5*a^5*b + 1815*A^2*B^
4*a^4*b^2 + 26620*A^3*B^3*a^3*b^3 + 219615*A^4*B^2*a^2*b^4 + 966306*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b^7))
^(1/6)*log(25*a^6*b^2*(-(B^6*a^6 + 66*A*B^5*a^5*b + 1815*A^2*B^4*a^4*b^2 + 26620*A^3*B^3*a^3*b^3 + 219615*A^4*
B^2*a^2*b^4 + 966306*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b^7))^(1/3) + 25*(B^2*a^2 + 22*A*B*a*b + 121*A^2*b^2
)*x - 25*(B*a^4*b + 11*A*a^3*b^2)*sqrt(x)*(-(B^6*a^6 + 66*A*B^5*a^5*b + 1815*A^2*B^4*a^4*b^2 + 26620*A^3*B^3*a
^3*b^3 + 219615*A^4*B^2*a^2*b^4 + 966306*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b^7))^(1/6)) + 10*(a^2*b^3*x^6 +
2*a^3*b^2*x^3 + a^4*b)*(-(B^6*a^6 + 66*A*B^5*a^5*b + 1815*A^2*B^4*a^4*b^2 + 26620*A^3*B^3*a^3*b^3 + 219615*A^
4*B^2*a^2*b^4 + 966306*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b^7))^(1/6)*log(5*a^3*b*(-(B^6*a^6 + 66*A*B^5*a^5*
b + 1815*A^2*B^4*a^4*b^2 + 26620*A^3*B^3*a^3*b^3 + 219615*A^4*B^2*a^2*b^4 + 966306*A^5*B*a*b^5 + 1771561*A^6*b
^6)/(a^17*b^7))^(1/6) + 5*(B*a + 11*A*b)*sqrt(x)) - 10*(a^2*b^3*x^6 + 2*a^3*b^2*x^3 + a^4*b)*(-(B^6*a^6 + 66*A
*B^5*a^5*b + 1815*A^2*B^4*a^4*b^2 + 26620*A^3*B^3*a^3*b^3 + 219615*A^4*B^2*a^2*b^4 + 966306*A^5*B*a*b^5 + 1771
561*A^6*b^6)/(a^17*b^7))^(1/6)*log(-5*a^3*b*(-(B^6*a^6 + 66*A*B^5*a^5*b + 1815*A^2*B^4*a^4*b^2 + 26620*A^3*B^3
*a^3*b^3 + 219615*A^4*B^2*a^2*b^4 + 966306*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b^7))^(1/6) + 5*(B*a + 11*A*b)
*sqrt(x)) + 12*((B*a*b + 11*A*b^2)*x^3 - 5*B*a^2 + 17*A*a*b)*sqrt(x))/(a^2*b^3*x^6 + 2*a^3*b^2*x^3 + a^4*b)

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giac [A]  time = 0.24, size = 322, normalized size = 1.00 \begin {gather*} \frac {5 \, \sqrt {3} {\left (\left (a b^{5}\right )^{\frac {1}{6}} B a + 11 \, \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \log \left (\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{432 \, a^{3} b^{2}} - \frac {5 \, \sqrt {3} {\left (\left (a b^{5}\right )^{\frac {1}{6}} B a + 11 \, \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \log \left (-\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{432 \, a^{3} b^{2}} + \frac {5 \, {\left (\left (a b^{5}\right )^{\frac {1}{6}} B a + 11 \, \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \arctan \left (\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} + 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{216 \, a^{3} b^{2}} + \frac {5 \, {\left (\left (a b^{5}\right )^{\frac {1}{6}} B a + 11 \, \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \arctan \left (-\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} - 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{216 \, a^{3} b^{2}} + \frac {5 \, {\left (\left (a b^{5}\right )^{\frac {1}{6}} B a + 11 \, \left (a b^{5}\right )^{\frac {1}{6}} A b\right )} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{108 \, a^{3} b^{2}} + \frac {B a b x^{\frac {7}{2}} + 11 \, A b^{2} x^{\frac {7}{2}} - 5 \, B a^{2} \sqrt {x} + 17 \, A a b \sqrt {x}}{36 \, {\left (b x^{3} + a\right )}^{2} a^{2} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^3+A)/(b*x^3+a)^3/x^(1/2),x, algorithm="giac")

[Out]

5/432*sqrt(3)*((a*b^5)^(1/6)*B*a + 11*(a*b^5)^(1/6)*A*b)*log(sqrt(3)*sqrt(x)*(a/b)^(1/6) + x + (a/b)^(1/3))/(a
^3*b^2) - 5/432*sqrt(3)*((a*b^5)^(1/6)*B*a + 11*(a*b^5)^(1/6)*A*b)*log(-sqrt(3)*sqrt(x)*(a/b)^(1/6) + x + (a/b
)^(1/3))/(a^3*b^2) + 5/216*((a*b^5)^(1/6)*B*a + 11*(a*b^5)^(1/6)*A*b)*arctan((sqrt(3)*(a/b)^(1/6) + 2*sqrt(x))
/(a/b)^(1/6))/(a^3*b^2) + 5/216*((a*b^5)^(1/6)*B*a + 11*(a*b^5)^(1/6)*A*b)*arctan(-(sqrt(3)*(a/b)^(1/6) - 2*sq
rt(x))/(a/b)^(1/6))/(a^3*b^2) + 5/108*((a*b^5)^(1/6)*B*a + 11*(a*b^5)^(1/6)*A*b)*arctan(sqrt(x)/(a/b)^(1/6))/(
a^3*b^2) + 1/36*(B*a*b*x^(7/2) + 11*A*b^2*x^(7/2) - 5*B*a^2*sqrt(x) + 17*A*a*b*sqrt(x))/((b*x^3 + a)^2*a^2*b)

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maple [A]  time = 0.17, size = 407, normalized size = 1.27 \begin {gather*} \frac {55 \left (\frac {a}{b}\right )^{\frac {1}{6}} A \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{108 a^{3}}+\frac {55 \left (\frac {a}{b}\right )^{\frac {1}{6}} A \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{216 a^{3}}+\frac {55 \left (\frac {a}{b}\right )^{\frac {1}{6}} A \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{216 a^{3}}+\frac {55 \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} A \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{432 a^{3}}-\frac {55 \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} A \ln \left (-x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{432 a^{3}}+\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{6}} B \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{108 a^{2} b}+\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{6}} B \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{216 a^{2} b}+\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{6}} B \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{216 a^{2} b}+\frac {5 \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} B \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{432 a^{2} b}-\frac {5 \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} B \ln \left (-x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{432 a^{2} b}+\frac {\frac {\left (11 A b +B a \right ) x^{\frac {7}{2}}}{36 a^{2}}+\frac {\left (17 A b -5 B a \right ) \sqrt {x}}{36 a b}}{\left (b \,x^{3}+a \right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^3+A)/(b*x^3+a)^3/x^(1/2),x)

[Out]

2*(1/72*(11*A*b+B*a)/a^2*x^(7/2)+1/72*(17*A*b-5*B*a)/a/b*x^(1/2))/(b*x^3+a)^2+55/108/a^3*(a/b)^(1/6)*arctan(1/
(a/b)^(1/6)*x^(1/2))*A+5/108/a^2/b*(a/b)^(1/6)*arctan(1/(a/b)^(1/6)*x^(1/2))*B-55/432/a^3*3^(1/2)*(a/b)^(1/6)*
ln(-x+3^(1/2)*(a/b)^(1/6)*x^(1/2)-(a/b)^(1/3))*A-5/432/a^2/b*3^(1/2)*(a/b)^(1/6)*ln(-x+3^(1/2)*(a/b)^(1/6)*x^(
1/2)-(a/b)^(1/3))*B+55/216/a^3*(a/b)^(1/6)*arctan(2/(a/b)^(1/6)*x^(1/2)-3^(1/2))*A+5/216/a^2/b*(a/b)^(1/6)*arc
tan(2/(a/b)^(1/6)*x^(1/2)-3^(1/2))*B+55/432/a^3*3^(1/2)*(a/b)^(1/6)*ln(x+3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/
3))*A+5/432/a^2/b*3^(1/2)*(a/b)^(1/6)*ln(x+3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))*B+55/216/a^3*(a/b)^(1/6)*a
rctan(2/(a/b)^(1/6)*x^(1/2)+3^(1/2))*A+5/216/a^2/b*(a/b)^(1/6)*arctan(2/(a/b)^(1/6)*x^(1/2)+3^(1/2))*B

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maxima [A]  time = 1.21, size = 336, normalized size = 1.05 \begin {gather*} \frac {{\left (B a b + 11 \, A b^{2}\right )} x^{\frac {7}{2}} - {\left (5 \, B a^{2} - 17 \, A a b\right )} \sqrt {x}}{36 \, {\left (a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{3} + a^{4} b\right )}} + \frac {5 \, {\left (\frac {\sqrt {3} {\left (B a + 11 \, A b\right )} \log \left (\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} b^{\frac {1}{6}}} - \frac {\sqrt {3} {\left (B a + 11 \, A b\right )} \log \left (-\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} b^{\frac {1}{6}}} + \frac {4 \, {\left (B a b^{\frac {1}{3}} + 11 \, A b^{\frac {4}{3}}\right )} \arctan \left (\frac {b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} b^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, {\left (B a^{\frac {4}{3}} b^{\frac {1}{3}} + 11 \, A a^{\frac {1}{3}} b^{\frac {4}{3}}\right )} \arctan \left (\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} + 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a b^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, {\left (B a^{\frac {4}{3}} b^{\frac {1}{3}} + 11 \, A a^{\frac {1}{3}} b^{\frac {4}{3}}\right )} \arctan \left (-\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} - 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a b^{\frac {1}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}}{432 \, a^{2} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^3+A)/(b*x^3+a)^3/x^(1/2),x, algorithm="maxima")

[Out]

1/36*((B*a*b + 11*A*b^2)*x^(7/2) - (5*B*a^2 - 17*A*a*b)*sqrt(x))/(a^2*b^3*x^6 + 2*a^3*b^2*x^3 + a^4*b) + 5/432
*(sqrt(3)*(B*a + 11*A*b)*log(sqrt(3)*a^(1/6)*b^(1/6)*sqrt(x) + b^(1/3)*x + a^(1/3))/(a^(5/6)*b^(1/6)) - sqrt(3
)*(B*a + 11*A*b)*log(-sqrt(3)*a^(1/6)*b^(1/6)*sqrt(x) + b^(1/3)*x + a^(1/3))/(a^(5/6)*b^(1/6)) + 4*(B*a*b^(1/3
) + 11*A*b^(4/3))*arctan(b^(1/3)*sqrt(x)/sqrt(a^(1/3)*b^(1/3)))/(a^(2/3)*b^(1/3)*sqrt(a^(1/3)*b^(1/3))) + 2*(B
*a^(4/3)*b^(1/3) + 11*A*a^(1/3)*b^(4/3))*arctan((sqrt(3)*a^(1/6)*b^(1/6) + 2*b^(1/3)*sqrt(x))/sqrt(a^(1/3)*b^(
1/3)))/(a*b^(1/3)*sqrt(a^(1/3)*b^(1/3))) + 2*(B*a^(4/3)*b^(1/3) + 11*A*a^(1/3)*b^(4/3))*arctan(-(sqrt(3)*a^(1/
6)*b^(1/6) - 2*b^(1/3)*sqrt(x))/sqrt(a^(1/3)*b^(1/3)))/(a*b^(1/3)*sqrt(a^(1/3)*b^(1/3))))/(a^2*b)

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mupad [B]  time = 2.95, size = 1952, normalized size = 6.08

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^3)/(x^(1/2)*(a + b*x^3)^3),x)

[Out]

((x^(7/2)*(11*A*b + B*a))/(36*a^2) + (x^(1/2)*(17*A*b - 5*B*a))/(36*a*b))/(a^2 + b^2*x^6 + 2*a*b*x^3) - (atan(
((((625*x^(1/2)*(14641*A^4*b^5 + B^4*a^4*b + 726*A^2*B^2*a^2*b^3 + 5324*A^3*B*a*b^4 + 44*A*B^3*a^3*b^2))/(2799
36*a^8) - (625*(11*A*b + B*a)*(1331*A^3*b^5 + B^3*a^3*b^2 + 363*A^2*B*a*b^4 + 33*A*B^2*a^2*b^3))/(279936*(-a)^
(47/6)*b^(7/6)))*(11*A*b + B*a)*5i)/(216*(-a)^(17/6)*b^(7/6)) + (((625*x^(1/2)*(14641*A^4*b^5 + B^4*a^4*b + 72
6*A^2*B^2*a^2*b^3 + 5324*A^3*B*a*b^4 + 44*A*B^3*a^3*b^2))/(279936*a^8) + (625*(11*A*b + B*a)*(1331*A^3*b^5 + B
^3*a^3*b^2 + 363*A^2*B*a*b^4 + 33*A*B^2*a^2*b^3))/(279936*(-a)^(47/6)*b^(7/6)))*(11*A*b + B*a)*5i)/(216*(-a)^(
17/6)*b^(7/6)))/((5*((625*x^(1/2)*(14641*A^4*b^5 + B^4*a^4*b + 726*A^2*B^2*a^2*b^3 + 5324*A^3*B*a*b^4 + 44*A*B
^3*a^3*b^2))/(279936*a^8) - (625*(11*A*b + B*a)*(1331*A^3*b^5 + B^3*a^3*b^2 + 363*A^2*B*a*b^4 + 33*A*B^2*a^2*b
^3))/(279936*(-a)^(47/6)*b^(7/6)))*(11*A*b + B*a))/(216*(-a)^(17/6)*b^(7/6)) - (5*((625*x^(1/2)*(14641*A^4*b^5
+ B^4*a^4*b + 726*A^2*B^2*a^2*b^3 + 5324*A^3*B*a*b^4 + 44*A*B^3*a^3*b^2))/(279936*a^8) + (625*(11*A*b + B*a)*
(1331*A^3*b^5 + B^3*a^3*b^2 + 363*A^2*B*a*b^4 + 33*A*B^2*a^2*b^3))/(279936*(-a)^(47/6)*b^(7/6)))*(11*A*b + B*a
))/(216*(-a)^(17/6)*b^(7/6))))*(11*A*b + B*a)*5i)/(108*(-a)^(17/6)*b^(7/6)) - (atan(((((3^(1/2)*1i)/2 - 1/2)*(
11*A*b + B*a)*((625*x^(1/2)*(14641*A^4*b^5 + B^4*a^4*b + 726*A^2*B^2*a^2*b^3 + 5324*A^3*B*a*b^4 + 44*A*B^3*a^3
*b^2))/(279936*a^8) - (625*((3^(1/2)*1i)/2 - 1/2)*(11*A*b + B*a)*(1331*A^3*b^5 + B^3*a^3*b^2 + 363*A^2*B*a*b^4
+ 33*A*B^2*a^2*b^3))/(279936*(-a)^(47/6)*b^(7/6)))*5i)/(216*(-a)^(17/6)*b^(7/6)) + (((3^(1/2)*1i)/2 - 1/2)*(1
1*A*b + B*a)*((625*x^(1/2)*(14641*A^4*b^5 + B^4*a^4*b + 726*A^2*B^2*a^2*b^3 + 5324*A^3*B*a*b^4 + 44*A*B^3*a^3*
b^2))/(279936*a^8) + (625*((3^(1/2)*1i)/2 - 1/2)*(11*A*b + B*a)*(1331*A^3*b^5 + B^3*a^3*b^2 + 363*A^2*B*a*b^4
+ 33*A*B^2*a^2*b^3))/(279936*(-a)^(47/6)*b^(7/6)))*5i)/(216*(-a)^(17/6)*b^(7/6)))/((5*((3^(1/2)*1i)/2 - 1/2)*(
11*A*b + B*a)*((625*x^(1/2)*(14641*A^4*b^5 + B^4*a^4*b + 726*A^2*B^2*a^2*b^3 + 5324*A^3*B*a*b^4 + 44*A*B^3*a^3
*b^2))/(279936*a^8) - (625*((3^(1/2)*1i)/2 - 1/2)*(11*A*b + B*a)*(1331*A^3*b^5 + B^3*a^3*b^2 + 363*A^2*B*a*b^4
+ 33*A*B^2*a^2*b^3))/(279936*(-a)^(47/6)*b^(7/6))))/(216*(-a)^(17/6)*b^(7/6)) - (5*((3^(1/2)*1i)/2 - 1/2)*(11
*A*b + B*a)*((625*x^(1/2)*(14641*A^4*b^5 + B^4*a^4*b + 726*A^2*B^2*a^2*b^3 + 5324*A^3*B*a*b^4 + 44*A*B^3*a^3*b
^2))/(279936*a^8) + (625*((3^(1/2)*1i)/2 - 1/2)*(11*A*b + B*a)*(1331*A^3*b^5 + B^3*a^3*b^2 + 363*A^2*B*a*b^4 +
33*A*B^2*a^2*b^3))/(279936*(-a)^(47/6)*b^(7/6))))/(216*(-a)^(17/6)*b^(7/6))))*((3^(1/2)*1i)/2 - 1/2)*(11*A*b
+ B*a)*5i)/(108*(-a)^(17/6)*b^(7/6)) - (atan(((((3^(1/2)*1i)/2 + 1/2)*(11*A*b + B*a)*((625*x^(1/2)*(14641*A^4*
b^5 + B^4*a^4*b + 726*A^2*B^2*a^2*b^3 + 5324*A^3*B*a*b^4 + 44*A*B^3*a^3*b^2))/(279936*a^8) - (625*((3^(1/2)*1i
)/2 + 1/2)*(11*A*b + B*a)*(1331*A^3*b^5 + B^3*a^3*b^2 + 363*A^2*B*a*b^4 + 33*A*B^2*a^2*b^3))/(279936*(-a)^(47/
6)*b^(7/6)))*5i)/(216*(-a)^(17/6)*b^(7/6)) + (((3^(1/2)*1i)/2 + 1/2)*(11*A*b + B*a)*((625*x^(1/2)*(14641*A^4*b
^5 + B^4*a^4*b + 726*A^2*B^2*a^2*b^3 + 5324*A^3*B*a*b^4 + 44*A*B^3*a^3*b^2))/(279936*a^8) + (625*((3^(1/2)*1i)
/2 + 1/2)*(11*A*b + B*a)*(1331*A^3*b^5 + B^3*a^3*b^2 + 363*A^2*B*a*b^4 + 33*A*B^2*a^2*b^3))/(279936*(-a)^(47/6
)*b^(7/6)))*5i)/(216*(-a)^(17/6)*b^(7/6)))/((5*((3^(1/2)*1i)/2 + 1/2)*(11*A*b + B*a)*((625*x^(1/2)*(14641*A^4*
b^5 + B^4*a^4*b + 726*A^2*B^2*a^2*b^3 + 5324*A^3*B*a*b^4 + 44*A*B^3*a^3*b^2))/(279936*a^8) - (625*((3^(1/2)*1i
)/2 + 1/2)*(11*A*b + B*a)*(1331*A^3*b^5 + B^3*a^3*b^2 + 363*A^2*B*a*b^4 + 33*A*B^2*a^2*b^3))/(279936*(-a)^(47/
6)*b^(7/6))))/(216*(-a)^(17/6)*b^(7/6)) - (5*((3^(1/2)*1i)/2 + 1/2)*(11*A*b + B*a)*((625*x^(1/2)*(14641*A^4*b^
5 + B^4*a^4*b + 726*A^2*B^2*a^2*b^3 + 5324*A^3*B*a*b^4 + 44*A*B^3*a^3*b^2))/(279936*a^8) + (625*((3^(1/2)*1i)/
2 + 1/2)*(11*A*b + B*a)*(1331*A^3*b^5 + B^3*a^3*b^2 + 363*A^2*B*a*b^4 + 33*A*B^2*a^2*b^3))/(279936*(-a)^(47/6)
*b^(7/6))))/(216*(-a)^(17/6)*b^(7/6))))*((3^(1/2)*1i)/2 + 1/2)*(11*A*b + B*a)*5i)/(108*(-a)^(17/6)*b^(7/6))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**3+A)/(b*x**3+a)**3/x**(1/2),x)

[Out]

Timed out

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