3.2.66 \(\int \frac {A+B x^3}{x^{7/2} (a+b x^3)^2} \, dx\)

Optimal. Leaf size=318 \[ \frac {(11 A b-5 a B) \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 a B) \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{17/6} \sqrt [6]{b}}+\frac {(11 A b-5 a B) \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{18 a^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 a B) \tan ^{-1}\left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}+\sqrt {3}\right )}{18 a^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{9 a^{17/6} \sqrt [6]{b}}-\frac {11 A b-5 a B}{15 a^2 b x^{5/2}}+\frac {A b-a B}{3 a b x^{5/2} \left (a+b x^3\right )} \]

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Rubi [A]  time = 0.50, antiderivative size = 318, 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, 325, 329, 209, 634, 618, 204, 628, 205} \begin {gather*} -\frac {11 A b-5 a B}{15 a^2 b x^{5/2}}+\frac {(11 A b-5 a B) \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 a B) \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{17/6} \sqrt [6]{b}}+\frac {(11 A b-5 a B) \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{18 a^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 a B) \tan ^{-1}\left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}+\sqrt {3}\right )}{18 a^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{9 a^{17/6} \sqrt [6]{b}}+\frac {A b-a B}{3 a b x^{5/2} \left (a+b x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x^3)/(x^(7/2)*(a + b*x^3)^2),x]

[Out]

-(11*A*b - 5*a*B)/(15*a^2*b*x^(5/2)) + (A*b - a*B)/(3*a*b*x^(5/2)*(a + b*x^3)) + ((11*A*b - 5*a*B)*ArcTan[Sqrt
[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(18*a^(17/6)*b^(1/6)) - ((11*A*b - 5*a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt
[x])/a^(1/6)])/(18*a^(17/6)*b^(1/6)) - ((11*A*b - 5*a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(9*a^(17/6)*b^(1/6
)) + ((11*A*b - 5*a*B)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(12*Sqrt[3]*a^(17/6)*b^(1/6
)) - ((11*A*b - 5*a*B)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(12*Sqrt[3]*a^(17/6)*b^(1/6
))

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 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 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}{x^{7/2} \left (a+b x^3\right )^2} \, dx &=\frac {A b-a B}{3 a b x^{5/2} \left (a+b x^3\right )}+\frac {\left (\frac {11 A b}{2}-\frac {5 a B}{2}\right ) \int \frac {1}{x^{7/2} \left (a+b x^3\right )} \, dx}{3 a b}\\ &=-\frac {11 A b-5 a B}{15 a^2 b x^{5/2}}+\frac {A b-a B}{3 a b x^{5/2} \left (a+b x^3\right )}-\frac {(11 A b-5 a B) \int \frac {1}{\sqrt {x} \left (a+b x^3\right )} \, dx}{6 a^2}\\ &=-\frac {11 A b-5 a B}{15 a^2 b x^{5/2}}+\frac {A b-a B}{3 a b x^{5/2} \left (a+b x^3\right )}-\frac {(11 A b-5 a B) \operatorname {Subst}\left (\int \frac {1}{a+b x^6} \, dx,x,\sqrt {x}\right )}{3 a^2}\\ &=-\frac {11 A b-5 a B}{15 a^2 b x^{5/2}}+\frac {A b-a B}{3 a b x^{5/2} \left (a+b x^3\right )}-\frac {(11 A b-5 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 )}{9 a^{17/6}}-\frac {(11 A b-5 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 )}{9 a^{17/6}}-\frac {(11 A b-5 a B) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx,x,\sqrt {x}\right )}{9 a^{8/3}}\\ &=-\frac {11 A b-5 a B}{15 a^2 b x^{5/2}}+\frac {A b-a B}{3 a b x^{5/2} \left (a+b x^3\right )}-\frac {(11 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{9 a^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 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 )}{36 a^{8/3}}-\frac {(11 A b-5 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 )}{36 a^{8/3}}+\frac {(11 A b-5 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 )}{12 \sqrt {3} a^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 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 )}{12 \sqrt {3} a^{17/6} \sqrt [6]{b}}\\ &=-\frac {11 A b-5 a B}{15 a^2 b x^{5/2}}+\frac {A b-a B}{3 a b x^{5/2} \left (a+b x^3\right )}-\frac {(11 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{9 a^{17/6} \sqrt [6]{b}}+\frac {(11 A b-5 a B) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 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 )}{18 \sqrt {3} a^{17/6} \sqrt [6]{b}}+\frac {(11 A b-5 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 )}{18 \sqrt {3} a^{17/6} \sqrt [6]{b}}\\ &=-\frac {11 A b-5 a B}{15 a^2 b x^{5/2}}+\frac {A b-a B}{3 a b x^{5/2} \left (a+b x^3\right )}+\frac {(11 A b-5 a B) \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{18 a^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 a B) \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{18 a^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{9 a^{17/6} \sqrt [6]{b}}+\frac {(11 A b-5 a B) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{17/6} \sqrt [6]{b}}-\frac {(11 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{17/6} \sqrt [6]{b}}\\ \end {align*}

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Mathematica [C]  time = 0.11, size = 74, normalized size = 0.23 \begin {gather*} \frac {5 x^3 (5 a B-11 A b) \, _2F_1\left (\frac {1}{6},1;\frac {7}{6};-\frac {b x^3}{a}\right )+\frac {a \left (-6 a A+5 a B x^3-11 A b x^3\right )}{a+b x^3}}{15 a^3 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^3)/(x^(7/2)*(a + b*x^3)^2),x]

[Out]

((a*(-6*a*A - 11*A*b*x^3 + 5*a*B*x^3))/(a + b*x^3) + 5*(-11*A*b + 5*a*B)*x^3*Hypergeometric2F1[1/6, 1, 7/6, -(
(b*x^3)/a)])/(15*a^3*x^(5/2))

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IntegrateAlgebraic [A]  time = 0.60, size = 200, normalized size = 0.63 \begin {gather*} \frac {(5 a B-11 A b) \tan ^{-1}\left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{9 a^{17/6} \sqrt [6]{b}}+\frac {(11 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}\right )}{18 a^{17/6} \sqrt [6]{b}}+\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 )}{6 \sqrt {3} a^{17/6} \sqrt [6]{b}}+\frac {-6 a A+5 a B x^3-11 A b x^3}{15 a^2 x^{5/2} \left (a+b x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x^3)/(x^(7/2)*(a + b*x^3)^2),x]

[Out]

(-6*a*A - 11*A*b*x^3 + 5*a*B*x^3)/(15*a^2*x^(5/2)*(a + b*x^3)) + ((-11*A*b + 5*a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a
^(1/6)])/(9*a^(17/6)*b^(1/6)) + ((11*A*b - 5*a*B)*ArcTan[(a^(1/3) - b^(1/3)*x)/(a^(1/6)*b^(1/6)*Sqrt[x])])/(18
*a^(17/6)*b^(1/6)) + ((-11*A*b + 5*a*B)*ArcTanh[(Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x])/(a^(1/3) + b^(1/3)*x)])/(6*S
qrt[3]*a^(17/6)*b^(1/6))

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fricas [B]  time = 1.09, size = 2584, normalized size = 8.13

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^3+A)/x^(7/2)/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

-1/180*(20*sqrt(3)*(a^2*b*x^6 + a^3*x^3)*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2 - 332
7500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/6)*arctan
(1/3*(2*sqrt(3)*sqrt(a^6*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3
*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/3) + (25*B^2*a^2 - 110*A*
B*a*b + 121*A^2*b^2)*x + (5*B*a^4 - 11*A*a^3*b)*sqrt(x)*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*B^
4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b
))^(1/6))*a^14*b*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5
490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(5/6) + 2*sqrt(3)*(5*B*a^15*b - 11*A
*a^14*b^2)*sqrt(x)*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 +
5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(5/6) - sqrt(3)*(15625*B^6*a^6 - 2
06250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*
B*a*b^5 + 1771561*A^6*b^6))/(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^
3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b^5 + 1771561*A^6*b^6)) + 20*sqrt(3)*(a^2*b*x^6 + a^3*x^3)*(
-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2
*b^4 - 4831530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/6)*arctan(1/3*(2*sqrt(3)*sqrt(a^6*(-(15625*B^6*a^6
- 206250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A
^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/3) + (25*B^2*a^2 - 110*A*B*a*b + 121*A^2*b^2)*x - (5*B*a^4 - 11*A*a
^3*b)*sqrt(x)*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490
375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/6))*a^14*b*(-(15625*B^6*a^6 - 206250
*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b
^5 + 1771561*A^6*b^6)/(a^17*b))^(5/6) + 2*sqrt(3)*(5*B*a^15*b - 11*A*a^14*b^2)*sqrt(x)*(-(15625*B^6*a^6 - 2062
50*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a
*b^5 + 1771561*A^6*b^6)/(a^17*b))^(5/6) + sqrt(3)*(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^
2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b^5 + 1771561*A^6*b^6))/(15625*B^6*a^6
- 206250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*
A^5*B*a*b^5 + 1771561*A^6*b^6)) - 5*(a^2*b*x^6 + a^3*x^3)*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*
B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17
*b))^(1/6)*log(a^6*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 +
5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/3) + (25*B^2*a^2 - 110*A*B*a*b
+ 121*A^2*b^2)*x + (5*B*a^4 - 11*A*a^3*b)*sqrt(x)*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*
b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/
6)) + 5*(a^2*b*x^6 + a^3*x^3)*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^
3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/6)*log(a^6*(-(15625*
B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4
831530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/3) + (25*B^2*a^2 - 110*A*B*a*b + 121*A^2*b^2)*x - (5*B*a^4
- 11*A*a^3*b)*sqrt(x)*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^
3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/6)) + 10*(a^2*b*x^6 + a^3*x^
3)*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2
*a^2*b^4 - 4831530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/6)*log(a^3*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*
b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b^5 + 177156
1*A^6*b^6)/(a^17*b))^(1/6) - (5*B*a - 11*A*b)*sqrt(x)) - 10*(a^2*b*x^6 + a^3*x^3)*(-(15625*B^6*a^6 - 206250*A*
B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b^5
+ 1771561*A^6*b^6)/(a^17*b))^(1/6)*log(-a^3*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2 -
3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/6) - (
5*B*a - 11*A*b)*sqrt(x)) - 12*((5*B*a - 11*A*b)*x^3 - 6*A*a)*sqrt(x))/(a^2*b*x^6 + a^3*x^3)

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giac [A]  time = 0.22, size = 313, normalized size = 0.98 \begin {gather*} \frac {\sqrt {3} {\left (5 \, \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 )}{36 \, a^{3} b} - \frac {\sqrt {3} {\left (5 \, \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 )}{36 \, a^{3} b} + \frac {B a \sqrt {x} - A b \sqrt {x}}{3 \, {\left (b x^{3} + a\right )} a^{2}} + \frac {{\left (5 \, \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 )}{18 \, a^{3} b} + \frac {{\left (5 \, \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 )}{18 \, a^{3} b} + \frac {{\left (5 \, \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 )}{9 \, a^{3} b} - \frac {2 \, A}{5 \, a^{2} x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^3+A)/x^(7/2)/(b*x^3+a)^2,x, algorithm="giac")

[Out]

1/36*sqrt(3)*(5*(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) - 1/36*sqrt(3)*(5*(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) + 1/3*(B*a*sqrt(x) - A*b*sqrt(x))/((b*x^3 + a)*a^2) + 1/18*(5*(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) + 1/18*(5*(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) + 1/9*(5*(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) - 2/5*A/(a^2*x^(5/2))

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maple [A]  time = 0.16, size = 395, normalized size = 1.24 \begin {gather*} -\frac {A b \sqrt {x}}{3 \left (b \,x^{3}+a \right ) a^{2}}+\frac {B \sqrt {x}}{3 \left (b \,x^{3}+a \right ) a}-\frac {11 \left (\frac {a}{b}\right )^{\frac {1}{6}} A b \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{9 a^{3}}-\frac {11 \left (\frac {a}{b}\right )^{\frac {1}{6}} A b \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{18 a^{3}}-\frac {11 \left (\frac {a}{b}\right )^{\frac {1}{6}} A b \arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{18 a^{3}}-\frac {11 \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} A 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 )}{36 a^{3}}+\frac {11 \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} A 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 )}{36 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 )}{9 a^{2}}+\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 )}{18 a^{2}}+\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 )}{18 a^{2}}+\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 )}{36 a^{2}}-\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 )}{36 a^{2}}-\frac {2 A}{5 a^{2} x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^3+A)/x^(7/2)/(b*x^3+a)^2,x)

[Out]

-1/3/a^2*x^(1/2)/(b*x^3+a)*A*b+1/3/a*x^(1/2)/(b*x^3+a)*B-11/9/a^3*A*b*(a/b)^(1/6)*arctan(1/(a/b)^(1/6)*x^(1/2)
)+11/36/a^3*A*b*3^(1/2)*(a/b)^(1/6)*ln(-x+3^(1/2)*(a/b)^(1/6)*x^(1/2)-(a/b)^(1/3))-11/18/a^3*A*b*(a/b)^(1/6)*a
rctan(2/(a/b)^(1/6)*x^(1/2)-3^(1/2))-11/36/a^3*A*b*3^(1/2)*(a/b)^(1/6)*ln(x+3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^
(1/3))-11/18/a^3*A*b*(a/b)^(1/6)*arctan(2/(a/b)^(1/6)*x^(1/2)+3^(1/2))+5/9/a^2*B*(a/b)^(1/6)*arctan(1/(a/b)^(1
/6)*x^(1/2))-5/36/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))+5/18/a^2*B*(a/b)^(1
/6)*arctan(2/(a/b)^(1/6)*x^(1/2)-3^(1/2))+5/36/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))+5/18/a^2*B*(a/b)^(1/6)*arctan(2/(a/b)^(1/6)*x^(1/2)+3^(1/2))-2/5/a^2*A/x^(5/2)

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maxima [A]  time = 1.19, size = 312, normalized size = 0.98 \begin {gather*} \frac {{\left (5 \, B a - 11 \, A b\right )} x^{3} - 6 \, A a}{15 \, {\left (a^{2} b x^{\frac {11}{2}} + a^{3} x^{\frac {5}{2}}\right )}} + \frac {\frac {\sqrt {3} {\left (5 \, 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 (5 \, 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 (5 \, 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 (5 \, 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 (5 \, 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}}}}}{36 \, a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^3+A)/x^(7/2)/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

1/15*((5*B*a - 11*A*b)*x^3 - 6*A*a)/(a^2*b*x^(11/2) + a^3*x^(5/2)) + 1/36*(sqrt(3)*(5*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)*(5*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*(5*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*(5*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*(5*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

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mupad [B]  time = 2.96, size = 2080, normalized size = 6.54

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^3)/(x^(7/2)*(a + b*x^3)^2),x)

[Out]

- ((2*A)/(5*a) + (x^3*(11*A*b - 5*B*a))/(15*a^2))/(a*x^(5/2) + b*x^(11/2)) - (atan((((x^(1/2)*(21346578*A^4*a^
10*b^9 + 911250*B^4*a^14*b^5 + 26462700*A^2*B^2*a^12*b^7 - 8019000*A*B^3*a^13*b^6 - 38811960*A^3*B*a^11*b^8) -
((11*A*b - 5*B*a)*(34930764*A^3*a^13*b^8 - 3280500*B^3*a^16*b^5 + 21651300*A*B^2*a^15*b^6 - 47632860*A^2*B*a^
14*b^7))/(18*(-a)^(17/6)*b^(1/6)))*(11*A*b - 5*B*a)*1i)/(18*(-a)^(17/6)*b^(1/6)) + ((x^(1/2)*(21346578*A^4*a^1
0*b^9 + 911250*B^4*a^14*b^5 + 26462700*A^2*B^2*a^12*b^7 - 8019000*A*B^3*a^13*b^6 - 38811960*A^3*B*a^11*b^8) +
((11*A*b - 5*B*a)*(34930764*A^3*a^13*b^8 - 3280500*B^3*a^16*b^5 + 21651300*A*B^2*a^15*b^6 - 47632860*A^2*B*a^1
4*b^7))/(18*(-a)^(17/6)*b^(1/6)))*(11*A*b - 5*B*a)*1i)/(18*(-a)^(17/6)*b^(1/6)))/(((x^(1/2)*(21346578*A^4*a^10
*b^9 + 911250*B^4*a^14*b^5 + 26462700*A^2*B^2*a^12*b^7 - 8019000*A*B^3*a^13*b^6 - 38811960*A^3*B*a^11*b^8) - (
(11*A*b - 5*B*a)*(34930764*A^3*a^13*b^8 - 3280500*B^3*a^16*b^5 + 21651300*A*B^2*a^15*b^6 - 47632860*A^2*B*a^14
*b^7))/(18*(-a)^(17/6)*b^(1/6)))*(11*A*b - 5*B*a))/(18*(-a)^(17/6)*b^(1/6)) - ((x^(1/2)*(21346578*A^4*a^10*b^9
+ 911250*B^4*a^14*b^5 + 26462700*A^2*B^2*a^12*b^7 - 8019000*A*B^3*a^13*b^6 - 38811960*A^3*B*a^11*b^8) + ((11*
A*b - 5*B*a)*(34930764*A^3*a^13*b^8 - 3280500*B^3*a^16*b^5 + 21651300*A*B^2*a^15*b^6 - 47632860*A^2*B*a^14*b^7
))/(18*(-a)^(17/6)*b^(1/6)))*(11*A*b - 5*B*a))/(18*(-a)^(17/6)*b^(1/6))))*(11*A*b - 5*B*a)*1i)/(9*(-a)^(17/6)*
b^(1/6)) - (atan(((((3^(1/2)*1i)/2 - 1/2)*(x^(1/2)*(21346578*A^4*a^10*b^9 + 911250*B^4*a^14*b^5 + 26462700*A^2
*B^2*a^12*b^7 - 8019000*A*B^3*a^13*b^6 - 38811960*A^3*B*a^11*b^8) - (((3^(1/2)*1i)/2 - 1/2)*(11*A*b - 5*B*a)*(
34930764*A^3*a^13*b^8 - 3280500*B^3*a^16*b^5 + 21651300*A*B^2*a^15*b^6 - 47632860*A^2*B*a^14*b^7))/(18*(-a)^(1
7/6)*b^(1/6)))*(11*A*b - 5*B*a)*1i)/(18*(-a)^(17/6)*b^(1/6)) + (((3^(1/2)*1i)/2 - 1/2)*(x^(1/2)*(21346578*A^4*
a^10*b^9 + 911250*B^4*a^14*b^5 + 26462700*A^2*B^2*a^12*b^7 - 8019000*A*B^3*a^13*b^6 - 38811960*A^3*B*a^11*b^8)
+ (((3^(1/2)*1i)/2 - 1/2)*(11*A*b - 5*B*a)*(34930764*A^3*a^13*b^8 - 3280500*B^3*a^16*b^5 + 21651300*A*B^2*a^1
5*b^6 - 47632860*A^2*B*a^14*b^7))/(18*(-a)^(17/6)*b^(1/6)))*(11*A*b - 5*B*a)*1i)/(18*(-a)^(17/6)*b^(1/6)))/(((
(3^(1/2)*1i)/2 - 1/2)*(x^(1/2)*(21346578*A^4*a^10*b^9 + 911250*B^4*a^14*b^5 + 26462700*A^2*B^2*a^12*b^7 - 8019
000*A*B^3*a^13*b^6 - 38811960*A^3*B*a^11*b^8) - (((3^(1/2)*1i)/2 - 1/2)*(11*A*b - 5*B*a)*(34930764*A^3*a^13*b^
8 - 3280500*B^3*a^16*b^5 + 21651300*A*B^2*a^15*b^6 - 47632860*A^2*B*a^14*b^7))/(18*(-a)^(17/6)*b^(1/6)))*(11*A
*b - 5*B*a))/(18*(-a)^(17/6)*b^(1/6)) - (((3^(1/2)*1i)/2 - 1/2)*(x^(1/2)*(21346578*A^4*a^10*b^9 + 911250*B^4*a
^14*b^5 + 26462700*A^2*B^2*a^12*b^7 - 8019000*A*B^3*a^13*b^6 - 38811960*A^3*B*a^11*b^8) + (((3^(1/2)*1i)/2 - 1
/2)*(11*A*b - 5*B*a)*(34930764*A^3*a^13*b^8 - 3280500*B^3*a^16*b^5 + 21651300*A*B^2*a^15*b^6 - 47632860*A^2*B*
a^14*b^7))/(18*(-a)^(17/6)*b^(1/6)))*(11*A*b - 5*B*a))/(18*(-a)^(17/6)*b^(1/6))))*((3^(1/2)*1i)/2 - 1/2)*(11*A
*b - 5*B*a)*1i)/(9*(-a)^(17/6)*b^(1/6)) - (atan(((((3^(1/2)*1i)/2 + 1/2)*(x^(1/2)*(21346578*A^4*a^10*b^9 + 911
250*B^4*a^14*b^5 + 26462700*A^2*B^2*a^12*b^7 - 8019000*A*B^3*a^13*b^6 - 38811960*A^3*B*a^11*b^8) - (((3^(1/2)*
1i)/2 + 1/2)*(11*A*b - 5*B*a)*(34930764*A^3*a^13*b^8 - 3280500*B^3*a^16*b^5 + 21651300*A*B^2*a^15*b^6 - 476328
60*A^2*B*a^14*b^7))/(18*(-a)^(17/6)*b^(1/6)))*(11*A*b - 5*B*a)*1i)/(18*(-a)^(17/6)*b^(1/6)) + (((3^(1/2)*1i)/2
+ 1/2)*(x^(1/2)*(21346578*A^4*a^10*b^9 + 911250*B^4*a^14*b^5 + 26462700*A^2*B^2*a^12*b^7 - 8019000*A*B^3*a^13
*b^6 - 38811960*A^3*B*a^11*b^8) + (((3^(1/2)*1i)/2 + 1/2)*(11*A*b - 5*B*a)*(34930764*A^3*a^13*b^8 - 3280500*B^
3*a^16*b^5 + 21651300*A*B^2*a^15*b^6 - 47632860*A^2*B*a^14*b^7))/(18*(-a)^(17/6)*b^(1/6)))*(11*A*b - 5*B*a)*1i
)/(18*(-a)^(17/6)*b^(1/6)))/((((3^(1/2)*1i)/2 + 1/2)*(x^(1/2)*(21346578*A^4*a^10*b^9 + 911250*B^4*a^14*b^5 + 2
6462700*A^2*B^2*a^12*b^7 - 8019000*A*B^3*a^13*b^6 - 38811960*A^3*B*a^11*b^8) - (((3^(1/2)*1i)/2 + 1/2)*(11*A*b
- 5*B*a)*(34930764*A^3*a^13*b^8 - 3280500*B^3*a^16*b^5 + 21651300*A*B^2*a^15*b^6 - 47632860*A^2*B*a^14*b^7))/
(18*(-a)^(17/6)*b^(1/6)))*(11*A*b - 5*B*a))/(18*(-a)^(17/6)*b^(1/6)) - (((3^(1/2)*1i)/2 + 1/2)*(x^(1/2)*(21346
578*A^4*a^10*b^9 + 911250*B^4*a^14*b^5 + 26462700*A^2*B^2*a^12*b^7 - 8019000*A*B^3*a^13*b^6 - 38811960*A^3*B*a
^11*b^8) + (((3^(1/2)*1i)/2 + 1/2)*(11*A*b - 5*B*a)*(34930764*A^3*a^13*b^8 - 3280500*B^3*a^16*b^5 + 21651300*A
*B^2*a^15*b^6 - 47632860*A^2*B*a^14*b^7))/(18*(-a)^(17/6)*b^(1/6)))*(11*A*b - 5*B*a))/(18*(-a)^(17/6)*b^(1/6))
))*((3^(1/2)*1i)/2 + 1/2)*(11*A*b - 5*B*a)*1i)/(9*(-a)^(17/6)*b^(1/6))

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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)/x**(7/2)/(b*x**3+a)**2,x)

[Out]

Timed out

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