3.2.65 \(\int \frac {x^{3/2} (A+B x^3)}{(a+b x^3)^2} \, dx\) [165]

3.2.65.1 Optimal result
3.2.65.2 Mathematica [A] (verified)
3.2.65.3 Rubi [A] (verified)
3.2.65.4 Maple [A] (verified)
3.2.65.5 Fricas [B] (verification not implemented)
3.2.65.6 Sympy [B] (verification not implemented)
3.2.65.7 Maxima [A] (verification not implemented)
3.2.65.8 Giac [A] (verification not implemented)
3.2.65.9 Mupad [B] (verification not implemented)

3.2.65.1 Optimal result

Integrand size = 22, antiderivative size = 289 \[ \int \frac {x^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx=\frac {(A b-a B) x^{5/2}}{3 a b \left (a+b x^3\right )}-\frac {(A b+5 a B) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{11/6}}+\frac {(A b+5 a B) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{18 a^{7/6} b^{11/6}}+\frac {(A b+5 a B) \arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{9 a^{7/6} b^{11/6}}+\frac {(A b+5 a B) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{7/6} b^{11/6}}-\frac {(A b+5 a B) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{b} x\right )}{12 \sqrt {3} a^{7/6} b^{11/6}} \]

output
1/3*(A*b-B*a)*x^(5/2)/a/b/(b*x^3+a)+1/9*(A*b+5*B*a)*arctan(b^(1/6)*x^(1/2) 
/a^(1/6))/a^(7/6)/b^(11/6)+1/18*(A*b+5*B*a)*arctan(-3^(1/2)+2*b^(1/6)*x^(1 
/2)/a^(1/6))/a^(7/6)/b^(11/6)+1/18*(A*b+5*B*a)*arctan(3^(1/2)+2*b^(1/6)*x^ 
(1/2)/a^(1/6))/a^(7/6)/b^(11/6)+1/36*(A*b+5*B*a)*ln(a^(1/3)+b^(1/3)*x-a^(1 
/6)*b^(1/6)*3^(1/2)*x^(1/2))/a^(7/6)/b^(11/6)*3^(1/2)-1/36*(A*b+5*B*a)*ln( 
a^(1/3)+b^(1/3)*x+a^(1/6)*b^(1/6)*3^(1/2)*x^(1/2))/a^(7/6)/b^(11/6)*3^(1/2 
)
 
3.2.65.2 Mathematica [A] (verified)

Time = 0.81 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.58 \[ \int \frac {x^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx=\frac {-\frac {6 \sqrt [6]{a} b^{5/6} (-A b+a B) x^{5/2}}{a+b x^3}+2 (A b+5 a B) \arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )-(A b+5 a B) \arctan \left (\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}\right )-\sqrt {3} (A b+5 a B) \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{a}+\sqrt [3]{b} x}\right )}{18 a^{7/6} b^{11/6}} \]

input
Integrate[(x^(3/2)*(A + B*x^3))/(a + b*x^3)^2,x]
 
output
((-6*a^(1/6)*b^(5/6)*(-(A*b) + a*B)*x^(5/2))/(a + b*x^3) + 2*(A*b + 5*a*B) 
*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)] - (A*b + 5*a*B)*ArcTan[(a^(1/3) - b^(1/ 
3)*x)/(a^(1/6)*b^(1/6)*Sqrt[x])] - Sqrt[3]*(A*b + 5*a*B)*ArcTanh[(Sqrt[3]* 
a^(1/6)*b^(1/6)*Sqrt[x])/(a^(1/3) + b^(1/3)*x)])/(18*a^(7/6)*b^(11/6))
 
3.2.65.3 Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {957, 851, 824, 27, 218, 1142, 25, 27, 1082, 217, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx\)

\(\Big \downarrow \) 957

\(\displaystyle \frac {(5 a B+A b) \int \frac {x^{3/2}}{b x^3+a}dx}{6 a b}+\frac {x^{5/2} (A b-a B)}{3 a b \left (a+b x^3\right )}\)

\(\Big \downarrow \) 851

\(\displaystyle \frac {(5 a B+A b) \int \frac {x^2}{b x^3+a}d\sqrt {x}}{3 a b}+\frac {x^{5/2} (A b-a B)}{3 a b \left (a+b x^3\right )}\)

\(\Big \downarrow \) 824

\(\displaystyle \frac {(5 a B+A b) \left (\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}d\sqrt {x}}{3 b^{2/3}}+\frac {\int -\frac {\sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} \sqrt {x}}{2 \left (\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}\right )}d\sqrt {x}}{3 \sqrt [6]{a} b^{2/3}}+\frac {\int -\frac {\sqrt {3} \sqrt [6]{b} \sqrt {x}+\sqrt [6]{a}}{2 \left (\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}\right )}d\sqrt {x}}{3 \sqrt [6]{a} b^{2/3}}\right )}{3 a b}+\frac {x^{5/2} (A b-a B)}{3 a b \left (a+b x^3\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(5 a B+A b) \left (\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}d\sqrt {x}}{3 b^{2/3}}-\frac {\int \frac {\sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\int \frac {\sqrt {3} \sqrt [6]{b} \sqrt {x}+\sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}\right )}{3 a b}+\frac {x^{5/2} (A b-a B)}{3 a b \left (a+b x^3\right )}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {(5 a B+A b) \left (-\frac {\int \frac {\sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\int \frac {\sqrt {3} \sqrt [6]{b} \sqrt {x}+\sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{3 a b}+\frac {x^{5/2} (A b-a B)}{3 a b \left (a+b x^3\right )}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {(5 a B+A b) \left (-\frac {-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {\sqrt {3} \int -\frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}\right )}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{3 a b}+\frac {x^{5/2} (A b-a B)}{3 a b \left (a+b x^3\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(5 a B+A b) \left (-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}\right )}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{3 a b}+\frac {x^{5/2} (A b-a B)}{3 a b \left (a+b x^3\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(5 a B+A b) \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{3 a b}+\frac {x^{5/2} (A b-a B)}{3 a b \left (a+b x^3\right )}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {(5 a B+A b) \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {\int \frac {1}{-x-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )}{\sqrt {3} \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {\int \frac {1}{-x-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}+1\right )}{\sqrt {3} \sqrt [6]{b}}+\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{3 a b}+\frac {x^{5/2} (A b-a B)}{3 a b \left (a+b x^3\right )}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {(5 a B+A b) \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} \sqrt {x}}{\sqrt [3]{b} x-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}+\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} \sqrt {x}+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}}d\sqrt {x}-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}+1\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{3 a b}+\frac {x^{5/2} (A b-a B)}{3 a b \left (a+b x^3\right )}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {(5 a B+A b) \left (\frac {\arctan \left (\frac {\sqrt [6]{b} \sqrt {x}}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}-\frac {\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}\right )\right )}{\sqrt [6]{b}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} \sqrt {x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 \sqrt [6]{b}}-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} \sqrt {x}}{\sqrt {3} \sqrt [6]{a}}+1\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}\right )}{3 a b}+\frac {x^{5/2} (A b-a B)}{3 a b \left (a+b x^3\right )}\)

input
Int[(x^(3/2)*(A + B*x^3))/(a + b*x^3)^2,x]
 
output
((A*b - a*B)*x^(5/2))/(3*a*b*(a + b*x^3)) + ((A*b + 5*a*B)*(ArcTan[(b^(1/6 
)*Sqrt[x])/a^(1/6)]/(3*a^(1/6)*b^(5/6)) - (ArcTan[Sqrt[3]*(1 - (2*b^(1/6)* 
Sqrt[x])/(Sqrt[3]*a^(1/6)))]/b^(1/6) - (Sqrt[3]*Log[a^(1/3) - Sqrt[3]*a^(1 
/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(2*b^(1/6)))/(6*a^(1/6)*b^(2/3)) - (-(Ar 
cTan[Sqrt[3]*(1 + (2*b^(1/6)*Sqrt[x])/(Sqrt[3]*a^(1/6)))]/b^(1/6)) + (Sqrt 
[3]*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(2*b^(1/6) 
))/(6*a^(1/6)*b^(2/3))))/(3*a*b)
 

3.2.65.3.1 Defintions of rubi rules used

rule 25
Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27
Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 218
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 824
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator 
[Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k 
- 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 
 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k 
- 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] 
; 2*(-1)^(m/2)*(r^(m + 2)/(a*n*s^m))   Int[1/(r^2 + s^2*x^2), x] + 2*(r^(m 
+ 1)/(a*n*s^m))   Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGt 
Q[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]
 

rule 851
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = 
 Denominator[m]}, Simp[k/c   Subst[Int[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 957
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] - Simp[(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] && N 
eQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1 
, m, (-n)*(p + 1)]))
 

rule 1082
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[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])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[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 1142
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 
3.2.65.4 Maple [A] (verified)

Time = 4.00 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.74

method result size
derivativedivides \(\frac {\left (A b -B a \right ) x^{\frac {5}{2}}}{3 a b \left (b \,x^{3}+a \right )}+\frac {\left (A b +5 B a \right ) \left (\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (-\sqrt {3}+\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {\arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 a b}\) \(213\)
default \(\frac {\left (A b -B a \right ) x^{\frac {5}{2}}}{3 a b \left (b \,x^{3}+a \right )}+\frac {\left (A b +5 B a \right ) \left (\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}-x -\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (-\sqrt {3}+\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (x +\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \sqrt {x}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {\arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 a b}\) \(213\)

input
int(x^(3/2)*(B*x^3+A)/(b*x^3+a)^2,x,method=_RETURNVERBOSE)
 
output
1/3*(A*b-B*a)*x^(5/2)/a/b/(b*x^3+a)+1/3*(A*b+5*B*a)/a/b*(1/12/a*3^(1/2)*(a 
/b)^(5/6)*ln(3^(1/2)*(a/b)^(1/6)*x^(1/2)-x-(a/b)^(1/3))+1/6/b/(a/b)^(1/6)* 
arctan(-3^(1/2)+2*x^(1/2)/(a/b)^(1/6))-1/12/a*3^(1/2)*(a/b)^(5/6)*ln(x+3^( 
1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))+1/6/b/(a/b)^(1/6)*arctan(2*x^(1/2)/( 
a/b)^(1/6)+3^(1/2))+1/3/b/(a/b)^(1/6)*arctan(x^(1/2)/(a/b)^(1/6)))
 
3.2.65.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1773 vs. \(2 (207) = 414\).

Time = 0.29 (sec) , antiderivative size = 1773, normalized size of antiderivative = 6.13 \[ \int \frac {x^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx=\text {Too large to display} \]

input
integrate(x^(3/2)*(B*x^3+A)/(b*x^3+a)^2,x, algorithm="fricas")
 
output
-1/36*(12*(B*a - A*b)*x^(5/2) - 2*(a*b^2*x^3 + a^2*b)*(-(15625*B^6*a^6 + 1 
8750*A*B^5*a^5*b + 9375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 375*A^4*B 
^2*a^2*b^4 + 30*A^5*B*a*b^5 + A^6*b^6)/(a^7*b^11))^(1/6)*log(a^6*b^9*(-(15 
625*B^6*a^6 + 18750*A*B^5*a^5*b + 9375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3* 
b^3 + 375*A^4*B^2*a^2*b^4 + 30*A^5*B*a*b^5 + A^6*b^6)/(a^7*b^11))^(5/6) + 
(3125*B^5*a^5 + 3125*A*B^4*a^4*b + 1250*A^2*B^3*a^3*b^2 + 250*A^3*B^2*a^2* 
b^3 + 25*A^4*B*a*b^4 + A^5*b^5)*sqrt(x)) + 2*(a*b^2*x^3 + a^2*b)*(-(15625* 
B^6*a^6 + 18750*A*B^5*a^5*b + 9375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 
+ 375*A^4*B^2*a^2*b^4 + 30*A^5*B*a*b^5 + A^6*b^6)/(a^7*b^11))^(1/6)*log(-a 
^6*b^9*(-(15625*B^6*a^6 + 18750*A*B^5*a^5*b + 9375*A^2*B^4*a^4*b^2 + 2500* 
A^3*B^3*a^3*b^3 + 375*A^4*B^2*a^2*b^4 + 30*A^5*B*a*b^5 + A^6*b^6)/(a^7*b^1 
1))^(5/6) + (3125*B^5*a^5 + 3125*A*B^4*a^4*b + 1250*A^2*B^3*a^3*b^2 + 250* 
A^3*B^2*a^2*b^3 + 25*A^4*B*a*b^4 + A^5*b^5)*sqrt(x)) - (a*b^2*x^3 + a^2*b 
- sqrt(-3)*(a*b^2*x^3 + a^2*b))*(-(15625*B^6*a^6 + 18750*A*B^5*a^5*b + 937 
5*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a^3*b^3 + 375*A^4*B^2*a^2*b^4 + 30*A^5*B* 
a*b^5 + A^6*b^6)/(a^7*b^11))^(1/6)*log(1/2*(sqrt(-3)*a^6*b^9 + a^6*b^9)*(- 
(15625*B^6*a^6 + 18750*A*B^5*a^5*b + 9375*A^2*B^4*a^4*b^2 + 2500*A^3*B^3*a 
^3*b^3 + 375*A^4*B^2*a^2*b^4 + 30*A^5*B*a*b^5 + A^6*b^6)/(a^7*b^11))^(5/6) 
 + (3125*B^5*a^5 + 3125*A*B^4*a^4*b + 1250*A^2*B^3*a^3*b^2 + 250*A^3*B^2*a 
^2*b^3 + 25*A^4*B*a*b^4 + A^5*b^5)*sqrt(x)) + (a*b^2*x^3 + a^2*b - sqrt...
 
3.2.65.6 Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1885 vs. \(2 (277) = 554\).

Time = 112.41 (sec) , antiderivative size = 1885, normalized size of antiderivative = 6.52 \[ \int \frac {x^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx=\text {Too large to display} \]

input
integrate(x**(3/2)*(B*x**3+A)/(b*x**3+a)**2,x)
 
output
Piecewise((zoo*(-2*A/(7*x**(7/2)) - 2*B/sqrt(x)), Eq(a, 0) & Eq(b, 0)), (( 
2*A*x**(5/2)/5 + 2*B*x**(11/2)/11)/a**2, Eq(b, 0)), ((-2*A/(7*x**(7/2)) - 
2*B/sqrt(x))/b**2, Eq(a, 0)), (2*A*a*b*log(sqrt(x) - (-a/b)**(1/6))/(36*a* 
*2*b**2*(-a/b)**(1/6) + 36*a*b**3*x**3*(-a/b)**(1/6)) - 2*A*a*b*log(sqrt(x 
) + (-a/b)**(1/6))/(36*a**2*b**2*(-a/b)**(1/6) + 36*a*b**3*x**3*(-a/b)**(1 
/6)) + A*a*b*log(-4*sqrt(x)*(-a/b)**(1/6) + 4*x + 4*(-a/b)**(1/3))/(36*a** 
2*b**2*(-a/b)**(1/6) + 36*a*b**3*x**3*(-a/b)**(1/6)) - A*a*b*log(4*sqrt(x) 
*(-a/b)**(1/6) + 4*x + 4*(-a/b)**(1/3))/(36*a**2*b**2*(-a/b)**(1/6) + 36*a 
*b**3*x**3*(-a/b)**(1/6)) + 2*sqrt(3)*A*a*b*atan(2*sqrt(3)*sqrt(x)/(3*(-a/ 
b)**(1/6)) - sqrt(3)/3)/(36*a**2*b**2*(-a/b)**(1/6) + 36*a*b**3*x**3*(-a/b 
)**(1/6)) + 2*sqrt(3)*A*a*b*atan(2*sqrt(3)*sqrt(x)/(3*(-a/b)**(1/6)) + sqr 
t(3)/3)/(36*a**2*b**2*(-a/b)**(1/6) + 36*a*b**3*x**3*(-a/b)**(1/6)) + 12*A 
*b**2*x**(5/2)*(-a/b)**(1/6)/(36*a**2*b**2*(-a/b)**(1/6) + 36*a*b**3*x**3* 
(-a/b)**(1/6)) + 2*A*b**2*x**3*log(sqrt(x) - (-a/b)**(1/6))/(36*a**2*b**2* 
(-a/b)**(1/6) + 36*a*b**3*x**3*(-a/b)**(1/6)) - 2*A*b**2*x**3*log(sqrt(x) 
+ (-a/b)**(1/6))/(36*a**2*b**2*(-a/b)**(1/6) + 36*a*b**3*x**3*(-a/b)**(1/6 
)) + A*b**2*x**3*log(-4*sqrt(x)*(-a/b)**(1/6) + 4*x + 4*(-a/b)**(1/3))/(36 
*a**2*b**2*(-a/b)**(1/6) + 36*a*b**3*x**3*(-a/b)**(1/6)) - A*b**2*x**3*log 
(4*sqrt(x)*(-a/b)**(1/6) + 4*x + 4*(-a/b)**(1/3))/(36*a**2*b**2*(-a/b)**(1 
/6) + 36*a*b**3*x**3*(-a/b)**(1/6)) + 2*sqrt(3)*A*b**2*x**3*atan(2*sqrt...
 
3.2.65.7 Maxima [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.81 \[ \int \frac {x^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx=-\frac {{\left (B a - A b\right )} x^{\frac {5}{2}}}{3 \, {\left (a b^{2} x^{3} + a^{2} b\right )}} - \frac {{\left (5 \, B a + A b\right )} {\left (\frac {\sqrt {3} \log \left (\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {\sqrt {3} \log \left (-\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} \sqrt {x} + b^{\frac {1}{3}} x + a^{\frac {1}{3}}\right )}{a^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {2 \, \arctan \left (\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} + 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {2 \, \arctan \left (-\frac {\sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} - 2 \, b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {4 \, \arctan \left (\frac {b^{\frac {1}{3}} \sqrt {x}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}}{36 \, a b} \]

input
integrate(x^(3/2)*(B*x^3+A)/(b*x^3+a)^2,x, algorithm="maxima")
 
output
-1/3*(B*a - A*b)*x^(5/2)/(a*b^2*x^3 + a^2*b) - 1/36*(5*B*a + A*b)*(sqrt(3) 
*log(sqrt(3)*a^(1/6)*b^(1/6)*sqrt(x) + b^(1/3)*x + a^(1/3))/(a^(1/6)*b^(5/ 
6)) - sqrt(3)*log(-sqrt(3)*a^(1/6)*b^(1/6)*sqrt(x) + b^(1/3)*x + a^(1/3))/ 
(a^(1/6)*b^(5/6)) - 2*arctan((sqrt(3)*a^(1/6)*b^(1/6) + 2*b^(1/3)*sqrt(x)) 
/sqrt(a^(1/3)*b^(1/3)))/(b^(2/3)*sqrt(a^(1/3)*b^(1/3))) - 2*arctan(-(sqrt( 
3)*a^(1/6)*b^(1/6) - 2*b^(1/3)*sqrt(x))/sqrt(a^(1/3)*b^(1/3)))/(b^(2/3)*sq 
rt(a^(1/3)*b^(1/3))) - 4*arctan(b^(1/3)*sqrt(x)/sqrt(a^(1/3)*b^(1/3)))/(b^ 
(2/3)*sqrt(a^(1/3)*b^(1/3))))/(a*b)
 
3.2.65.8 Giac [A] (verification not implemented)

Time = 0.65 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.00 \[ \int \frac {x^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx=\frac {{\left (5 \, B a + A b\right )} \arctan \left (\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} + 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 \, \left (a b^{5}\right )^{\frac {1}{6}} a b} + \frac {{\left (5 \, B a + A b\right )} \arctan \left (-\frac {\sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}} - 2 \, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 \, \left (a b^{5}\right )^{\frac {1}{6}} a b} + \frac {{\left (5 \, B a \left (\frac {a}{b}\right )^{\frac {5}{6}} + A b \left (\frac {a}{b}\right )^{\frac {5}{6}}\right )} \arctan \left (\frac {\sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{9 \, a^{2} b} - \frac {B a x^{\frac {5}{2}} - A b x^{\frac {5}{2}}}{3 \, {\left (b x^{3} + a\right )} a b} - \frac {\sqrt {3} {\left (5 \, \left (a b^{5}\right )^{\frac {5}{6}} B a + \left (a b^{5}\right )^{\frac {5}{6}} A b\right )} \log \left (\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{36 \, a^{2} b^{6}} + \frac {\sqrt {3} {\left (5 \, \left (a b^{5}\right )^{\frac {5}{6}} B a + \left (a b^{5}\right )^{\frac {5}{6}} A b\right )} \log \left (-\sqrt {3} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{6}} + x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{36 \, a^{2} b^{6}} \]

input
integrate(x^(3/2)*(B*x^3+A)/(b*x^3+a)^2,x, algorithm="giac")
 
output
1/18*(5*B*a + A*b)*arctan((sqrt(3)*(a/b)^(1/6) + 2*sqrt(x))/(a/b)^(1/6))/( 
(a*b^5)^(1/6)*a*b) + 1/18*(5*B*a + A*b)*arctan(-(sqrt(3)*(a/b)^(1/6) - 2*s 
qrt(x))/(a/b)^(1/6))/((a*b^5)^(1/6)*a*b) + 1/9*(5*B*a*(a/b)^(5/6) + A*b*(a 
/b)^(5/6))*arctan(sqrt(x)/(a/b)^(1/6))/(a^2*b) - 1/3*(B*a*x^(5/2) - A*b*x^ 
(5/2))/((b*x^3 + a)*a*b) - 1/36*sqrt(3)*(5*(a*b^5)^(5/6)*B*a + (a*b^5)^(5/ 
6)*A*b)*log(sqrt(3)*sqrt(x)*(a/b)^(1/6) + x + (a/b)^(1/3))/(a^2*b^6) + 1/3 
6*sqrt(3)*(5*(a*b^5)^(5/6)*B*a + (a*b^5)^(5/6)*A*b)*log(-sqrt(3)*sqrt(x)*( 
a/b)^(1/6) + x + (a/b)^(1/3))/(a^2*b^6)
 
3.2.65.9 Mupad [B] (verification not implemented)

Time = 7.29 (sec) , antiderivative size = 1578, normalized size of antiderivative = 5.46 \[ \int \frac {x^{3/2} \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx=\text {Too large to display} \]

input
int((x^(3/2)*(A + B*x^3))/(a + b*x^3)^2,x)
 
output
(x^(5/2)*(A*b - B*a))/(3*a*b*(a + b*x^3)) - (atan(((((3^(1/2)*1i)/2 - 1/2) 
^2*(A*b + 5*B*a)^2*((4*A^3*b^3)/3 + (500*B^3*a^3)/3 + 100*A*B^2*a^2*b + 20 
*A^2*B*a*b^2 - (x^(1/2)*((3^(1/2)*1i)/2 - 1/2)*(A*b + 5*B*a)*(24*A^2*a*b^4 
 + 600*B^2*a^3*b^2 + 240*A*B*a^2*b^3))/(18*(-a)^(7/6)*b^(11/6)))*1i)/(324* 
(-a)^(7/3)*b^(11/3)) - (((3^(1/2)*1i)/2 - 1/2)^2*(A*b + 5*B*a)^2*((4*A^3*b 
^3)/3 + (500*B^3*a^3)/3 + 100*A*B^2*a^2*b + 20*A^2*B*a*b^2 + (x^(1/2)*((3^ 
(1/2)*1i)/2 - 1/2)*(A*b + 5*B*a)*(24*A^2*a*b^4 + 600*B^2*a^3*b^2 + 240*A*B 
*a^2*b^3))/(18*(-a)^(7/6)*b^(11/6)))*1i)/(324*(-a)^(7/3)*b^(11/3)))/((((3^ 
(1/2)*1i)/2 - 1/2)^2*(A*b + 5*B*a)^2*((4*A^3*b^3)/3 + (500*B^3*a^3)/3 + 10 
0*A*B^2*a^2*b + 20*A^2*B*a*b^2 - (x^(1/2)*((3^(1/2)*1i)/2 - 1/2)*(A*b + 5* 
B*a)*(24*A^2*a*b^4 + 600*B^2*a^3*b^2 + 240*A*B*a^2*b^3))/(18*(-a)^(7/6)*b^ 
(11/6))))/(324*(-a)^(7/3)*b^(11/3)) + (((3^(1/2)*1i)/2 - 1/2)^2*(A*b + 5*B 
*a)^2*((4*A^3*b^3)/3 + (500*B^3*a^3)/3 + 100*A*B^2*a^2*b + 20*A^2*B*a*b^2 
+ (x^(1/2)*((3^(1/2)*1i)/2 - 1/2)*(A*b + 5*B*a)*(24*A^2*a*b^4 + 600*B^2*a^ 
3*b^2 + 240*A*B*a^2*b^3))/(18*(-a)^(7/6)*b^(11/6))))/(324*(-a)^(7/3)*b^(11 
/3))))*((3^(1/2)*1i)/2 - 1/2)*(A*b + 5*B*a)*1i)/(9*(-a)^(7/6)*b^(11/6)) - 
(atan(((((3^(1/2)*1i)/2 + 1/2)^2*(A*b + 5*B*a)^2*((4*A^3*b^3)/3 + (500*B^3 
*a^3)/3 + 100*A*B^2*a^2*b + 20*A^2*B*a*b^2 - (x^(1/2)*((3^(1/2)*1i)/2 + 1/ 
2)*(A*b + 5*B*a)*(24*A^2*a*b^4 + 600*B^2*a^3*b^2 + 240*A*B*a^2*b^3))/(18*( 
-a)^(7/6)*b^(11/6)))*1i)/(324*(-a)^(7/3)*b^(11/3)) - (((3^(1/2)*1i)/2 +...