\(\int \frac {x^{5/2} (A+B x^2)}{(a+b x^2)^3} \, dx\) [178]

Optimal result

Integrand size = 22, antiderivative size = 230 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {(A b-a B) x^{3/2}}{4 b^2 \left (a+b x^2\right )^2}+\frac {(3 A b-11 a B) x^{3/2}}{16 a b^2 \left (a+b x^2\right )}-\frac {3 (A b+7 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{5/4} b^{11/4}}+\frac {3 (A b+7 a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{5/4} b^{11/4}}-\frac {3 (A b+7 a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{32 \sqrt {2} a^{5/4} b^{11/4}} \] Output:

-1/4*(A*b-B*a)*x^(3/2)/b^2/(b*x^2+a)^2+1/16*(3*A*b-11*B*a)*x^(3/2)/a/b^2/( 
b*x^2+a)-3/64*(A*b+7*B*a)*arctan(1-2^(1/2)*b^(1/4)*x^(1/2)/a^(1/4))*2^(1/2 
)/a^(5/4)/b^(11/4)+3/64*(A*b+7*B*a)*arctan(1+2^(1/2)*b^(1/4)*x^(1/2)/a^(1/ 
4))*2^(1/2)/a^(5/4)/b^(11/4)-3/64*(A*b+7*B*a)*arctanh(2^(1/2)*a^(1/4)*b^(1 
/4)*x^(1/2)/(a^(1/2)+b^(1/2)*x))*2^(1/2)/a^(5/4)/b^(11/4)
 

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.74 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {-\frac {4 \sqrt [4]{a} b^{3/4} x^{3/2} \left (7 a^2 B-3 A b^2 x^2+a b \left (A+11 B x^2\right )\right )}{\left (a+b x^2\right )^2}-3 \sqrt {2} (A b+7 a B) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-3 \sqrt {2} (A b+7 a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{64 a^{5/4} b^{11/4}} \] Input:

Integrate[(x^(5/2)*(A + B*x^2))/(a + b*x^2)^3,x]
 

Output:

((-4*a^(1/4)*b^(3/4)*x^(3/2)*(7*a^2*B - 3*A*b^2*x^2 + a*b*(A + 11*B*x^2))) 
/(a + b*x^2)^2 - 3*Sqrt[2]*(A*b + 7*a*B)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqr 
t[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - 3*Sqrt[2]*(A*b + 7*a*B)*ArcTanh[(Sqrt[2]* 
a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(64*a^(5/4)*b^(11/4))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.27, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {362, 252, 266, 826, 1476, 1082, 217, 1479, 25, 27, 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.

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

\(\Big \downarrow \) 362

\(\displaystyle \frac {(7 a B+A b) \int \frac {x^{5/2}}{\left (b x^2+a\right )^2}dx}{8 a b}+\frac {x^{7/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 252

\(\displaystyle \frac {(7 a B+A b) \left (\frac {3 \int \frac {\sqrt {x}}{b x^2+a}dx}{4 b}-\frac {x^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{7/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {(7 a B+A b) \left (\frac {3 \int \frac {x}{b x^2+a}d\sqrt {x}}{2 b}-\frac {x^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{7/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {(7 a B+A b) \left (\frac {3 \left (\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}\right )}{2 b}-\frac {x^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{7/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {(7 a B+A b) \left (\frac {3 \left (\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}\right )}{2 b}-\frac {x^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{7/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {(7 a B+A b) \left (\frac {3 \left (\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}\right )}{2 b}-\frac {x^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{7/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {(7 a B+A b) \left (\frac {3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}\right )}{2 b}-\frac {x^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{7/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {(7 a B+A b) \left (\frac {3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{2 b}-\frac {x^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{7/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(7 a B+A b) \left (\frac {3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{2 b}-\frac {x^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{7/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(7 a B+A b) \left (\frac {3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {b}}\right )}{2 b}-\frac {x^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{7/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {(7 a B+A b) \left (\frac {3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{2 b}-\frac {x^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{7/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\)

Input:

Int[(x^(5/2)*(A + B*x^2))/(a + b*x^2)^3,x]
 

Output:

((A*b - a*B)*x^(7/2))/(4*a*b*(a + b*x^2)^2) + ((A*b + 7*a*B)*(-1/2*x^(3/2) 
/(b*(a + b*x^2)) + (3*((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(S 
qrt[2]*a^(1/4)*b^(1/4))) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/( 
Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[b]) - (-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4 
)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(Sqrt[2]*a^(1/4)*b^(1/4)) + Log[Sqrt[a] + S 
qrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(2*Sqrt[2]*a^(1/4)*b^(1/4)))/( 
2*Sqrt[b])))/(2*b)))/(8*a*b)
 

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 252

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x 
)^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* 
(p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c 
}, x] && LtQ[p, -1] && GtQ[m, 1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi 
alQ[a, b, c, 2, m, p, x]
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 362

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*b*e 
*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(2*a*b*(p + 1))   I 
nt[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && N 
eQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || 
  !RationalQ[m] || (ILtQ[p + 1/2, 0] && LeQ[-1, m, -2*(p + 1)]))
 

rule 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ 
a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] 
 && AtomQ[SplitProduct[SumBaseQ, b]]))
 

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 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.72

method	result	size
derivativedivides	\(\frac {\frac {\left (3 A b -11 B a \right ) x^{\frac {7}{2}}}{16 a b}-\frac {\left (A b +7 B a \right ) x^{\frac {3}{2}}}{16 b^{2}}}{\left (b \,x^{2}+a \right )^{2}}+\frac {3 \left (A b +7 B a \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b^{3} a \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(166\)
default	\(\frac {\frac {\left (3 A b -11 B a \right ) x^{\frac {7}{2}}}{16 a b}-\frac {\left (A b +7 B a \right ) x^{\frac {3}{2}}}{16 b^{2}}}{\left (b \,x^{2}+a \right )^{2}}+\frac {3 \left (A b +7 B a \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b^{3} a \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(166\)

Input:

int(x^(5/2)*(B*x^2+A)/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

2*(1/32*(3*A*b-11*B*a)/a/b*x^(7/2)-1/32*(A*b+7*B*a)/b^2*x^(3/2))/(b*x^2+a) 
^2+3/128*(A*b+7*B*a)/b^3/a/(a/b)^(1/4)*2^(1/2)*(ln((x-(a/b)^(1/4)*x^(1/2)* 
2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))+2*arctan 
(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1))
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.15 (sec) , antiderivative size = 871, normalized size of antiderivative = 3.79 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

integrate(x^(5/2)*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="fricas")
 

Output:

1/64*(3*(a*b^4*x^4 + 2*a^2*b^3*x^2 + a^3*b^2)*(-(2401*B^4*a^4 + 1372*A*B^3 
*a^3*b + 294*A^2*B^2*a^2*b^2 + 28*A^3*B*a*b^3 + A^4*b^4)/(a^5*b^11))^(1/4) 
*log(27*a^4*b^8*(-(2401*B^4*a^4 + 1372*A*B^3*a^3*b + 294*A^2*B^2*a^2*b^2 + 
 28*A^3*B*a*b^3 + A^4*b^4)/(a^5*b^11))^(3/4) + 27*(343*B^3*a^3 + 147*A*B^2 
*a^2*b + 21*A^2*B*a*b^2 + A^3*b^3)*sqrt(x)) - 3*(I*a*b^4*x^4 + 2*I*a^2*b^3 
*x^2 + I*a^3*b^2)*(-(2401*B^4*a^4 + 1372*A*B^3*a^3*b + 294*A^2*B^2*a^2*b^2 
 + 28*A^3*B*a*b^3 + A^4*b^4)/(a^5*b^11))^(1/4)*log(27*I*a^4*b^8*(-(2401*B^ 
4*a^4 + 1372*A*B^3*a^3*b + 294*A^2*B^2*a^2*b^2 + 28*A^3*B*a*b^3 + A^4*b^4) 
/(a^5*b^11))^(3/4) + 27*(343*B^3*a^3 + 147*A*B^2*a^2*b + 21*A^2*B*a*b^2 + 
A^3*b^3)*sqrt(x)) - 3*(-I*a*b^4*x^4 - 2*I*a^2*b^3*x^2 - I*a^3*b^2)*(-(2401 
*B^4*a^4 + 1372*A*B^3*a^3*b + 294*A^2*B^2*a^2*b^2 + 28*A^3*B*a*b^3 + A^4*b 
^4)/(a^5*b^11))^(1/4)*log(-27*I*a^4*b^8*(-(2401*B^4*a^4 + 1372*A*B^3*a^3*b 
 + 294*A^2*B^2*a^2*b^2 + 28*A^3*B*a*b^3 + A^4*b^4)/(a^5*b^11))^(3/4) + 27* 
(343*B^3*a^3 + 147*A*B^2*a^2*b + 21*A^2*B*a*b^2 + A^3*b^3)*sqrt(x)) - 3*(a 
*b^4*x^4 + 2*a^2*b^3*x^2 + a^3*b^2)*(-(2401*B^4*a^4 + 1372*A*B^3*a^3*b + 2 
94*A^2*B^2*a^2*b^2 + 28*A^3*B*a*b^3 + A^4*b^4)/(a^5*b^11))^(1/4)*log(-27*a 
^4*b^8*(-(2401*B^4*a^4 + 1372*A*B^3*a^3*b + 294*A^2*B^2*a^2*b^2 + 28*A^3*B 
*a*b^3 + A^4*b^4)/(a^5*b^11))^(3/4) + 27*(343*B^3*a^3 + 147*A*B^2*a^2*b + 
21*A^2*B*a*b^2 + A^3*b^3)*sqrt(x)) - 4*((11*B*a*b - 3*A*b^2)*x^3 + (7*B*a^ 
2 + A*a*b)*x)*sqrt(x))/(a*b^4*x^4 + 2*a^2*b^3*x^2 + a^3*b^2)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x**(5/2)*(B*x**2+A)/(b*x**2+a)**3,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.09 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {{\left (11 \, B a b - 3 \, A b^{2}\right )} x^{\frac {7}{2}} + {\left (7 \, B a^{2} + A a b\right )} x^{\frac {3}{2}}}{16 \, {\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} + \frac {3 \, {\left (7 \, B a + A b\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{128 \, a b^{2}} \] Input:

integrate(x^(5/2)*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="maxima")
 

Output:

-1/16*((11*B*a*b - 3*A*b^2)*x^(7/2) + (7*B*a^2 + A*a*b)*x^(3/2))/(a*b^4*x^ 
4 + 2*a^2*b^3*x^2 + a^3*b^2) + 3/128*(7*B*a + A*b)*(2*sqrt(2)*arctan(1/2*s 
qrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)) 
)/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) 
*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a) 
*sqrt(b))*sqrt(b)) - sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b) 
*x + sqrt(a))/(a^(1/4)*b^(3/4)) + sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqr 
t(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)))/(a*b^2)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.27 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {11 \, B a b x^{\frac {7}{2}} - 3 \, A b^{2} x^{\frac {7}{2}} + 7 \, B a^{2} x^{\frac {3}{2}} + A a b x^{\frac {3}{2}}}{16 \, {\left (b x^{2} + a\right )}^{2} a b^{2}} + \frac {3 \, \sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{2} b^{5}} + \frac {3 \, \sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{2} b^{5}} - \frac {3 \, \sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{2} b^{5}} + \frac {3 \, \sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{2} b^{5}} \] Input:

integrate(x^(5/2)*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

-1/16*(11*B*a*b*x^(7/2) - 3*A*b^2*x^(7/2) + 7*B*a^2*x^(3/2) + A*a*b*x^(3/2 
))/((b*x^2 + a)^2*a*b^2) + 3/64*sqrt(2)*(7*(a*b^3)^(3/4)*B*a + (a*b^3)^(3/ 
4)*A*b)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/ 
(a^2*b^5) + 3/64*sqrt(2)*(7*(a*b^3)^(3/4)*B*a + (a*b^3)^(3/4)*A*b)*arctan( 
-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a^2*b^5) - 3/ 
128*sqrt(2)*(7*(a*b^3)^(3/4)*B*a + (a*b^3)^(3/4)*A*b)*log(sqrt(2)*sqrt(x)* 
(a/b)^(1/4) + x + sqrt(a/b))/(a^2*b^5) + 3/128*sqrt(2)*(7*(a*b^3)^(3/4)*B* 
a + (a*b^3)^(3/4)*A*b)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/( 
a^2*b^5)
 

Mupad [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.53 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {3\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )\,\left (A\,b+7\,B\,a\right )}{32\,{\left (-a\right )}^{5/4}\,b^{11/4}}-\frac {3\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )\,\left (A\,b+7\,B\,a\right )}{32\,{\left (-a\right )}^{5/4}\,b^{11/4}}-\frac {\frac {x^{3/2}\,\left (A\,b+7\,B\,a\right )}{16\,b^2}-\frac {x^{7/2}\,\left (3\,A\,b-11\,B\,a\right )}{16\,a\,b}}{a^2+2\,a\,b\,x^2+b^2\,x^4} \] Input:

int((x^(5/2)*(A + B*x^2))/(a + b*x^2)^3,x)
 

Output:

(3*atanh((b^(1/4)*x^(1/2))/(-a)^(1/4))*(A*b + 7*B*a))/(32*(-a)^(5/4)*b^(11 
/4)) - (3*atan((b^(1/4)*x^(1/2))/(-a)^(1/4))*(A*b + 7*B*a))/(32*(-a)^(5/4) 
*b^(11/4)) - ((x^(3/2)*(A*b + 7*B*a))/(16*b^2) - (x^(7/2)*(3*A*b - 11*B*a) 
)/(16*a*b))/(a^2 + b^2*x^4 + 2*a*b*x^2)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.33 \[ \int \frac {x^{5/2} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {-6 b^{\frac {1}{4}} a^{\frac {7}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )-6 b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{2}+6 b^{\frac {1}{4}} a^{\frac {7}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )+6 b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{2}+3 b^{\frac {1}{4}} a^{\frac {7}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right )+3 b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) x^{2}-3 b^{\frac {1}{4}} a^{\frac {7}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right )-3 b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) x^{2}-8 \sqrt {x}\, a b x}{16 a \,b^{2} \left (b \,x^{2}+a \right )} \] Input:

int(x^(5/2)*(B*x^2+A)/(b*x^2+a)^3,x)
 

Output:

( - 6*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x 
)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a - 6*b**(1/4)*a**(3/4)*sqrt(2)*at 
an((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt 
(2)))*b*x**2 + 6*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) 
 + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a + 6*b**(1/4)*a**(3/4) 
*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a* 
*(1/4)*sqrt(2)))*b*x**2 + 3*b**(1/4)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1 
/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a + 3*b**(1/4)*a**(3/4)*sqrt(2 
)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b*x**2 - 
 3*b**(1/4)*a**(3/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt( 
a) + sqrt(b)*x)*a - 3*b**(1/4)*a**(3/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1 
/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b*x**2 - 8*sqrt(x)*a*b*x)/(16*a*b**2*(a 
 + b*x**2))