\(\int \frac {\sqrt {x} (A+B x^2)}{(a+b x^2)^3} \, dx\) [180]

Optimal result

Integrand size = 22, antiderivative size = 236 \[ \int \frac {\sqrt {x} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {(A b-a B) x^{3/2}}{4 a b \left (a+b x^2\right )^2}+\frac {(5 A b+3 a B) x^{3/2}}{16 a^2 b \left (a+b x^2\right )}-\frac {(5 A b+3 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{9/4} b^{7/4}}+\frac {(5 A b+3 a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{9/4} b^{7/4}}-\frac {(5 A b+3 a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{32 \sqrt {2} a^{9/4} b^{7/4}} \] Output:

1/4*(A*b-B*a)*x^(3/2)/a/b/(b*x^2+a)^2+1/16*(5*A*b+3*B*a)*x^(3/2)/a^2/b/(b* 
x^2+a)-1/64*(5*A*b+3*B*a)*arctan(1-2^(1/2)*b^(1/4)*x^(1/2)/a^(1/4))*2^(1/2 
)/a^(9/4)/b^(7/4)+1/64*(5*A*b+3*B*a)*arctan(1+2^(1/2)*b^(1/4)*x^(1/2)/a^(1 
/4))*2^(1/2)/a^(9/4)/b^(7/4)-1/64*(5*A*b+3*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^(9/4)/b^(7/4)
 

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {x} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {\frac {4 \sqrt [4]{a} b^{3/4} x^{3/2} \left (9 a A b-a^2 B+5 A b^2 x^2+3 a b B x^2\right )}{\left (a+b x^2\right )^2}-\sqrt {2} (5 A b+3 a B) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-\sqrt {2} (5 A b+3 a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{64 a^{9/4} b^{7/4}} \] Input:

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

Output:

((4*a^(1/4)*b^(3/4)*x^(3/2)*(9*a*A*b - a^2*B + 5*A*b^2*x^2 + 3*a*b*B*x^2)) 
/(a + b*x^2)^2 - Sqrt[2]*(5*A*b + 3*a*B)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqr 
t[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - Sqrt[2]*(5*A*b + 3*a*B)*ArcTanh[(Sqrt[2]* 
a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(64*a^(9/4)*b^(7/4))
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.24, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {362, 253, 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.

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

\(\Big \downarrow \) 362

\(\displaystyle \frac {(3 a B+5 A b) \int \frac {\sqrt {x}}{\left (b x^2+a\right )^2}dx}{8 a b}+\frac {x^{3/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 253

\(\displaystyle \frac {(3 a B+5 A b) \left (\frac {\int \frac {\sqrt {x}}{b x^2+a}dx}{4 a}+\frac {x^{3/2}}{2 a \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{3/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {(3 a B+5 A b) \left (\frac {\int \frac {x}{b x^2+a}d\sqrt {x}}{2 a}+\frac {x^{3/2}}{2 a \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{3/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {(3 a B+5 A b) \left (\frac {\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}}{2 a}+\frac {x^{3/2}}{2 a \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{3/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {(3 a B+5 A b) \left (\frac {\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}}{2 a}+\frac {x^{3/2}}{2 a \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{3/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {(3 a B+5 A b) \left (\frac {\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}}{2 a}+\frac {x^{3/2}}{2 a \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{3/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {(3 a B+5 A b) \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}}{2 a}+\frac {x^{3/2}}{2 a \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{3/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {(3 a B+5 A b) \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}}{2 a}+\frac {x^{3/2}}{2 a \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{3/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(3 a B+5 A b) \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}}{2 a}+\frac {x^{3/2}}{2 a \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{3/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(3 a B+5 A b) \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {b}}}{2 a}+\frac {x^{3/2}}{2 a \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{3/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {(3 a B+5 A b) \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}}{2 a}+\frac {x^{3/2}}{2 a \left (a+b x^2\right )}\right )}{8 a b}+\frac {x^{3/2} (A b-a B)}{4 a b \left (a+b x^2\right )^2}\)

Input:

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

Output:

((A*b - a*B)*x^(3/2))/(4*a*b*(a + b*x^2)^2) + ((5*A*b + 3*a*B)*(x^(3/2)/(2 
*a*(a + b*x^2)) + ((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[ 
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] + Sqrt[ 
2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(2*Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sq 
rt[b]))/(2*a)))/(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 253

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x 
)^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 
2*a*(p + 1))   Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m 
}, x] && LtQ[p, -1] && IntBinomialQ[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 = 0.38 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.71

method	result	size
derivativedivides	\(\frac {\frac {\left (5 A b +3 B a \right ) x^{\frac {7}{2}}}{16 a^{2}}+\frac {\left (9 A b -B a \right ) x^{\frac {3}{2}}}{16 a b}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (5 A b +3 B a \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(168\)
default	\(\frac {\frac {\left (5 A b +3 B a \right ) x^{\frac {7}{2}}}{16 a^{2}}+\frac {\left (9 A b -B a \right ) x^{\frac {3}{2}}}{16 a b}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (5 A b +3 B a \right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(168\)

Input:

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

Output:

2*(1/32*(5*A*b+3*B*a)/a^2*x^(7/2)+1/32*(9*A*b-B*a)/a/b*x^(3/2))/(b*x^2+a)^ 
2+1/128*(5*A*b+3*B*a)/a^2/b^2/(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*arc 
tan(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 = 0.11 (sec) , antiderivative size = 878, normalized size of antiderivative = 3.72 \[ \int \frac {\sqrt {x} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

1/64*((a^2*b^3*x^4 + 2*a^3*b^2*x^2 + a^4*b)*(-(81*B^4*a^4 + 540*A*B^3*a^3* 
b + 1350*A^2*B^2*a^2*b^2 + 1500*A^3*B*a*b^3 + 625*A^4*b^4)/(a^9*b^7))^(1/4 
)*log(a^7*b^5*(-(81*B^4*a^4 + 540*A*B^3*a^3*b + 1350*A^2*B^2*a^2*b^2 + 150 
0*A^3*B*a*b^3 + 625*A^4*b^4)/(a^9*b^7))^(3/4) + (27*B^3*a^3 + 135*A*B^2*a^ 
2*b + 225*A^2*B*a*b^2 + 125*A^3*b^3)*sqrt(x)) - (I*a^2*b^3*x^4 + 2*I*a^3*b 
^2*x^2 + I*a^4*b)*(-(81*B^4*a^4 + 540*A*B^3*a^3*b + 1350*A^2*B^2*a^2*b^2 + 
 1500*A^3*B*a*b^3 + 625*A^4*b^4)/(a^9*b^7))^(1/4)*log(I*a^7*b^5*(-(81*B^4* 
a^4 + 540*A*B^3*a^3*b + 1350*A^2*B^2*a^2*b^2 + 1500*A^3*B*a*b^3 + 625*A^4* 
b^4)/(a^9*b^7))^(3/4) + (27*B^3*a^3 + 135*A*B^2*a^2*b + 225*A^2*B*a*b^2 + 
125*A^3*b^3)*sqrt(x)) - (-I*a^2*b^3*x^4 - 2*I*a^3*b^2*x^2 - I*a^4*b)*(-(81 
*B^4*a^4 + 540*A*B^3*a^3*b + 1350*A^2*B^2*a^2*b^2 + 1500*A^3*B*a*b^3 + 625 
*A^4*b^4)/(a^9*b^7))^(1/4)*log(-I*a^7*b^5*(-(81*B^4*a^4 + 540*A*B^3*a^3*b 
+ 1350*A^2*B^2*a^2*b^2 + 1500*A^3*B*a*b^3 + 625*A^4*b^4)/(a^9*b^7))^(3/4) 
+ (27*B^3*a^3 + 135*A*B^2*a^2*b + 225*A^2*B*a*b^2 + 125*A^3*b^3)*sqrt(x)) 
- (a^2*b^3*x^4 + 2*a^3*b^2*x^2 + a^4*b)*(-(81*B^4*a^4 + 540*A*B^3*a^3*b + 
1350*A^2*B^2*a^2*b^2 + 1500*A^3*B*a*b^3 + 625*A^4*b^4)/(a^9*b^7))^(1/4)*lo 
g(-a^7*b^5*(-(81*B^4*a^4 + 540*A*B^3*a^3*b + 1350*A^2*B^2*a^2*b^2 + 1500*A 
^3*B*a*b^3 + 625*A^4*b^4)/(a^9*b^7))^(3/4) + (27*B^3*a^3 + 135*A*B^2*a^2*b 
 + 225*A^2*B*a*b^2 + 125*A^3*b^3)*sqrt(x)) + 4*((3*B*a*b + 5*A*b^2)*x^3 - 
(B*a^2 - 9*A*a*b)*x)*sqrt(x))/(a^2*b^3*x^4 + 2*a^3*b^2*x^2 + a^4*b)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {x} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {{\left (3 \, B a b + 5 \, A b^{2}\right )} x^{\frac {7}{2}} - {\left (B a^{2} - 9 \, A a b\right )} x^{\frac {3}{2}}}{16 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}} + \frac {{\left (3 \, B a + 5 \, A b\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{128 \, a^{2} b} \] Input:

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

Output:

1/16*((3*B*a*b + 5*A*b^2)*x^(7/2) - (B*a^2 - 9*A*a*b)*x^(3/2))/(a^2*b^3*x^ 
4 + 2*a^3*b^2*x^2 + a^4*b) + 1/128*(3*B*a + 5*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^2*b)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt {x} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {3 \, B a b x^{\frac {7}{2}} + 5 \, A b^{2} x^{\frac {7}{2}} - B a^{2} x^{\frac {3}{2}} + 9 \, A a b x^{\frac {3}{2}}}{16 \, {\left (b x^{2} + a\right )}^{2} a^{2} b} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{3} b^{4}} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{3} b^{4}} - \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{3} b^{4}} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {3}{4}} B a + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{3} b^{4}} \] Input:

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

Output:

1/16*(3*B*a*b*x^(7/2) + 5*A*b^2*x^(7/2) - B*a^2*x^(3/2) + 9*A*a*b*x^(3/2)) 
/((b*x^2 + a)^2*a^2*b) + 1/64*sqrt(2)*(3*(a*b^3)^(3/4)*B*a + 5*(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^3*b^4) + 1/64*sqrt(2)*(3*(a*b^3)^(3/4)*B*a + 5*(a*b^3)^(3/4)*A*b)*arcta 
n(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a^3*b^4) - 
1/128*sqrt(2)*(3*(a*b^3)^(3/4)*B*a + 5*(a*b^3)^(3/4)*A*b)*log(sqrt(2)*sqrt 
(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^3*b^4) + 1/128*sqrt(2)*(3*(a*b^3)^(3/4 
)*B*a + 5*(a*b^3)^(3/4)*A*b)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a 
/b))/(a^3*b^4)
 

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {x} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {\frac {x^{7/2}\,\left (5\,A\,b+3\,B\,a\right )}{16\,a^2}+\frac {x^{3/2}\,\left (9\,A\,b-B\,a\right )}{16\,a\,b}}{a^2+2\,a\,b\,x^2+b^2\,x^4}+\frac {\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )\,\left (5\,A\,b+3\,B\,a\right )}{32\,{\left (-a\right )}^{9/4}\,b^{7/4}}-\frac {\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )\,\left (5\,A\,b+3\,B\,a\right )}{32\,{\left (-a\right )}^{9/4}\,b^{7/4}} \] Input:

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

Output:

((x^(7/2)*(5*A*b + 3*B*a))/(16*a^2) + (x^(3/2)*(9*A*b - B*a))/(16*a*b))/(a 
^2 + b^2*x^4 + 2*a*b*x^2) + (atan((b^(1/4)*x^(1/2))/(-a)^(1/4))*(5*A*b + 3 
*B*a))/(32*(-a)^(9/4)*b^(7/4)) - (atanh((b^(1/4)*x^(1/2))/(-a)^(1/4))*(5*A 
*b + 3*B*a))/(32*(-a)^(9/4)*b^(7/4))
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt {x} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {-2 b^{\frac {1}{4}} a^{\frac {7}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )-2 b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{2}+2 b^{\frac {1}{4}} a^{\frac {7}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )+2 b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{2}+b^{\frac {1}{4}} a^{\frac {7}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right )+b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) x^{2}-b^{\frac {1}{4}} a^{\frac {7}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right )-b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) x^{2}+8 \sqrt {x}\, a b x}{16 a^{2} b \left (b \,x^{2}+a \right )} \] Input:

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

Output:

( - 2*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 - 2*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 + 2*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 + 2*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 + 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 + b**(1/4)*a**(3/4)*sqrt(2)*lo 
g( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b*x**2 - b** 
(1/4)*a**(3/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + s 
qrt(b)*x)*a - 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**2*b*(a + b*x** 
2))