3.4.82 \(\int \frac {A+B x^2}{x^{7/2} (a+b x^2)^2} \, dx\) [382]

3.4.82.1 Optimal result

Integrand size = 22, antiderivative size = 310 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^2} \, dx=-\frac {9 A b-5 a B}{10 a^2 b x^{5/2}}+\frac {9 A b-5 a B}{2 a^3 \sqrt {x}}+\frac {A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )}-\frac {\sqrt [4]{b} (9 A b-5 a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4}}+\frac {\sqrt [4]{b} (9 A b-5 a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{13/4}}+\frac {\sqrt [4]{b} (9 A b-5 a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4}}-\frac {\sqrt [4]{b} (9 A b-5 a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{8 \sqrt {2} a^{13/4}} \]

output

1/10*(-9*A*b+5*B*a)/a^2/b/x^(5/2)+1/2*(A*b-B*a)/a/b/x^(5/2)/(b*x^2+a)-1/8* 
b^(1/4)*(9*A*b-5*B*a)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(13/4)*2 
^(1/2)+1/8*b^(1/4)*(9*A*b-5*B*a)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4)) 
/a^(13/4)*2^(1/2)+1/16*b^(1/4)*(9*A*b-5*B*a)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)* 
b^(1/4)*2^(1/2)*x^(1/2))/a^(13/4)*2^(1/2)-1/16*b^(1/4)*(9*A*b-5*B*a)*ln(a^ 
(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(13/4)*2^(1/2)+1/2*(9*A 
*b-5*B*a)/a^3/x^(1/2)
 

3.4.82.2 Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.60 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^2} \, dx=\frac {-\frac {4 \sqrt [4]{a} \left (-45 A b^2 x^4+4 a^2 \left (A+5 B x^2\right )+a \left (-36 A b x^2+25 b B x^4\right )\right )}{x^{5/2} \left (a+b x^2\right )}+5 \sqrt {2} \sqrt [4]{b} (-9 A b+5 a B) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+5 \sqrt {2} \sqrt [4]{b} (-9 A b+5 a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{40 a^{13/4}} \]

input

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

output

((-4*a^(1/4)*(-45*A*b^2*x^4 + 4*a^2*(A + 5*B*x^2) + a*(-36*A*b*x^2 + 25*b* 
B*x^4)))/(x^(5/2)*(a + b*x^2)) + 5*Sqrt[2]*b^(1/4)*(-9*A*b + 5*a*B)*ArcTan 
[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 5*Sqrt[2]*b^(1 
/4)*(-9*A*b + 5*a*B)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + 
Sqrt[b]*x)])/(40*a^(13/4))
 

3.4.82.3 Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.97, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {362, 264, 264, 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.

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

\(\Big \downarrow \) 362

\(\displaystyle \frac {(9 A b-5 a B) \int \frac {1}{x^{7/2} \left (b x^2+a\right )}dx}{4 a b}+\frac {A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {(9 A b-5 a B) \left (-\frac {b \int \frac {1}{x^{3/2} \left (b x^2+a\right )}dx}{a}-\frac {2}{5 a x^{5/2}}\right )}{4 a b}+\frac {A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )}\)

\(\Big \downarrow \) 264

\(\displaystyle \frac {(9 A b-5 a B) \left (-\frac {b \left (-\frac {b \int \frac {\sqrt {x}}{b x^2+a}dx}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{4 a b}+\frac {A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {(9 A b-5 a B) \left (-\frac {b \left (-\frac {2 b \int \frac {x}{b x^2+a}d\sqrt {x}}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{4 a b}+\frac {A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )}\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {(9 A b-5 a B) \left (-\frac {b \left (-\frac {2 b \left (\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{4 a b}+\frac {A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {(9 A b-5 a B) \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{4 a b}+\frac {A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {(9 A b-5 a B) \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{4 a b}+\frac {A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {(9 A b-5 a B) \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{4 a b}+\frac {A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {(9 A b-5 a B) \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{4 a b}+\frac {A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {(9 A b-5 a B) \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{4 a b}+\frac {A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(9 A b-5 a B) \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{4 a b}+\frac {A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {(9 A b-5 a B) \left (-\frac {b \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{4 a b}+\frac {A b-a B}{2 a b x^{5/2} \left (a+b x^2\right )}\)

input

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

output

(A*b - a*B)/(2*a*b*x^(5/2)*(a + b*x^2)) + ((9*A*b - 5*a*B)*(-2/(5*a*x^(5/2 
)) - (b*(-2/(a*Sqrt[x]) - (2*b*((-(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[Sq 
rt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(2*Sqrt[2]*a^(1/4)*b^ 
(1/4)))/(2*Sqrt[b])))/a))/a))/(4*a*b)
 

3.4.82.3.1 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 264

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

3.4.82.4 Maple [A] (verified)

Time = 2.80 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.55

method	result	size
derivativedivides	\(-\frac {2 A}{5 a^{2} x^{\frac {5}{2}}}-\frac {2 \left (-2 A b +B a \right )}{a^{3} \sqrt {x}}+\frac {2 b \left (\frac {\left (\frac {A b}{4}-\frac {B a}{4}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {9 A b}{4}-\frac {5 B a}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{3}}\)	\(170\)
default	\(-\frac {2 A}{5 a^{2} x^{\frac {5}{2}}}-\frac {2 \left (-2 A b +B a \right )}{a^{3} \sqrt {x}}+\frac {2 b \left (\frac {\left (\frac {A b}{4}-\frac {B a}{4}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {9 A b}{4}-\frac {5 B a}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{3}}\)	\(170\)
risch	\(-\frac {2 \left (-10 A b \,x^{2}+5 B a \,x^{2}+A a \right )}{5 a^{3} x^{\frac {5}{2}}}+\frac {b \left (\frac {2 \left (\frac {A b}{4}-\frac {B a}{4}\right ) x^{\frac {3}{2}}}{b \,x^{2}+a}+\frac {\left (\frac {9 A b}{4}-\frac {5 B a}{4}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{3}}\)	\(171\)

input

int((B*x^2+A)/x^(7/2)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
 

output

-2/5*A/a^2/x^(5/2)-2*(-2*A*b+B*a)/a^3/x^(1/2)+2/a^3*b*((1/4*A*b-1/4*B*a)*x 
^(3/2)/(b*x^2+a)+1/8*(9/4*A*b-5/4*B*a)/b/(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)))
 

3.4.82.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 838, normalized size of antiderivative = 2.70 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^2} \, dx=\frac {5 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )} \left (-\frac {625 \, B^{4} a^{4} b - 4500 \, A B^{3} a^{3} b^{2} + 12150 \, A^{2} B^{2} a^{2} b^{3} - 14580 \, A^{3} B a b^{4} + 6561 \, A^{4} b^{5}}{a^{13}}\right )^{\frac {1}{4}} \log \left (a^{10} \left (-\frac {625 \, B^{4} a^{4} b - 4500 \, A B^{3} a^{3} b^{2} + 12150 \, A^{2} B^{2} a^{2} b^{3} - 14580 \, A^{3} B a b^{4} + 6561 \, A^{4} b^{5}}{a^{13}}\right )^{\frac {3}{4}} - {\left (125 \, B^{3} a^{3} b - 675 \, A B^{2} a^{2} b^{2} + 1215 \, A^{2} B a b^{3} - 729 \, A^{3} b^{4}\right )} \sqrt {x}\right ) - 5 \, {\left (i \, a^{3} b x^{5} + i \, a^{4} x^{3}\right )} \left (-\frac {625 \, B^{4} a^{4} b - 4500 \, A B^{3} a^{3} b^{2} + 12150 \, A^{2} B^{2} a^{2} b^{3} - 14580 \, A^{3} B a b^{4} + 6561 \, A^{4} b^{5}}{a^{13}}\right )^{\frac {1}{4}} \log \left (i \, a^{10} \left (-\frac {625 \, B^{4} a^{4} b - 4500 \, A B^{3} a^{3} b^{2} + 12150 \, A^{2} B^{2} a^{2} b^{3} - 14580 \, A^{3} B a b^{4} + 6561 \, A^{4} b^{5}}{a^{13}}\right )^{\frac {3}{4}} - {\left (125 \, B^{3} a^{3} b - 675 \, A B^{2} a^{2} b^{2} + 1215 \, A^{2} B a b^{3} - 729 \, A^{3} b^{4}\right )} \sqrt {x}\right ) - 5 \, {\left (-i \, a^{3} b x^{5} - i \, a^{4} x^{3}\right )} \left (-\frac {625 \, B^{4} a^{4} b - 4500 \, A B^{3} a^{3} b^{2} + 12150 \, A^{2} B^{2} a^{2} b^{3} - 14580 \, A^{3} B a b^{4} + 6561 \, A^{4} b^{5}}{a^{13}}\right )^{\frac {1}{4}} \log \left (-i \, a^{10} \left (-\frac {625 \, B^{4} a^{4} b - 4500 \, A B^{3} a^{3} b^{2} + 12150 \, A^{2} B^{2} a^{2} b^{3} - 14580 \, A^{3} B a b^{4} + 6561 \, A^{4} b^{5}}{a^{13}}\right )^{\frac {3}{4}} - {\left (125 \, B^{3} a^{3} b - 675 \, A B^{2} a^{2} b^{2} + 1215 \, A^{2} B a b^{3} - 729 \, A^{3} b^{4}\right )} \sqrt {x}\right ) - 5 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )} \left (-\frac {625 \, B^{4} a^{4} b - 4500 \, A B^{3} a^{3} b^{2} + 12150 \, A^{2} B^{2} a^{2} b^{3} - 14580 \, A^{3} B a b^{4} + 6561 \, A^{4} b^{5}}{a^{13}}\right )^{\frac {1}{4}} \log \left (-a^{10} \left (-\frac {625 \, B^{4} a^{4} b - 4500 \, A B^{3} a^{3} b^{2} + 12150 \, A^{2} B^{2} a^{2} b^{3} - 14580 \, A^{3} B a b^{4} + 6561 \, A^{4} b^{5}}{a^{13}}\right )^{\frac {3}{4}} - {\left (125 \, B^{3} a^{3} b - 675 \, A B^{2} a^{2} b^{2} + 1215 \, A^{2} B a b^{3} - 729 \, A^{3} b^{4}\right )} \sqrt {x}\right ) - 4 \, {\left (5 \, {\left (5 \, B a b - 9 \, A b^{2}\right )} x^{4} + 4 \, A a^{2} + 4 \, {\left (5 \, B a^{2} - 9 \, A a b\right )} x^{2}\right )} \sqrt {x}}{40 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}} \]

input

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

output

1/40*(5*(a^3*b*x^5 + a^4*x^3)*(-(625*B^4*a^4*b - 4500*A*B^3*a^3*b^2 + 1215 
0*A^2*B^2*a^2*b^3 - 14580*A^3*B*a*b^4 + 6561*A^4*b^5)/a^13)^(1/4)*log(a^10 
*(-(625*B^4*a^4*b - 4500*A*B^3*a^3*b^2 + 12150*A^2*B^2*a^2*b^3 - 14580*A^3 
*B*a*b^4 + 6561*A^4*b^5)/a^13)^(3/4) - (125*B^3*a^3*b - 675*A*B^2*a^2*b^2 
+ 1215*A^2*B*a*b^3 - 729*A^3*b^4)*sqrt(x)) - 5*(I*a^3*b*x^5 + I*a^4*x^3)*( 
-(625*B^4*a^4*b - 4500*A*B^3*a^3*b^2 + 12150*A^2*B^2*a^2*b^3 - 14580*A^3*B 
*a*b^4 + 6561*A^4*b^5)/a^13)^(1/4)*log(I*a^10*(-(625*B^4*a^4*b - 4500*A*B^ 
3*a^3*b^2 + 12150*A^2*B^2*a^2*b^3 - 14580*A^3*B*a*b^4 + 6561*A^4*b^5)/a^13 
)^(3/4) - (125*B^3*a^3*b - 675*A*B^2*a^2*b^2 + 1215*A^2*B*a*b^3 - 729*A^3* 
b^4)*sqrt(x)) - 5*(-I*a^3*b*x^5 - I*a^4*x^3)*(-(625*B^4*a^4*b - 4500*A*B^3 
*a^3*b^2 + 12150*A^2*B^2*a^2*b^3 - 14580*A^3*B*a*b^4 + 6561*A^4*b^5)/a^13) 
^(1/4)*log(-I*a^10*(-(625*B^4*a^4*b - 4500*A*B^3*a^3*b^2 + 12150*A^2*B^2*a 
^2*b^3 - 14580*A^3*B*a*b^4 + 6561*A^4*b^5)/a^13)^(3/4) - (125*B^3*a^3*b - 
675*A*B^2*a^2*b^2 + 1215*A^2*B*a*b^3 - 729*A^3*b^4)*sqrt(x)) - 5*(a^3*b*x^ 
5 + a^4*x^3)*(-(625*B^4*a^4*b - 4500*A*B^3*a^3*b^2 + 12150*A^2*B^2*a^2*b^3 
 - 14580*A^3*B*a*b^4 + 6561*A^4*b^5)/a^13)^(1/4)*log(-a^10*(-(625*B^4*a^4* 
b - 4500*A*B^3*a^3*b^2 + 12150*A^2*B^2*a^2*b^3 - 14580*A^3*B*a*b^4 + 6561* 
A^4*b^5)/a^13)^(3/4) - (125*B^3*a^3*b - 675*A*B^2*a^2*b^2 + 1215*A^2*B*a*b 
^3 - 729*A^3*b^4)*sqrt(x)) - 4*(5*(5*B*a*b - 9*A*b^2)*x^4 + 4*A*a^2 + 4*(5 
*B*a^2 - 9*A*a*b)*x^2)*sqrt(x))/(a^3*b*x^5 + a^4*x^3)
 

3.4.82.6 Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^2} \, dx=\text {Timed out} \]

input

integrate((B*x**2+A)/x**(7/2)/(b*x**2+a)**2,x)
 

output

Timed out
 

3.4.82.7 Maxima [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.81 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^2} \, dx=-\frac {5 \, {\left (5 \, B a b - 9 \, A b^{2}\right )} x^{4} + 4 \, A a^{2} + 4 \, {\left (5 \, B a^{2} - 9 \, A a b\right )} x^{2}}{10 \, {\left (a^{3} b x^{\frac {9}{2}} + a^{4} x^{\frac {5}{2}}\right )}} - \frac {{\left (5 \, B a b - 9 \, A b^{2}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{16 \, a^{3}} \]

input

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

output

-1/10*(5*(5*B*a*b - 9*A*b^2)*x^4 + 4*A*a^2 + 4*(5*B*a^2 - 9*A*a*b)*x^2)/(a 
^3*b*x^(9/2) + a^4*x^(5/2)) - 1/16*(5*B*a*b - 9*A*b^2)*(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)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqr 
t(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqr 
t(a)*sqrt(b))*sqrt(b)) - sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqr 
t(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)) + sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1/4) 
*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)))/a^3
 

3.4.82.8 Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^2} \, dx=-\frac {B a b x^{\frac {3}{2}} - A b^{2} x^{\frac {3}{2}}}{2 \, {\left (b x^{2} + a\right )} a^{3}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{4} b^{2}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a^{4} b^{2}} + \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{4} b^{2}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a^{4} b^{2}} - \frac {2 \, {\left (5 \, B a x^{2} - 10 \, A b x^{2} + A a\right )}}{5 \, a^{3} x^{\frac {5}{2}}} \]

input

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

output

-1/2*(B*a*b*x^(3/2) - A*b^2*x^(3/2))/((b*x^2 + a)*a^3) - 1/8*sqrt(2)*(5*(a 
*b^3)^(3/4)*B*a - 9*(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^4*b^2) - 1/8*sqrt(2)*(5*(a*b^3)^(3/4)*B* 
a - 9*(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^4*b^2) + 1/16*sqrt(2)*(5*(a*b^3)^(3/4)*B*a - 9*(a*b^3 
)^(3/4)*A*b)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^4*b^2) - 
1/16*sqrt(2)*(5*(a*b^3)^(3/4)*B*a - 9*(a*b^3)^(3/4)*A*b)*log(-sqrt(2)*sqrt 
(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^4*b^2) - 2/5*(5*B*a*x^2 - 10*A*b*x^2 + 
 A*a)/(a^3*x^(5/2))
 

3.4.82.9 Mupad [B] (verification not implemented)

Time = 5.14 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.39 \[ \int \frac {A+B x^2}{x^{7/2} \left (a+b x^2\right )^2} \, dx=\frac {\frac {2\,x^2\,\left (9\,A\,b-5\,B\,a\right )}{5\,a^2}-\frac {2\,A}{5\,a}+\frac {b\,x^4\,\left (9\,A\,b-5\,B\,a\right )}{2\,a^3}}{a\,x^{5/2}+b\,x^{9/2}}+\frac {{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}}{a^{1/4}}\right )\,\left (9\,A\,b-5\,B\,a\right )}{4\,a^{13/4}}-\frac {{\left (-b\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}}{a^{1/4}}\right )\,\left (9\,A\,b-5\,B\,a\right )}{4\,a^{13/4}} \]

input

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

output

((2*x^2*(9*A*b - 5*B*a))/(5*a^2) - (2*A)/(5*a) + (b*x^4*(9*A*b - 5*B*a))/( 
2*a^3))/(a*x^(5/2) + b*x^(9/2)) + ((-b)^(1/4)*atan(((-b)^(1/4)*x^(1/2))/a^ 
(1/4))*(9*A*b - 5*B*a))/(4*a^(13/4)) - ((-b)^(1/4)*atanh(((-b)^(1/4)*x^(1/ 
2))/a^(1/4))*(9*A*b - 5*B*a))/(4*a^(13/4))