\(\int \frac {A+B x^2}{x^{3/2} (a+b x^2)^3} \, dx\) [388]

Optimal result

Integrand size = 22, antiderivative size = 322 \[ \int \frac {A+B x^2}{x^{3/2} \left (a+b x^2\right )^3} \, dx=-\frac {5 (9 A b-a B)}{16 a^3 b \sqrt {x}}+\frac {A b-a B}{4 a b \sqrt {x} \left (a+b x^2\right )^2}+\frac {9 A b-a B}{16 a^2 b \sqrt {x} \left (a+b x^2\right )}+\frac {5 (9 A b-a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{13/4} b^{3/4}}-\frac {5 (9 A b-a B) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{13/4} b^{3/4}}-\frac {5 (9 A b-a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{13/4} b^{3/4}}+\frac {5 (9 A b-a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{13/4} b^{3/4}} \]

[Out]

5/64*(9*A*b-B*a)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(13/4)/b^(3/4)*2^(1/2)-5/64*(9*A*b-B*a)*arctan(1+
b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(13/4)/b^(3/4)*2^(1/2)-5/128*(9*A*b-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)/b^(3/4)*2^(1/2)+5/128*(9*A*b-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)/b^(3/4)*2^(1/2)-5/16*(9*A*b-B*a)/a^3/b/x^(1/2)+1/4*(A*b-B*a)/a/b/(b*x^2+a)^2/x^(1/2)+1/16*(9*A
*b-B*a)/a^2/b/(b*x^2+a)/x^(1/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {468, 296, 331, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {A+B x^2}{x^{3/2} \left (a+b x^2\right )^3} \, dx=\frac {5 (9 A b-a B) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{13/4} b^{3/4}}-\frac {5 (9 A b-a B) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt {2} a^{13/4} b^{3/4}}-\frac {5 (9 A b-a B) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{13/4} b^{3/4}}+\frac {5 (9 A b-a B) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{64 \sqrt {2} a^{13/4} b^{3/4}}-\frac {5 (9 A b-a B)}{16 a^3 b \sqrt {x}}+\frac {9 A b-a B}{16 a^2 b \sqrt {x} \left (a+b x^2\right )}+\frac {A b-a B}{4 a b \sqrt {x} \left (a+b x^2\right )^2} \]

[In]

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

[Out]

(-5*(9*A*b - a*B))/(16*a^3*b*Sqrt[x]) + (A*b - a*B)/(4*a*b*Sqrt[x]*(a + b*x^2)^2) + (9*A*b - a*B)/(16*a^2*b*Sq
rt[x]*(a + b*x^2)) + (5*(9*A*b - a*B)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(13/4)*b^(3
/4)) - (5*(9*A*b - a*B)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(13/4)*b^(3/4)) - (5*(9*A
*b - a*B)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(13/4)*b^(3/4)) + (5*(9*A*
b - a*B)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(64*Sqrt[2]*a^(13/4)*b^(3/4))

Rule 210

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 296

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 303

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

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 468

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

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[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 1179

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

Rubi steps \begin{align*} \text {integral}& = \frac {A b-a B}{4 a b \sqrt {x} \left (a+b x^2\right )^2}+\frac {\left (\frac {9 A b}{2}-\frac {a B}{2}\right ) \int \frac {1}{x^{3/2} \left (a+b x^2\right )^2} \, dx}{4 a b} \\ & = \frac {A b-a B}{4 a b \sqrt {x} \left (a+b x^2\right )^2}+\frac {9 A b-a B}{16 a^2 b \sqrt {x} \left (a+b x^2\right )}+\frac {(5 (9 A b-a B)) \int \frac {1}{x^{3/2} \left (a+b x^2\right )} \, dx}{32 a^2 b} \\ & = -\frac {5 (9 A b-a B)}{16 a^3 b \sqrt {x}}+\frac {A b-a B}{4 a b \sqrt {x} \left (a+b x^2\right )^2}+\frac {9 A b-a B}{16 a^2 b \sqrt {x} \left (a+b x^2\right )}-\frac {(5 (9 A b-a B)) \int \frac {\sqrt {x}}{a+b x^2} \, dx}{32 a^3} \\ & = -\frac {5 (9 A b-a B)}{16 a^3 b \sqrt {x}}+\frac {A b-a B}{4 a b \sqrt {x} \left (a+b x^2\right )^2}+\frac {9 A b-a B}{16 a^2 b \sqrt {x} \left (a+b x^2\right )}-\frac {(5 (9 A b-a B)) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{16 a^3} \\ & = -\frac {5 (9 A b-a B)}{16 a^3 b \sqrt {x}}+\frac {A b-a B}{4 a b \sqrt {x} \left (a+b x^2\right )^2}+\frac {9 A b-a B}{16 a^2 b \sqrt {x} \left (a+b x^2\right )}+\frac {(5 (9 A b-a B)) \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a^3 \sqrt {b}}-\frac {(5 (9 A b-a B)) \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{32 a^3 \sqrt {b}} \\ & = -\frac {5 (9 A b-a B)}{16 a^3 b \sqrt {x}}+\frac {A b-a B}{4 a b \sqrt {x} \left (a+b x^2\right )^2}+\frac {9 A b-a B}{16 a^2 b \sqrt {x} \left (a+b x^2\right )}-\frac {(5 (9 A b-a B)) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{64 a^3 b}-\frac {(5 (9 A b-a B)) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {x}\right )}{64 a^3 b}-\frac {(5 (9 A b-a B)) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a}}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} a^{13/4} b^{3/4}}-\frac {(5 (9 A b-a B)) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {x}\right )}{64 \sqrt {2} a^{13/4} b^{3/4}} \\ & = -\frac {5 (9 A b-a B)}{16 a^3 b \sqrt {x}}+\frac {A b-a B}{4 a b \sqrt {x} \left (a+b x^2\right )^2}+\frac {9 A b-a B}{16 a^2 b \sqrt {x} \left (a+b x^2\right )}-\frac {5 (9 A b-a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{13/4} b^{3/4}}+\frac {5 (9 A b-a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{13/4} b^{3/4}}-\frac {(5 (9 A b-a B)) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{13/4} b^{3/4}}+\frac {(5 (9 A b-a B)) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{13/4} b^{3/4}} \\ & = -\frac {5 (9 A b-a B)}{16 a^3 b \sqrt {x}}+\frac {A b-a B}{4 a b \sqrt {x} \left (a+b x^2\right )^2}+\frac {9 A b-a B}{16 a^2 b \sqrt {x} \left (a+b x^2\right )}+\frac {5 (9 A b-a B) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{13/4} b^{3/4}}-\frac {5 (9 A b-a B) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{13/4} b^{3/4}}-\frac {5 (9 A b-a B) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{13/4} b^{3/4}}+\frac {5 (9 A b-a B) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{13/4} b^{3/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.58 \[ \int \frac {A+B x^2}{x^{3/2} \left (a+b x^2\right )^3} \, dx=\frac {-\frac {4 \sqrt [4]{a} \left (45 A b^2 x^4+a^2 \left (32 A-9 B x^2\right )+a b x^2 \left (81 A-5 B x^2\right )\right )}{\sqrt {x} \left (a+b x^2\right )^2}+\frac {5 \sqrt {2} (9 A b-a B) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{b^{3/4}}+\frac {5 \sqrt {2} (9 A b-a B) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{b^{3/4}}}{64 a^{13/4}} \]

[In]

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

[Out]

((-4*a^(1/4)*(45*A*b^2*x^4 + a^2*(32*A - 9*B*x^2) + a*b*x^2*(81*A - 5*B*x^2)))/(Sqrt[x]*(a + b*x^2)^2) + (5*Sq
rt[2]*(9*A*b - a*B)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/b^(3/4) + (5*Sqrt[2]*(9*A
*b - a*B)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/b^(3/4))/(64*a^(13/4))

Maple [A] (verified)

Time = 2.70 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.54

method	result	size
derivativedivides	\(-\frac {2 A}{a^{3} \sqrt {x}}-\frac {2 \left (\frac {\left (\frac {13}{32} b^{2} A -\frac {5}{32} a b B \right ) x^{\frac {7}{2}}+\frac {a \left (17 A b -9 B a \right ) x^{\frac {3}{2}}}{32}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (\frac {45 A b}{32}-\frac {5 B a}{32}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{3}}\)	\(173\)
default	\(-\frac {2 A}{a^{3} \sqrt {x}}-\frac {2 \left (\frac {\left (\frac {13}{32} b^{2} A -\frac {5}{32} a b B \right ) x^{\frac {7}{2}}+\frac {a \left (17 A b -9 B a \right ) x^{\frac {3}{2}}}{32}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (\frac {45 A b}{32}-\frac {5 B a}{32}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{3}}\)	\(173\)
risch	\(-\frac {2 A}{a^{3} \sqrt {x}}-\frac {\frac {2 \left (\frac {13}{32} b^{2} A -\frac {5}{32} a b B \right ) x^{\frac {7}{2}}+\frac {a \left (17 A b -9 B a \right ) x^{\frac {3}{2}}}{16}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (\frac {45 A b}{32}-\frac {5 B a}{32}\right ) \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{a^{3}}\)	\(174\)

[In]

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

[Out]

-2*A/a^3/x^(1/2)-2/a^3*(((13/32*b^2*A-5/32*a*b*B)*x^(7/2)+1/32*a*(17*A*b-9*B*a)*x^(3/2))/(b*x^2+a)^2+1/8*(45/3
2*A*b-5/32*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)))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 870, normalized size of antiderivative = 2.70 \[ \int \frac {A+B x^2}{x^{3/2} \left (a+b x^2\right )^3} \, dx=-\frac {5 \, {\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )} \left (-\frac {B^{4} a^{4} - 36 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 2916 \, A^{3} B a b^{3} + 6561 \, A^{4} b^{4}}{a^{13} b^{3}}\right )^{\frac {1}{4}} \log \left (125 \, a^{10} b^{2} \left (-\frac {B^{4} a^{4} - 36 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 2916 \, A^{3} B a b^{3} + 6561 \, A^{4} b^{4}}{a^{13} b^{3}}\right )^{\frac {3}{4}} - 125 \, {\left (B^{3} a^{3} - 27 \, A B^{2} a^{2} b + 243 \, A^{2} B a b^{2} - 729 \, A^{3} b^{3}\right )} \sqrt {x}\right ) + 5 \, {\left (-i \, a^{3} b^{2} x^{5} - 2 i \, a^{4} b x^{3} - i \, a^{5} x\right )} \left (-\frac {B^{4} a^{4} - 36 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 2916 \, A^{3} B a b^{3} + 6561 \, A^{4} b^{4}}{a^{13} b^{3}}\right )^{\frac {1}{4}} \log \left (125 i \, a^{10} b^{2} \left (-\frac {B^{4} a^{4} - 36 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 2916 \, A^{3} B a b^{3} + 6561 \, A^{4} b^{4}}{a^{13} b^{3}}\right )^{\frac {3}{4}} - 125 \, {\left (B^{3} a^{3} - 27 \, A B^{2} a^{2} b + 243 \, A^{2} B a b^{2} - 729 \, A^{3} b^{3}\right )} \sqrt {x}\right ) + 5 \, {\left (i \, a^{3} b^{2} x^{5} + 2 i \, a^{4} b x^{3} + i \, a^{5} x\right )} \left (-\frac {B^{4} a^{4} - 36 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 2916 \, A^{3} B a b^{3} + 6561 \, A^{4} b^{4}}{a^{13} b^{3}}\right )^{\frac {1}{4}} \log \left (-125 i \, a^{10} b^{2} \left (-\frac {B^{4} a^{4} - 36 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 2916 \, A^{3} B a b^{3} + 6561 \, A^{4} b^{4}}{a^{13} b^{3}}\right )^{\frac {3}{4}} - 125 \, {\left (B^{3} a^{3} - 27 \, A B^{2} a^{2} b + 243 \, A^{2} B a b^{2} - 729 \, A^{3} b^{3}\right )} \sqrt {x}\right ) - 5 \, {\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )} \left (-\frac {B^{4} a^{4} - 36 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 2916 \, A^{3} B a b^{3} + 6561 \, A^{4} b^{4}}{a^{13} b^{3}}\right )^{\frac {1}{4}} \log \left (-125 \, a^{10} b^{2} \left (-\frac {B^{4} a^{4} - 36 \, A B^{3} a^{3} b + 486 \, A^{2} B^{2} a^{2} b^{2} - 2916 \, A^{3} B a b^{3} + 6561 \, A^{4} b^{4}}{a^{13} b^{3}}\right )^{\frac {3}{4}} - 125 \, {\left (B^{3} a^{3} - 27 \, A B^{2} a^{2} b + 243 \, A^{2} B a b^{2} - 729 \, A^{3} b^{3}\right )} \sqrt {x}\right ) - 4 \, {\left (5 \, {\left (B a b - 9 \, A b^{2}\right )} x^{4} - 32 \, A a^{2} + 9 \, {\left (B a^{2} - 9 \, A a b\right )} x^{2}\right )} \sqrt {x}}{64 \, {\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )}} \]

[In]

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

[Out]

-1/64*(5*(a^3*b^2*x^5 + 2*a^4*b*x^3 + a^5*x)*(-(B^4*a^4 - 36*A*B^3*a^3*b + 486*A^2*B^2*a^2*b^2 - 2916*A^3*B*a*
b^3 + 6561*A^4*b^4)/(a^13*b^3))^(1/4)*log(125*a^10*b^2*(-(B^4*a^4 - 36*A*B^3*a^3*b + 486*A^2*B^2*a^2*b^2 - 291
6*A^3*B*a*b^3 + 6561*A^4*b^4)/(a^13*b^3))^(3/4) - 125*(B^3*a^3 - 27*A*B^2*a^2*b + 243*A^2*B*a*b^2 - 729*A^3*b^
3)*sqrt(x)) + 5*(-I*a^3*b^2*x^5 - 2*I*a^4*b*x^3 - I*a^5*x)*(-(B^4*a^4 - 36*A*B^3*a^3*b + 486*A^2*B^2*a^2*b^2 -
2916*A^3*B*a*b^3 + 6561*A^4*b^4)/(a^13*b^3))^(1/4)*log(125*I*a^10*b^2*(-(B^4*a^4 - 36*A*B^3*a^3*b + 486*A^2*B
^2*a^2*b^2 - 2916*A^3*B*a*b^3 + 6561*A^4*b^4)/(a^13*b^3))^(3/4) - 125*(B^3*a^3 - 27*A*B^2*a^2*b + 243*A^2*B*a*
b^2 - 729*A^3*b^3)*sqrt(x)) + 5*(I*a^3*b^2*x^5 + 2*I*a^4*b*x^3 + I*a^5*x)*(-(B^4*a^4 - 36*A*B^3*a^3*b + 486*A^
2*B^2*a^2*b^2 - 2916*A^3*B*a*b^3 + 6561*A^4*b^4)/(a^13*b^3))^(1/4)*log(-125*I*a^10*b^2*(-(B^4*a^4 - 36*A*B^3*a
^3*b + 486*A^2*B^2*a^2*b^2 - 2916*A^3*B*a*b^3 + 6561*A^4*b^4)/(a^13*b^3))^(3/4) - 125*(B^3*a^3 - 27*A*B^2*a^2*
b + 243*A^2*B*a*b^2 - 729*A^3*b^3)*sqrt(x)) - 5*(a^3*b^2*x^5 + 2*a^4*b*x^3 + a^5*x)*(-(B^4*a^4 - 36*A*B^3*a^3*
b + 486*A^2*B^2*a^2*b^2 - 2916*A^3*B*a*b^3 + 6561*A^4*b^4)/(a^13*b^3))^(1/4)*log(-125*a^10*b^2*(-(B^4*a^4 - 36
*A*B^3*a^3*b + 486*A^2*B^2*a^2*b^2 - 2916*A^3*B*a*b^3 + 6561*A^4*b^4)/(a^13*b^3))^(3/4) - 125*(B^3*a^3 - 27*A*
B^2*a^2*b + 243*A^2*B*a*b^2 - 729*A^3*b^3)*sqrt(x)) - 4*(5*(B*a*b - 9*A*b^2)*x^4 - 32*A*a^2 + 9*(B*a^2 - 9*A*a
*b)*x^2)*sqrt(x))/(a^3*b^2*x^5 + 2*a^4*b*x^3 + a^5*x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x^2}{x^{3/2} \left (a+b x^2\right )^3} \, dx=\frac {5 \, {\left (B a b - 9 \, A b^{2}\right )} x^{4} - 32 \, A a^{2} + 9 \, {\left (B a^{2} - 9 \, A a b\right )} x^{2}}{16 \, {\left (a^{3} b^{2} x^{\frac {9}{2}} + 2 \, a^{4} b x^{\frac {5}{2}} + a^{5} \sqrt {x}\right )}} + \frac {5 \, {\left (B a - 9 \, A b\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{128 \, a^{3}} \]

[In]

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

[Out]

1/16*(5*(B*a*b - 9*A*b^2)*x^4 - 32*A*a^2 + 9*(B*a^2 - 9*A*a*b)*x^2)/(a^3*b^2*x^(9/2) + 2*a^4*b*x^(5/2) + a^5*s
qrt(x)) + 5/128*(B*a - 9*A*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)) + 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)*sqrt(x) + sqrt(b)*x
+ sqrt(a))/(a^(1/4)*b^(3/4)))/a^3

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x^2}{x^{3/2} \left (a+b x^2\right )^3} \, dx=-\frac {2 \, A}{a^{3} \sqrt {x}} + \frac {5 \, B a b x^{\frac {7}{2}} - 13 \, A b^{2} x^{\frac {7}{2}} + 9 \, B a^{2} x^{\frac {3}{2}} - 17 \, A a b x^{\frac {3}{2}}}{16 \, {\left (b x^{2} + a\right )}^{2} a^{3}} + \frac {5 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{4} b^{3}} + \frac {5 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{4} b^{3}} - \frac {5 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{4} b^{3}} + \frac {5 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {3}{4}} B a - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} A b\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{4} b^{3}} \]

[In]

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

[Out]

-2*A/(a^3*sqrt(x)) + 1/16*(5*B*a*b*x^(7/2) - 13*A*b^2*x^(7/2) + 9*B*a^2*x^(3/2) - 17*A*a*b*x^(3/2))/((b*x^2 +
a)^2*a^3) + 5/64*sqrt(2)*((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^3) + 5/64*sqrt(2)*((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^3) - 5/128*sqrt(2)*((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^3) + 5/128*sqrt(2)*((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^3)

Mupad [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.41 \[ \int \frac {A+B x^2}{x^{3/2} \left (a+b x^2\right )^3} \, dx=\frac {5\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )\,\left (9\,A\,b-B\,a\right )}{32\,{\left (-a\right )}^{13/4}\,b^{3/4}}-\frac {\frac {2\,A}{a}+\frac {9\,x^2\,\left (9\,A\,b-B\,a\right )}{16\,a^2}+\frac {5\,b\,x^4\,\left (9\,A\,b-B\,a\right )}{16\,a^3}}{a^2\,\sqrt {x}+b^2\,x^{9/2}+2\,a\,b\,x^{5/2}}-\frac {5\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )\,\left (9\,A\,b-B\,a\right )}{32\,{\left (-a\right )}^{13/4}\,b^{3/4}} \]

[In]

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

[Out]

(5*atan((b^(1/4)*x^(1/2))/(-a)^(1/4))*(9*A*b - B*a))/(32*(-a)^(13/4)*b^(3/4)) - ((2*A)/a + (9*x^2*(9*A*b - B*a
))/(16*a^2) + (5*b*x^4*(9*A*b - B*a))/(16*a^3))/(a^2*x^(1/2) + b^2*x^(9/2) + 2*a*b*x^(5/2)) - (5*atanh((b^(1/4
)*x^(1/2))/(-a)^(1/4))*(9*A*b - B*a))/(32*(-a)^(13/4)*b^(3/4))