\(\int \frac {(d x)^{15/2}}{(a^2+2 a b x^2+b^2 x^4)^{3/2}} \, dx\) [757]

Optimal result

Integrand size = 30, antiderivative size = 551 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {13 d^3 (d x)^{9/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a d^7 \sqrt {d x} \left (a+b x^2\right )}{16 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 d^5 (d x)^{5/2} \left (a+b x^2\right )}{80 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

[Out]

-13/16*d^3*(d*x)^(9/2)/b^2/((b*x^2+a)^2)^(1/2)-1/4*d*(d*x)^(13/2)/b/(b*x^2+a)/((b*x^2+a)^2)^(1/2)+117/80*d^5*(
d*x)^(5/2)*(b*x^2+a)/b^3/((b*x^2+a)^2)^(1/2)-117/64*a^(5/4)*d^(15/2)*(b*x^2+a)*arctan(1-b^(1/4)*2^(1/2)*(d*x)^
(1/2)/a^(1/4)/d^(1/2))/b^(17/4)*2^(1/2)/((b*x^2+a)^2)^(1/2)+117/64*a^(5/4)*d^(15/2)*(b*x^2+a)*arctan(1+b^(1/4)
*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/b^(17/4)*2^(1/2)/((b*x^2+a)^2)^(1/2)-117/128*a^(5/4)*d^(15/2)*(b*x^2+a)*
ln(a^(1/2)*d^(1/2)+x*b^(1/2)*d^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*(d*x)^(1/2))/b^(17/4)*2^(1/2)/((b*x^2+a)^2)^(1/2)
+117/128*a^(5/4)*d^(15/2)*(b*x^2+a)*ln(a^(1/2)*d^(1/2)+x*b^(1/2)*d^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*(d*x)^(1/2))/
b^(17/4)*2^(1/2)/((b*x^2+a)^2)^(1/2)-117/16*a*d^7*(b*x^2+a)*(d*x)^(1/2)/b^4/((b*x^2+a)^2)^(1/2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1126, 294, 327, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=-\frac {13 d^3 (d x)^{9/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a d^7 \sqrt {d x} \left (a+b x^2\right )}{16 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 d^5 (d x)^{5/2} \left (a+b x^2\right )}{80 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{32 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

[In]

Int[(d*x)^(15/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

(-13*d^3*(d*x)^(9/2))/(16*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(13/2))/(4*b*(a + b*x^2)*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4]) - (117*a*d^7*Sqrt[d*x]*(a + b*x^2))/(16*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (117*d^5*
(d*x)^(5/2)*(a + b*x^2))/(80*b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (117*a^(5/4)*d^(15/2)*(a + b*x^2)*ArcTan[1
- (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*b^(17/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (11
7*a^(5/4)*d^(15/2)*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*b^(17/4)
*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (117*a^(5/4)*d^(15/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x
- Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(64*Sqrt[2]*b^(17/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (117*a^(5/4)*d^(
15/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(64*Sqrt[2]*b^
(17/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^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 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), 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 294

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

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && 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 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 1126

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + b*x^2 + c*x^4)^FracPa
rt[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c,
d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

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 {\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{15/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{11/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{8 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {13 d^3 (d x)^{9/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{7/2}}{a b+b^2 x^2} \, dx}{32 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {13 d^3 (d x)^{9/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 d^5 (d x)^{5/2} \left (a+b x^2\right )}{80 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (117 a d^6 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{3/2}}{a b+b^2 x^2} \, dx}{32 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {13 d^3 (d x)^{9/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a d^7 \sqrt {d x} \left (a+b x^2\right )}{16 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 d^5 (d x)^{5/2} \left (a+b x^2\right )}{80 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 a^2 d^8 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{32 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {13 d^3 (d x)^{9/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a d^7 \sqrt {d x} \left (a+b x^2\right )}{16 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 d^5 (d x)^{5/2} \left (a+b x^2\right )}{80 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 a^2 d^7 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{16 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {13 d^3 (d x)^{9/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a d^7 \sqrt {d x} \left (a+b x^2\right )}{16 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 d^5 (d x)^{5/2} \left (a+b x^2\right )}{80 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 a^{3/2} d^6 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a} d-\sqrt {b} x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{32 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 a^{3/2} d^6 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {a} d+\sqrt {b} x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{32 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {13 d^3 (d x)^{9/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a d^7 \sqrt {d x} \left (a+b x^2\right )}{16 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 d^5 (d x)^{5/2} \left (a+b x^2\right )}{80 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (117 a^{5/4} d^{15/2} \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}}{\sqrt [4]{b}}+2 x}{-\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {d x}\right )}{64 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (117 a^{5/4} d^{15/2} \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}}{\sqrt [4]{b}}-2 x}{-\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt {d x}\right )}{64 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 a^{3/2} d^8 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {d x}\right )}{64 b^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 a^{3/2} d^8 \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt {d x}\right )}{64 b^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {13 d^3 (d x)^{9/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a d^7 \sqrt {d x} \left (a+b x^2\right )}{16 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 d^5 (d x)^{5/2} \left (a+b x^2\right )}{80 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 a^{5/4} d^{15/2} \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (117 a^{5/4} d^{15/2} \left (a b+b^2 x^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ & = -\frac {13 d^3 (d x)^{9/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a d^7 \sqrt {d x} \left (a+b x^2\right )}{16 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 d^5 (d x)^{5/2} \left (a+b x^2\right )}{80 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.37 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {d^7 \sqrt {d x} \left (4 \sqrt [4]{b} \sqrt {x} \left (-585 a^3-1053 a^2 b x^2-416 a b^2 x^4+32 b^3 x^6\right )-585 \sqrt {2} a^{5/4} \left (a+b x^2\right )^2 \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+585 \sqrt {2} a^{5/4} \left (a+b x^2\right )^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{320 b^{17/4} \sqrt {x} \left (a+b x^2\right ) \sqrt {\left (a+b x^2\right )^2}} \]

[In]

Integrate[(d*x)^(15/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^(3/2),x]

[Out]

(d^7*Sqrt[d*x]*(4*b^(1/4)*Sqrt[x]*(-585*a^3 - 1053*a^2*b*x^2 - 416*a*b^2*x^4 + 32*b^3*x^6) - 585*Sqrt[2]*a^(5/
4)*(a + b*x^2)^2*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 585*Sqrt[2]*a^(5/4)*(a + b*
x^2)^2*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(320*b^(17/4)*Sqrt[x]*(a + b*x^2)*Sq
rt[(a + b*x^2)^2])

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.46

method	result	size
risch	\(-\frac {2 \left (-b \,x^{2}+15 a \right ) x \,d^{8} \sqrt {\left (b \,x^{2}+a \right )^{2}}}{5 b^{4} \sqrt {d x}\, \left (b \,x^{2}+a \right )}+\frac {2 a^{2} d^{9} \left (\frac {-\frac {25 b \left (d x \right )^{\frac {5}{2}}}{32}-\frac {21 a \,d^{2} \sqrt {d x}}{32}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{2}}+\frac {117 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a \,d^{2}}\right ) \sqrt {\left (b \,x^{2}+a \right )^{2}}}{b^{4} \left (b \,x^{2}+a \right )}\)	\(256\)
default	\(-\frac {\left (-256 \left (d x \right )^{\frac {5}{2}} b^{3} x^{4}-585 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (-\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}-d x -\sqrt {\frac {a \,d^{2}}{b}}}\right ) a \,b^{2} d^{2} x^{4}-1170 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) a \,b^{2} d^{2} x^{4}-1170 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) a \,b^{2} d^{2} x^{4}-512 \left (d x \right )^{\frac {5}{2}} a \,b^{2} x^{2}+3840 \sqrt {d x}\, a \,b^{2} d^{2} x^{4}-1170 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (-\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}-d x -\sqrt {\frac {a \,d^{2}}{b}}}\right ) a^{2} b \,d^{2} x^{2}-2340 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) a^{2} b \,d^{2} x^{2}-2340 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) a^{2} b \,d^{2} x^{2}+744 \left (d x \right )^{\frac {5}{2}} a^{2} b +7680 \sqrt {d x}\, a^{2} b \,d^{2} x^{2}-585 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (-\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}-d x -\sqrt {\frac {a \,d^{2}}{b}}}\right ) a^{3} d^{2}-1170 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) a^{3} d^{2}-1170 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right ) a^{3} d^{2}+4680 \sqrt {d x}\, a^{3} d^{2}\right ) d^{5} \left (b \,x^{2}+a \right )}{640 b^{4} {\left (\left (b \,x^{2}+a \right )^{2}\right )}^{\frac {3}{2}}}\)	\(746\)

[In]

int((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/5*(-b*x^2+15*a)*x/b^4/(d*x)^(1/2)*d^8*((b*x^2+a)^2)^(1/2)/(b*x^2+a)+2*a^2/b^4*d^9*((-25/32*b*(d*x)^(5/2)-21
/32*a*d^2*(d*x)^(1/2))/(b*d^2*x^2+a*d^2)^2+117/256*(a*d^2/b)^(1/4)/a/d^2*2^(1/2)*(ln((d*x+(a*d^2/b)^(1/4)*(d*x
)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2))/(d*x-(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))+2*arctan(2^(1/2)/(
a*d^2/b)^(1/4)*(d*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)-1)))*((b*x^2+a)^2)^(1/2)/(b*x^2+a)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.66 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {585 \, \left (-\frac {a^{5} d^{30}}{b^{17}}\right )^{\frac {1}{4}} {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )} \log \left (117 \, \sqrt {d x} a d^{7} + 117 \, \left (-\frac {a^{5} d^{30}}{b^{17}}\right )^{\frac {1}{4}} b^{4}\right ) - 585 \, \left (-\frac {a^{5} d^{30}}{b^{17}}\right )^{\frac {1}{4}} {\left (-i \, b^{6} x^{4} - 2 i \, a b^{5} x^{2} - i \, a^{2} b^{4}\right )} \log \left (117 \, \sqrt {d x} a d^{7} + 117 i \, \left (-\frac {a^{5} d^{30}}{b^{17}}\right )^{\frac {1}{4}} b^{4}\right ) - 585 \, \left (-\frac {a^{5} d^{30}}{b^{17}}\right )^{\frac {1}{4}} {\left (i \, b^{6} x^{4} + 2 i \, a b^{5} x^{2} + i \, a^{2} b^{4}\right )} \log \left (117 \, \sqrt {d x} a d^{7} - 117 i \, \left (-\frac {a^{5} d^{30}}{b^{17}}\right )^{\frac {1}{4}} b^{4}\right ) - 585 \, \left (-\frac {a^{5} d^{30}}{b^{17}}\right )^{\frac {1}{4}} {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )} \log \left (117 \, \sqrt {d x} a d^{7} - 117 \, \left (-\frac {a^{5} d^{30}}{b^{17}}\right )^{\frac {1}{4}} b^{4}\right ) + 4 \, {\left (32 \, b^{3} d^{7} x^{6} - 416 \, a b^{2} d^{7} x^{4} - 1053 \, a^{2} b d^{7} x^{2} - 585 \, a^{3} d^{7}\right )} \sqrt {d x}}{320 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} \]

[In]

integrate((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="fricas")

[Out]

1/320*(585*(-a^5*d^30/b^17)^(1/4)*(b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4)*log(117*sqrt(d*x)*a*d^7 + 117*(-a^5*d^30/b
^17)^(1/4)*b^4) - 585*(-a^5*d^30/b^17)^(1/4)*(-I*b^6*x^4 - 2*I*a*b^5*x^2 - I*a^2*b^4)*log(117*sqrt(d*x)*a*d^7
+ 117*I*(-a^5*d^30/b^17)^(1/4)*b^4) - 585*(-a^5*d^30/b^17)^(1/4)*(I*b^6*x^4 + 2*I*a*b^5*x^2 + I*a^2*b^4)*log(1
17*sqrt(d*x)*a*d^7 - 117*I*(-a^5*d^30/b^17)^(1/4)*b^4) - 585*(-a^5*d^30/b^17)^(1/4)*(b^6*x^4 + 2*a*b^5*x^2 + a
^2*b^4)*log(117*sqrt(d*x)*a*d^7 - 117*(-a^5*d^30/b^17)^(1/4)*b^4) + 4*(32*b^3*d^7*x^6 - 416*a*b^2*d^7*x^4 - 10
53*a^2*b*d^7*x^2 - 585*a^3*d^7)*sqrt(d*x))/(b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4)

Sympy [F(-1)]

Timed out. \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate((d*x)**(15/2)/(b**2*x**4+2*a*b*x**2+a**2)**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\int { \frac {\left (d x\right )^{\frac {15}{2}}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="maxima")

[Out]

-1/2*a^2*d^(15/2)*x^(5/2)/(a*b^4*x^2 + a^2*b^3 + (b^5*x^2 + a*b^4)*x^2) - 2*a*d^(15/2)*integrate(x^(3/2)/(b^4*
x^2 + a*b^3), x) + d^(15/2)*integrate(x^(7/2)/(b^3*x^2 + a*b^2), x) + 21/128*(2*sqrt(2)*a^(3/2)*arctan(1/2*sqr
t(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/sqrt(sqrt(a)*sqrt(b)) + 2*sqrt(2)*a^
(3/2)*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)*sq
rt(b)) + sqrt(2)*a^(5/4)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/b^(1/4) - sqrt(2)*a^(5/4)*
log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/b^(1/4))*d^(15/2)/b^4 - 1/16*(17*a^2*b*d^(15/2)*x^
(5/2) + 21*a^3*d^(15/2)*sqrt(x))/(b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 377, normalized size of antiderivative = 0.68 \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\frac {1}{640} \, d^{7} {\left (\frac {1170 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{5} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {1170 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{b^{5} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {585 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{5} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {585 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{b^{5} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {40 \, {\left (25 \, \sqrt {d x} a^{2} b d^{4} x^{2} + 21 \, \sqrt {d x} a^{3} d^{4}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} b^{4} \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {256 \, {\left (\sqrt {d x} b^{12} d^{10} x^{2} - 15 \, \sqrt {d x} a b^{11} d^{10}\right )}}{b^{15} d^{10} \mathrm {sgn}\left (b x^{2} + a\right )}\right )} \]

[In]

integrate((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^(3/2),x, algorithm="giac")

[Out]

1/640*d^7*(1170*sqrt(2)*(a*b^3*d^2)^(1/4)*a*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/
b)^(1/4))/(b^5*sgn(b*x^2 + a)) + 1170*sqrt(2)*(a*b^3*d^2)^(1/4)*a*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4)
- 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(b^5*sgn(b*x^2 + a)) + 585*sqrt(2)*(a*b^3*d^2)^(1/4)*a*log(d*x + sqrt(2)*(a*d
^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(b^5*sgn(b*x^2 + a)) - 585*sqrt(2)*(a*b^3*d^2)^(1/4)*a*log(d*x - sqrt(2
)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(b^5*sgn(b*x^2 + a)) - 40*(25*sqrt(d*x)*a^2*b*d^4*x^2 + 21*sqrt(d
*x)*a^3*d^4)/((b*d^2*x^2 + a*d^2)^2*b^4*sgn(b*x^2 + a)) + 256*(sqrt(d*x)*b^12*d^10*x^2 - 15*sqrt(d*x)*a*b^11*d
^10)/(b^15*d^10*sgn(b*x^2 + a)))

Mupad [F(-1)]

Timed out. \[ \int \frac {(d x)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx=\int \frac {{\left (d\,x\right )}^{15/2}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}} \,d x \]

[In]

int((d*x)^(15/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^(3/2),x)

[Out]

int((d*x)^(15/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^(3/2), x)