3.6.39 \(\int \frac {(d x)^{13/2}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=391 \[ \frac {77 d^{13/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{9/4} b^{15/4}}-\frac {77 d^{13/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{9/4} b^{15/4}}-\frac {77 d^{13/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{9/4} b^{15/4}}+\frac {77 d^{13/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{9/4} b^{15/4}}+\frac {77 d^5 (d x)^{3/2}}{4096 a^2 b^3 \left (a+b x^2\right )}+\frac {77 d^5 (d x)^{3/2}}{5120 a b^3 \left (a+b x^2\right )^2}-\frac {77 d^5 (d x)^{3/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {11 d^3 (d x)^{7/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5} \]

________________________________________________________________________________________

Rubi [A]  time = 0.48, antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {28, 288, 290, 329, 297, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {77 d^5 (d x)^{3/2}}{4096 a^2 b^3 \left (a+b x^2\right )}+\frac {77 d^{13/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{9/4} b^{15/4}}-\frac {77 d^{13/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{9/4} b^{15/4}}-\frac {77 d^{13/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{9/4} b^{15/4}}+\frac {77 d^{13/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{9/4} b^{15/4}}+\frac {77 d^5 (d x)^{3/2}}{5120 a b^3 \left (a+b x^2\right )^2}-\frac {77 d^5 (d x)^{3/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {11 d^3 (d x)^{7/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*x)^(13/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-(d*(d*x)^(11/2))/(10*b*(a + b*x^2)^5) - (11*d^3*(d*x)^(7/2))/(160*b^2*(a + b*x^2)^4) - (77*d^5*(d*x)^(3/2))/(
1920*b^3*(a + b*x^2)^3) + (77*d^5*(d*x)^(3/2))/(5120*a*b^3*(a + b*x^2)^2) + (77*d^5*(d*x)^(3/2))/(4096*a^2*b^3
*(a + b*x^2)) - (77*d^(13/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(9/4)*
b^(15/4)) + (77*d^(13/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(9/4)*b^(1
5/4)) + (77*d^(13/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16384*Sqrt
[2]*a^(9/4)*b^(15/4)) - (77*d^(13/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*
x]])/(16384*Sqrt[2]*a^(9/4)*b^(15/4))

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 288

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 290

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] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 297

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 329

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 617

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 628

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 1162

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 1165

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*} \int \frac {(d x)^{13/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {(d x)^{13/2}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac {d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5}+\frac {1}{20} \left (11 b^4 d^2\right ) \int \frac {(d x)^{9/2}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac {d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5}-\frac {11 d^3 (d x)^{7/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac {1}{320} \left (77 b^2 d^4\right ) \int \frac {(d x)^{5/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5}-\frac {11 d^3 (d x)^{7/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {77 d^5 (d x)^{3/2}}{1920 b^3 \left (a+b x^2\right )^3}+\frac {\left (77 d^6\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^3} \, dx}{1280}\\ &=-\frac {d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5}-\frac {11 d^3 (d x)^{7/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {77 d^5 (d x)^{3/2}}{1920 b^3 \left (a+b x^2\right )^3}+\frac {77 d^5 (d x)^{3/2}}{5120 a b^3 \left (a+b x^2\right )^2}+\frac {\left (77 d^6\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^2} \, dx}{2048 a b}\\ &=-\frac {d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5}-\frac {11 d^3 (d x)^{7/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {77 d^5 (d x)^{3/2}}{1920 b^3 \left (a+b x^2\right )^3}+\frac {77 d^5 (d x)^{3/2}}{5120 a b^3 \left (a+b x^2\right )^2}+\frac {77 d^5 (d x)^{3/2}}{4096 a^2 b^3 \left (a+b x^2\right )}+\frac {\left (77 d^6\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{8192 a^2 b^2}\\ &=-\frac {d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5}-\frac {11 d^3 (d x)^{7/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {77 d^5 (d x)^{3/2}}{1920 b^3 \left (a+b x^2\right )^3}+\frac {77 d^5 (d x)^{3/2}}{5120 a b^3 \left (a+b x^2\right )^2}+\frac {77 d^5 (d x)^{3/2}}{4096 a^2 b^3 \left (a+b x^2\right )}+\frac {\left (77 d^5\right ) \operatorname {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 a^2 b^2}\\ &=-\frac {d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5}-\frac {11 d^3 (d x)^{7/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {77 d^5 (d x)^{3/2}}{1920 b^3 \left (a+b x^2\right )^3}+\frac {77 d^5 (d x)^{3/2}}{5120 a b^3 \left (a+b x^2\right )^2}+\frac {77 d^5 (d x)^{3/2}}{4096 a^2 b^3 \left (a+b x^2\right )}-\frac {\left (77 d^5\right ) \operatorname {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 )}{8192 a^2 b^{5/2}}+\frac {\left (77 d^5\right ) \operatorname {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 )}{8192 a^2 b^{5/2}}\\ &=-\frac {d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5}-\frac {11 d^3 (d x)^{7/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {77 d^5 (d x)^{3/2}}{1920 b^3 \left (a+b x^2\right )^3}+\frac {77 d^5 (d x)^{3/2}}{5120 a b^3 \left (a+b x^2\right )^2}+\frac {77 d^5 (d x)^{3/2}}{4096 a^2 b^3 \left (a+b x^2\right )}+\frac {\left (77 d^{13/2}\right ) \operatorname {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 )}{16384 \sqrt {2} a^{9/4} b^{15/4}}+\frac {\left (77 d^{13/2}\right ) \operatorname {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 )}{16384 \sqrt {2} a^{9/4} b^{15/4}}+\frac {\left (77 d^7\right ) \operatorname {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 )}{16384 a^2 b^4}+\frac {\left (77 d^7\right ) \operatorname {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 )}{16384 a^2 b^4}\\ &=-\frac {d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5}-\frac {11 d^3 (d x)^{7/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {77 d^5 (d x)^{3/2}}{1920 b^3 \left (a+b x^2\right )^3}+\frac {77 d^5 (d x)^{3/2}}{5120 a b^3 \left (a+b x^2\right )^2}+\frac {77 d^5 (d x)^{3/2}}{4096 a^2 b^3 \left (a+b x^2\right )}+\frac {77 d^{13/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{9/4} b^{15/4}}-\frac {77 d^{13/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{9/4} b^{15/4}}+\frac {\left (77 d^{13/2}\right ) \operatorname {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 )}{8192 \sqrt {2} a^{9/4} b^{15/4}}-\frac {\left (77 d^{13/2}\right ) \operatorname {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 )}{8192 \sqrt {2} a^{9/4} b^{15/4}}\\ &=-\frac {d (d x)^{11/2}}{10 b \left (a+b x^2\right )^5}-\frac {11 d^3 (d x)^{7/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {77 d^5 (d x)^{3/2}}{1920 b^3 \left (a+b x^2\right )^3}+\frac {77 d^5 (d x)^{3/2}}{5120 a b^3 \left (a+b x^2\right )^2}+\frac {77 d^5 (d x)^{3/2}}{4096 a^2 b^3 \left (a+b x^2\right )}-\frac {77 d^{13/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{9/4} b^{15/4}}+\frac {77 d^{13/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{9/4} b^{15/4}}+\frac {77 d^{13/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{9/4} b^{15/4}}-\frac {77 d^{13/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{9/4} b^{15/4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.03, size = 85, normalized size = 0.22 \begin {gather*} \frac {2 d^6 x \sqrt {d x} \left (77 \left (a+b x^2\right )^5 \, _2F_1\left (\frac {3}{4},6;\frac {7}{4};-\frac {b x^2}{a}\right )-a^3 \left (77 a^2+187 a b x^2+221 b^2 x^4\right )\right )}{1989 a^3 b^3 \left (a+b x^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(13/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(2*d^6*x*Sqrt[d*x]*(-(a^3*(77*a^2 + 187*a*b*x^2 + 221*b^2*x^4)) + 77*(a + b*x^2)^5*Hypergeometric2F1[3/4, 6, 7
/4, -((b*x^2)/a)]))/(1989*a^3*b^3*(a + b*x^2)^5)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 1.30, size = 244, normalized size = 0.62 \begin {gather*} -\frac {77 d^{13/2} \tan ^{-1}\left (\frac {\frac {\sqrt [4]{a} \sqrt {d}}{\sqrt {2} \sqrt [4]{b}}-\frac {\sqrt [4]{b} \sqrt {d} x}{\sqrt {2} \sqrt [4]{a}}}{\sqrt {d x}}\right )}{8192 \sqrt {2} a^{9/4} b^{15/4}}-\frac {77 d^{13/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}}{\sqrt {a} d+\sqrt {b} d x}\right )}{8192 \sqrt {2} a^{9/4} b^{15/4}}-\frac {d^7 (d x)^{3/2} \left (385 a^4 d^8+1760 a^3 b d^8 x^2+3130 a^2 b^2 d^8 x^4-5544 a b^3 d^8 x^6-1155 b^4 d^8 x^8\right )}{61440 a^2 b^3 \left (a d^2+b d^2 x^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d*x)^(13/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-1/61440*(d^7*(d*x)^(3/2)*(385*a^4*d^8 + 1760*a^3*b*d^8*x^2 + 3130*a^2*b^2*d^8*x^4 - 5544*a*b^3*d^8*x^6 - 1155
*b^4*d^8*x^8))/(a^2*b^3*(a*d^2 + b*d^2*x^2)^5) - (77*d^(13/2)*ArcTan[((a^(1/4)*Sqrt[d])/(Sqrt[2]*b^(1/4)) - (b
^(1/4)*Sqrt[d]*x)/(Sqrt[2]*a^(1/4)))/Sqrt[d*x]])/(8192*Sqrt[2]*a^(9/4)*b^(15/4)) - (77*d^(13/2)*ArcTanh[(Sqrt[
2]*a^(1/4)*b^(1/4)*Sqrt[d]*Sqrt[d*x])/(Sqrt[a]*d + Sqrt[b]*d*x)])/(8192*Sqrt[2]*a^(9/4)*b^(15/4))

________________________________________________________________________________________

fricas [A]  time = 1.76, size = 518, normalized size = 1.32 \begin {gather*} -\frac {4620 \, {\left (a^{2} b^{8} x^{10} + 5 \, a^{3} b^{7} x^{8} + 10 \, a^{4} b^{6} x^{6} + 10 \, a^{5} b^{5} x^{4} + 5 \, a^{6} b^{4} x^{2} + a^{7} b^{3}\right )} \left (-\frac {d^{26}}{a^{9} b^{15}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\left (-\frac {d^{26}}{a^{9} b^{15}}\right )^{\frac {1}{4}} \sqrt {d x} a^{2} b^{4} d^{19} - \sqrt {d^{39} x - \sqrt {-\frac {d^{26}}{a^{9} b^{15}}} a^{5} b^{7} d^{26}} \left (-\frac {d^{26}}{a^{9} b^{15}}\right )^{\frac {1}{4}} a^{2} b^{4}}{d^{26}}\right ) - 1155 \, {\left (a^{2} b^{8} x^{10} + 5 \, a^{3} b^{7} x^{8} + 10 \, a^{4} b^{6} x^{6} + 10 \, a^{5} b^{5} x^{4} + 5 \, a^{6} b^{4} x^{2} + a^{7} b^{3}\right )} \left (-\frac {d^{26}}{a^{9} b^{15}}\right )^{\frac {1}{4}} \log \left (456533 \, \sqrt {d x} d^{19} + 456533 \, \left (-\frac {d^{26}}{a^{9} b^{15}}\right )^{\frac {3}{4}} a^{7} b^{11}\right ) + 1155 \, {\left (a^{2} b^{8} x^{10} + 5 \, a^{3} b^{7} x^{8} + 10 \, a^{4} b^{6} x^{6} + 10 \, a^{5} b^{5} x^{4} + 5 \, a^{6} b^{4} x^{2} + a^{7} b^{3}\right )} \left (-\frac {d^{26}}{a^{9} b^{15}}\right )^{\frac {1}{4}} \log \left (456533 \, \sqrt {d x} d^{19} - 456533 \, \left (-\frac {d^{26}}{a^{9} b^{15}}\right )^{\frac {3}{4}} a^{7} b^{11}\right ) - 4 \, {\left (1155 \, b^{4} d^{6} x^{9} + 5544 \, a b^{3} d^{6} x^{7} - 3130 \, a^{2} b^{2} d^{6} x^{5} - 1760 \, a^{3} b d^{6} x^{3} - 385 \, a^{4} d^{6} x\right )} \sqrt {d x}}{245760 \, {\left (a^{2} b^{8} x^{10} + 5 \, a^{3} b^{7} x^{8} + 10 \, a^{4} b^{6} x^{6} + 10 \, a^{5} b^{5} x^{4} + 5 \, a^{6} b^{4} x^{2} + a^{7} b^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(13/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

-1/245760*(4620*(a^2*b^8*x^10 + 5*a^3*b^7*x^8 + 10*a^4*b^6*x^6 + 10*a^5*b^5*x^4 + 5*a^6*b^4*x^2 + a^7*b^3)*(-d
^26/(a^9*b^15))^(1/4)*arctan(-((-d^26/(a^9*b^15))^(1/4)*sqrt(d*x)*a^2*b^4*d^19 - sqrt(d^39*x - sqrt(-d^26/(a^9
*b^15))*a^5*b^7*d^26)*(-d^26/(a^9*b^15))^(1/4)*a^2*b^4)/d^26) - 1155*(a^2*b^8*x^10 + 5*a^3*b^7*x^8 + 10*a^4*b^
6*x^6 + 10*a^5*b^5*x^4 + 5*a^6*b^4*x^2 + a^7*b^3)*(-d^26/(a^9*b^15))^(1/4)*log(456533*sqrt(d*x)*d^19 + 456533*
(-d^26/(a^9*b^15))^(3/4)*a^7*b^11) + 1155*(a^2*b^8*x^10 + 5*a^3*b^7*x^8 + 10*a^4*b^6*x^6 + 10*a^5*b^5*x^4 + 5*
a^6*b^4*x^2 + a^7*b^3)*(-d^26/(a^9*b^15))^(1/4)*log(456533*sqrt(d*x)*d^19 - 456533*(-d^26/(a^9*b^15))^(3/4)*a^
7*b^11) - 4*(1155*b^4*d^6*x^9 + 5544*a*b^3*d^6*x^7 - 3130*a^2*b^2*d^6*x^5 - 1760*a^3*b*d^6*x^3 - 385*a^4*d^6*x
)*sqrt(d*x))/(a^2*b^8*x^10 + 5*a^3*b^7*x^8 + 10*a^4*b^6*x^6 + 10*a^5*b^5*x^4 + 5*a^6*b^4*x^2 + a^7*b^3)

________________________________________________________________________________________

giac [A]  time = 0.23, size = 355, normalized size = 0.91 \begin {gather*} \frac {1}{491520} \, d^{6} {\left (\frac {2310 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \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 )}{a^{3} b^{6} d} + \frac {2310 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \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 )}{a^{3} b^{6} d} - \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \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 )}{a^{3} b^{6} d} + \frac {1155 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \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 )}{a^{3} b^{6} d} + \frac {8 \, {\left (1155 \, \sqrt {d x} b^{4} d^{10} x^{9} + 5544 \, \sqrt {d x} a b^{3} d^{10} x^{7} - 3130 \, \sqrt {d x} a^{2} b^{2} d^{10} x^{5} - 1760 \, \sqrt {d x} a^{3} b d^{10} x^{3} - 385 \, \sqrt {d x} a^{4} d^{10} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{2} b^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(13/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

1/491520*d^6*(2310*sqrt(2)*(a*b^3*d^2)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2
/b)^(1/4))/(a^3*b^6*d) + 2310*sqrt(2)*(a*b^3*d^2)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt(
d*x))/(a*d^2/b)^(1/4))/(a^3*b^6*d) - 1155*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x
) + sqrt(a*d^2/b))/(a^3*b^6*d) + 1155*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) +
sqrt(a*d^2/b))/(a^3*b^6*d) + 8*(1155*sqrt(d*x)*b^4*d^10*x^9 + 5544*sqrt(d*x)*a*b^3*d^10*x^7 - 3130*sqrt(d*x)*a
^2*b^2*d^10*x^5 - 1760*sqrt(d*x)*a^3*b*d^10*x^3 - 385*sqrt(d*x)*a^4*d^10*x)/((b*d^2*x^2 + a*d^2)^5*a^2*b^3))

________________________________________________________________________________________

maple [A]  time = 0.02, size = 339, normalized size = 0.87 \begin {gather*} -\frac {77 \left (d x \right )^{\frac {3}{2}} a^{2} d^{15}}{12288 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{3}}-\frac {11 \left (d x \right )^{\frac {7}{2}} a \,d^{13}}{384 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{2}}-\frac {313 \left (d x \right )^{\frac {11}{2}} d^{11}}{6144 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b}+\frac {231 \left (d x \right )^{\frac {15}{2}} d^{9}}{2560 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a}+\frac {77 \left (d x \right )^{\frac {19}{2}} b \,d^{7}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{2}}+\frac {77 \sqrt {2}\, d^{7} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{16384 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{2} b^{4}}+\frac {77 \sqrt {2}\, d^{7} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{16384 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{2} b^{4}}+\frac {77 \sqrt {2}\, d^{7} \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 )}{32768 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{2} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(13/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

-77/12288*d^15/(b*d^2*x^2+a*d^2)^5/b^3*a^2*(d*x)^(3/2)-11/384*d^13/(b*d^2*x^2+a*d^2)^5/b^2*a*(d*x)^(7/2)-313/6
144*d^11/(b*d^2*x^2+a*d^2)^5/b*(d*x)^(11/2)+231/2560*d^9/(b*d^2*x^2+a*d^2)^5/a*(d*x)^(15/2)+77/4096*d^7/(b*d^2
*x^2+a*d^2)^5/a^2*b*(d*x)^(19/2)+77/32768*d^7/a^2/b^4/(a/b*d^2)^(1/4)*2^(1/2)*ln((d*x-(a/b*d^2)^(1/4)*(d*x)^(1
/2)*2^(1/2)+(a/b*d^2)^(1/2))/(d*x+(a/b*d^2)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a/b*d^2)^(1/2)))+77/16384*d^7/a^2/b^4/(
a/b*d^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b*d^2)^(1/4)*(d*x)^(1/2)+1)+77/16384*d^7/a^2/b^4/(a/b*d^2)^(1/4)*2^(1
/2)*arctan(2^(1/2)/(a/b*d^2)^(1/4)*(d*x)^(1/2)-1)

________________________________________________________________________________________

maxima [A]  time = 3.00, size = 385, normalized size = 0.98 \begin {gather*} \frac {\frac {1155 \, d^{8} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{a^{2} b^{3}} + \frac {8 \, {\left (1155 \, \left (d x\right )^{\frac {19}{2}} b^{4} d^{8} + 5544 \, \left (d x\right )^{\frac {15}{2}} a b^{3} d^{10} - 3130 \, \left (d x\right )^{\frac {11}{2}} a^{2} b^{2} d^{12} - 1760 \, \left (d x\right )^{\frac {7}{2}} a^{3} b d^{14} - 385 \, \left (d x\right )^{\frac {3}{2}} a^{4} d^{16}\right )}}{a^{2} b^{8} d^{10} x^{10} + 5 \, a^{3} b^{7} d^{10} x^{8} + 10 \, a^{4} b^{6} d^{10} x^{6} + 10 \, a^{5} b^{5} d^{10} x^{4} + 5 \, a^{6} b^{4} d^{10} x^{2} + a^{7} b^{3} d^{10}}}{491520 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(13/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

1/491520*(1155*d^8*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) + 2*sqrt(d*x)*sqrt(b))/sqrt(sq
rt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(
1/4) - 2*sqrt(d*x)*sqrt(b))/sqrt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(b)) - sqrt(2)*log(sqrt(b)*d
*x + sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(1/4)*b^(3/4)) + sqrt(2)*log(sqrt(b)*d*x -
sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(1/4)*b^(3/4)))/(a^2*b^3) + 8*(1155*(d*x)^(19/2)
*b^4*d^8 + 5544*(d*x)^(15/2)*a*b^3*d^10 - 3130*(d*x)^(11/2)*a^2*b^2*d^12 - 1760*(d*x)^(7/2)*a^3*b*d^14 - 385*(
d*x)^(3/2)*a^4*d^16)/(a^2*b^8*d^10*x^10 + 5*a^3*b^7*d^10*x^8 + 10*a^4*b^6*d^10*x^6 + 10*a^5*b^5*d^10*x^4 + 5*a
^6*b^4*d^10*x^2 + a^7*b^3*d^10))/d

________________________________________________________________________________________

mupad [B]  time = 4.32, size = 208, normalized size = 0.53 \begin {gather*} \frac {77\,d^{13/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{9/4}\,b^{15/4}}-\frac {\frac {313\,d^{11}\,{\left (d\,x\right )}^{11/2}}{6144\,b}-\frac {231\,d^9\,{\left (d\,x\right )}^{15/2}}{2560\,a}+\frac {77\,a^2\,d^{15}\,{\left (d\,x\right )}^{3/2}}{12288\,b^3}+\frac {11\,a\,d^{13}\,{\left (d\,x\right )}^{7/2}}{384\,b^2}-\frac {77\,b\,d^7\,{\left (d\,x\right )}^{19/2}}{4096\,a^2}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}}-\frac {77\,d^{13/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{9/4}\,b^{15/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(13/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)

[Out]

(77*d^(13/2)*atan((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(8192*(-a)^(9/4)*b^(15/4)) - ((313*d^11*(d*x)^(
11/2))/(6144*b) - (231*d^9*(d*x)^(15/2))/(2560*a) + (77*a^2*d^15*(d*x)^(3/2))/(12288*b^3) + (11*a*d^13*(d*x)^(
7/2))/(384*b^2) - (77*b*d^7*(d*x)^(19/2))/(4096*a^2))/(a^5*d^10 + b^5*d^10*x^10 + 5*a^4*b*d^10*x^2 + 5*a*b^4*d
^10*x^8 + 10*a^3*b^2*d^10*x^4 + 10*a^2*b^3*d^10*x^6) - (77*d^(13/2)*atanh((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^
(1/2))))/(8192*(-a)^(9/4)*b^(15/4))

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**(13/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

________________________________________________________________________________________