3.8.15 \(\int \frac {(d x)^{17/2}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\) [715]

Optimal. Leaf size=388 \[ -\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}-\frac {3 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {11 d^5 (d x)^{7/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {77 d^7 (d x)^{3/2}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {231 d^7 (d x)^{3/2}}{4096 a b^4 \left (a+b x^2\right )}-\frac {231 d^{17/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{5/4} b^{19/4}}+\frac {231 d^{17/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{5/4} b^{19/4}}+\frac {231 d^{17/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{5/4} b^{19/4}}-\frac {231 d^{17/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{16384 \sqrt {2} a^{5/4} b^{19/4}} \]

[Out]

-1/10*d*(d*x)^(15/2)/b/(b*x^2+a)^5-3/32*d^3*(d*x)^(11/2)/b^2/(b*x^2+a)^4-11/128*d^5*(d*x)^(7/2)/b^3/(b*x^2+a)^
3-77/1024*d^7*(d*x)^(3/2)/b^4/(b*x^2+a)^2+231/4096*d^7*(d*x)^(3/2)/a/b^4/(b*x^2+a)-231/16384*d^(17/2)*arctan(1
-b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(5/4)/b^(19/4)*2^(1/2)+231/16384*d^(17/2)*arctan(1+b^(1/4)*2^(
1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(5/4)/b^(19/4)*2^(1/2)+231/32768*d^(17/2)*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))/a^(5/4)/b^(19/4)*2^(1/2)-231/32768*d^(17/2)*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))/a^(5/4)/b^(19/4)*2^(1/2)

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Rubi [A]
time = 0.29, antiderivative size = 388, 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, 294, 296, 335, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {231 d^{17/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{5/4} b^{19/4}}+\frac {231 d^{17/2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{5/4} b^{19/4}}+\frac {231 d^{17/2} \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^{5/4} b^{19/4}}-\frac {231 d^{17/2} \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^{5/4} b^{19/4}}+\frac {231 d^7 (d x)^{3/2}}{4096 a b^4 \left (a+b x^2\right )}-\frac {77 d^7 (d x)^{3/2}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {11 d^5 (d x)^{7/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {3 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*x)^(17/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-1/10*(d*(d*x)^(15/2))/(b*(a + b*x^2)^5) - (3*d^3*(d*x)^(11/2))/(32*b^2*(a + b*x^2)^4) - (11*d^5*(d*x)^(7/2))/
(128*b^3*(a + b*x^2)^3) - (77*d^7*(d*x)^(3/2))/(1024*b^4*(a + b*x^2)^2) + (231*d^7*(d*x)^(3/2))/(4096*a*b^4*(a
+ b*x^2)) - (231*d^(17/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(5/4)*b^
(19/4)) + (231*d^(17/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(5/4)*b^(19
/4)) + (231*d^(17/2)*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^(5/4)*b^(19/4)) - (231*d^(17/2)*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^(5/4)*b^(19/4))

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 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 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 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 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 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*} \int \frac {(d x)^{17/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {(d x)^{17/2}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}+\frac {1}{4} \left (3 b^4 d^2\right ) \int \frac {(d x)^{13/2}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}-\frac {3 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^4}+\frac {1}{64} \left (33 b^2 d^4\right ) \int \frac {(d x)^{9/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}-\frac {3 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {11 d^5 (d x)^{7/2}}{128 b^3 \left (a+b x^2\right )^3}+\frac {1}{256} \left (77 d^6\right ) \int \frac {(d x)^{5/2}}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}-\frac {3 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {11 d^5 (d x)^{7/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {77 d^7 (d x)^{3/2}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {\left (231 d^8\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^2} \, dx}{2048 b^2}\\ &=-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}-\frac {3 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {11 d^5 (d x)^{7/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {77 d^7 (d x)^{3/2}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {231 d^7 (d x)^{3/2}}{4096 a b^4 \left (a+b x^2\right )}+\frac {\left (231 d^8\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{8192 a b^3}\\ &=-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}-\frac {3 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {11 d^5 (d x)^{7/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {77 d^7 (d x)^{3/2}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {231 d^7 (d x)^{3/2}}{4096 a b^4 \left (a+b x^2\right )}+\frac {\left (231 d^7\right ) \text {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 a b^3}\\ &=-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}-\frac {3 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {11 d^5 (d x)^{7/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {77 d^7 (d x)^{3/2}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {231 d^7 (d x)^{3/2}}{4096 a b^4 \left (a+b x^2\right )}-\frac {\left (231 d^7\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 )}{8192 a b^{7/2}}+\frac {\left (231 d^7\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 )}{8192 a b^{7/2}}\\ &=-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}-\frac {3 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {11 d^5 (d x)^{7/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {77 d^7 (d x)^{3/2}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {231 d^7 (d x)^{3/2}}{4096 a b^4 \left (a+b x^2\right )}+\frac {\left (231 d^{17/2}\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 )}{16384 \sqrt {2} a^{5/4} b^{19/4}}+\frac {\left (231 d^{17/2}\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 )}{16384 \sqrt {2} a^{5/4} b^{19/4}}+\frac {\left (231 d^9\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 )}{16384 a b^5}+\frac {\left (231 d^9\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 )}{16384 a b^5}\\ &=-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}-\frac {3 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {11 d^5 (d x)^{7/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {77 d^7 (d x)^{3/2}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {231 d^7 (d x)^{3/2}}{4096 a b^4 \left (a+b x^2\right )}+\frac {231 d^{17/2} \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^{5/4} b^{19/4}}-\frac {231 d^{17/2} \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^{5/4} b^{19/4}}+\frac {\left (231 d^{17/2}\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 )}{8192 \sqrt {2} a^{5/4} b^{19/4}}-\frac {\left (231 d^{17/2}\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 )}{8192 \sqrt {2} a^{5/4} b^{19/4}}\\ &=-\frac {d (d x)^{15/2}}{10 b \left (a+b x^2\right )^5}-\frac {3 d^3 (d x)^{11/2}}{32 b^2 \left (a+b x^2\right )^4}-\frac {11 d^5 (d x)^{7/2}}{128 b^3 \left (a+b x^2\right )^3}-\frac {77 d^7 (d x)^{3/2}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {231 d^7 (d x)^{3/2}}{4096 a b^4 \left (a+b x^2\right )}-\frac {231 d^{17/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{5/4} b^{19/4}}+\frac {231 d^{17/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{5/4} b^{19/4}}+\frac {231 d^{17/2} \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^{5/4} b^{19/4}}-\frac {231 d^{17/2} \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^{5/4} b^{19/4}}\\ \end {align*}

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Mathematica [A]
time = 0.74, size = 205, normalized size = 0.53 \begin {gather*} \frac {d^8 \sqrt {d x} \left (4 \sqrt [4]{a} b^{3/4} x^{3/2} \left (-385 a^4-1760 a^3 b x^2-3130 a^2 b^2 x^4-2648 a b^3 x^6+1155 b^4 x^8\right )+1155 \sqrt {2} \left (a+b x^2\right )^5 \tan ^{-1}\left (\frac {-\sqrt {a}+\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-1155 \sqrt {2} \left (a+b x^2\right )^5 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{81920 a^{5/4} b^{19/4} \sqrt {x} \left (a+b x^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(17/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(d^8*Sqrt[d*x]*(4*a^(1/4)*b^(3/4)*x^(3/2)*(-385*a^4 - 1760*a^3*b*x^2 - 3130*a^2*b^2*x^4 - 2648*a*b^3*x^6 + 115
5*b^4*x^8) + 1155*Sqrt[2]*(a + b*x^2)^5*ArcTan[(-Sqrt[a] + Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - 115
5*Sqrt[2]*(a + b*x^2)^5*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(81920*a^(5/4)*b^(1
9/4)*Sqrt[x]*(a + b*x^2)^5)

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Maple [A]
time = 0.10, size = 238, normalized size = 0.61

method	result	size
derivativedivides	\(2 d^{11} \left (\frac {-\frac {77 a^{3} d^{6} \left (d x \right )^{\frac {3}{2}}}{8192 b^{4}}-\frac {11 a^{2} d^{4} \left (d x \right )^{\frac {7}{2}}}{256 b^{3}}-\frac {313 a \,d^{2} \left (d x \right )^{\frac {11}{2}}}{4096 b^{2}}-\frac {331 \left (d x \right )^{\frac {15}{2}}}{5120 b}+\frac {231 \left (d x \right )^{\frac {19}{2}}}{8192 a \,d^{2}}}{\left (d^{2} x^{2} b +a \,d^{2}\right )^{5}}+\frac {231 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a \,d^{2} b^{5} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\)	\(238\)
default	\(2 d^{11} \left (\frac {-\frac {77 a^{3} d^{6} \left (d x \right )^{\frac {3}{2}}}{8192 b^{4}}-\frac {11 a^{2} d^{4} \left (d x \right )^{\frac {7}{2}}}{256 b^{3}}-\frac {313 a \,d^{2} \left (d x \right )^{\frac {11}{2}}}{4096 b^{2}}-\frac {331 \left (d x \right )^{\frac {15}{2}}}{5120 b}+\frac {231 \left (d x \right )^{\frac {19}{2}}}{8192 a \,d^{2}}}{\left (d^{2} x^{2} b +a \,d^{2}\right )^{5}}+\frac {231 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{65536 a \,d^{2} b^{5} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\)	\(238\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(17/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

2*d^11*((-77/8192*a^3*d^6/b^4*(d*x)^(3/2)-11/256/b^3*a^2*d^4*(d*x)^(7/2)-313/4096/b^2*a*d^2*(d*x)^(11/2)-331/5
120/b*(d*x)^(15/2)+231/8192/a/d^2*(d*x)^(19/2))/(b*d^2*x^2+a*d^2)^5+231/65536/a/d^2/b^5/(a*d^2/b)^(1/4)*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)))

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Maxima [A]
time = 0.52, size = 383, normalized size = 0.99 \begin {gather*} \frac {\frac {1155 \, d^{10} {\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 b^{4}} + \frac {8 \, {\left (1155 \, \left (d x\right )^{\frac {19}{2}} b^{4} d^{10} - 2648 \, \left (d x\right )^{\frac {15}{2}} a b^{3} d^{12} - 3130 \, \left (d x\right )^{\frac {11}{2}} a^{2} b^{2} d^{14} - 1760 \, \left (d x\right )^{\frac {7}{2}} a^{3} b d^{16} - 385 \, \left (d x\right )^{\frac {3}{2}} a^{4} d^{18}\right )}}{a b^{9} d^{10} x^{10} + 5 \, a^{2} b^{8} d^{10} x^{8} + 10 \, a^{3} b^{7} d^{10} x^{6} + 10 \, a^{4} b^{6} d^{10} x^{4} + 5 \, a^{5} b^{5} d^{10} x^{2} + a^{6} b^{4} d^{10}}}{163840 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(17/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

1/163840*(1155*d^10*(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(s
qrt(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*b^4) + 8*(1155*(d*x)^(19/2)*
b^4*d^10 - 2648*(d*x)^(15/2)*a*b^3*d^12 - 3130*(d*x)^(11/2)*a^2*b^2*d^14 - 1760*(d*x)^(7/2)*a^3*b*d^16 - 385*(
d*x)^(3/2)*a^4*d^18)/(a*b^9*d^10*x^10 + 5*a^2*b^8*d^10*x^8 + 10*a^3*b^7*d^10*x^6 + 10*a^4*b^6*d^10*x^4 + 5*a^5
*b^5*d^10*x^2 + a^6*b^4*d^10))/d

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Fricas [A]
time = 0.35, size = 506, normalized size = 1.30 \begin {gather*} -\frac {4620 \, {\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )} \left (-\frac {d^{34}}{a^{5} b^{19}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\left (-\frac {d^{34}}{a^{5} b^{19}}\right )^{\frac {1}{4}} \sqrt {d x} a b^{5} d^{25} - \sqrt {d^{51} x - \sqrt {-\frac {d^{34}}{a^{5} b^{19}}} a^{3} b^{9} d^{34}} \left (-\frac {d^{34}}{a^{5} b^{19}}\right )^{\frac {1}{4}} a b^{5}}{d^{34}}\right ) - 1155 \, {\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )} \left (-\frac {d^{34}}{a^{5} b^{19}}\right )^{\frac {1}{4}} \log \left (12326391 \, \sqrt {d x} d^{25} + 12326391 \, \left (-\frac {d^{34}}{a^{5} b^{19}}\right )^{\frac {3}{4}} a^{4} b^{14}\right ) + 1155 \, {\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )} \left (-\frac {d^{34}}{a^{5} b^{19}}\right )^{\frac {1}{4}} \log \left (12326391 \, \sqrt {d x} d^{25} - 12326391 \, \left (-\frac {d^{34}}{a^{5} b^{19}}\right )^{\frac {3}{4}} a^{4} b^{14}\right ) - 4 \, {\left (1155 \, b^{4} d^{8} x^{9} - 2648 \, a b^{3} d^{8} x^{7} - 3130 \, a^{2} b^{2} d^{8} x^{5} - 1760 \, a^{3} b d^{8} x^{3} - 385 \, a^{4} d^{8} x\right )} \sqrt {d x}}{81920 \, {\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(17/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

-1/81920*(4620*(a*b^9*x^10 + 5*a^2*b^8*x^8 + 10*a^3*b^7*x^6 + 10*a^4*b^6*x^4 + 5*a^5*b^5*x^2 + a^6*b^4)*(-d^34
/(a^5*b^19))^(1/4)*arctan(-((-d^34/(a^5*b^19))^(1/4)*sqrt(d*x)*a*b^5*d^25 - sqrt(d^51*x - sqrt(-d^34/(a^5*b^19
))*a^3*b^9*d^34)*(-d^34/(a^5*b^19))^(1/4)*a*b^5)/d^34) - 1155*(a*b^9*x^10 + 5*a^2*b^8*x^8 + 10*a^3*b^7*x^6 + 1
0*a^4*b^6*x^4 + 5*a^5*b^5*x^2 + a^6*b^4)*(-d^34/(a^5*b^19))^(1/4)*log(12326391*sqrt(d*x)*d^25 + 12326391*(-d^3
4/(a^5*b^19))^(3/4)*a^4*b^14) + 1155*(a*b^9*x^10 + 5*a^2*b^8*x^8 + 10*a^3*b^7*x^6 + 10*a^4*b^6*x^4 + 5*a^5*b^5
*x^2 + a^6*b^4)*(-d^34/(a^5*b^19))^(1/4)*log(12326391*sqrt(d*x)*d^25 - 12326391*(-d^34/(a^5*b^19))^(3/4)*a^4*b
^14) - 4*(1155*b^4*d^8*x^9 - 2648*a*b^3*d^8*x^7 - 3130*a^2*b^2*d^8*x^5 - 1760*a^3*b*d^8*x^3 - 385*a^4*d^8*x)*s
qrt(d*x))/(a*b^9*x^10 + 5*a^2*b^8*x^8 + 10*a^3*b^7*x^6 + 10*a^4*b^6*x^4 + 5*a^5*b^5*x^2 + a^6*b^4)

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Sympy [F(-1)] Timed out
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)**(17/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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Giac [A]
time = 11.85, size = 355, normalized size = 0.91 \begin {gather*} \frac {1}{163840} \, d^{8} {\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^{2} b^{7} 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^{2} b^{7} 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^{2} b^{7} 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^{2} b^{7} d} + \frac {8 \, {\left (1155 \, \sqrt {d x} b^{4} d^{10} x^{9} - 2648 \, \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 b^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(17/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

1/163840*d^8*(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^2*b^7*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^2*b^7*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^2*b^7*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^2*b^7*d) + 8*(1155*sqrt(d*x)*b^4*d^10*x^9 - 2648*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*b^4))

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Mupad [B]
time = 4.29, size = 210, normalized size = 0.54 \begin {gather*} \frac {231\,d^{17/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{5/4}\,b^{19/4}}-\frac {231\,d^{17/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{5/4}\,b^{19/4}}-\frac {\frac {331\,d^{11}\,{\left (d\,x\right )}^{15/2}}{2560\,b}-\frac {231\,d^9\,{\left (d\,x\right )}^{19/2}}{4096\,a}+\frac {11\,a^2\,d^{15}\,{\left (d\,x\right )}^{7/2}}{128\,b^3}+\frac {77\,a^3\,d^{17}\,{\left (d\,x\right )}^{3/2}}{4096\,b^4}+\frac {313\,a\,d^{13}\,{\left (d\,x\right )}^{11/2}}{2048\,b^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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(17/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)

[Out]

(231*d^(17/2)*atanh((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(8192*(-a)^(5/4)*b^(19/4)) - (231*d^(17/2)*at
an((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(8192*(-a)^(5/4)*b^(19/4)) - ((331*d^11*(d*x)^(15/2))/(2560*b)
- (231*d^9*(d*x)^(19/2))/(4096*a) + (11*a^2*d^15*(d*x)^(7/2))/(128*b^3) + (77*a^3*d^17*(d*x)^(3/2))/(4096*b^4
) + (313*a*d^13*(d*x)^(11/2))/(2048*b^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)

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