3.6.40 \(\int \frac {(d x)^{11/2}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=391 \[ -\frac {63 d^{11/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{11/4} b^{13/4}}+\frac {63 d^{11/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{11/4} b^{13/4}}-\frac {63 d^{11/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{11/4} b^{13/4}}+\frac {63 d^{11/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{11/4} b^{13/4}}+\frac {21 d^5 \sqrt {d x}}{4096 a^2 b^3 \left (a+b x^2\right )}+\frac {3 d^5 \sqrt {d x}}{1024 a b^3 \left (a+b x^2\right )^2}-\frac {3 d^5 \sqrt {d x}}{128 b^3 \left (a+b x^2\right )^3}-\frac {9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5} \]

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Rubi [A]  time = 0.47, 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, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {21 d^5 \sqrt {d x}}{4096 a^2 b^3 \left (a+b x^2\right )}-\frac {63 d^{11/2} \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^{11/4} b^{13/4}}+\frac {63 d^{11/2} \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^{11/4} b^{13/4}}-\frac {63 d^{11/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{11/4} b^{13/4}}+\frac {63 d^{11/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{11/4} b^{13/4}}+\frac {3 d^5 \sqrt {d x}}{1024 a b^3 \left (a+b x^2\right )^2}-\frac {3 d^5 \sqrt {d x}}{128 b^3 \left (a+b x^2\right )^3}-\frac {9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*x)^(11/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-(d*(d*x)^(9/2))/(10*b*(a + b*x^2)^5) - (9*d^3*(d*x)^(5/2))/(160*b^2*(a + b*x^2)^4) - (3*d^5*Sqrt[d*x])/(128*b
^3*(a + b*x^2)^3) + (3*d^5*Sqrt[d*x])/(1024*a*b^3*(a + b*x^2)^2) + (21*d^5*Sqrt[d*x])/(4096*a^2*b^3*(a + b*x^2
)) - (63*d^(11/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(11/4)*b^(13/4))
+ (63*d^(11/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(11/4)*b^(13/4)) - (
63*d^(11/2)*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^(11
/4)*b^(13/4)) + (63*d^(11/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(16
384*Sqrt[2]*a^(11/4)*b^(13/4))

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 211

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 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 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)^{11/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {(d x)^{11/2}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}+\frac {1}{20} \left (9 b^4 d^2\right ) \int \frac {(d x)^{7/2}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac {9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac {1}{64} \left (9 b^2 d^4\right ) \int \frac {(d x)^{3/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac {9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {3 d^5 \sqrt {d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac {1}{256} \left (3 d^6\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac {9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {3 d^5 \sqrt {d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac {3 d^5 \sqrt {d x}}{1024 a b^3 \left (a+b x^2\right )^2}+\frac {\left (21 d^6\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^2} \, dx}{2048 a b}\\ &=-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac {9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {3 d^5 \sqrt {d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac {3 d^5 \sqrt {d x}}{1024 a b^3 \left (a+b x^2\right )^2}+\frac {21 d^5 \sqrt {d x}}{4096 a^2 b^3 \left (a+b x^2\right )}+\frac {\left (63 d^6\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{8192 a^2 b^2}\\ &=-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac {9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {3 d^5 \sqrt {d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac {3 d^5 \sqrt {d x}}{1024 a b^3 \left (a+b x^2\right )^2}+\frac {21 d^5 \sqrt {d x}}{4096 a^2 b^3 \left (a+b x^2\right )}+\frac {\left (63 d^5\right ) \operatorname {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 a^2 b^2}\\ &=-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac {9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {3 d^5 \sqrt {d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac {3 d^5 \sqrt {d x}}{1024 a b^3 \left (a+b x^2\right )^2}+\frac {21 d^5 \sqrt {d x}}{4096 a^2 b^3 \left (a+b x^2\right )}+\frac {\left (63 d^4\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^{5/2} b^2}+\frac {\left (63 d^4\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^{5/2} b^2}\\ &=-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac {9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {3 d^5 \sqrt {d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac {3 d^5 \sqrt {d x}}{1024 a b^3 \left (a+b x^2\right )^2}+\frac {21 d^5 \sqrt {d x}}{4096 a^2 b^3 \left (a+b x^2\right )}-\frac {\left (63 d^{11/2}\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^{11/4} b^{13/4}}-\frac {\left (63 d^{11/2}\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^{11/4} b^{13/4}}+\frac {\left (63 d^6\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^{5/2} b^{7/2}}+\frac {\left (63 d^6\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^{5/2} b^{7/2}}\\ &=-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac {9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {3 d^5 \sqrt {d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac {3 d^5 \sqrt {d x}}{1024 a b^3 \left (a+b x^2\right )^2}+\frac {21 d^5 \sqrt {d x}}{4096 a^2 b^3 \left (a+b x^2\right )}-\frac {63 d^{11/2} \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^{11/4} b^{13/4}}+\frac {63 d^{11/2} \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^{11/4} b^{13/4}}+\frac {\left (63 d^{11/2}\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^{11/4} b^{13/4}}-\frac {\left (63 d^{11/2}\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^{11/4} b^{13/4}}\\ &=-\frac {d (d x)^{9/2}}{10 b \left (a+b x^2\right )^5}-\frac {9 d^3 (d x)^{5/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {3 d^5 \sqrt {d x}}{128 b^3 \left (a+b x^2\right )^3}+\frac {3 d^5 \sqrt {d x}}{1024 a b^3 \left (a+b x^2\right )^2}+\frac {21 d^5 \sqrt {d x}}{4096 a^2 b^3 \left (a+b x^2\right )}-\frac {63 d^{11/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{11/4} b^{13/4}}+\frac {63 d^{11/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{11/4} b^{13/4}}-\frac {63 d^{11/2} \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^{11/4} b^{13/4}}+\frac {63 d^{11/2} \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^{11/4} b^{13/4}}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 337, normalized size = 0.86 \begin {gather*} \frac {d^5 \sqrt {d x} \left (-\frac {3465 \sqrt {2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{11/4} \sqrt {x}}+\frac {3465 \sqrt {2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{11/4} \sqrt {x}}-\frac {6930 \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{a^{11/4} \sqrt {x}}+\frac {6930 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{a^{11/4} \sqrt {x}}+\frac {9240 \sqrt [4]{b}}{a^2 \left (a+b x^2\right )}-\frac {49152 a^2 \sqrt [4]{b}}{\left (a+b x^2\right )^5}-\frac {196608 a b^{5/4} x^2}{\left (a+b x^2\right )^5}-\frac {327680 b^{9/4} x^4}{\left (a+b x^2\right )^5}+\frac {5280 \sqrt [4]{b}}{a \left (a+b x^2\right )^2}+\frac {3840 \sqrt [4]{b}}{\left (a+b x^2\right )^3}+\frac {3072 a \sqrt [4]{b}}{\left (a+b x^2\right )^4}\right )}{1802240 b^{13/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(11/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(d^5*Sqrt[d*x]*((-49152*a^2*b^(1/4))/(a + b*x^2)^5 - (196608*a*b^(5/4)*x^2)/(a + b*x^2)^5 - (327680*b^(9/4)*x^
4)/(a + b*x^2)^5 + (3072*a*b^(1/4))/(a + b*x^2)^4 + (3840*b^(1/4))/(a + b*x^2)^3 + (5280*b^(1/4))/(a*(a + b*x^
2)^2) + (9240*b^(1/4))/(a^2*(a + b*x^2)) - (6930*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(11
/4)*Sqrt[x]) + (6930*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(11/4)*Sqrt[x]) - (3465*Sqrt[2]
*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(11/4)*Sqrt[x]) + (3465*Sqrt[2]*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(11/4)*Sqrt[x])))/(1802240*b^(13/4))

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IntegrateAlgebraic [A]  time = 1.27, size = 244, normalized size = 0.62 \begin {gather*} -\frac {63 d^{11/2} \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^{11/4} b^{13/4}}+\frac {63 d^{11/2} \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^{11/4} b^{13/4}}-\frac {d^7 \sqrt {d x} \left (315 a^4 d^8+1512 a^3 b d^8 x^2+2870 a^2 b^2 d^8 x^4-480 a b^3 d^8 x^6-105 b^4 d^8 x^8\right )}{20480 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)^(11/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-1/20480*(d^7*Sqrt[d*x]*(315*a^4*d^8 + 1512*a^3*b*d^8*x^2 + 2870*a^2*b^2*d^8*x^4 - 480*a*b^3*d^8*x^6 - 105*b^4
*d^8*x^8))/(a^2*b^3*(a*d^2 + b*d^2*x^2)^5) - (63*d^(11/2)*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^(11/4)*b^(13/4)) + (63*d^(11/2)*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^(11/4)*b^(13/4))

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fricas [A]  time = 3.18, size = 513, normalized size = 1.31 \begin {gather*} \frac {1260 \, {\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^{22}}{a^{11} b^{13}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {d x} a^{8} b^{10} d^{5} \left (-\frac {d^{22}}{a^{11} b^{13}}\right )^{\frac {3}{4}} - \sqrt {a^{6} b^{6} \sqrt {-\frac {d^{22}}{a^{11} b^{13}}} + d^{11} x} a^{8} b^{10} \left (-\frac {d^{22}}{a^{11} b^{13}}\right )^{\frac {3}{4}}}{d^{22}}\right ) + 315 \, {\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^{22}}{a^{11} b^{13}}\right )^{\frac {1}{4}} \log \left (63 \, a^{3} b^{3} \left (-\frac {d^{22}}{a^{11} b^{13}}\right )^{\frac {1}{4}} + 63 \, \sqrt {d x} d^{5}\right ) - 315 \, {\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^{22}}{a^{11} b^{13}}\right )^{\frac {1}{4}} \log \left (-63 \, a^{3} b^{3} \left (-\frac {d^{22}}{a^{11} b^{13}}\right )^{\frac {1}{4}} + 63 \, \sqrt {d x} d^{5}\right ) + 4 \, {\left (105 \, b^{4} d^{5} x^{8} + 480 \, a b^{3} d^{5} x^{6} - 2870 \, a^{2} b^{2} d^{5} x^{4} - 1512 \, a^{3} b d^{5} x^{2} - 315 \, a^{4} d^{5}\right )} \sqrt {d x}}{81920 \, {\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)^(11/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

1/81920*(1260*(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^2
2/(a^11*b^13))^(1/4)*arctan(-(sqrt(d*x)*a^8*b^10*d^5*(-d^22/(a^11*b^13))^(3/4) - sqrt(a^6*b^6*sqrt(-d^22/(a^11
*b^13)) + d^11*x)*a^8*b^10*(-d^22/(a^11*b^13))^(3/4))/d^22) + 315*(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^22/(a^11*b^13))^(1/4)*log(63*a^3*b^3*(-d^22/(a^11*b^13))^(1
/4) + 63*sqrt(d*x)*d^5) - 315*(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^22/(a^11*b^13))^(1/4)*log(-63*a^3*b^3*(-d^22/(a^11*b^13))^(1/4) + 63*sqrt(d*x)*d^5) + 4*(105*b^
4*d^5*x^8 + 480*a*b^3*d^5*x^6 - 2870*a^2*b^2*d^5*x^4 - 1512*a^3*b*d^5*x^2 - 315*a^4*d^5)*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)

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giac [A]  time = 0.21, size = 342, normalized size = 0.87 \begin {gather*} \frac {1}{163840} \, d^{5} {\left (\frac {630 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{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^{4}} + \frac {630 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{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^{4}} + \frac {315 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{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^{4}} - \frac {315 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{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^{4}} + \frac {8 \, {\left (105 \, \sqrt {d x} b^{4} d^{10} x^{8} + 480 \, \sqrt {d x} a b^{3} d^{10} x^{6} - 2870 \, \sqrt {d x} a^{2} b^{2} d^{10} x^{4} - 1512 \, \sqrt {d x} a^{3} b d^{10} x^{2} - 315 \, \sqrt {d x} a^{4} d^{10}\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)^(11/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

1/163840*d^5*(630*sqrt(2)*(a*b^3*d^2)^(1/4)*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^4) + 630*sqrt(2)*(a*b^3*d^2)^(1/4)*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^4) + 315*sqrt(2)*(a*b^3*d^2)^(1/4)*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqr
t(a*d^2/b))/(a^3*b^4) - 315*sqrt(2)*(a*b^3*d^2)^(1/4)*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2
/b))/(a^3*b^4) + 8*(105*sqrt(d*x)*b^4*d^10*x^8 + 480*sqrt(d*x)*a*b^3*d^10*x^6 - 2870*sqrt(d*x)*a^2*b^2*d^10*x^
4 - 1512*sqrt(d*x)*a^3*b*d^10*x^2 - 315*sqrt(d*x)*a^4*d^10)/((b*d^2*x^2 + a*d^2)^5*a^2*b^3))

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maple [A]  time = 0.02, size = 339, normalized size = 0.87 \begin {gather*} -\frac {63 \sqrt {d x}\, a^{2} d^{15}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{3}}-\frac {189 \left (d x \right )^{\frac {5}{2}} a \,d^{13}}{2560 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{2}}-\frac {287 \left (d x \right )^{\frac {9}{2}} d^{11}}{2048 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b}+\frac {3 \left (d x \right )^{\frac {13}{2}} d^{9}}{128 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a}+\frac {21 \left (d x \right )^{\frac {17}{2}} b \,d^{7}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{2}}+\frac {63 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{5} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{16384 a^{3} b^{3}}+\frac {63 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{5} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{16384 a^{3} b^{3}}+\frac {63 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{5} \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 a^{3} b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(11/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

-63/4096*d^15/(b*d^2*x^2+a*d^2)^5/b^3*a^2*(d*x)^(1/2)-189/2560*d^13/(b*d^2*x^2+a*d^2)^5/b^2*a*(d*x)^(5/2)-287/
2048*d^11/(b*d^2*x^2+a*d^2)^5/b*(d*x)^(9/2)+3/128*d^9/(b*d^2*x^2+a*d^2)^5/a*(d*x)^(13/2)+21/4096*d^7/(b*d^2*x^
2+a*d^2)^5/a^2*b*(d*x)^(17/2)+63/32768*d^5/a^3/b^3*(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)))+63/16384*d^5/a^3/b^3*(a/b
*d^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b*d^2)^(1/4)*(d*x)^(1/2)+1)+63/16384*d^5/a^3/b^3*(a/b*d^2)^(1/4)*2^(1/2)
*arctan(2^(1/2)/(a/b*d^2)^(1/4)*(d*x)^(1/2)-1)

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maxima [A]  time = 3.16, size = 394, normalized size = 1.01 \begin {gather*} \frac {\frac {8 \, {\left (105 \, \left (d x\right )^{\frac {17}{2}} b^{4} d^{8} + 480 \, \left (d x\right )^{\frac {13}{2}} a b^{3} d^{10} - 2870 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2} d^{12} - 1512 \, \left (d x\right )^{\frac {5}{2}} a^{3} b d^{14} - 315 \, \sqrt {d x} 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}} + \frac {315 \, {\left (\frac {\sqrt {2} d^{8} \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 {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{8} \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 {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{7} \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 {a}} + \frac {2 \, \sqrt {2} d^{7} \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 {a}}\right )}}{a^{2} b^{3}}}{163840 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(11/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

1/163840*(8*(105*(d*x)^(17/2)*b^4*d^8 + 480*(d*x)^(13/2)*a*b^3*d^10 - 2870*(d*x)^(9/2)*a^2*b^2*d^12 - 1512*(d*
x)^(5/2)*a^3*b*d^14 - 315*sqrt(d*x)*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) + 315*(sqrt(2)*d^8*log(sqrt(b)*d*x + sqrt(2)*(a*d^2)^
(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(3/4)*b^(1/4)) - sqrt(2)*d^8*log(sqrt(b)*d*x - sqrt(2)*(a*d^2)^(
1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(3/4)*b^(1/4)) + 2*sqrt(2)*d^7*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(a)) + 2*sqrt(2)*
d^7*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(s
qrt(a)*sqrt(b)*d)*sqrt(a)))/(a^2*b^3))/d

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mupad [B]  time = 4.23, size = 208, normalized size = 0.53 \begin {gather*} -\frac {\frac {287\,d^{11}\,{\left (d\,x\right )}^{9/2}}{2048\,b}-\frac {3\,d^9\,{\left (d\,x\right )}^{13/2}}{128\,a}+\frac {63\,a^2\,d^{15}\,\sqrt {d\,x}}{4096\,b^3}+\frac {189\,a\,d^{13}\,{\left (d\,x\right )}^{5/2}}{2560\,b^2}-\frac {21\,b\,d^7\,{\left (d\,x\right )}^{17/2}}{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 {63\,d^{11/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{11/4}\,b^{13/4}}-\frac {63\,d^{11/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{11/4}\,b^{13/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(11/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)

[Out]

- ((287*d^11*(d*x)^(9/2))/(2048*b) - (3*d^9*(d*x)^(13/2))/(128*a) + (63*a^2*d^15*(d*x)^(1/2))/(4096*b^3) + (18
9*a*d^13*(d*x)^(5/2))/(2560*b^2) - (21*b*d^7*(d*x)^(17/2))/(4096*a^2))/(a^5*d^10 + b^5*d^10*x^10 + 5*a^4*b*d^1
0*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) - (63*d^(11/2)*atan((b^(1/4)*(d*x)^(1/2)
)/((-a)^(1/4)*d^(1/2))))/(8192*(-a)^(11/4)*b^(13/4)) - (63*d^(11/2)*atanh((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^
(1/2))))/(8192*(-a)^(11/4)*b^(13/4))

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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)**(11/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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