3.6.36 \(\int \frac {(d x)^{19/2}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=385 \[ -\frac {663 d^{19/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{3/4} b^{21/4}}+\frac {663 d^{19/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{3/4} b^{21/4}}-\frac {663 d^{19/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{3/4} b^{21/4}}+\frac {663 d^{19/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{3/4} b^{21/4}}-\frac {663 d^9 \sqrt {d x}}{4096 b^5 \left (a+b x^2\right )}-\frac {663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5} \]

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Rubi [A]  time = 0.45, antiderivative size = 385, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {28, 288, 329, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {663 d^{19/2} \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^{3/4} b^{21/4}}+\frac {663 d^{19/2} \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^{3/4} b^{21/4}}-\frac {663 d^{19/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{3/4} b^{21/4}}+\frac {663 d^{19/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{3/4} b^{21/4}}-\frac {663 d^9 \sqrt {d x}}{4096 b^5 \left (a+b x^2\right )}-\frac {663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d*x)^(19/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-(d*(d*x)^(17/2))/(10*b*(a + b*x^2)^5) - (17*d^3*(d*x)^(13/2))/(160*b^2*(a + b*x^2)^4) - (221*d^5*(d*x)^(9/2))
/(1920*b^3*(a + b*x^2)^3) - (663*d^7*(d*x)^(5/2))/(5120*b^4*(a + b*x^2)^2) - (663*d^9*Sqrt[d*x])/(4096*b^5*(a
+ b*x^2)) - (663*d^(19/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(3/4)*b^(
21/4)) + (663*d^(19/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(3/4)*b^(21/
4)) - (663*d^(19/2)*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^(3/4)*b^(21/4)) + (663*d^(19/2)*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^(3/4)*b^(21/4))

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 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)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {(d x)^{19/2}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}+\frac {1}{20} \left (17 b^4 d^2\right ) \int \frac {(d x)^{15/2}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac {17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac {1}{320} \left (221 b^2 d^4\right ) \int \frac {(d x)^{11/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac {17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}+\frac {\left (663 d^6\right ) \int \frac {(d x)^{7/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{1280}\\ &=-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac {17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}+\frac {\left (663 d^8\right ) \int \frac {(d x)^{3/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{2048 b^2}\\ &=-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac {17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {663 d^9 \sqrt {d x}}{4096 b^5 \left (a+b x^2\right )}+\frac {\left (663 d^{10}\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{8192 b^4}\\ &=-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac {17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {663 d^9 \sqrt {d x}}{4096 b^5 \left (a+b x^2\right )}+\frac {\left (663 d^9\right ) \operatorname {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 b^4}\\ &=-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac {17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {663 d^9 \sqrt {d x}}{4096 b^5 \left (a+b x^2\right )}+\frac {\left (663 d^8\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 \sqrt {a} b^4}+\frac {\left (663 d^8\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 \sqrt {a} b^4}\\ &=-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac {17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {663 d^9 \sqrt {d x}}{4096 b^5 \left (a+b x^2\right )}-\frac {\left (663 d^{19/2}\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^{3/4} b^{21/4}}-\frac {\left (663 d^{19/2}\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^{3/4} b^{21/4}}+\frac {\left (663 d^{10}\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 \sqrt {a} b^{11/2}}+\frac {\left (663 d^{10}\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 \sqrt {a} b^{11/2}}\\ &=-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac {17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {663 d^9 \sqrt {d x}}{4096 b^5 \left (a+b x^2\right )}-\frac {663 d^{19/2} \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^{3/4} b^{21/4}}+\frac {663 d^{19/2} \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^{3/4} b^{21/4}}+\frac {\left (663 d^{19/2}\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^{3/4} b^{21/4}}-\frac {\left (663 d^{19/2}\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^{3/4} b^{21/4}}\\ &=-\frac {d (d x)^{17/2}}{10 b \left (a+b x^2\right )^5}-\frac {17 d^3 (d x)^{13/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {221 d^5 (d x)^{9/2}}{1920 b^3 \left (a+b x^2\right )^3}-\frac {663 d^7 (d x)^{5/2}}{5120 b^4 \left (a+b x^2\right )^2}-\frac {663 d^9 \sqrt {d x}}{4096 b^5 \left (a+b x^2\right )}-\frac {663 d^{19/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{3/4} b^{21/4}}+\frac {663 d^{19/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{3/4} b^{21/4}}-\frac {663 d^{19/2} \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^{3/4} b^{21/4}}+\frac {663 d^{19/2} \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^{3/4} b^{21/4}}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 381, normalized size = 0.99 \begin {gather*} \frac {d^9 \sqrt {d x} \left (-\frac {765765 \sqrt {2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{3/4} \sqrt {x}}+\frac {765765 \sqrt {2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{3/4} \sqrt {x}}-\frac {1531530 \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{a^{3/4} \sqrt {x}}+\frac {1531530 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{a^{3/4} \sqrt {x}}-\frac {10862592 a^4 \sqrt [4]{b}}{\left (a+b x^2\right )^5}-\frac {43450368 a^3 b^{5/4} x^2}{\left (a+b x^2\right )^5}+\frac {678912 a^3 \sqrt [4]{b}}{\left (a+b x^2\right )^4}-\frac {72417280 a^2 b^{9/4} x^4}{\left (a+b x^2\right )^5}+\frac {848640 a^2 \sqrt [4]{b}}{\left (a+b x^2\right )^3}-\frac {25231360 b^{17/4} x^8}{\left (a+b x^2\right )^5}-\frac {61276160 a b^{13/4} x^6}{\left (a+b x^2\right )^5}+\frac {2042040 \sqrt [4]{b}}{a+b x^2}+\frac {1166880 a \sqrt [4]{b}}{\left (a+b x^2\right )^2}\right )}{37847040 b^{21/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(19/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(d^9*Sqrt[d*x]*((-10862592*a^4*b^(1/4))/(a + b*x^2)^5 - (43450368*a^3*b^(5/4)*x^2)/(a + b*x^2)^5 - (72417280*a
^2*b^(9/4)*x^4)/(a + b*x^2)^5 - (61276160*a*b^(13/4)*x^6)/(a + b*x^2)^5 - (25231360*b^(17/4)*x^8)/(a + b*x^2)^
5 + (678912*a^3*b^(1/4))/(a + b*x^2)^4 + (848640*a^2*b^(1/4))/(a + b*x^2)^3 + (1166880*a*b^(1/4))/(a + b*x^2)^
2 + (2042040*b^(1/4))/(a + b*x^2) - (1531530*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(3/4)*S
qrt[x]) + (1531530*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(3/4)*Sqrt[x]) - (765765*Sqrt[2]*
Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(3/4)*Sqrt[x]) + (765765*Sqrt[2]*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(3/4)*Sqrt[x])))/(37847040*b^(21/4))

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IntegrateAlgebraic [A]  time = 1.22, size = 222, normalized size = 0.58 \begin {gather*} -\frac {663 d^{19/2} \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^{3/4} b^{21/4}}+\frac {663 d^{19/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x}\right )}{8192 \sqrt {2} a^{3/4} b^{21/4}}-\frac {d^9 \sqrt {d x} \left (9945 a^4+47736 a^3 b x^2+90610 a^2 b^2 x^4+84320 a b^3 x^6+37645 b^4 x^8\right )}{61440 b^5 \left (a+b x^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(d*x)^(19/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-1/61440*(d^9*Sqrt[d*x]*(9945*a^4 + 47736*a^3*b*x^2 + 90610*a^2*b^2*x^4 + 84320*a*b^3*x^6 + 37645*b^4*x^8))/(b
^5*(a + b*x^2)^5) - (663*d^(19/2)*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^(3/4)*b^(21/4)) + (663*d^(19/2)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x
])/(Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x)])/(8192*Sqrt[2]*a^(3/4)*b^(21/4))

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fricas [A]  time = 1.69, size = 489, normalized size = 1.27 \begin {gather*} \frac {39780 \, {\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )} \left (-\frac {d^{38}}{a^{3} b^{21}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\left (-\frac {d^{38}}{a^{3} b^{21}}\right )^{\frac {3}{4}} \sqrt {d x} a^{2} b^{16} d^{9} - \sqrt {d^{19} x + \sqrt {-\frac {d^{38}}{a^{3} b^{21}}} a^{2} b^{10}} \left (-\frac {d^{38}}{a^{3} b^{21}}\right )^{\frac {3}{4}} a^{2} b^{16}}{d^{38}}\right ) + 9945 \, {\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )} \left (-\frac {d^{38}}{a^{3} b^{21}}\right )^{\frac {1}{4}} \log \left (663 \, \sqrt {d x} d^{9} + 663 \, \left (-\frac {d^{38}}{a^{3} b^{21}}\right )^{\frac {1}{4}} a b^{5}\right ) - 9945 \, {\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )} \left (-\frac {d^{38}}{a^{3} b^{21}}\right )^{\frac {1}{4}} \log \left (663 \, \sqrt {d x} d^{9} - 663 \, \left (-\frac {d^{38}}{a^{3} b^{21}}\right )^{\frac {1}{4}} a b^{5}\right ) - 4 \, {\left (37645 \, b^{4} d^{9} x^{8} + 84320 \, a b^{3} d^{9} x^{6} + 90610 \, a^{2} b^{2} d^{9} x^{4} + 47736 \, a^{3} b d^{9} x^{2} + 9945 \, a^{4} d^{9}\right )} \sqrt {d x}}{245760 \, {\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(19/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

1/245760*(39780*(b^10*x^10 + 5*a*b^9*x^8 + 10*a^2*b^8*x^6 + 10*a^3*b^7*x^4 + 5*a^4*b^6*x^2 + a^5*b^5)*(-d^38/(
a^3*b^21))^(1/4)*arctan(-((-d^38/(a^3*b^21))^(3/4)*sqrt(d*x)*a^2*b^16*d^9 - sqrt(d^19*x + sqrt(-d^38/(a^3*b^21
))*a^2*b^10)*(-d^38/(a^3*b^21))^(3/4)*a^2*b^16)/d^38) + 9945*(b^10*x^10 + 5*a*b^9*x^8 + 10*a^2*b^8*x^6 + 10*a^
3*b^7*x^4 + 5*a^4*b^6*x^2 + a^5*b^5)*(-d^38/(a^3*b^21))^(1/4)*log(663*sqrt(d*x)*d^9 + 663*(-d^38/(a^3*b^21))^(
1/4)*a*b^5) - 9945*(b^10*x^10 + 5*a*b^9*x^8 + 10*a^2*b^8*x^6 + 10*a^3*b^7*x^4 + 5*a^4*b^6*x^2 + a^5*b^5)*(-d^3
8/(a^3*b^21))^(1/4)*log(663*sqrt(d*x)*d^9 - 663*(-d^38/(a^3*b^21))^(1/4)*a*b^5) - 4*(37645*b^4*d^9*x^8 + 84320
*a*b^3*d^9*x^6 + 90610*a^2*b^2*d^9*x^4 + 47736*a^3*b*d^9*x^2 + 9945*a^4*d^9)*sqrt(d*x))/(b^10*x^10 + 5*a*b^9*x
^8 + 10*a^2*b^8*x^6 + 10*a^3*b^7*x^4 + 5*a^4*b^6*x^2 + a^5*b^5)

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giac [A]  time = 0.24, size = 339, normalized size = 0.88 \begin {gather*} \frac {1}{491520} \, d^{9} {\left (\frac {19890 \, \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 b^{6}} + \frac {19890 \, \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 b^{6}} + \frac {9945 \, \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 b^{6}} - \frac {9945 \, \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 b^{6}} - \frac {8 \, {\left (37645 \, \sqrt {d x} b^{4} d^{10} x^{8} + 84320 \, \sqrt {d x} a b^{3} d^{10} x^{6} + 90610 \, \sqrt {d x} a^{2} b^{2} d^{10} x^{4} + 47736 \, \sqrt {d x} a^{3} b d^{10} x^{2} + 9945 \, \sqrt {d x} a^{4} d^{10}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} b^{5}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(19/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

1/491520*d^9*(19890*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*b^6) + 19890*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*b^6) + 9945*sqrt(2)*(a*b^3*d^2)^(1/4)*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sq
rt(a*d^2/b))/(a*b^6) - 9945*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*b^6) - 8*(37645*sqrt(d*x)*b^4*d^10*x^8 + 84320*sqrt(d*x)*a*b^3*d^10*x^6 + 90610*sqrt(d*x)*a^2*b^2*d^10
*x^4 + 47736*sqrt(d*x)*a^3*b*d^10*x^2 + 9945*sqrt(d*x)*a^4*d^10)/((b*d^2*x^2 + a*d^2)^5*b^5))

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maple [A]  time = 0.03, size = 344, normalized size = 0.89 \begin {gather*} -\frac {663 \sqrt {d x}\, a^{4} d^{19}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{5}}-\frac {1989 \left (d x \right )^{\frac {5}{2}} a^{3} d^{17}}{2560 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{4}}-\frac {9061 \left (d x \right )^{\frac {9}{2}} a^{2} d^{15}}{6144 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{3}}-\frac {527 \left (d x \right )^{\frac {13}{2}} a \,d^{13}}{384 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{2}}-\frac {7529 \left (d x \right )^{\frac {17}{2}} d^{11}}{12288 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b}+\frac {663 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{9} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{16384 a \,b^{5}}+\frac {663 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{9} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{16384 a \,b^{5}}+\frac {663 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{9} \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 \,b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(19/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

-663/4096*d^19/(b*d^2*x^2+a*d^2)^5/b^5*a^4*(d*x)^(1/2)-1989/2560*d^17/(b*d^2*x^2+a*d^2)^5/b^4*a^3*(d*x)^(5/2)-
9061/6144*d^15/(b*d^2*x^2+a*d^2)^5/b^3*a^2*(d*x)^(9/2)-527/384*d^13/(b*d^2*x^2+a*d^2)^5/b^2*a*(d*x)^(13/2)-752
9/12288*d^11/(b*d^2*x^2+a*d^2)^5/b*(d*x)^(17/2)+663/32768*d^9/b^5*(a/b*d^2)^(1/4)/a*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)))+663/1638
4*d^9/b^5*(a/b*d^2)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/b*d^2)^(1/4)*(d*x)^(1/2)+1)+663/16384*d^9/b^5*(a/b*d^2)^
(1/4)/a*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 = 3.11, size = 386, normalized size = 1.00 \begin {gather*} -\frac {\frac {8 \, {\left (37645 \, \left (d x\right )^{\frac {17}{2}} b^{4} d^{12} + 84320 \, \left (d x\right )^{\frac {13}{2}} a b^{3} d^{14} + 90610 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2} d^{16} + 47736 \, \left (d x\right )^{\frac {5}{2}} a^{3} b d^{18} + 9945 \, \sqrt {d x} a^{4} d^{20}\right )}}{b^{10} d^{10} x^{10} + 5 \, a b^{9} d^{10} x^{8} + 10 \, a^{2} b^{8} d^{10} x^{6} + 10 \, a^{3} b^{7} d^{10} x^{4} + 5 \, a^{4} b^{6} d^{10} x^{2} + a^{5} b^{5} d^{10}} - \frac {9945 \, {\left (\frac {\sqrt {2} d^{12} \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^{12} \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^{11} \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^{11} \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 )}}{b^{5}}}{491520 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(19/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

-1/491520*(8*(37645*(d*x)^(17/2)*b^4*d^12 + 84320*(d*x)^(13/2)*a*b^3*d^14 + 90610*(d*x)^(9/2)*a^2*b^2*d^16 + 4
7736*(d*x)^(5/2)*a^3*b*d^18 + 9945*sqrt(d*x)*a^4*d^20)/(b^10*d^10*x^10 + 5*a*b^9*d^10*x^8 + 10*a^2*b^8*d^10*x^
6 + 10*a^3*b^7*d^10*x^4 + 5*a^4*b^6*d^10*x^2 + a^5*b^5*d^10) - 9945*(sqrt(2)*d^12*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^12*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^11*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^11*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)))/b^5)/d

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mupad [B]  time = 4.27, size = 213, normalized size = 0.55 \begin {gather*} -\frac {\frac {7529\,d^{11}\,{\left (d\,x\right )}^{17/2}}{12288\,b}+\frac {9061\,a^2\,d^{15}\,{\left (d\,x\right )}^{9/2}}{6144\,b^3}+\frac {1989\,a^3\,d^{17}\,{\left (d\,x\right )}^{5/2}}{2560\,b^4}+\frac {663\,a^4\,d^{19}\,\sqrt {d\,x}}{4096\,b^5}+\frac {527\,a\,d^{13}\,{\left (d\,x\right )}^{13/2}}{384\,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}}-\frac {663\,d^{19/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{3/4}\,b^{21/4}}-\frac {663\,d^{19/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{3/4}\,b^{21/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(19/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)

[Out]

- ((7529*d^11*(d*x)^(17/2))/(12288*b) + (9061*a^2*d^15*(d*x)^(9/2))/(6144*b^3) + (1989*a^3*d^17*(d*x)^(5/2))/(
2560*b^4) + (663*a^4*d^19*(d*x)^(1/2))/(4096*b^5) + (527*a*d^13*(d*x)^(13/2))/(384*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) - (663*d^(19/2)*atan((
b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(8192*(-a)^(3/4)*b^(21/4)) - (663*d^(19/2)*atanh((b^(1/4)*(d*x)^(1
/2))/((-a)^(1/4)*d^(1/2))))/(8192*(-a)^(3/4)*b^(21/4))

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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)**(19/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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