3.6.48 \(\int \frac {1}{(d x)^{5/2} (a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=404 \[ \frac {33649 b^{3/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{27/4} d^{5/2}}-\frac {33649 b^{3/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{27/4} d^{5/2}}+\frac {33649 b^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{27/4} d^{5/2}}-\frac {33649 b^{3/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{27/4} d^{5/2}}-\frac {33649}{12288 a^6 d (d x)^{3/2}}+\frac {4807}{4096 a^5 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {437}{1024 a^4 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {437}{1920 a^3 d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5} \]

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Rubi [A]  time = 0.51, antiderivative size = 404, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {28, 290, 325, 329, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {33649 b^{3/4} \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^{27/4} d^{5/2}}-\frac {33649 b^{3/4} \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^{27/4} d^{5/2}}+\frac {33649 b^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{27/4} d^{5/2}}-\frac {33649 b^{3/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{27/4} d^{5/2}}+\frac {4807}{4096 a^5 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {437}{1024 a^4 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {437}{1920 a^3 d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}-\frac {33649}{12288 a^6 d (d x)^{3/2}}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]

[Out]

-33649/(12288*a^6*d*(d*x)^(3/2)) + 1/(10*a*d*(d*x)^(3/2)*(a + b*x^2)^5) + 23/(160*a^2*d*(d*x)^(3/2)*(a + b*x^2
)^4) + 437/(1920*a^3*d*(d*x)^(3/2)*(a + b*x^2)^3) + 437/(1024*a^4*d*(d*x)^(3/2)*(a + b*x^2)^2) + 4807/(4096*a^
5*d*(d*x)^(3/2)*(a + b*x^2)) + (33649*b^(3/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192
*Sqrt[2]*a^(27/4)*d^(5/2)) - (33649*b^(3/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*S
qrt[2]*a^(27/4)*d^(5/2)) + (33649*b^(3/4)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sq
rt[d*x]])/(16384*Sqrt[2]*a^(27/4)*d^(5/2)) - (33649*b^(3/4)*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^(27/4)*d^(5/2))

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 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 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -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 {1}{(d x)^{5/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {\left (23 b^5\right ) \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^5} \, dx}{20 a}\\ &=\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {\left (437 b^4\right ) \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^4} \, dx}{320 a^2}\\ &=\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {437}{1920 a^3 d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {\left (437 b^3\right ) \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^3} \, dx}{256 a^3}\\ &=\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {437}{1920 a^3 d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {437}{1024 a^4 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {\left (4807 b^2\right ) \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )^2} \, dx}{2048 a^4}\\ &=\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {437}{1920 a^3 d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {437}{1024 a^4 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {4807}{4096 a^5 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {(33649 b) \int \frac {1}{(d x)^{5/2} \left (a b+b^2 x^2\right )} \, dx}{8192 a^5}\\ &=-\frac {33649}{12288 a^6 d (d x)^{3/2}}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {437}{1920 a^3 d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {437}{1024 a^4 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {4807}{4096 a^5 d (d x)^{3/2} \left (a+b x^2\right )}-\frac {\left (33649 b^2\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{8192 a^6 d^2}\\ &=-\frac {33649}{12288 a^6 d (d x)^{3/2}}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {437}{1920 a^3 d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {437}{1024 a^4 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {4807}{4096 a^5 d (d x)^{3/2} \left (a+b x^2\right )}-\frac {\left (33649 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 a^6 d^3}\\ &=-\frac {33649}{12288 a^6 d (d x)^{3/2}}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {437}{1920 a^3 d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {437}{1024 a^4 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {4807}{4096 a^5 d (d x)^{3/2} \left (a+b x^2\right )}-\frac {\left (33649 b^2\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^{13/2} d^4}-\frac {\left (33649 b^2\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^{13/2} d^4}\\ &=-\frac {33649}{12288 a^6 d (d x)^{3/2}}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {437}{1920 a^3 d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {437}{1024 a^4 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {4807}{4096 a^5 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {\left (33649 b^{3/4}\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^{27/4} d^{5/2}}+\frac {\left (33649 b^{3/4}\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^{27/4} d^{5/2}}-\frac {\left (33649 \sqrt {b}\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^{13/2} d^2}-\frac {\left (33649 \sqrt {b}\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^{13/2} d^2}\\ &=-\frac {33649}{12288 a^6 d (d x)^{3/2}}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {437}{1920 a^3 d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {437}{1024 a^4 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {4807}{4096 a^5 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {33649 b^{3/4} \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^{27/4} d^{5/2}}-\frac {33649 b^{3/4} \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^{27/4} d^{5/2}}-\frac {\left (33649 b^{3/4}\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^{27/4} d^{5/2}}+\frac {\left (33649 b^{3/4}\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^{27/4} d^{5/2}}\\ &=-\frac {33649}{12288 a^6 d (d x)^{3/2}}+\frac {1}{10 a d (d x)^{3/2} \left (a+b x^2\right )^5}+\frac {23}{160 a^2 d (d x)^{3/2} \left (a+b x^2\right )^4}+\frac {437}{1920 a^3 d (d x)^{3/2} \left (a+b x^2\right )^3}+\frac {437}{1024 a^4 d (d x)^{3/2} \left (a+b x^2\right )^2}+\frac {4807}{4096 a^5 d (d x)^{3/2} \left (a+b x^2\right )}+\frac {33649 b^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{27/4} d^{5/2}}-\frac {33649 b^{3/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{27/4} d^{5/2}}+\frac {33649 b^{3/4} \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^{27/4} d^{5/2}}-\frac {33649 b^{3/4} \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^{27/4} d^{5/2}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 32, normalized size = 0.08 \begin {gather*} -\frac {2 x \, _2F_1\left (-\frac {3}{4},6;\frac {1}{4};-\frac {b x^2}{a}\right )}{3 a^6 (d x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]

[Out]

(-2*x*Hypergeometric2F1[-3/4, 6, 1/4, -((b*x^2)/a)])/(3*a^6*(d*x)^(5/2))

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IntegrateAlgebraic [A]  time = 1.33, size = 255, normalized size = 0.63 \begin {gather*} \frac {33649 b^{3/4} \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^{27/4} d^{5/2}}-\frac {33649 b^{3/4} \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^{27/4} d^{5/2}}+\frac {-40960 a^5 d^{10}-437345 a^4 b d^{10} x^2-1157176 a^3 b^2 d^{10} x^4-1367810 a^2 b^3 d^{10} x^6-769120 a b^4 d^{10} x^8-168245 b^5 d^{10} x^{10}}{61440 a^6 d (d x)^{3/2} \left (a d^2+b d^2 x^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((d*x)^(5/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]

[Out]

(-40960*a^5*d^10 - 437345*a^4*b*d^10*x^2 - 1157176*a^3*b^2*d^10*x^4 - 1367810*a^2*b^3*d^10*x^6 - 769120*a*b^4*
d^10*x^8 - 168245*b^5*d^10*x^10)/(61440*a^6*d*(d*x)^(3/2)*(a*d^2 + b*d^2*x^2)^5) + (33649*b^(3/4)*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^(27/4)*d^
(5/2)) - (33649*b^(3/4)*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^(27/4)*d^(5/2))

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fricas [A]  time = 1.46, size = 568, normalized size = 1.41 \begin {gather*} -\frac {2018940 \, {\left (a^{6} b^{5} d^{3} x^{12} + 5 \, a^{7} b^{4} d^{3} x^{10} + 10 \, a^{8} b^{3} d^{3} x^{8} + 10 \, a^{9} b^{2} d^{3} x^{6} + 5 \, a^{10} b d^{3} x^{4} + a^{11} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{27} d^{10}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {d x} a^{20} b d^{7} \left (-\frac {b^{3}}{a^{27} d^{10}}\right )^{\frac {3}{4}} - \sqrt {a^{14} d^{6} \sqrt {-\frac {b^{3}}{a^{27} d^{10}}} + b^{2} d x} a^{20} d^{7} \left (-\frac {b^{3}}{a^{27} d^{10}}\right )^{\frac {3}{4}}}{b^{3}}\right ) + 504735 \, {\left (a^{6} b^{5} d^{3} x^{12} + 5 \, a^{7} b^{4} d^{3} x^{10} + 10 \, a^{8} b^{3} d^{3} x^{8} + 10 \, a^{9} b^{2} d^{3} x^{6} + 5 \, a^{10} b d^{3} x^{4} + a^{11} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{27} d^{10}}\right )^{\frac {1}{4}} \log \left (33649 \, a^{7} d^{3} \left (-\frac {b^{3}}{a^{27} d^{10}}\right )^{\frac {1}{4}} + 33649 \, \sqrt {d x} b\right ) - 504735 \, {\left (a^{6} b^{5} d^{3} x^{12} + 5 \, a^{7} b^{4} d^{3} x^{10} + 10 \, a^{8} b^{3} d^{3} x^{8} + 10 \, a^{9} b^{2} d^{3} x^{6} + 5 \, a^{10} b d^{3} x^{4} + a^{11} d^{3} x^{2}\right )} \left (-\frac {b^{3}}{a^{27} d^{10}}\right )^{\frac {1}{4}} \log \left (-33649 \, a^{7} d^{3} \left (-\frac {b^{3}}{a^{27} d^{10}}\right )^{\frac {1}{4}} + 33649 \, \sqrt {d x} b\right ) + 4 \, {\left (168245 \, b^{5} x^{10} + 769120 \, a b^{4} x^{8} + 1367810 \, a^{2} b^{3} x^{6} + 1157176 \, a^{3} b^{2} x^{4} + 437345 \, a^{4} b x^{2} + 40960 \, a^{5}\right )} \sqrt {d x}}{245760 \, {\left (a^{6} b^{5} d^{3} x^{12} + 5 \, a^{7} b^{4} d^{3} x^{10} + 10 \, a^{8} b^{3} d^{3} x^{8} + 10 \, a^{9} b^{2} d^{3} x^{6} + 5 \, a^{10} b d^{3} x^{4} + a^{11} d^{3} x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

-1/245760*(2018940*(a^6*b^5*d^3*x^12 + 5*a^7*b^4*d^3*x^10 + 10*a^8*b^3*d^3*x^8 + 10*a^9*b^2*d^3*x^6 + 5*a^10*b
*d^3*x^4 + a^11*d^3*x^2)*(-b^3/(a^27*d^10))^(1/4)*arctan(-(sqrt(d*x)*a^20*b*d^7*(-b^3/(a^27*d^10))^(3/4) - sqr
t(a^14*d^6*sqrt(-b^3/(a^27*d^10)) + b^2*d*x)*a^20*d^7*(-b^3/(a^27*d^10))^(3/4))/b^3) + 504735*(a^6*b^5*d^3*x^1
2 + 5*a^7*b^4*d^3*x^10 + 10*a^8*b^3*d^3*x^8 + 10*a^9*b^2*d^3*x^6 + 5*a^10*b*d^3*x^4 + a^11*d^3*x^2)*(-b^3/(a^2
7*d^10))^(1/4)*log(33649*a^7*d^3*(-b^3/(a^27*d^10))^(1/4) + 33649*sqrt(d*x)*b) - 504735*(a^6*b^5*d^3*x^12 + 5*
a^7*b^4*d^3*x^10 + 10*a^8*b^3*d^3*x^8 + 10*a^9*b^2*d^3*x^6 + 5*a^10*b*d^3*x^4 + a^11*d^3*x^2)*(-b^3/(a^27*d^10
))^(1/4)*log(-33649*a^7*d^3*(-b^3/(a^27*d^10))^(1/4) + 33649*sqrt(d*x)*b) + 4*(168245*b^5*x^10 + 769120*a*b^4*
x^8 + 1367810*a^2*b^3*x^6 + 1157176*a^3*b^2*x^4 + 437345*a^4*b*x^2 + 40960*a^5)*sqrt(d*x))/(a^6*b^5*d^3*x^12 +
5*a^7*b^4*d^3*x^10 + 10*a^8*b^3*d^3*x^8 + 10*a^9*b^2*d^3*x^6 + 5*a^10*b*d^3*x^4 + a^11*d^3*x^2)

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giac [A]  time = 0.20, size = 356, normalized size = 0.88 \begin {gather*} -\frac {33649 \, \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 )}{16384 \, a^{7} d^{3}} - \frac {33649 \, \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 )}{16384 \, a^{7} d^{3}} - \frac {33649 \, \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 )}{32768 \, a^{7} d^{3}} + \frac {33649 \, \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 )}{32768 \, a^{7} d^{3}} - \frac {2}{3 \, \sqrt {d x} a^{6} d^{2} x} - \frac {127285 \, \sqrt {d x} b^{5} d^{8} x^{8} + 564320 \, \sqrt {d x} a b^{4} d^{8} x^{6} + 958210 \, \sqrt {d x} a^{2} b^{3} d^{8} x^{4} + 747576 \, \sqrt {d x} a^{3} b^{2} d^{8} x^{2} + 232545 \, \sqrt {d x} a^{4} b d^{8}}{61440 \, {\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{6} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

-33649/16384*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^7*d^3) - 33649/16384*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^7*d^3) - 33649/32768*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^7*d^3) + 33649/32768*sqrt(2)*(a*b^3*d^2)^(1/4)*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqr
t(d*x) + sqrt(a*d^2/b))/(a^7*d^3) - 2/3/(sqrt(d*x)*a^6*d^2*x) - 1/61440*(127285*sqrt(d*x)*b^5*d^8*x^8 + 564320
*sqrt(d*x)*a*b^4*d^8*x^6 + 958210*sqrt(d*x)*a^2*b^3*d^8*x^4 + 747576*sqrt(d*x)*a^3*b^2*d^8*x^2 + 232545*sqrt(d
*x)*a^4*b*d^8)/((b*d^2*x^2 + a*d^2)^5*a^6*d)

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maple [A]  time = 0.03, size = 352, normalized size = 0.87 \begin {gather*} -\frac {15503 \sqrt {d x}\, b \,d^{7}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{2}}-\frac {31149 \left (d x \right )^{\frac {5}{2}} b^{2} d^{5}}{2560 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{3}}-\frac {95821 \left (d x \right )^{\frac {9}{2}} b^{3} d^{3}}{6144 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{4}}-\frac {3527 \left (d x \right )^{\frac {13}{2}} b^{4} d}{384 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{5}}-\frac {25457 \left (d x \right )^{\frac {17}{2}} b^{5}}{12288 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{6} d}-\frac {2}{3 \left (d x \right )^{\frac {3}{2}} a^{6} d}-\frac {33649 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{16384 a^{7} d^{3}}-\frac {33649 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{16384 a^{7} d^{3}}-\frac {33649 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, b \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^{7} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

-15503/4096*d^7/a^2*b/(b*d^2*x^2+a*d^2)^5*(d*x)^(1/2)-31149/2560*d^5/a^3*b^2/(b*d^2*x^2+a*d^2)^5*(d*x)^(5/2)-9
5821/6144*d^3/a^4*b^3/(b*d^2*x^2+a*d^2)^5*(d*x)^(9/2)-3527/384*d/a^5*b^4/(b*d^2*x^2+a*d^2)^5*(d*x)^(13/2)-2545
7/12288/d/a^6*b^5/(b*d^2*x^2+a*d^2)^5*(d*x)^(17/2)-33649/32768/d^3/a^7*b*(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)))-336
49/16384/d^3/a^7*b*(a/b*d^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b*d^2)^(1/4)*(d*x)^(1/2)+1)-33649/16384/d^3/a^7*b
*(a/b*d^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b*d^2)^(1/4)*(d*x)^(1/2)-1)-2/3/a^6/d/(d*x)^(3/2)

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maxima [A]  time = 3.23, size = 395, normalized size = 0.98 \begin {gather*} -\frac {\frac {8 \, {\left (168245 \, b^{5} d^{10} x^{10} + 769120 \, a b^{4} d^{10} x^{8} + 1367810 \, a^{2} b^{3} d^{10} x^{6} + 1157176 \, a^{3} b^{2} d^{10} x^{4} + 437345 \, a^{4} b d^{10} x^{2} + 40960 \, a^{5} d^{10}\right )}}{\left (d x\right )^{\frac {23}{2}} a^{6} b^{5} + 5 \, \left (d x\right )^{\frac {19}{2}} a^{7} b^{4} d^{2} + 10 \, \left (d x\right )^{\frac {15}{2}} a^{8} b^{3} d^{4} + 10 \, \left (d x\right )^{\frac {11}{2}} a^{9} b^{2} d^{6} + 5 \, \left (d x\right )^{\frac {7}{2}} a^{10} b d^{8} + \left (d x\right )^{\frac {3}{2}} a^{11} d^{10}} + \frac {504735 \, {\left (\frac {\sqrt {2} b^{\frac {3}{4}} \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}}} - \frac {\sqrt {2} b^{\frac {3}{4}} \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}}} + \frac {2 \, \sqrt {2} b \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} d} + \frac {2 \, \sqrt {2} b \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} d}\right )}}{a^{6}}}{491520 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x)^(5/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

-1/491520*(8*(168245*b^5*d^10*x^10 + 769120*a*b^4*d^10*x^8 + 1367810*a^2*b^3*d^10*x^6 + 1157176*a^3*b^2*d^10*x
^4 + 437345*a^4*b*d^10*x^2 + 40960*a^5*d^10)/((d*x)^(23/2)*a^6*b^5 + 5*(d*x)^(19/2)*a^7*b^4*d^2 + 10*(d*x)^(15
/2)*a^8*b^3*d^4 + 10*(d*x)^(11/2)*a^9*b^2*d^6 + 5*(d*x)^(7/2)*a^10*b*d^8 + (d*x)^(3/2)*a^11*d^10) + 504735*(sq
rt(2)*b^(3/4)*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) - sqrt(2)*b
^(3/4)*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) + 2*sqrt(2)*b*arct
an(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)*sq
rt(b)*d)*sqrt(a)*d) + 2*sqrt(2)*b*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) - 2*sqrt(d*x)*sqrt(b))/sq
rt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(a)*d))/a^6)/d

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mupad [B]  time = 4.46, size = 226, normalized size = 0.56 \begin {gather*} \frac {33649\,{\left (-b\right )}^{3/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{8192\,a^{27/4}\,d^{5/2}}-\frac {\frac {2\,d^9}{3\,a}+\frac {87469\,b\,d^9\,x^2}{12288\,a^2}+\frac {144647\,b^2\,d^9\,x^4}{7680\,a^3}+\frac {136781\,b^3\,d^9\,x^6}{6144\,a^4}+\frac {4807\,b^4\,d^9\,x^8}{384\,a^5}+\frac {33649\,b^5\,d^9\,x^{10}}{12288\,a^6}}{b^5\,{\left (d\,x\right )}^{23/2}+a^5\,d^{10}\,{\left (d\,x\right )}^{3/2}+10\,a^3\,b^2\,d^6\,{\left (d\,x\right )}^{11/2}+10\,a^2\,b^3\,d^4\,{\left (d\,x\right )}^{15/2}+5\,a^4\,b\,d^8\,{\left (d\,x\right )}^{7/2}+5\,a\,b^4\,d^2\,{\left (d\,x\right )}^{19/2}}+\frac {33649\,{\left (-b\right )}^{3/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{8192\,a^{27/4}\,d^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d*x)^(5/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^3),x)

[Out]

(33649*(-b)^(3/4)*atan(((-b)^(1/4)*(d*x)^(1/2))/(a^(1/4)*d^(1/2))))/(8192*a^(27/4)*d^(5/2)) - ((2*d^9)/(3*a) +
(87469*b*d^9*x^2)/(12288*a^2) + (144647*b^2*d^9*x^4)/(7680*a^3) + (136781*b^3*d^9*x^6)/(6144*a^4) + (4807*b^4
*d^9*x^8)/(384*a^5) + (33649*b^5*d^9*x^10)/(12288*a^6))/(b^5*(d*x)^(23/2) + a^5*d^10*(d*x)^(3/2) + 10*a^3*b^2*
d^6*(d*x)^(11/2) + 10*a^2*b^3*d^4*(d*x)^(15/2) + 5*a^4*b*d^8*(d*x)^(7/2) + 5*a*b^4*d^2*(d*x)^(19/2)) + (33649*
(-b)^(3/4)*atanh(((-b)^(1/4)*(d*x)^(1/2))/(a^(1/4)*d^(1/2))))/(8192*a^(27/4)*d^(5/2))

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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(1/(d*x)**(5/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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