3.6.49 \(\int \frac {1}{(d x)^{7/2} (a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=422 \[ \frac {69615 b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{29/4} d^{7/2}}-\frac {69615 b^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{29/4} d^{7/2}}-\frac {69615 b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{29/4} d^{7/2}}+\frac {69615 b^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{29/4} d^{7/2}}+\frac {69615 b}{4096 a^7 d^3 \sqrt {d x}}-\frac {13923}{4096 a^6 d (d x)^{5/2}}+\frac {7735}{4096 a^5 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5} \]

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Rubi [A]  time = 0.55, antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {28, 290, 325, 329, 297, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {69615 b^{5/4} \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^{29/4} d^{7/2}}-\frac {69615 b^{5/4} \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^{29/4} d^{7/2}}-\frac {69615 b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{29/4} d^{7/2}}+\frac {69615 b^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{29/4} d^{7/2}}+\frac {69615 b}{4096 a^7 d^3 \sqrt {d x}}+\frac {7735}{4096 a^5 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}-\frac {13923}{4096 a^6 d (d x)^{5/2}}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]

[Out]

-13923/(4096*a^6*d*(d*x)^(5/2)) + (69615*b)/(4096*a^7*d^3*Sqrt[d*x]) + 1/(10*a*d*(d*x)^(5/2)*(a + b*x^2)^5) +
5/(32*a^2*d*(d*x)^(5/2)*(a + b*x^2)^4) + 35/(128*a^3*d*(d*x)^(5/2)*(a + b*x^2)^3) + 595/(1024*a^4*d*(d*x)^(5/2
)*(a + b*x^2)^2) + 7735/(4096*a^5*d*(d*x)^(5/2)*(a + b*x^2)) - (69615*b^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt
[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(29/4)*d^(7/2)) + (69615*b^(5/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d
*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(29/4)*d^(7/2)) + (69615*b^(5/4)*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^(29/4)*d^(7/2)) - (69615*b^(5/4)*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^(29/4)*d^(7/2))

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 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 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)^{7/2} \left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {\left (5 b^5\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^5} \, dx}{4 a}\\ &=\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {\left (105 b^4\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^4} \, dx}{64 a^2}\\ &=\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {\left (595 b^3\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^3} \, dx}{256 a^3}\\ &=\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {\left (7735 b^2\right ) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )^2} \, dx}{2048 a^4}\\ &=\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {7735}{4096 a^5 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {(69615 b) \int \frac {1}{(d x)^{7/2} \left (a b+b^2 x^2\right )} \, dx}{8192 a^5}\\ &=-\frac {13923}{4096 a^6 d (d x)^{5/2}}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {7735}{4096 a^5 d (d x)^{5/2} \left (a+b x^2\right )}-\frac {\left (69615 b^2\right ) \int \frac {1}{(d x)^{3/2} \left (a b+b^2 x^2\right )} \, dx}{8192 a^6 d^2}\\ &=-\frac {13923}{4096 a^6 d (d x)^{5/2}}+\frac {69615 b}{4096 a^7 d^3 \sqrt {d x}}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {7735}{4096 a^5 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {\left (69615 b^3\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{8192 a^7 d^4}\\ &=-\frac {13923}{4096 a^6 d (d x)^{5/2}}+\frac {69615 b}{4096 a^7 d^3 \sqrt {d x}}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {7735}{4096 a^5 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {\left (69615 b^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 a^7 d^5}\\ &=-\frac {13923}{4096 a^6 d (d x)^{5/2}}+\frac {69615 b}{4096 a^7 d^3 \sqrt {d x}}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {7735}{4096 a^5 d (d x)^{5/2} \left (a+b x^2\right )}-\frac {\left (69615 b^{5/2}\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^7 d^5}+\frac {\left (69615 b^{5/2}\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^7 d^5}\\ &=-\frac {13923}{4096 a^6 d (d x)^{5/2}}+\frac {69615 b}{4096 a^7 d^3 \sqrt {d x}}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {7735}{4096 a^5 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {\left (69615 b^{5/4}\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^{29/4} d^{7/2}}+\frac {\left (69615 b^{5/4}\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^{29/4} d^{7/2}}+\frac {(69615 b) \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^7 d^3}+\frac {(69615 b) \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^7 d^3}\\ &=-\frac {13923}{4096 a^6 d (d x)^{5/2}}+\frac {69615 b}{4096 a^7 d^3 \sqrt {d x}}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {7735}{4096 a^5 d (d x)^{5/2} \left (a+b x^2\right )}+\frac {69615 b^{5/4} \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^{29/4} d^{7/2}}-\frac {69615 b^{5/4} \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^{29/4} d^{7/2}}+\frac {\left (69615 b^{5/4}\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^{29/4} d^{7/2}}-\frac {\left (69615 b^{5/4}\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^{29/4} d^{7/2}}\\ &=-\frac {13923}{4096 a^6 d (d x)^{5/2}}+\frac {69615 b}{4096 a^7 d^3 \sqrt {d x}}+\frac {1}{10 a d (d x)^{5/2} \left (a+b x^2\right )^5}+\frac {5}{32 a^2 d (d x)^{5/2} \left (a+b x^2\right )^4}+\frac {35}{128 a^3 d (d x)^{5/2} \left (a+b x^2\right )^3}+\frac {595}{1024 a^4 d (d x)^{5/2} \left (a+b x^2\right )^2}+\frac {7735}{4096 a^5 d (d x)^{5/2} \left (a+b x^2\right )}-\frac {69615 b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{29/4} d^{7/2}}+\frac {69615 b^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{29/4} d^{7/2}}+\frac {69615 b^{5/4} \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^{29/4} d^{7/2}}-\frac {69615 b^{5/4} \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^{29/4} d^{7/2}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 37, normalized size = 0.09 \begin {gather*} -\frac {2 \sqrt {d x} \, _2F_1\left (-\frac {5}{4},6;-\frac {1}{4};-\frac {b x^2}{a}\right )}{5 a^6 d^4 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 1.37, size = 269, normalized size = 0.64 \begin {gather*} -\frac {69615 b^{5/4} \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^{29/4} d^{7/2}}-\frac {69615 b^{5/4} \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^{29/4} d^{7/2}}+\frac {-8192 a^6 d^{12}+204800 a^5 b d^{12} x^2+1317575 a^4 b^2 d^{12} x^4+2951200 a^3 b^3 d^{12} x^6+3171350 a^2 b^4 d^{12} x^8+1670760 a b^5 d^{12} x^{10}+348075 b^6 d^{12} x^{12}}{20480 a^7 d^3 (d x)^{5/2} \left (a d^2+b d^2 x^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((d*x)^(7/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)^3),x]

[Out]

(-8192*a^6*d^12 + 204800*a^5*b*d^12*x^2 + 1317575*a^4*b^2*d^12*x^4 + 2951200*a^3*b^3*d^12*x^6 + 3171350*a^2*b^
4*d^12*x^8 + 1670760*a*b^5*d^12*x^10 + 348075*b^6*d^12*x^12)/(20480*a^7*d^3*(d*x)^(5/2)*(a*d^2 + b*d^2*x^2)^5)
- (69615*b^(5/4)*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^(29/4)*d^(7/2)) - (69615*b^(5/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]*Sqrt[d*x])/(Sqr
t[a]*d + Sqrt[b]*d*x)])/(8192*Sqrt[2]*a^(29/4)*d^(7/2))

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fricas [A]  time = 2.52, size = 591, normalized size = 1.40 \begin {gather*} -\frac {1392300 \, {\left (a^{7} b^{5} d^{4} x^{13} + 5 \, a^{8} b^{4} d^{4} x^{11} + 10 \, a^{9} b^{3} d^{4} x^{9} + 10 \, a^{10} b^{2} d^{4} x^{7} + 5 \, a^{11} b d^{4} x^{5} + a^{12} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{29} d^{14}}\right )^{\frac {1}{4}} \arctan \left (-\frac {337371570183375 \, \sqrt {d x} a^{7} b^{4} d^{3} \left (-\frac {b^{5}}{a^{29} d^{14}}\right )^{\frac {1}{4}} - \sqrt {-113819576367995923331126390625 \, a^{15} b^{5} d^{8} \sqrt {-\frac {b^{5}}{a^{29} d^{14}}} + 113819576367995923331126390625 \, b^{8} d x} a^{7} d^{3} \left (-\frac {b^{5}}{a^{29} d^{14}}\right )^{\frac {1}{4}}}{337371570183375 \, b^{5}}\right ) - 348075 \, {\left (a^{7} b^{5} d^{4} x^{13} + 5 \, a^{8} b^{4} d^{4} x^{11} + 10 \, a^{9} b^{3} d^{4} x^{9} + 10 \, a^{10} b^{2} d^{4} x^{7} + 5 \, a^{11} b d^{4} x^{5} + a^{12} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{29} d^{14}}\right )^{\frac {1}{4}} \log \left (337371570183375 \, a^{22} d^{11} \left (-\frac {b^{5}}{a^{29} d^{14}}\right )^{\frac {3}{4}} + 337371570183375 \, \sqrt {d x} b^{4}\right ) + 348075 \, {\left (a^{7} b^{5} d^{4} x^{13} + 5 \, a^{8} b^{4} d^{4} x^{11} + 10 \, a^{9} b^{3} d^{4} x^{9} + 10 \, a^{10} b^{2} d^{4} x^{7} + 5 \, a^{11} b d^{4} x^{5} + a^{12} d^{4} x^{3}\right )} \left (-\frac {b^{5}}{a^{29} d^{14}}\right )^{\frac {1}{4}} \log \left (-337371570183375 \, a^{22} d^{11} \left (-\frac {b^{5}}{a^{29} d^{14}}\right )^{\frac {3}{4}} + 337371570183375 \, \sqrt {d x} b^{4}\right ) - 4 \, {\left (348075 \, b^{6} x^{12} + 1670760 \, a b^{5} x^{10} + 3171350 \, a^{2} b^{4} x^{8} + 2951200 \, a^{3} b^{3} x^{6} + 1317575 \, a^{4} b^{2} x^{4} + 204800 \, a^{5} b x^{2} - 8192 \, a^{6}\right )} \sqrt {d x}}{81920 \, {\left (a^{7} b^{5} d^{4} x^{13} + 5 \, a^{8} b^{4} d^{4} x^{11} + 10 \, a^{9} b^{3} d^{4} x^{9} + 10 \, a^{10} b^{2} d^{4} x^{7} + 5 \, a^{11} b d^{4} x^{5} + a^{12} d^{4} x^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

-1/81920*(1392300*(a^7*b^5*d^4*x^13 + 5*a^8*b^4*d^4*x^11 + 10*a^9*b^3*d^4*x^9 + 10*a^10*b^2*d^4*x^7 + 5*a^11*b
*d^4*x^5 + a^12*d^4*x^3)*(-b^5/(a^29*d^14))^(1/4)*arctan(-1/337371570183375*(337371570183375*sqrt(d*x)*a^7*b^4
*d^3*(-b^5/(a^29*d^14))^(1/4) - sqrt(-113819576367995923331126390625*a^15*b^5*d^8*sqrt(-b^5/(a^29*d^14)) + 113
819576367995923331126390625*b^8*d*x)*a^7*d^3*(-b^5/(a^29*d^14))^(1/4))/b^5) - 348075*(a^7*b^5*d^4*x^13 + 5*a^8
*b^4*d^4*x^11 + 10*a^9*b^3*d^4*x^9 + 10*a^10*b^2*d^4*x^7 + 5*a^11*b*d^4*x^5 + a^12*d^4*x^3)*(-b^5/(a^29*d^14))
^(1/4)*log(337371570183375*a^22*d^11*(-b^5/(a^29*d^14))^(3/4) + 337371570183375*sqrt(d*x)*b^4) + 348075*(a^7*b
^5*d^4*x^13 + 5*a^8*b^4*d^4*x^11 + 10*a^9*b^3*d^4*x^9 + 10*a^10*b^2*d^4*x^7 + 5*a^11*b*d^4*x^5 + a^12*d^4*x^3)
*(-b^5/(a^29*d^14))^(1/4)*log(-337371570183375*a^22*d^11*(-b^5/(a^29*d^14))^(3/4) + 337371570183375*sqrt(d*x)*
b^4) - 4*(348075*b^6*x^12 + 1670760*a*b^5*x^10 + 3171350*a^2*b^4*x^8 + 2951200*a^3*b^3*x^6 + 1317575*a^4*b^2*x
^4 + 204800*a^5*b*x^2 - 8192*a^6)*sqrt(d*x))/(a^7*b^5*d^4*x^13 + 5*a^8*b^4*d^4*x^11 + 10*a^9*b^3*d^4*x^9 + 10*
a^10*b^2*d^4*x^7 + 5*a^11*b*d^4*x^5 + a^12*d^4*x^3)

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giac [A]  time = 0.20, size = 362, normalized size = 0.86 \begin {gather*} \frac {69615 \, \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 )}{16384 \, a^{8} b d^{5}} + \frac {69615 \, \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 )}{16384 \, a^{8} b d^{5}} - \frac {69615 \, \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 )}{32768 \, a^{8} b d^{5}} + \frac {69615 \, \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 )}{32768 \, a^{8} b d^{5}} + \frac {348075 \, b^{6} d^{12} x^{12} + 1670760 \, a b^{5} d^{12} x^{10} + 3171350 \, a^{2} b^{4} d^{12} x^{8} + 2951200 \, a^{3} b^{3} d^{12} x^{6} + 1317575 \, a^{4} b^{2} d^{12} x^{4} + 204800 \, a^{5} b d^{12} x^{2} - 8192 \, a^{6} d^{12}}{20480 \, {\left (\sqrt {d x} b d^{2} x^{2} + \sqrt {d x} a d^{2}\right )}^{5} a^{7} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

69615/16384*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^8*b*d^5) + 69615/16384*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^8*b*d^5) - 69615/32768*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*s
qrt(d*x) + sqrt(a*d^2/b))/(a^8*b*d^5) + 69615/32768*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^8*b*d^5) + 1/20480*(348075*b^6*d^12*x^12 + 1670760*a*b^5*d^12*x^10 + 3171350*a
^2*b^4*d^12*x^8 + 2951200*a^3*b^3*d^12*x^6 + 1317575*a^4*b^2*d^12*x^4 + 204800*a^5*b*d^12*x^2 - 8192*a^6*d^12)
/((sqrt(d*x)*b*d^2*x^2 + sqrt(d*x)*a*d^2)^5*a^7*d^3)

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maple [A]  time = 0.04, size = 368, normalized size = 0.87 \begin {gather*} \frac {34139 \left (d x \right )^{\frac {3}{2}} b^{2} d^{5}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{3}}+\frac {3597 \left (d x \right )^{\frac {7}{2}} b^{3} d^{3}}{128 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{4}}+\frac {75471 \left (d x \right )^{\frac {11}{2}} b^{4} d}{2048 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{5}}+\frac {56269 \left (d x \right )^{\frac {15}{2}} b^{5}}{2560 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{6} d}+\frac {20463 \left (d x \right )^{\frac {19}{2}} b^{6}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{7} d^{3}}-\frac {2}{5 \left (d x \right )^{\frac {5}{2}} a^{6} d}+\frac {69615 \sqrt {2}\, b \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{16384 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{7} d^{3}}+\frac {69615 \sqrt {2}\, b \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{16384 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{7} d^{3}}+\frac {69615 \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 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} a^{7} d^{3}}+\frac {12 b}{\sqrt {d x}\, a^{7} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

34139/4096*d^5*b^2/a^3/(b*d^2*x^2+a*d^2)^5*(d*x)^(3/2)+3597/128*d^3*b^3/a^4/(b*d^2*x^2+a*d^2)^5*(d*x)^(7/2)+75
471/2048*d*b^4/a^5/(b*d^2*x^2+a*d^2)^5*(d*x)^(11/2)+56269/2560/d*b^5/a^6/(b*d^2*x^2+a*d^2)^5*(d*x)^(15/2)+2046
3/4096/d^3*b^6/a^7/(b*d^2*x^2+a*d^2)^5*(d*x)^(19/2)+69615/32768/d^3*b/a^7/(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)))+69
615/16384/d^3*b/a^7/(a/b*d^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b*d^2)^(1/4)*(d*x)^(1/2)+1)+69615/16384/d^3*b/a^
7/(a/b*d^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b*d^2)^(1/4)*(d*x)^(1/2)-1)-2/5/a^6/d/(d*x)^(5/2)+12*b/a^7/d^3/(d*
x)^(1/2)

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maxima [A]  time = 3.32, size = 410, normalized size = 0.97 \begin {gather*} \frac {\frac {8 \, {\left (348075 \, b^{6} d^{12} x^{12} + 1670760 \, a b^{5} d^{12} x^{10} + 3171350 \, a^{2} b^{4} d^{12} x^{8} + 2951200 \, a^{3} b^{3} d^{12} x^{6} + 1317575 \, a^{4} b^{2} d^{12} x^{4} + 204800 \, a^{5} b d^{12} x^{2} - 8192 \, a^{6} d^{12}\right )}}{\left (d x\right )^{\frac {25}{2}} a^{7} b^{5} d^{2} + 5 \, \left (d x\right )^{\frac {21}{2}} a^{8} b^{4} d^{4} + 10 \, \left (d x\right )^{\frac {17}{2}} a^{9} b^{3} d^{6} + 10 \, \left (d x\right )^{\frac {13}{2}} a^{10} b^{2} d^{8} + 5 \, \left (d x\right )^{\frac {9}{2}} a^{11} b d^{10} + \left (d x\right )^{\frac {5}{2}} a^{12} d^{12}} + \frac {348075 \, b^{2} {\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^{7} d^{2}}}{163840 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x)^(7/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

1/163840*(8*(348075*b^6*d^12*x^12 + 1670760*a*b^5*d^12*x^10 + 3171350*a^2*b^4*d^12*x^8 + 2951200*a^3*b^3*d^12*
x^6 + 1317575*a^4*b^2*d^12*x^4 + 204800*a^5*b*d^12*x^2 - 8192*a^6*d^12)/((d*x)^(25/2)*a^7*b^5*d^2 + 5*(d*x)^(2
1/2)*a^8*b^4*d^4 + 10*(d*x)^(17/2)*a^9*b^3*d^6 + 10*(d*x)^(13/2)*a^10*b^2*d^8 + 5*(d*x)^(9/2)*a^11*b*d^10 + (d
*x)^(5/2)*a^12*d^12) + 348075*b^2*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) + 2*sqrt(d*x)*s
qrt(b))/sqrt(sqrt(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^7*d^2))/d

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mupad [B]  time = 0.27, size = 239, normalized size = 0.57 \begin {gather*} \frac {\frac {10\,b\,d^9\,x^2}{a^2}-\frac {2\,d^9}{5\,a}+\frac {263515\,b^2\,d^9\,x^4}{4096\,a^3}+\frac {18445\,b^3\,d^9\,x^6}{128\,a^4}+\frac {317135\,b^4\,d^9\,x^8}{2048\,a^5}+\frac {41769\,b^5\,d^9\,x^{10}}{512\,a^6}+\frac {69615\,b^6\,d^9\,x^{12}}{4096\,a^7}}{b^5\,{\left (d\,x\right )}^{25/2}+a^5\,d^{10}\,{\left (d\,x\right )}^{5/2}+10\,a^3\,b^2\,d^6\,{\left (d\,x\right )}^{13/2}+10\,a^2\,b^3\,d^4\,{\left (d\,x\right )}^{17/2}+5\,a^4\,b\,d^8\,{\left (d\,x\right )}^{9/2}+5\,a\,b^4\,d^2\,{\left (d\,x\right )}^{21/2}}-\frac {69615\,{\left (-b\right )}^{5/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{8192\,a^{29/4}\,d^{7/2}}+\frac {69615\,{\left (-b\right )}^{5/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{8192\,a^{29/4}\,d^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d*x)^(7/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)^3),x)

[Out]

((10*b*d^9*x^2)/a^2 - (2*d^9)/(5*a) + (263515*b^2*d^9*x^4)/(4096*a^3) + (18445*b^3*d^9*x^6)/(128*a^4) + (31713
5*b^4*d^9*x^8)/(2048*a^5) + (41769*b^5*d^9*x^10)/(512*a^6) + (69615*b^6*d^9*x^12)/(4096*a^7))/(b^5*(d*x)^(25/2
) + a^5*d^10*(d*x)^(5/2) + 10*a^3*b^2*d^6*(d*x)^(13/2) + 10*a^2*b^3*d^4*(d*x)^(17/2) + 5*a^4*b*d^8*(d*x)^(9/2)
+ 5*a*b^4*d^2*(d*x)^(21/2)) - (69615*(-b)^(5/4)*atan(((-b)^(1/4)*(d*x)^(1/2))/(a^(1/4)*d^(1/2))))/(8192*a^(29
/4)*d^(7/2)) + (69615*(-b)^(5/4)*atanh(((-b)^(1/4)*(d*x)^(1/2))/(a^(1/4)*d^(1/2))))/(8192*a^(29/4)*d^(7/2))

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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)**(7/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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