3.716 \(\int \frac {(d x)^{15/2}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=388 \[ -\frac {117 d^{15/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{7/4} b^{17/4}}+\frac {117 d^{15/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{7/4} b^{17/4}}-\frac {117 d^{15/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{7/4} b^{17/4}}+\frac {117 d^{15/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{7/4} b^{17/4}}+\frac {39 d^7 \sqrt {d x}}{4096 a b^4 \left (a+b x^2\right )}-\frac {39 d^7 \sqrt {d x}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {39 d^5 (d x)^{5/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {13 d^3 (d x)^{9/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5} \]

[Out]

-1/10*d*(d*x)^(13/2)/b/(b*x^2+a)^5-13/160*d^3*(d*x)^(9/2)/b^2/(b*x^2+a)^4-39/640*d^5*(d*x)^(5/2)/b^3/(b*x^2+a)
^3-117/16384*d^(15/2)*arctan(1-b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(7/4)/b^(17/4)*2^(1/2)+117/16384
*d^(15/2)*arctan(1+b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(7/4)/b^(17/4)*2^(1/2)-117/32768*d^(15/2)*ln
(a^(1/2)*d^(1/2)+x*b^(1/2)*d^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*(d*x)^(1/2))/a^(7/4)/b^(17/4)*2^(1/2)+117/32768*d^(
15/2)*ln(a^(1/2)*d^(1/2)+x*b^(1/2)*d^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*(d*x)^(1/2))/a^(7/4)/b^(17/4)*2^(1/2)-39/10
24*d^7*(d*x)^(1/2)/b^4/(b*x^2+a)^2+39/4096*d^7*(d*x)^(1/2)/a/b^4/(b*x^2+a)

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Rubi [A]  time = 0.45, antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {28, 288, 290, 329, 211, 1165, 628, 1162, 617, 204} \[ -\frac {117 d^{15/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{7/4} b^{17/4}}+\frac {117 d^{15/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{7/4} b^{17/4}}-\frac {117 d^{15/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{7/4} b^{17/4}}+\frac {117 d^{15/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{7/4} b^{17/4}}+\frac {39 d^7 \sqrt {d x}}{4096 a b^4 \left (a+b x^2\right )}-\frac {39 d^7 \sqrt {d x}}{1024 b^4 \left (a+b x^2\right )^2}-\frac {39 d^5 (d x)^{5/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {13 d^3 (d x)^{9/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^(15/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-(d*(d*x)^(13/2))/(10*b*(a + b*x^2)^5) - (13*d^3*(d*x)^(9/2))/(160*b^2*(a + b*x^2)^4) - (39*d^5*(d*x)^(5/2))/(
640*b^3*(a + b*x^2)^3) - (39*d^7*Sqrt[d*x])/(1024*b^4*(a + b*x^2)^2) + (39*d^7*Sqrt[d*x])/(4096*a*b^4*(a + b*x
^2)) - (117*d^(15/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(7/4)*b^(17/4)
) + (117*d^(15/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(7/4)*b^(17/4)) -
(117*d^(15/2)*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^
(7/4)*b^(17/4)) + (117*d^(15/2)*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^(7/4)*b^(17/4))

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)^{15/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {(d x)^{15/2}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}+\frac {1}{20} \left (13 b^4 d^2\right ) \int \frac {(d x)^{11/2}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}-\frac {13 d^3 (d x)^{9/2}}{160 b^2 \left (a+b x^2\right )^4}+\frac {1}{320} \left (117 b^2 d^4\right ) \int \frac {(d x)^{7/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}-\frac {13 d^3 (d x)^{9/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {39 d^5 (d x)^{5/2}}{640 b^3 \left (a+b x^2\right )^3}+\frac {1}{256} \left (39 d^6\right ) \int \frac {(d x)^{3/2}}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}-\frac {13 d^3 (d x)^{9/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {39 d^5 (d x)^{5/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {39 d^7 \sqrt {d x}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {\left (39 d^8\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )^2} \, dx}{2048 b^2}\\ &=-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}-\frac {13 d^3 (d x)^{9/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {39 d^5 (d x)^{5/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {39 d^7 \sqrt {d x}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {39 d^7 \sqrt {d x}}{4096 a b^4 \left (a+b x^2\right )}+\frac {\left (117 d^8\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{8192 a b^3}\\ &=-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}-\frac {13 d^3 (d x)^{9/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {39 d^5 (d x)^{5/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {39 d^7 \sqrt {d x}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {39 d^7 \sqrt {d x}}{4096 a b^4 \left (a+b x^2\right )}+\frac {\left (117 d^7\right ) \operatorname {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 a b^3}\\ &=-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}-\frac {13 d^3 (d x)^{9/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {39 d^5 (d x)^{5/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {39 d^7 \sqrt {d x}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {39 d^7 \sqrt {d x}}{4096 a b^4 \left (a+b x^2\right )}+\frac {\left (117 d^6\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^{3/2} b^3}+\frac {\left (117 d^6\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^{3/2} b^3}\\ &=-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}-\frac {13 d^3 (d x)^{9/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {39 d^5 (d x)^{5/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {39 d^7 \sqrt {d x}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {39 d^7 \sqrt {d x}}{4096 a b^4 \left (a+b x^2\right )}-\frac {\left (117 d^{15/2}\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^{7/4} b^{17/4}}-\frac {\left (117 d^{15/2}\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^{7/4} b^{17/4}}+\frac {\left (117 d^8\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^{3/2} b^{9/2}}+\frac {\left (117 d^8\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^{3/2} b^{9/2}}\\ &=-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}-\frac {13 d^3 (d x)^{9/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {39 d^5 (d x)^{5/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {39 d^7 \sqrt {d x}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {39 d^7 \sqrt {d x}}{4096 a b^4 \left (a+b x^2\right )}-\frac {117 d^{15/2} \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^{7/4} b^{17/4}}+\frac {117 d^{15/2} \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^{7/4} b^{17/4}}+\frac {\left (117 d^{15/2}\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^{7/4} b^{17/4}}-\frac {\left (117 d^{15/2}\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^{7/4} b^{17/4}}\\ &=-\frac {d (d x)^{13/2}}{10 b \left (a+b x^2\right )^5}-\frac {13 d^3 (d x)^{9/2}}{160 b^2 \left (a+b x^2\right )^4}-\frac {39 d^5 (d x)^{5/2}}{640 b^3 \left (a+b x^2\right )^3}-\frac {39 d^7 \sqrt {d x}}{1024 b^4 \left (a+b x^2\right )^2}+\frac {39 d^7 \sqrt {d x}}{4096 a b^4 \left (a+b x^2\right )}-\frac {117 d^{15/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{7/4} b^{17/4}}+\frac {117 d^{15/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{7/4} b^{17/4}}-\frac {117 d^{15/2} \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^{7/4} b^{17/4}}+\frac {117 d^{15/2} \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^{7/4} b^{17/4}}\\ \end {align*}

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Mathematica [A]  time = 0.26, size = 359, normalized size = 0.93 \[ \frac {d^7 \sqrt {d x} \left (-\frac {45045 \sqrt {2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{7/4} \sqrt {x}}+\frac {45045 \sqrt {2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{7/4} \sqrt {x}}-\frac {90090 \sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{a^{7/4} \sqrt {x}}+\frac {90090 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{a^{7/4} \sqrt {x}}-\frac {638976 a^3 \sqrt [4]{b}}{\left (a+b x^2\right )^5}-\frac {2555904 a^2 b^{5/4} x^2}{\left (a+b x^2\right )^5}+\frac {120120 \sqrt [4]{b}}{a^2+a b x^2}+\frac {39936 a^2 \sqrt [4]{b}}{\left (a+b x^2\right )^4}-\frac {3604480 b^{13/4} x^6}{\left (a+b x^2\right )^5}-\frac {4259840 a b^{9/4} x^4}{\left (a+b x^2\right )^5}+\frac {68640 \sqrt [4]{b}}{\left (a+b x^2\right )^2}+\frac {49920 a \sqrt [4]{b}}{\left (a+b x^2\right )^3}\right )}{12615680 b^{17/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^(15/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(d^7*Sqrt[d*x]*((-638976*a^3*b^(1/4))/(a + b*x^2)^5 - (2555904*a^2*b^(5/4)*x^2)/(a + b*x^2)^5 - (4259840*a*b^(
9/4)*x^4)/(a + b*x^2)^5 - (3604480*b^(13/4)*x^6)/(a + b*x^2)^5 + (39936*a^2*b^(1/4))/(a + b*x^2)^4 + (49920*a*
b^(1/4))/(a + b*x^2)^3 + (68640*b^(1/4))/(a + b*x^2)^2 + (120120*b^(1/4))/(a^2 + a*b*x^2) - (90090*Sqrt[2]*Arc
Tan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(a^(7/4)*Sqrt[x]) + (90090*Sqrt[2]*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqr
t[x])/a^(1/4)])/(a^(7/4)*Sqrt[x]) - (45045*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])
/(a^(7/4)*Sqrt[x]) + (45045*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(a^(7/4)*Sqrt[
x])))/(12615680*b^(17/4))

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fricas [A]  time = 1.14, size = 505, normalized size = 1.30 \[ \frac {2340 \, {\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )} \left (-\frac {d^{30}}{a^{7} b^{17}}\right )^{\frac {1}{4}} \arctan \left (-\frac {\left (-\frac {d^{30}}{a^{7} b^{17}}\right )^{\frac {3}{4}} \sqrt {d x} a^{5} b^{13} d^{7} - \sqrt {d^{15} x + \sqrt {-\frac {d^{30}}{a^{7} b^{17}}} a^{4} b^{8}} \left (-\frac {d^{30}}{a^{7} b^{17}}\right )^{\frac {3}{4}} a^{5} b^{13}}{d^{30}}\right ) + 585 \, {\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )} \left (-\frac {d^{30}}{a^{7} b^{17}}\right )^{\frac {1}{4}} \log \left (117 \, \sqrt {d x} d^{7} + 117 \, \left (-\frac {d^{30}}{a^{7} b^{17}}\right )^{\frac {1}{4}} a^{2} b^{4}\right ) - 585 \, {\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )} \left (-\frac {d^{30}}{a^{7} b^{17}}\right )^{\frac {1}{4}} \log \left (117 \, \sqrt {d x} d^{7} - 117 \, \left (-\frac {d^{30}}{a^{7} b^{17}}\right )^{\frac {1}{4}} a^{2} b^{4}\right ) + 4 \, {\left (195 \, b^{4} d^{7} x^{8} - 4960 \, a b^{3} d^{7} x^{6} - 5330 \, a^{2} b^{2} d^{7} x^{4} - 2808 \, a^{3} b d^{7} x^{2} - 585 \, a^{4} d^{7}\right )} \sqrt {d x}}{81920 \, {\left (a b^{9} x^{10} + 5 \, a^{2} b^{8} x^{8} + 10 \, a^{3} b^{7} x^{6} + 10 \, a^{4} b^{6} x^{4} + 5 \, a^{5} b^{5} x^{2} + a^{6} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

1/81920*(2340*(a*b^9*x^10 + 5*a^2*b^8*x^8 + 10*a^3*b^7*x^6 + 10*a^4*b^6*x^4 + 5*a^5*b^5*x^2 + a^6*b^4)*(-d^30/
(a^7*b^17))^(1/4)*arctan(-((-d^30/(a^7*b^17))^(3/4)*sqrt(d*x)*a^5*b^13*d^7 - sqrt(d^15*x + sqrt(-d^30/(a^7*b^1
7))*a^4*b^8)*(-d^30/(a^7*b^17))^(3/4)*a^5*b^13)/d^30) + 585*(a*b^9*x^10 + 5*a^2*b^8*x^8 + 10*a^3*b^7*x^6 + 10*
a^4*b^6*x^4 + 5*a^5*b^5*x^2 + a^6*b^4)*(-d^30/(a^7*b^17))^(1/4)*log(117*sqrt(d*x)*d^7 + 117*(-d^30/(a^7*b^17))
^(1/4)*a^2*b^4) - 585*(a*b^9*x^10 + 5*a^2*b^8*x^8 + 10*a^3*b^7*x^6 + 10*a^4*b^6*x^4 + 5*a^5*b^5*x^2 + a^6*b^4)
*(-d^30/(a^7*b^17))^(1/4)*log(117*sqrt(d*x)*d^7 - 117*(-d^30/(a^7*b^17))^(1/4)*a^2*b^4) + 4*(195*b^4*d^7*x^8 -
4960*a*b^3*d^7*x^6 - 5330*a^2*b^2*d^7*x^4 - 2808*a^3*b*d^7*x^2 - 585*a^4*d^7)*sqrt(d*x))/(a*b^9*x^10 + 5*a^2*
b^8*x^8 + 10*a^3*b^7*x^6 + 10*a^4*b^6*x^4 + 5*a^5*b^5*x^2 + a^6*b^4)

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giac [A]  time = 0.22, size = 342, normalized size = 0.88 \[ \frac {1}{163840} \, d^{7} {\left (\frac {1170 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} b^{5}} + \frac {1170 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} b^{5}} + \frac {585 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{2} b^{5}} - \frac {585 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{2} b^{5}} + \frac {8 \, {\left (195 \, \sqrt {d x} b^{4} d^{10} x^{8} - 4960 \, \sqrt {d x} a b^{3} d^{10} x^{6} - 5330 \, \sqrt {d x} a^{2} b^{2} d^{10} x^{4} - 2808 \, \sqrt {d x} a^{3} b d^{10} x^{2} - 585 \, \sqrt {d x} a^{4} d^{10}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a b^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

1/163840*d^7*(1170*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^2*b^5) + 1170*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^2*b^5) + 585*sqrt(2)*(a*b^3*d^2)^(1/4)*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + s
qrt(a*d^2/b))/(a^2*b^5) - 585*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^2*b^5) + 8*(195*sqrt(d*x)*b^4*d^10*x^8 - 4960*sqrt(d*x)*a*b^3*d^10*x^6 - 5330*sqrt(d*x)*a^2*b^2*d^10
*x^4 - 2808*sqrt(d*x)*a^3*b*d^10*x^2 - 585*sqrt(d*x)*a^4*d^10)/((b*d^2*x^2 + a*d^2)^5*a*b^4))

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maple [A]  time = 0.02, size = 341, normalized size = 0.88 \[ -\frac {117 \sqrt {d x}\, a^{3} d^{17}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{4}}-\frac {351 \left (d x \right )^{\frac {5}{2}} a^{2} d^{15}}{2560 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{3}}-\frac {533 \left (d x \right )^{\frac {9}{2}} a \,d^{13}}{2048 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b^{2}}-\frac {31 \left (d x \right )^{\frac {13}{2}} d^{11}}{128 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b}+\frac {39 \left (d x \right )^{\frac {17}{2}} d^{9}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a}+\frac {117 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{7} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{16384 a^{2} b^{4}}+\frac {117 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{7} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{16384 a^{2} b^{4}}+\frac {117 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, d^{7} \ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{32768 a^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

-117/4096*d^17/(b*d^2*x^2+a*d^2)^5/b^4*a^3*(d*x)^(1/2)-351/2560*d^15/(b*d^2*x^2+a*d^2)^5/b^3*a^2*(d*x)^(5/2)-5
33/2048*d^13/(b*d^2*x^2+a*d^2)^5/b^2*a*(d*x)^(9/2)-31/128*d^11/(b*d^2*x^2+a*d^2)^5/b*(d*x)^(13/2)+39/4096*d^9/
(b*d^2*x^2+a*d^2)^5/a*(d*x)^(17/2)+117/32768*d^7/a^2/b^4*(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)))+117/16384*d^7/a^2/b
^4*(a/b*d^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b*d^2)^(1/4)*(d*x)^(1/2)+1)+117/16384*d^7/a^2/b^4*(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 = 3.22, size = 392, normalized size = 1.01 \[ \frac {\frac {8 \, {\left (195 \, \left (d x\right )^{\frac {17}{2}} b^{4} d^{10} - 4960 \, \left (d x\right )^{\frac {13}{2}} a b^{3} d^{12} - 5330 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2} d^{14} - 2808 \, \left (d x\right )^{\frac {5}{2}} a^{3} b d^{16} - 585 \, \sqrt {d x} a^{4} d^{18}\right )}}{a b^{9} d^{10} x^{10} + 5 \, a^{2} b^{8} d^{10} x^{8} + 10 \, a^{3} b^{7} d^{10} x^{6} + 10 \, a^{4} b^{6} d^{10} x^{4} + 5 \, a^{5} b^{5} d^{10} x^{2} + a^{6} b^{4} d^{10}} + \frac {585 \, {\left (\frac {\sqrt {2} d^{10} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{10} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{9} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{9} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )}}{a b^{4}}}{163840 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^(15/2)/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

1/163840*(8*(195*(d*x)^(17/2)*b^4*d^10 - 4960*(d*x)^(13/2)*a*b^3*d^12 - 5330*(d*x)^(9/2)*a^2*b^2*d^14 - 2808*(
d*x)^(5/2)*a^3*b*d^16 - 585*sqrt(d*x)*a^4*d^18)/(a*b^9*d^10*x^10 + 5*a^2*b^8*d^10*x^8 + 10*a^3*b^7*d^10*x^6 +
10*a^4*b^6*d^10*x^4 + 5*a^5*b^5*d^10*x^2 + a^6*b^4*d^10) + 585*(sqrt(2)*d^10*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^10*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^9*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^9*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)))/(a*b^4))/d

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mupad [B]  time = 0.13, size = 210, normalized size = 0.54 \[ \frac {117\,d^{15/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{7/4}\,b^{17/4}}-\frac {\frac {31\,d^{11}\,{\left (d\,x\right )}^{13/2}}{128\,b}-\frac {39\,d^9\,{\left (d\,x\right )}^{17/2}}{4096\,a}+\frac {351\,a^2\,d^{15}\,{\left (d\,x\right )}^{5/2}}{2560\,b^3}+\frac {117\,a^3\,d^{17}\,\sqrt {d\,x}}{4096\,b^4}+\frac {533\,a\,d^{13}\,{\left (d\,x\right )}^{9/2}}{2048\,b^2}}{a^5\,d^{10}+5\,a^4\,b\,d^{10}\,x^2+10\,a^3\,b^2\,d^{10}\,x^4+10\,a^2\,b^3\,d^{10}\,x^6+5\,a\,b^4\,d^{10}\,x^8+b^5\,d^{10}\,x^{10}}+\frac {117\,d^{15/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{7/4}\,b^{17/4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(15/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)

[Out]

(117*d^(15/2)*atan((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(8192*(-a)^(7/4)*b^(17/4)) - ((31*d^11*(d*x)^(
13/2))/(128*b) - (39*d^9*(d*x)^(17/2))/(4096*a) + (351*a^2*d^15*(d*x)^(5/2))/(2560*b^3) + (117*a^3*d^17*(d*x)^
(1/2))/(4096*b^4) + (533*a*d^13*(d*x)^(9/2))/(2048*b^2))/(a^5*d^10 + b^5*d^10*x^10 + 5*a^4*b*d^10*x^2 + 5*a*b^
4*d^10*x^8 + 10*a^3*b^2*d^10*x^4 + 10*a^2*b^3*d^10*x^6) + (117*d^(15/2)*atanh((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4
)*d^(1/2))))/(8192*(-a)^(7/4)*b^(17/4))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**(15/2)/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

Timed out

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