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

Optimal. Leaf size=551 \[ -\frac {13 d^3 (d x)^{9/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a d^7 \sqrt {d x} \left (a+b x^2\right )}{16 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 d^5 (d x)^{5/2} \left (a+b x^2\right )}{80 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{32 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]

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Rubi [A]  time = 0.40, antiderivative size = 551, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1112, 288, 321, 329, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {117 a d^7 \sqrt {d x} \left (a+b x^2\right )}{16 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 d^5 (d x)^{5/2} \left (a+b x^2\right )}{80 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {13 d^3 (d x)^{9/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{64 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{32 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-13*d^3*(d*x)^(9/2))/(16*b^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (d*(d*x)^(13/2))/(4*b*(a + b*x^2)*Sqrt[a^2 +
2*a*b*x^2 + b^2*x^4]) - (117*a*d^7*Sqrt[d*x]*(a + b*x^2))/(16*b^4*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (117*d^5*
(d*x)^(5/2)*(a + b*x^2))/(80*b^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (117*a^(5/4)*d^(15/2)*(a + b*x^2)*ArcTan[1
- (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*b^(17/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (11
7*a^(5/4)*d^(15/2)*(a + b*x^2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(32*Sqrt[2]*b^(17/4)
*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (117*a^(5/4)*d^(15/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x
- Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(64*Sqrt[2]*b^(17/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (117*a^(5/4)*d^(
15/2)*(a + b*x^2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(64*Sqrt[2]*b^
(17/4)*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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 321

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*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 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 1112

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(a + b*x^2 + c*x^4)^FracPa
rt[p]/(c^IntPart[p]*(b/2 + c*x^2)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^2)^(2*p), x], x] /; FreeQ[{a, b, c,
d, m, p}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

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 )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{15/2}}{\left (a b+b^2 x^2\right )^3} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (13 d^2 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{11/2}}{\left (a b+b^2 x^2\right )^2} \, dx}{8 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {13 d^3 (d x)^{9/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 d^4 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{7/2}}{a b+b^2 x^2} \, dx}{32 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {13 d^3 (d x)^{9/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 d^5 (d x)^{5/2} \left (a+b x^2\right )}{80 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (117 a d^6 \left (a b+b^2 x^2\right )\right ) \int \frac {(d x)^{3/2}}{a b+b^2 x^2} \, dx}{32 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {13 d^3 (d x)^{9/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a d^7 \sqrt {d x} \left (a+b x^2\right )}{16 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 d^5 (d x)^{5/2} \left (a+b x^2\right )}{80 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 a^2 d^8 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{32 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {13 d^3 (d x)^{9/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a d^7 \sqrt {d x} \left (a+b x^2\right )}{16 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 d^5 (d x)^{5/2} \left (a+b x^2\right )}{80 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 a^2 d^7 \left (a b+b^2 x^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{16 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {13 d^3 (d x)^{9/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a d^7 \sqrt {d x} \left (a+b x^2\right )}{16 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 d^5 (d x)^{5/2} \left (a+b x^2\right )}{80 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 a^{3/2} d^6 \left (a b+b^2 x^2\right )\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 )}{32 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 a^{3/2} d^6 \left (a b+b^2 x^2\right )\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 )}{32 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {13 d^3 (d x)^{9/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a d^7 \sqrt {d x} \left (a+b x^2\right )}{16 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 d^5 (d x)^{5/2} \left (a+b x^2\right )}{80 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (117 a^{5/4} d^{15/2} \left (a b+b^2 x^2\right )\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 )}{64 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (117 a^{5/4} d^{15/2} \left (a b+b^2 x^2\right )\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 )}{64 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 a^{3/2} d^8 \left (a b+b^2 x^2\right )\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 )}{64 b^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 a^{3/2} d^8 \left (a b+b^2 x^2\right )\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 )}{64 b^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {13 d^3 (d x)^{9/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a d^7 \sqrt {d x} \left (a+b x^2\right )}{16 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 d^5 (d x)^{5/2} \left (a+b x^2\right )}{80 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (117 a^{5/4} d^{15/2} \left (a b+b^2 x^2\right )\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 )}{32 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (117 a^{5/4} d^{15/2} \left (a b+b^2 x^2\right )\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 )}{32 \sqrt {2} b^{21/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac {13 d^3 (d x)^{9/2}}{16 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {d (d x)^{13/2}}{4 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a d^7 \sqrt {d x} \left (a+b x^2\right )}{16 b^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 d^5 (d x)^{5/2} \left (a+b x^2\right )}{80 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{32 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {117 a^{5/4} d^{15/2} \left (a+b x^2\right ) \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{64 \sqrt {2} b^{17/4} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 498, normalized size = 0.90 \begin {gather*} \frac {d^7 \sqrt {d x} \left (-585 \sqrt {2} a^{5/4} b^2 x^4 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+585 \sqrt {2} a^{5/4} b^2 x^4 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-1170 \sqrt {2} a^{9/4} b x^2 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+1170 \sqrt {2} a^{9/4} b x^2 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-1170 \sqrt {2} a^{5/4} \left (a+b x^2\right )^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )+1170 \sqrt {2} a^{5/4} \left (a+b x^2\right )^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )-585 \sqrt {2} a^{13/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )+585 \sqrt {2} a^{13/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )-4680 a^3 \sqrt [4]{b} \sqrt {x}-8424 a^2 b^{5/4} x^{5/2}-3328 a b^{9/4} x^{9/2}+256 b^{13/4} x^{13/2}\right )}{640 b^{17/4} \sqrt {x} \left (a+b x^2\right ) \sqrt {\left (a+b x^2\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^7*Sqrt[d*x]*(-4680*a^3*b^(1/4)*Sqrt[x] - 8424*a^2*b^(5/4)*x^(5/2) - 3328*a*b^(9/4)*x^(9/2) + 256*b^(13/4)*x
^(13/2) - 1170*Sqrt[2]*a^(5/4)*(a + b*x^2)^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] + 1170*Sqrt[2]*a^(5
/4)*(a + b*x^2)^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)] - 585*Sqrt[2]*a^(13/4)*Log[Sqrt[a] - Sqrt[2]*a
^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] - 1170*Sqrt[2]*a^(9/4)*b*x^2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x]
+ Sqrt[b]*x] - 585*Sqrt[2]*a^(5/4)*b^2*x^4*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 585*S
qrt[2]*a^(13/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 1170*Sqrt[2]*a^(9/4)*b*x^2*Log[Sq
rt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x] + 585*Sqrt[2]*a^(5/4)*b^2*x^4*Log[Sqrt[a] + Sqrt[2]*a^(1/
4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]))/(640*b^(17/4)*Sqrt[x]*(a + b*x^2)*Sqrt[(a + b*x^2)^2])

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IntegrateAlgebraic [A]  time = 111.37, size = 269, normalized size = 0.49 \begin {gather*} \frac {\left (a d^2+b d^2 x^2\right ) \left (-\frac {117 a^{5/4} d^{15/2} \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 )}{32 \sqrt {2} b^{17/4}}+\frac {117 a^{5/4} d^{15/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}}{\sqrt {a} d+\sqrt {b} d x}\right )}{32 \sqrt {2} b^{17/4}}+\frac {d^5 \sqrt {d x} \left (-585 a^3 d^6-1053 a^2 b d^6 x^2-416 a b^2 d^6 x^4+32 b^3 d^6 x^6\right )}{80 b^4 \left (a d^2+b d^2 x^2\right )^2}\right )}{d^2 \sqrt {\frac {\left (a d^2+b d^2 x^2\right )^2}{d^4}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a*d^2 + b*d^2*x^2)*((d^5*Sqrt[d*x]*(-585*a^3*d^6 - 1053*a^2*b*d^6*x^2 - 416*a*b^2*d^6*x^4 + 32*b^3*d^6*x^6))
/(80*b^4*(a*d^2 + b*d^2*x^2)^2) - (117*a^(5/4)*d^(15/2)*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]])/(32*Sqrt[2]*b^(17/4)) + (117*a^(5/4)*d^(15/2)*ArcTanh[(Sqrt[2]*a^(1
/4)*b^(1/4)*Sqrt[d]*Sqrt[d*x])/(Sqrt[a]*d + Sqrt[b]*d*x)])/(32*Sqrt[2]*b^(17/4))))/(d^2*Sqrt[(a*d^2 + b*d^2*x^
2)^2/d^4])

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fricas [A]  time = 1.02, size = 341, normalized size = 0.62 \begin {gather*} \frac {2340 \, \left (-\frac {a^{5} d^{30}}{b^{17}}\right )^{\frac {1}{4}} {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )} \arctan \left (-\frac {\left (-\frac {a^{5} d^{30}}{b^{17}}\right )^{\frac {3}{4}} \sqrt {d x} a b^{13} d^{7} - \left (-\frac {a^{5} d^{30}}{b^{17}}\right )^{\frac {3}{4}} \sqrt {a^{2} d^{15} x + \sqrt {-\frac {a^{5} d^{30}}{b^{17}}} b^{8}} b^{13}}{a^{5} d^{30}}\right ) + 585 \, \left (-\frac {a^{5} d^{30}}{b^{17}}\right )^{\frac {1}{4}} {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )} \log \left (117 \, \sqrt {d x} a d^{7} + 117 \, \left (-\frac {a^{5} d^{30}}{b^{17}}\right )^{\frac {1}{4}} b^{4}\right ) - 585 \, \left (-\frac {a^{5} d^{30}}{b^{17}}\right )^{\frac {1}{4}} {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )} \log \left (117 \, \sqrt {d x} a d^{7} - 117 \, \left (-\frac {a^{5} d^{30}}{b^{17}}\right )^{\frac {1}{4}} b^{4}\right ) + 4 \, {\left (32 \, b^{3} d^{7} x^{6} - 416 \, a b^{2} d^{7} x^{4} - 1053 \, a^{2} b d^{7} x^{2} - 585 \, a^{3} d^{7}\right )} \sqrt {d x}}{320 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} \end {gather*}

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)^(3/2),x, algorithm="fricas")

[Out]

1/320*(2340*(-a^5*d^30/b^17)^(1/4)*(b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4)*arctan(-((-a^5*d^30/b^17)^(3/4)*sqrt(d*x)
*a*b^13*d^7 - (-a^5*d^30/b^17)^(3/4)*sqrt(a^2*d^15*x + sqrt(-a^5*d^30/b^17)*b^8)*b^13)/(a^5*d^30)) + 585*(-a^5
*d^30/b^17)^(1/4)*(b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4)*log(117*sqrt(d*x)*a*d^7 + 117*(-a^5*d^30/b^17)^(1/4)*b^4)
- 585*(-a^5*d^30/b^17)^(1/4)*(b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4)*log(117*sqrt(d*x)*a*d^7 - 117*(-a^5*d^30/b^17)^
(1/4)*b^4) + 4*(32*b^3*d^7*x^6 - 416*a*b^2*d^7*x^4 - 1053*a^2*b*d^7*x^2 - 585*a^3*d^7)*sqrt(d*x))/(b^6*x^4 + 2
*a*b^5*x^2 + a^2*b^4)

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giac [A]  time = 0.35, size = 419, normalized size = 0.76 \begin {gather*} \frac {1}{640} \, d^{7} {\left (\frac {1170 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \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 )}{b^{5} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {1170 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \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 )}{b^{5} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {585 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \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 )}{b^{5} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {585 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} a \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 )}{b^{5} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} - \frac {40 \, {\left (25 \, \sqrt {d x} a^{2} b d^{4} x^{2} + 21 \, \sqrt {d x} a^{3} d^{4}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{2} b^{4} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )} + \frac {256 \, {\left (\sqrt {d x} b^{12} d^{10} x^{2} - 15 \, \sqrt {d x} a b^{11} d^{10}\right )}}{b^{15} d^{10} \mathrm {sgn}\left (b d^{4} x^{2} + a d^{4}\right )}\right )} \end {gather*}

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)^(3/2),x, algorithm="giac")

[Out]

1/640*d^7*(1170*sqrt(2)*(a*b^3*d^2)^(1/4)*a*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/
b)^(1/4))/(b^5*sgn(b*d^4*x^2 + a*d^4)) + 1170*sqrt(2)*(a*b^3*d^2)^(1/4)*a*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/
b)^(1/4) - 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(b^5*sgn(b*d^4*x^2 + a*d^4)) + 585*sqrt(2)*(a*b^3*d^2)^(1/4)*a*log(d*
x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(b^5*sgn(b*d^4*x^2 + a*d^4)) - 585*sqrt(2)*(a*b^3*d^2)^
(1/4)*a*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(b^5*sgn(b*d^4*x^2 + a*d^4)) - 40*(25*sqr
t(d*x)*a^2*b*d^4*x^2 + 21*sqrt(d*x)*a^3*d^4)/((b*d^2*x^2 + a*d^2)^2*b^4*sgn(b*d^4*x^2 + a*d^4)) + 256*(sqrt(d*
x)*b^12*d^10*x^2 - 15*sqrt(d*x)*a*b^11*d^10)/(b^15*d^10*sgn(b*d^4*x^2 + a*d^4)))

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maple [B]  time = 0.02, size = 737, normalized size = 1.34 \begin {gather*} \frac {\left (1170 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a \,b^{2} d^{2} x^{4} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+1170 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a \,b^{2} d^{2} x^{4} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+585 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a \,b^{2} d^{2} x^{4} \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 )-3840 \sqrt {d x}\, a \,b^{2} d^{2} x^{4}+2340 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} b \,d^{2} x^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2340 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} b \,d^{2} x^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+1170 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} b \,d^{2} x^{2} \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 )-7680 \sqrt {d x}\, a^{2} b \,d^{2} x^{2}+256 \left (d x \right )^{\frac {5}{2}} b^{3} x^{4}+1170 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{3} d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+1170 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{3} d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+585 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{3} d^{2} \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 )-4680 \sqrt {d x}\, a^{3} d^{2}+512 \left (d x \right )^{\frac {5}{2}} a \,b^{2} x^{2}-744 \left (d x \right )^{\frac {5}{2}} a^{2} b \right ) \left (b \,x^{2}+a \right ) d^{5}}{640 \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} b^{4}} \end {gather*}

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)^(3/2),x)

[Out]

1/640*(1170*arctan((2^(1/2)*(d*x)^(1/2)-(a/b*d^2)^(1/4))/(a/b*d^2)^(1/4))*(a/b*d^2)^(1/4)*2^(1/2)*x^4*a*b^2*d^
2+585*(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)))*x^4*a*b^2*d^2+1170*(a/b*d^2)^(1/4)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)
+(a/b*d^2)^(1/4))/(a/b*d^2)^(1/4))*x^4*a*b^2*d^2+256*(d*x)^(5/2)*x^4*b^3+2340*arctan((2^(1/2)*(d*x)^(1/2)-(a/b
*d^2)^(1/4))/(a/b*d^2)^(1/4))*(a/b*d^2)^(1/4)*2^(1/2)*x^2*a^2*b*d^2+1170*(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)))*x^2
*a^2*b*d^2+2340*(a/b*d^2)^(1/4)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a/b*d^2)^(1/4))/(a/b*d^2)^(1/4))*x^2*a^2*
b*d^2+512*(d*x)^(5/2)*x^2*a*b^2-3840*(d*x)^(1/2)*x^4*a*b^2*d^2+1170*arctan((2^(1/2)*(d*x)^(1/2)-(a/b*d^2)^(1/4
))/(a/b*d^2)^(1/4))*(a/b*d^2)^(1/4)*2^(1/2)*a^3*d^2+585*(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)))*a^3*d^2+1170*(a/b*d^
2)^(1/4)*2^(1/2)*arctan((2^(1/2)*(d*x)^(1/2)+(a/b*d^2)^(1/4))/(a/b*d^2)^(1/4))*a^3*d^2-744*(d*x)^(5/2)*a^2*b-7
680*(d*x)^(1/2)*x^2*a^2*b*d^2-4680*(d*x)^(1/2)*a^3*d^2)*d^5*(b*x^2+a)/b^4/((b*x^2+a)^2)^(3/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {a^{2} d^{\frac {15}{2}} x^{\frac {5}{2}}}{2 \, {\left (a b^{4} x^{2} + a^{2} b^{3} + {\left (b^{5} x^{2} + a b^{4}\right )} x^{2}\right )}} - 2 \, a d^{\frac {15}{2}} \int \frac {x^{\frac {3}{2}}}{b^{4} x^{2} + a b^{3}}\,{d x} + d^{\frac {15}{2}} \int \frac {x^{\frac {7}{2}}}{b^{3} x^{2} + a b^{2}}\,{d x} + \frac {21 \, {\left (\frac {2 \, \sqrt {2} a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} a^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} a^{\frac {5}{4}} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} a^{\frac {5}{4}} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{b^{\frac {1}{4}}}\right )} d^{\frac {15}{2}}}{128 \, b^{4}} - \frac {17 \, a^{2} b d^{\frac {15}{2}} x^{\frac {5}{2}} + 21 \, a^{3} d^{\frac {15}{2}} \sqrt {x}}{16 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} \end {gather*}

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)^(3/2),x, algorithm="maxima")

[Out]

-1/2*a^2*d^(15/2)*x^(5/2)/(a*b^4*x^2 + a^2*b^3 + (b^5*x^2 + a*b^4)*x^2) - 2*a*d^(15/2)*integrate(x^(3/2)/(b^4*
x^2 + a*b^3), x) + d^(15/2)*integrate(x^(7/2)/(b^3*x^2 + a*b^2), x) + 21/128*(2*sqrt(2)*a^(3/2)*arctan(1/2*sqr
t(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/sqrt(sqrt(a)*sqrt(b)) + 2*sqrt(2)*a^
(3/2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/sqrt(sqrt(a)*sq
rt(b)) + sqrt(2)*a^(5/4)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/b^(1/4) - sqrt(2)*a^(5/4)*
log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/b^(1/4))*d^(15/2)/b^4 - 1/16*(17*a^2*b*d^(15/2)*x^
(5/2) + 21*a^3*d^(15/2)*sqrt(x))/(b^6*x^4 + 2*a*b^5*x^2 + a^2*b^4)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d\,x\right )}^{15/2}}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}} \,d x \end {gather*}

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)^(3/2),x)

[Out]

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

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

[Out]

Timed out

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