3.7.96 \(\int \frac {(d x)^{19/2}}{(a^2+2 a b x^2+b^2 x^4)^2} \, dx\) [696]

Optimal. Leaf size=368 \[ -\frac {663 a d^9 \sqrt {d x}}{64 b^5}+\frac {663 d^7 (d x)^{5/2}}{320 b^4}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac {17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}-\frac {663 a^{5/4} d^{19/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} b^{21/4}}+\frac {663 a^{5/4} d^{19/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} b^{21/4}}-\frac {663 a^{5/4} d^{19/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} b^{21/4}}+\frac {663 a^{5/4} d^{19/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} b^{21/4}} \]

[Out]

663/320*d^7*(d*x)^(5/2)/b^4-1/6*d*(d*x)^(17/2)/b/(b*x^2+a)^3-17/48*d^3*(d*x)^(13/2)/b^2/(b*x^2+a)^2-221/192*d^
5*(d*x)^(9/2)/b^3/(b*x^2+a)-663/256*a^(5/4)*d^(19/2)*arctan(1-b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/b^(
21/4)*2^(1/2)+663/256*a^(5/4)*d^(19/2)*arctan(1+b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/b^(21/4)*2^(1/2)-
663/512*a^(5/4)*d^(19/2)*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))/b^(21/4)*2^
(1/2)+663/512*a^(5/4)*d^(19/2)*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))/b^(21
/4)*2^(1/2)-663/64*a*d^9*(d*x)^(1/2)/b^5

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Rubi [A]
time = 0.28, antiderivative size = 368, 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, 294, 327, 335, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {663 a^{5/4} d^{19/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} b^{21/4}}+\frac {663 a^{5/4} d^{19/2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{128 \sqrt {2} b^{21/4}}-\frac {663 a^{5/4} d^{19/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} b^{21/4}}+\frac {663 a^{5/4} d^{19/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{256 \sqrt {2} b^{21/4}}-\frac {663 a d^9 \sqrt {d x}}{64 b^5}-\frac {221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}-\frac {17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}+\frac {663 d^7 (d x)^{5/2}}{320 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-663*a*d^9*Sqrt[d*x])/(64*b^5) + (663*d^7*(d*x)^(5/2))/(320*b^4) - (d*(d*x)^(17/2))/(6*b*(a + b*x^2)^3) - (17
*d^3*(d*x)^(13/2))/(48*b^2*(a + b*x^2)^2) - (221*d^5*(d*x)^(9/2))/(192*b^3*(a + b*x^2)) - (663*a^(5/4)*d^(19/2
)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*b^(21/4)) + (663*a^(5/4)*d^(19/2)*Ar
cTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(128*Sqrt[2]*b^(21/4)) - (663*a^(5/4)*d^(19/2)*Log[Sq
rt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(256*Sqrt[2]*b^(21/4)) + (663*a^(5/4)*
d^(19/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(256*Sqrt[2]*b^(21/4))

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 217

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 294

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 327

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 335

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 631

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 642

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 1176

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 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rubi steps

\begin {align*} \int \frac {(d x)^{19/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {(d x)^{19/2}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}+\frac {1}{12} \left (17 b^2 d^2\right ) \int \frac {(d x)^{15/2}}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac {17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}+\frac {1}{96} \left (221 d^4\right ) \int \frac {(d x)^{11/2}}{\left (a b+b^2 x^2\right )^2} \, dx\\ &=-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac {17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}+\frac {\left (663 d^6\right ) \int \frac {(d x)^{7/2}}{a b+b^2 x^2} \, dx}{128 b^2}\\ &=\frac {663 d^7 (d x)^{5/2}}{320 b^4}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac {17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}-\frac {\left (663 a d^8\right ) \int \frac {(d x)^{3/2}}{a b+b^2 x^2} \, dx}{128 b^3}\\ &=-\frac {663 a d^9 \sqrt {d x}}{64 b^5}+\frac {663 d^7 (d x)^{5/2}}{320 b^4}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac {17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}+\frac {\left (663 a^2 d^{10}\right ) \int \frac {1}{\sqrt {d x} \left (a b+b^2 x^2\right )} \, dx}{128 b^4}\\ &=-\frac {663 a d^9 \sqrt {d x}}{64 b^5}+\frac {663 d^7 (d x)^{5/2}}{320 b^4}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac {17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}+\frac {\left (663 a^2 d^9\right ) \text {Subst}\left (\int \frac {1}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{64 b^4}\\ &=-\frac {663 a d^9 \sqrt {d x}}{64 b^5}+\frac {663 d^7 (d x)^{5/2}}{320 b^4}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac {17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}+\frac {\left (663 a^{3/2} d^8\right ) \text {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 )}{128 b^4}+\frac {\left (663 a^{3/2} d^8\right ) \text {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 )}{128 b^4}\\ &=-\frac {663 a d^9 \sqrt {d x}}{64 b^5}+\frac {663 d^7 (d x)^{5/2}}{320 b^4}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac {17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}-\frac {\left (663 a^{5/4} d^{19/2}\right ) \text {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 )}{256 \sqrt {2} b^{21/4}}-\frac {\left (663 a^{5/4} d^{19/2}\right ) \text {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 )}{256 \sqrt {2} b^{21/4}}+\frac {\left (663 a^{3/2} d^{10}\right ) \text {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 )}{256 b^{11/2}}+\frac {\left (663 a^{3/2} d^{10}\right ) \text {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 )}{256 b^{11/2}}\\ &=-\frac {663 a d^9 \sqrt {d x}}{64 b^5}+\frac {663 d^7 (d x)^{5/2}}{320 b^4}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac {17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}-\frac {663 a^{5/4} d^{19/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} b^{21/4}}+\frac {663 a^{5/4} d^{19/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} b^{21/4}}+\frac {\left (663 a^{5/4} d^{19/2}\right ) \text {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 )}{128 \sqrt {2} b^{21/4}}-\frac {\left (663 a^{5/4} d^{19/2}\right ) \text {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 )}{128 \sqrt {2} b^{21/4}}\\ &=-\frac {663 a d^9 \sqrt {d x}}{64 b^5}+\frac {663 d^7 (d x)^{5/2}}{320 b^4}-\frac {d (d x)^{17/2}}{6 b \left (a+b x^2\right )^3}-\frac {17 d^3 (d x)^{13/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {221 d^5 (d x)^{9/2}}{192 b^3 \left (a+b x^2\right )}-\frac {663 a^{5/4} d^{19/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} b^{21/4}}+\frac {663 a^{5/4} d^{19/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} b^{21/4}}-\frac {663 a^{5/4} d^{19/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} b^{21/4}}+\frac {663 a^{5/4} d^{19/2} \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} b^{21/4}}\\ \end {align*}

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Mathematica [A]
time = 0.57, size = 205, normalized size = 0.56 \begin {gather*} \frac {d^9 \sqrt {d x} \left (4 \sqrt [4]{b} \sqrt {x} \left (-9945 a^4-27846 a^3 b x^2-24973 a^2 b^2 x^4-6528 a b^3 x^6+384 b^4 x^8\right )+9945 \sqrt {2} a^{5/4} \left (a+b x^2\right )^3 \tan ^{-1}\left (\frac {-\sqrt {a}+\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+9945 \sqrt {2} a^{5/4} \left (a+b x^2\right )^3 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{3840 b^{21/4} \sqrt {x} \left (a+b x^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^9*Sqrt[d*x]*(4*b^(1/4)*Sqrt[x]*(-9945*a^4 - 27846*a^3*b*x^2 - 24973*a^2*b^2*x^4 - 6528*a*b^3*x^6 + 384*b^4*
x^8) + 9945*Sqrt[2]*a^(5/4)*(a + b*x^2)^3*ArcTan[(-Sqrt[a] + Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 9
945*Sqrt[2]*a^(5/4)*(a + b*x^2)^3*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)]))/(3840*b^(
21/4)*Sqrt[x]*(a + b*x^2)^3)

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Maple [A]
time = 0.10, size = 237, normalized size = 0.64

method	result	size
derivativedivides	\(2 d^{7} \left (-\frac {-\frac {\left (d x \right )^{\frac {5}{2}} b}{5}+4 a \,d^{2} \sqrt {d x}}{b^{5}}+\frac {a^{2} d^{4} \left (\frac {-\frac {617 b^{2} \left (d x \right )^{\frac {9}{2}}}{384}-\frac {173 a b \,d^{2} \left (d x \right )^{\frac {5}{2}}}{64}-\frac {151 a^{2} d^{4} \sqrt {d x}}{128}}{\left (d^{2} x^{2} b +a \,d^{2}\right )^{3}}+\frac {663 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a \,d^{2}}\right )}{b^{5}}\right )\)	\(237\)
default	\(2 d^{7} \left (-\frac {-\frac {\left (d x \right )^{\frac {5}{2}} b}{5}+4 a \,d^{2} \sqrt {d x}}{b^{5}}+\frac {a^{2} d^{4} \left (\frac {-\frac {617 b^{2} \left (d x \right )^{\frac {9}{2}}}{384}-\frac {173 a b \,d^{2} \left (d x \right )^{\frac {5}{2}}}{64}-\frac {151 a^{2} d^{4} \sqrt {d x}}{128}}{\left (d^{2} x^{2} b +a \,d^{2}\right )^{3}}+\frac {663 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a \,d^{2}}\right )}{b^{5}}\right )\)	\(237\)
risch	\(-\frac {2 \left (-b \,x^{2}+20 a \right ) x \,d^{10}}{5 b^{5} \sqrt {d x}}+\left (-\frac {617 a^{2} d \left (d x \right )^{\frac {9}{2}}}{192 b^{3} \left (d^{2} x^{2} b +a \,d^{2}\right )^{3}}-\frac {173 a^{3} d^{3} \left (d x \right )^{\frac {5}{2}}}{32 b^{4} \left (d^{2} x^{2} b +a \,d^{2}\right )^{3}}-\frac {151 a^{4} d^{5} \sqrt {d x}}{64 b^{5} \left (d^{2} x^{2} b +a \,d^{2}\right )^{3}}+\frac {663 a \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )}{512 b^{5} d}+\frac {663 a \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )}{256 b^{5} d}+\frac {663 a \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )}{256 b^{5} d}\right ) d^{10}\)	\(306\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^(19/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x,method=_RETURNVERBOSE)

[Out]

2*d^7*(-1/b^5*(-1/5*(d*x)^(5/2)*b+4*a*d^2*(d*x)^(1/2))+1/b^5*a^2*d^4*((-617/384*b^2*(d*x)^(9/2)-173/64*a*b*d^2
*(d*x)^(5/2)-151/128*a^2*d^4*(d*x)^(1/2))/(b*d^2*x^2+a*d^2)^3+663/1024*(a*d^2/b)^(1/4)/a/d^2*2^(1/2)*(ln((d*x+
(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2))/(d*x-(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2))
)+2*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)-1))))

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Maxima [A]
time = 0.52, size = 361, normalized size = 0.98 \begin {gather*} -\frac {\frac {40 \, {\left (617 \, \left (d x\right )^{\frac {9}{2}} a^{2} b^{2} d^{12} + 1038 \, \left (d x\right )^{\frac {5}{2}} a^{3} b d^{14} + 453 \, \sqrt {d x} a^{4} d^{16}\right )}}{b^{8} d^{6} x^{6} + 3 \, a b^{7} d^{6} x^{4} + 3 \, a^{2} b^{6} d^{6} x^{2} + a^{3} b^{5} d^{6}} - \frac {9945 \, {\left (\frac {\sqrt {2} d^{12} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{12} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d^{11} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d^{11} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )} a^{2}}{b^{5}} - \frac {3072 \, {\left (\left (d x\right )^{\frac {5}{2}} b d^{8} - 20 \, \sqrt {d x} a d^{10}\right )}}{b^{5}}}{7680 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7680*(40*(617*(d*x)^(9/2)*a^2*b^2*d^12 + 1038*(d*x)^(5/2)*a^3*b*d^14 + 453*sqrt(d*x)*a^4*d^16)/(b^8*d^6*x^6
+ 3*a*b^7*d^6*x^4 + 3*a^2*b^6*d^6*x^2 + a^3*b^5*d^6) - 9945*(sqrt(2)*d^12*log(sqrt(b)*d*x + sqrt(2)*(a*d^2)^(
1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(3/4)*b^(1/4)) - sqrt(2)*d^12*log(sqrt(b)*d*x - sqrt(2)*(a*d^2)^(
1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(3/4)*b^(1/4)) + 2*sqrt(2)*d^11*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^
2)^(1/4)*b^(1/4) + 2*sqrt(d*x)*sqrt(b))/sqrt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(a)) + 2*sqrt(2)
*d^11*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) - 2*sqrt(d*x)*sqrt(b))/sqrt(sqrt(a)*sqrt(b)*d))/(sqrt
(sqrt(a)*sqrt(b)*d)*sqrt(a)))*a^2/b^5 - 3072*((d*x)^(5/2)*b*d^8 - 20*sqrt(d*x)*a*d^10)/b^5)/d

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Fricas [A]
time = 0.40, size = 399, normalized size = 1.08 \begin {gather*} \frac {39780 \, \left (-\frac {a^{5} d^{38}}{b^{21}}\right )^{\frac {1}{4}} {\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )} \arctan \left (-\frac {\left (-\frac {a^{5} d^{38}}{b^{21}}\right )^{\frac {3}{4}} \sqrt {d x} a b^{16} d^{9} - \left (-\frac {a^{5} d^{38}}{b^{21}}\right )^{\frac {3}{4}} \sqrt {a^{2} d^{19} x + \sqrt {-\frac {a^{5} d^{38}}{b^{21}}} b^{10}} b^{16}}{a^{5} d^{38}}\right ) + 9945 \, \left (-\frac {a^{5} d^{38}}{b^{21}}\right )^{\frac {1}{4}} {\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )} \log \left (663 \, \sqrt {d x} a d^{9} + 663 \, \left (-\frac {a^{5} d^{38}}{b^{21}}\right )^{\frac {1}{4}} b^{5}\right ) - 9945 \, \left (-\frac {a^{5} d^{38}}{b^{21}}\right )^{\frac {1}{4}} {\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )} \log \left (663 \, \sqrt {d x} a d^{9} - 663 \, \left (-\frac {a^{5} d^{38}}{b^{21}}\right )^{\frac {1}{4}} b^{5}\right ) + 4 \, {\left (384 \, b^{4} d^{9} x^{8} - 6528 \, a b^{3} d^{9} x^{6} - 24973 \, a^{2} b^{2} d^{9} x^{4} - 27846 \, a^{3} b d^{9} x^{2} - 9945 \, a^{4} d^{9}\right )} \sqrt {d x}}{3840 \, {\left (b^{8} x^{6} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{2} + a^{3} b^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3840*(39780*(-a^5*d^38/b^21)^(1/4)*(b^8*x^6 + 3*a*b^7*x^4 + 3*a^2*b^6*x^2 + a^3*b^5)*arctan(-((-a^5*d^38/b^2
1)^(3/4)*sqrt(d*x)*a*b^16*d^9 - (-a^5*d^38/b^21)^(3/4)*sqrt(a^2*d^19*x + sqrt(-a^5*d^38/b^21)*b^10)*b^16)/(a^5
*d^38)) + 9945*(-a^5*d^38/b^21)^(1/4)*(b^8*x^6 + 3*a*b^7*x^4 + 3*a^2*b^6*x^2 + a^3*b^5)*log(663*sqrt(d*x)*a*d^
9 + 663*(-a^5*d^38/b^21)^(1/4)*b^5) - 9945*(-a^5*d^38/b^21)^(1/4)*(b^8*x^6 + 3*a*b^7*x^4 + 3*a^2*b^6*x^2 + a^3
*b^5)*log(663*sqrt(d*x)*a*d^9 - 663*(-a^5*d^38/b^21)^(1/4)*b^5) + 4*(384*b^4*d^9*x^8 - 6528*a*b^3*d^9*x^6 - 24
973*a^2*b^2*d^9*x^4 - 27846*a^3*b*d^9*x^2 - 9945*a^4*d^9)*sqrt(d*x))/(b^8*x^6 + 3*a*b^7*x^4 + 3*a^2*b^6*x^2 +
a^3*b^5)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d x\right )^{\frac {19}{2}}}{\left (a + b x^{2}\right )^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 4.02, size = 336, normalized size = 0.91 \begin {gather*} \frac {1}{7680} \, d^{9} {\left (\frac {19890 \, \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^{6}} + \frac {19890 \, \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^{6}} + \frac {9945 \, \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^{6}} - \frac {9945 \, \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^{6}} - \frac {40 \, {\left (617 \, \sqrt {d x} a^{2} b^{2} d^{6} x^{4} + 1038 \, \sqrt {d x} a^{3} b d^{6} x^{2} + 453 \, \sqrt {d x} a^{4} d^{6}\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} b^{5}} + \frac {3072 \, {\left (\sqrt {d x} b^{16} d^{10} x^{2} - 20 \, \sqrt {d x} a b^{15} d^{10}\right )}}{b^{20} d^{10}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7680*d^9*(19890*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^6 + 19890*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^6 + 9945*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^6 - 9945*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
^6 - 40*(617*sqrt(d*x)*a^2*b^2*d^6*x^4 + 1038*sqrt(d*x)*a^3*b*d^6*x^2 + 453*sqrt(d*x)*a^4*d^6)/((b*d^2*x^2 + a
*d^2)^3*b^5) + 3072*(sqrt(d*x)*b^16*d^10*x^2 - 20*sqrt(d*x)*a*b^15*d^10)/(b^20*d^10))

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Mupad [B]
time = 0.13, size = 188, normalized size = 0.51 \begin {gather*} \frac {2\,d^7\,{\left (d\,x\right )}^{5/2}}{5\,b^4}-\frac {\frac {151\,a^4\,d^{15}\,\sqrt {d\,x}}{64}+\frac {617\,a^2\,b^2\,d^{11}\,{\left (d\,x\right )}^{9/2}}{192}+\frac {173\,a^3\,b\,d^{13}\,{\left (d\,x\right )}^{5/2}}{32}}{a^3\,b^5\,d^6+3\,a^2\,b^6\,d^6\,x^2+3\,a\,b^7\,d^6\,x^4+b^8\,d^6\,x^6}-\frac {663\,{\left (-a\right )}^{5/4}\,d^{19/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{128\,b^{21/4}}-\frac {8\,a\,d^9\,\sqrt {d\,x}}{b^5}+\frac {{\left (-a\right )}^{5/4}\,d^{19/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )\,663{}\mathrm {i}}{128\,b^{21/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*d^7*(d*x)^(5/2))/(5*b^4) - ((151*a^4*d^15*(d*x)^(1/2))/64 + (617*a^2*b^2*d^11*(d*x)^(9/2))/192 + (173*a^3*b
*d^13*(d*x)^(5/2))/32)/(a^3*b^5*d^6 + b^8*d^6*x^6 + 3*a*b^7*d^6*x^4 + 3*a^2*b^6*d^6*x^2) - (663*(-a)^(5/4)*d^(
19/2)*atan((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(128*b^(21/4)) + ((-a)^(5/4)*d^(19/2)*atan((b^(1/4)*(d
*x)^(1/2)*1i)/((-a)^(1/4)*d^(1/2)))*663i)/(128*b^(21/4)) - (8*a*d^9*(d*x)^(1/2))/b^5

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