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

Optimal. Leaf size=389 \[ \frac {117 d^{5/2} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{17/4} b^{7/4}}-\frac {117 d^{5/2} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}+\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x\right )}{16384 \sqrt {2} a^{17/4} b^{7/4}}-\frac {117 d^{5/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{17/4} b^{7/4}}+\frac {117 d^{5/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{17/4} b^{7/4}}+\frac {117 d (d x)^{3/2}}{4096 a^4 b \left (a+b x^2\right )}+\frac {117 d (d x)^{3/2}}{5120 a^3 b \left (a+b x^2\right )^2}+\frac {13 d (d x)^{3/2}}{640 a^2 b \left (a+b x^2\right )^3}+\frac {3 d (d x)^{3/2}}{160 a b \left (a+b x^2\right )^4}-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5} \]

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Rubi [A]  time = 0.46, antiderivative size = 389, 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, 297, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {117 d^{5/2} \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^{17/4} b^{7/4}}-\frac {117 d^{5/2} \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^{17/4} b^{7/4}}-\frac {117 d^{5/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{17/4} b^{7/4}}+\frac {117 d^{5/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{8192 \sqrt {2} a^{17/4} b^{7/4}}+\frac {117 d (d x)^{3/2}}{4096 a^4 b \left (a+b x^2\right )}+\frac {117 d (d x)^{3/2}}{5120 a^3 b \left (a+b x^2\right )^2}+\frac {13 d (d x)^{3/2}}{640 a^2 b \left (a+b x^2\right )^3}+\frac {3 d (d x)^{3/2}}{160 a b \left (a+b x^2\right )^4}-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*(d*x)^(3/2))/(10*b*(a + b*x^2)^5) + (3*d*(d*x)^(3/2))/(160*a*b*(a + b*x^2)^4) + (13*d*(d*x)^(3/2))/(640*a^
2*b*(a + b*x^2)^3) + (117*d*(d*x)^(3/2))/(5120*a^3*b*(a + b*x^2)^2) + (117*d*(d*x)^(3/2))/(4096*a^4*b*(a + b*x
^2)) - (117*d^(5/2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(17/4)*b^(7/4))
+ (117*d^(5/2)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])])/(8192*Sqrt[2]*a^(17/4)*b^(7/4)) + (
117*d^(5/2)*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^(17
/4)*b^(7/4)) - (117*d^(5/2)*Log[Sqrt[a]*Sqrt[d] + Sqrt[b]*Sqrt[d]*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d*x]])/(163
84*Sqrt[2]*a^(17/4)*b^(7/4))

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 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 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 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)^{5/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {(d x)^{5/2}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}+\frac {1}{20} \left (3 b^4 d^2\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}+\frac {3 d (d x)^{3/2}}{160 a b \left (a+b x^2\right )^4}+\frac {\left (39 b^3 d^2\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^4} \, dx}{320 a}\\ &=-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}+\frac {3 d (d x)^{3/2}}{160 a b \left (a+b x^2\right )^4}+\frac {13 d (d x)^{3/2}}{640 a^2 b \left (a+b x^2\right )^3}+\frac {\left (117 b^2 d^2\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^3} \, dx}{1280 a^2}\\ &=-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}+\frac {3 d (d x)^{3/2}}{160 a b \left (a+b x^2\right )^4}+\frac {13 d (d x)^{3/2}}{640 a^2 b \left (a+b x^2\right )^3}+\frac {117 d (d x)^{3/2}}{5120 a^3 b \left (a+b x^2\right )^2}+\frac {\left (117 b d^2\right ) \int \frac {\sqrt {d x}}{\left (a b+b^2 x^2\right )^2} \, dx}{2048 a^3}\\ &=-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}+\frac {3 d (d x)^{3/2}}{160 a b \left (a+b x^2\right )^4}+\frac {13 d (d x)^{3/2}}{640 a^2 b \left (a+b x^2\right )^3}+\frac {117 d (d x)^{3/2}}{5120 a^3 b \left (a+b x^2\right )^2}+\frac {117 d (d x)^{3/2}}{4096 a^4 b \left (a+b x^2\right )}+\frac {\left (117 d^2\right ) \int \frac {\sqrt {d x}}{a b+b^2 x^2} \, dx}{8192 a^4}\\ &=-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}+\frac {3 d (d x)^{3/2}}{160 a b \left (a+b x^2\right )^4}+\frac {13 d (d x)^{3/2}}{640 a^2 b \left (a+b x^2\right )^3}+\frac {117 d (d x)^{3/2}}{5120 a^3 b \left (a+b x^2\right )^2}+\frac {117 d (d x)^{3/2}}{4096 a^4 b \left (a+b x^2\right )}+\frac {(117 d) \operatorname {Subst}\left (\int \frac {x^2}{a b+\frac {b^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{4096 a^4}\\ &=-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}+\frac {3 d (d x)^{3/2}}{160 a b \left (a+b x^2\right )^4}+\frac {13 d (d x)^{3/2}}{640 a^2 b \left (a+b x^2\right )^3}+\frac {117 d (d x)^{3/2}}{5120 a^3 b \left (a+b x^2\right )^2}+\frac {117 d (d x)^{3/2}}{4096 a^4 b \left (a+b x^2\right )}-\frac {(117 d) \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^4 \sqrt {b}}+\frac {(117 d) \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^4 \sqrt {b}}\\ &=-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}+\frac {3 d (d x)^{3/2}}{160 a b \left (a+b x^2\right )^4}+\frac {13 d (d x)^{3/2}}{640 a^2 b \left (a+b x^2\right )^3}+\frac {117 d (d x)^{3/2}}{5120 a^3 b \left (a+b x^2\right )^2}+\frac {117 d (d x)^{3/2}}{4096 a^4 b \left (a+b x^2\right )}+\frac {\left (117 d^{5/2}\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^{17/4} b^{7/4}}+\frac {\left (117 d^{5/2}\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^{17/4} b^{7/4}}+\frac {\left (117 d^3\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^4 b^2}+\frac {\left (117 d^3\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^4 b^2}\\ &=-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}+\frac {3 d (d x)^{3/2}}{160 a b \left (a+b x^2\right )^4}+\frac {13 d (d x)^{3/2}}{640 a^2 b \left (a+b x^2\right )^3}+\frac {117 d (d x)^{3/2}}{5120 a^3 b \left (a+b x^2\right )^2}+\frac {117 d (d x)^{3/2}}{4096 a^4 b \left (a+b x^2\right )}+\frac {117 d^{5/2} \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^{17/4} b^{7/4}}-\frac {117 d^{5/2} \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^{17/4} b^{7/4}}+\frac {\left (117 d^{5/2}\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^{17/4} b^{7/4}}-\frac {\left (117 d^{5/2}\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^{17/4} b^{7/4}}\\ &=-\frac {d (d x)^{3/2}}{10 b \left (a+b x^2\right )^5}+\frac {3 d (d x)^{3/2}}{160 a b \left (a+b x^2\right )^4}+\frac {13 d (d x)^{3/2}}{640 a^2 b \left (a+b x^2\right )^3}+\frac {117 d (d x)^{3/2}}{5120 a^3 b \left (a+b x^2\right )^2}+\frac {117 d (d x)^{3/2}}{4096 a^4 b \left (a+b x^2\right )}-\frac {117 d^{5/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{17/4} b^{7/4}}+\frac {117 d^{5/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{8192 \sqrt {2} a^{17/4} b^{7/4}}+\frac {117 d^{5/2} \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^{17/4} b^{7/4}}-\frac {117 d^{5/2} \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^{17/4} b^{7/4}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 48, normalized size = 0.12 \begin {gather*} \frac {2 d (d x)^{3/2} \left (\frac {\, _2F_1\left (\frac {3}{4},6;\frac {7}{4};-\frac {b x^2}{a}\right )}{a^5}-\frac {1}{\left (a+b x^2\right )^5}\right )}{17 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*d*(d*x)^(3/2)*(-(a + b*x^2)^(-5) + Hypergeometric2F1[3/4, 6, 7/4, -((b*x^2)/a)]/a^5))/(17*b)

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IntegrateAlgebraic [A]  time = 1.11, size = 241, normalized size = 0.62 \begin {gather*} -\frac {117 d^{5/2} \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 )}{8192 \sqrt {2} a^{17/4} b^{7/4}}-\frac {117 d^{5/2} \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 )}{8192 \sqrt {2} a^{17/4} b^{7/4}}-\frac {(d x)^{3/2} \left (195 a^4 d^{11}-4960 a^3 b d^{11} x^2-5330 a^2 b^2 d^{11} x^4-2808 a b^3 d^{11} x^6-585 b^4 d^{11} x^8\right )}{20480 a^4 b \left (a d^2+b d^2 x^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/20480*((d*x)^(3/2)*(195*a^4*d^11 - 4960*a^3*b*d^11*x^2 - 5330*a^2*b^2*d^11*x^4 - 2808*a*b^3*d^11*x^6 - 585*
b^4*d^11*x^8))/(a^4*b*(a*d^2 + b*d^2*x^2)^5) - (117*d^(5/2)*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]])/(8192*Sqrt[2]*a^(17/4)*b^(7/4)) - (117*d^(5/2)*ArcTanh[(Sqrt[2]
*a^(1/4)*b^(1/4)*Sqrt[d]*Sqrt[d*x])/(Sqrt[a]*d + Sqrt[b]*d*x)])/(8192*Sqrt[2]*a^(17/4)*b^(7/4))

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fricas [A]  time = 0.70, size = 512, normalized size = 1.32 \begin {gather*} -\frac {2340 \, {\left (a^{4} b^{6} x^{10} + 5 \, a^{5} b^{5} x^{8} + 10 \, a^{6} b^{4} x^{6} + 10 \, a^{7} b^{3} x^{4} + 5 \, a^{8} b^{2} x^{2} + a^{9} b\right )} \left (-\frac {d^{10}}{a^{17} b^{7}}\right )^{\frac {1}{4}} \arctan \left (-\frac {1601613 \, \sqrt {d x} a^{4} b^{2} d^{7} \left (-\frac {d^{10}}{a^{17} b^{7}}\right )^{\frac {1}{4}} - \sqrt {-2565164201769 \, a^{9} b^{3} d^{10} \sqrt {-\frac {d^{10}}{a^{17} b^{7}}} + 2565164201769 \, d^{15} x} a^{4} b^{2} \left (-\frac {d^{10}}{a^{17} b^{7}}\right )^{\frac {1}{4}}}{1601613 \, d^{10}}\right ) - 585 \, {\left (a^{4} b^{6} x^{10} + 5 \, a^{5} b^{5} x^{8} + 10 \, a^{6} b^{4} x^{6} + 10 \, a^{7} b^{3} x^{4} + 5 \, a^{8} b^{2} x^{2} + a^{9} b\right )} \left (-\frac {d^{10}}{a^{17} b^{7}}\right )^{\frac {1}{4}} \log \left (1601613 \, a^{13} b^{5} \left (-\frac {d^{10}}{a^{17} b^{7}}\right )^{\frac {3}{4}} + 1601613 \, \sqrt {d x} d^{7}\right ) + 585 \, {\left (a^{4} b^{6} x^{10} + 5 \, a^{5} b^{5} x^{8} + 10 \, a^{6} b^{4} x^{6} + 10 \, a^{7} b^{3} x^{4} + 5 \, a^{8} b^{2} x^{2} + a^{9} b\right )} \left (-\frac {d^{10}}{a^{17} b^{7}}\right )^{\frac {1}{4}} \log \left (-1601613 \, a^{13} b^{5} \left (-\frac {d^{10}}{a^{17} b^{7}}\right )^{\frac {3}{4}} + 1601613 \, \sqrt {d x} d^{7}\right ) - 4 \, {\left (585 \, b^{4} d^{2} x^{9} + 2808 \, a b^{3} d^{2} x^{7} + 5330 \, a^{2} b^{2} d^{2} x^{5} + 4960 \, a^{3} b d^{2} x^{3} - 195 \, a^{4} d^{2} x\right )} \sqrt {d x}}{81920 \, {\left (a^{4} b^{6} x^{10} + 5 \, a^{5} b^{5} x^{8} + 10 \, a^{6} b^{4} x^{6} + 10 \, a^{7} b^{3} x^{4} + 5 \, a^{8} b^{2} x^{2} + a^{9} b\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/81920*(2340*(a^4*b^6*x^10 + 5*a^5*b^5*x^8 + 10*a^6*b^4*x^6 + 10*a^7*b^3*x^4 + 5*a^8*b^2*x^2 + a^9*b)*(-d^10
/(a^17*b^7))^(1/4)*arctan(-1/1601613*(1601613*sqrt(d*x)*a^4*b^2*d^7*(-d^10/(a^17*b^7))^(1/4) - sqrt(-256516420
1769*a^9*b^3*d^10*sqrt(-d^10/(a^17*b^7)) + 2565164201769*d^15*x)*a^4*b^2*(-d^10/(a^17*b^7))^(1/4))/d^10) - 585
*(a^4*b^6*x^10 + 5*a^5*b^5*x^8 + 10*a^6*b^4*x^6 + 10*a^7*b^3*x^4 + 5*a^8*b^2*x^2 + a^9*b)*(-d^10/(a^17*b^7))^(
1/4)*log(1601613*a^13*b^5*(-d^10/(a^17*b^7))^(3/4) + 1601613*sqrt(d*x)*d^7) + 585*(a^4*b^6*x^10 + 5*a^5*b^5*x^
8 + 10*a^6*b^4*x^6 + 10*a^7*b^3*x^4 + 5*a^8*b^2*x^2 + a^9*b)*(-d^10/(a^17*b^7))^(1/4)*log(-1601613*a^13*b^5*(-
d^10/(a^17*b^7))^(3/4) + 1601613*sqrt(d*x)*d^7) - 4*(585*b^4*d^2*x^9 + 2808*a*b^3*d^2*x^7 + 5330*a^2*b^2*d^2*x
^5 + 4960*a^3*b*d^2*x^3 - 195*a^4*d^2*x)*sqrt(d*x))/(a^4*b^6*x^10 + 5*a^5*b^5*x^8 + 10*a^6*b^4*x^6 + 10*a^7*b^
3*x^4 + 5*a^8*b^2*x^2 + a^9*b)

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giac [A]  time = 0.22, size = 355, normalized size = 0.91 \begin {gather*} \frac {1}{163840} \, d^{2} {\left (\frac {1170 \, \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 )}{a^{5} b^{4} d} + \frac {1170 \, \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 )}{a^{5} b^{4} d} - \frac {585 \, \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 )}{a^{5} b^{4} d} + \frac {585 \, \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 )}{a^{5} b^{4} d} + \frac {8 \, {\left (585 \, \sqrt {d x} b^{4} d^{10} x^{9} + 2808 \, \sqrt {d x} a b^{3} d^{10} x^{7} + 5330 \, \sqrt {d x} a^{2} b^{2} d^{10} x^{5} + 4960 \, \sqrt {d x} a^{3} b d^{10} x^{3} - 195 \, \sqrt {d x} a^{4} d^{10} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{5} a^{4} b}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/163840*d^2*(1170*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^5*b^4*d) + 1170*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^5*b^4*d) - 585*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^5*b^4*d) + 585*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sq
rt(a*d^2/b))/(a^5*b^4*d) + 8*(585*sqrt(d*x)*b^4*d^10*x^9 + 2808*sqrt(d*x)*a*b^3*d^10*x^7 + 5330*sqrt(d*x)*a^2*
b^2*d^10*x^5 + 4960*sqrt(d*x)*a^3*b*d^10*x^3 - 195*sqrt(d*x)*a^4*d^10*x)/((b*d^2*x^2 + a*d^2)^5*a^4*b))

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maple [A]  time = 0.02, size = 341, normalized size = 0.88 \begin {gather*} -\frac {39 \left (d x \right )^{\frac {3}{2}} d^{11}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} b}+\frac {31 \left (d x \right )^{\frac {7}{2}} d^{9}}{128 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a}+\frac {533 \left (d x \right )^{\frac {11}{2}} b \,d^{7}}{2048 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{2}}+\frac {351 \left (d x \right )^{\frac {15}{2}} b^{2} d^{5}}{2560 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{3}}+\frac {117 \left (d x \right )^{\frac {19}{2}} b^{3} d^{3}}{4096 \left (b \,d^{2} x^{2}+d^{2} a \right )^{5} a^{4}}+\frac {117 \sqrt {2}\, d^{3} \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^{4} b^{2}}+\frac {117 \sqrt {2}\, d^{3} \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^{4} b^{2}}+\frac {117 \sqrt {2}\, d^{3} \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^{4} b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-39/4096*d^11/(b*d^2*x^2+a*d^2)^5/b*(d*x)^(3/2)+31/128*d^9/(b*d^2*x^2+a*d^2)^5/a*(d*x)^(7/2)+533/2048*d^7/(b*d
^2*x^2+a*d^2)^5/a^2*b*(d*x)^(11/2)+351/2560*d^5/(b*d^2*x^2+a*d^2)^5/a^3*b^2*(d*x)^(15/2)+117/4096*d^3/(b*d^2*x
^2+a*d^2)^5/a^4*b^3*(d*x)^(19/2)+117/32768*d^3/a^4/b^2/(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^3/a^4/b^2
/(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^3/a^4/b^2/(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 = 3.16, size = 383, normalized size = 0.98 \begin {gather*} \frac {\frac {8 \, {\left (585 \, \left (d x\right )^{\frac {19}{2}} b^{4} d^{4} + 2808 \, \left (d x\right )^{\frac {15}{2}} a b^{3} d^{6} + 5330 \, \left (d x\right )^{\frac {11}{2}} a^{2} b^{2} d^{8} + 4960 \, \left (d x\right )^{\frac {7}{2}} a^{3} b d^{10} - 195 \, \left (d x\right )^{\frac {3}{2}} a^{4} d^{12}\right )}}{a^{4} b^{6} d^{10} x^{10} + 5 \, a^{5} b^{5} d^{10} x^{8} + 10 \, a^{6} b^{4} d^{10} x^{6} + 10 \, a^{7} b^{3} d^{10} x^{4} + 5 \, a^{8} b^{2} d^{10} x^{2} + a^{9} b d^{10}} + \frac {585 \, d^{4} {\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^{4} b}}{163840 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/163840*(8*(585*(d*x)^(19/2)*b^4*d^4 + 2808*(d*x)^(15/2)*a*b^3*d^6 + 5330*(d*x)^(11/2)*a^2*b^2*d^8 + 4960*(d*
x)^(7/2)*a^3*b*d^10 - 195*(d*x)^(3/2)*a^4*d^12)/(a^4*b^6*d^10*x^10 + 5*a^5*b^5*d^10*x^8 + 10*a^6*b^4*d^10*x^6
+ 10*a^7*b^3*d^10*x^4 + 5*a^8*b^2*d^10*x^2 + a^9*b*d^10) + 585*d^4*(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)) + 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(sqr
t(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^4*b))/d

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mupad [B]  time = 4.28, size = 209, normalized size = 0.54 \begin {gather*} \frac {\frac {31\,d^9\,{\left (d\,x\right )}^{7/2}}{128\,a}-\frac {39\,d^{11}\,{\left (d\,x\right )}^{3/2}}{4096\,b}+\frac {351\,b^2\,d^5\,{\left (d\,x\right )}^{15/2}}{2560\,a^3}+\frac {117\,b^3\,d^3\,{\left (d\,x\right )}^{19/2}}{4096\,a^4}+\frac {533\,b\,d^7\,{\left (d\,x\right )}^{11/2}}{2048\,a^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^{5/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{17/4}\,b^{7/4}}-\frac {117\,d^{5/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{8192\,{\left (-a\right )}^{17/4}\,b^{7/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((31*d^9*(d*x)^(7/2))/(128*a) - (39*d^11*(d*x)^(3/2))/(4096*b) + (351*b^2*d^5*(d*x)^(15/2))/(2560*a^3) + (117*
b^3*d^3*(d*x)^(19/2))/(4096*a^4) + (533*b*d^7*(d*x)^(11/2))/(2048*a^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^(5/2)*atan((b^(1/4)*(d*x)^(1/2
))/((-a)^(1/4)*d^(1/2))))/(8192*(-a)^(17/4)*b^(7/4)) - (117*d^(5/2)*atanh((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^
(1/2))))/(8192*(-a)^(17/4)*b^(7/4))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d x\right )^{\frac {5}{2}}}{\left (a + b x^{2}\right )^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((d*x)**(5/2)/(a + b*x**2)**6, x)

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