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

Optimal. Leaf size=310 \[ \frac{\left (6 a^2 c^2+12 a b c+7 b^2\right ) \left (c+d x^2\right )^3 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{6 a^2 b^2 d^3}-\frac{\left (24 a^2 c^2+60 a b c+35 b^2\right ) \left (c+d x^2\right )^2 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{24 a^3 b d^3}+\frac{\left (24 a^2 c^2+60 a b c+35 b^2\right ) \left (c+d x^2\right ) \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{16 a^4 d^3}-\frac{b \left (24 a^2 c^2+60 a b c+35 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{\frac{a c+a d x^2+b}{c+d x^2}}}{\sqrt{a}}\right )}{16 a^{9/2} d^3}-\frac{(a c+b)^2 \left (c+d x^2\right )^3}{a b^2 d^3 \sqrt{\frac{a c+a d x^2+b}{c+d x^2}}} \]

[Out]

-(((b + a*c)^2*(c + d*x^2)^3)/(a*b^2*d^3*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])) + ((35*b^2 + 60*a*b*c + 24*a^
2*c^2)*(c + d*x^2)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(16*a^4*d^3) - ((35*b^2 + 60*a*b*c + 24*a^2*c^2)*(c
+ d*x^2)^2*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(24*a^3*b*d^3) + ((7*b^2 + 12*a*b*c + 6*a^2*c^2)*(c + d*x^2)
^3*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/(6*a^2*b^2*d^3) - (b*(35*b^2 + 60*a*b*c + 24*a^2*c^2)*ArcTanh[Sqrt[(
b + a*c + a*d*x^2)/(c + d*x^2)]/Sqrt[a]])/(16*a^(9/2)*d^3)

________________________________________________________________________________________

Rubi [A]  time = 0.782944, antiderivative size = 323, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6722, 1975, 446, 89, 80, 50, 63, 217, 206} \[ -\frac{\left (24 a^2 c^2+60 a b c+35 b^2\right ) \left (c+d x^2\right ) \left (a \left (c+d x^2\right )+b\right )}{24 a^3 b d^3 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left (24 a^2 c^2+60 a b c+35 b^2\right ) \left (a \left (c+d x^2\right )+b\right )}{16 a^4 d^3 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{b \left (24 a^2 c^2+60 a b c+35 b^2\right ) \sqrt{a \left (c+d x^2\right )+b} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{c+d x^2}}{\sqrt{a \left (c+d x^2\right )+b}}\right )}{16 a^{9/2} d^3 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left (c+d x^2\right )^2 \left (a \left (c+d x^2\right )+b\right )}{6 a^2 d^3 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{(a c+b)^2 \left (c+d x^2\right )^2}{a^2 b d^3 \sqrt{a+\frac{b}{c+d x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b/(c + d*x^2))^(3/2),x]

[Out]

((b + a*c)^2*(c + d*x^2)^2)/(a^2*b*d^3*Sqrt[a + b/(c + d*x^2)]) + ((35*b^2 + 60*a*b*c + 24*a^2*c^2)*(b + a*(c
+ d*x^2)))/(16*a^4*d^3*Sqrt[a + b/(c + d*x^2)]) - ((35*b^2 + 60*a*b*c + 24*a^2*c^2)*(c + d*x^2)*(b + a*(c + d*
x^2)))/(24*a^3*b*d^3*Sqrt[a + b/(c + d*x^2)]) + ((c + d*x^2)^2*(b + a*(c + d*x^2)))/(6*a^2*d^3*Sqrt[a + b/(c +
d*x^2)]) - (b*(35*b^2 + 60*a*b*c + 24*a^2*c^2)*Sqrt[b + a*(c + d*x^2)]*ArcTanh[(Sqrt[a]*Sqrt[c + d*x^2])/Sqrt
[b + a*(c + d*x^2)]])/(16*a^(9/2)*d^3*Sqrt[c + d*x^2]*Sqrt[a + b/(c + d*x^2)])

Rule 6722

Int[(u_.)*((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[(a + b*v^n)^FracPart[p]/(v^(n*FracPart[p])*(b + a/
v^n)^FracPart[p]), Int[u*v^(n*p)*(b + a/v^n)^p, x], x] /; FreeQ[{a, b, p}, x] &&  !IntegerQ[p] && ILtQ[n, 0] &
& BinomialQ[v, x] &&  !LinearQ[v, x]

Rule 1975

Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*ExpandToSum[u, x]^p*ExpandToSum[v, x]^q
, x] /; FreeQ[{e, m, p, q}, x] && BinomialQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0]
&&  !BinomialMatchQ[{u, v}, x]

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

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

Rubi steps

\begin{align*} \int \frac{x^5}{\left (a+\frac{b}{c+d x^2}\right )^{3/2}} \, dx &=\frac{\sqrt{b+a \left (c+d x^2\right )} \int \frac{x^5 \left (c+d x^2\right )^{3/2}}{\left (b+a \left (c+d x^2\right )\right )^{3/2}} \, dx}{\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=\frac{\sqrt{b+a \left (c+d x^2\right )} \int \frac{x^5 \left (c+d x^2\right )^{3/2}}{\left (b+a c+a d x^2\right )^{3/2}} \, dx}{\sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=\frac{\sqrt{b+a \left (c+d x^2\right )} \operatorname{Subst}\left (\int \frac{x^2 (c+d x)^{3/2}}{(b+a c+a d x)^{3/2}} \, dx,x,x^2\right )}{2 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=\frac{(b+a c)^2 \left (c+d x^2\right )^2}{a^2 b d^3 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\sqrt{b+a \left (c+d x^2\right )} \operatorname{Subst}\left (\int \frac{(c+d x)^{3/2} \left (\frac{1}{2} (b+a c) (5 b+4 a c) d-\frac{1}{2} a b d^2 x\right )}{\sqrt{b+a c+a d x}} \, dx,x,x^2\right )}{a^2 b d^3 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=\frac{(b+a c)^2 \left (c+d x^2\right )^2}{a^2 b d^3 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left (c+d x^2\right )^2 \left (b+a \left (c+d x^2\right )\right )}{6 a^2 d^3 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\left (\left (35 b^2+60 a b c+24 a^2 c^2\right ) \sqrt{b+a \left (c+d x^2\right )}\right ) \operatorname{Subst}\left (\int \frac{(c+d x)^{3/2}}{\sqrt{b+a c+a d x}} \, dx,x,x^2\right )}{12 a^2 b d^2 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=\frac{(b+a c)^2 \left (c+d x^2\right )^2}{a^2 b d^3 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\left (35 b^2+60 a b c+24 a^2 c^2\right ) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{24 a^3 b d^3 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left (c+d x^2\right )^2 \left (b+a \left (c+d x^2\right )\right )}{6 a^2 d^3 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left (\left (35 b^2+60 a b c+24 a^2 c^2\right ) \sqrt{b+a \left (c+d x^2\right )}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c+d x}}{\sqrt{b+a c+a d x}} \, dx,x,x^2\right )}{16 a^3 d^2 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=\frac{(b+a c)^2 \left (c+d x^2\right )^2}{a^2 b d^3 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left (35 b^2+60 a b c+24 a^2 c^2\right ) \left (b+a \left (c+d x^2\right )\right )}{16 a^4 d^3 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\left (35 b^2+60 a b c+24 a^2 c^2\right ) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{24 a^3 b d^3 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left (c+d x^2\right )^2 \left (b+a \left (c+d x^2\right )\right )}{6 a^2 d^3 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\left (b \left (35 b^2+60 a b c+24 a^2 c^2\right ) \sqrt{b+a \left (c+d x^2\right )}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+d x} \sqrt{b+a c+a d x}} \, dx,x,x^2\right )}{32 a^4 d^2 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=\frac{(b+a c)^2 \left (c+d x^2\right )^2}{a^2 b d^3 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left (35 b^2+60 a b c+24 a^2 c^2\right ) \left (b+a \left (c+d x^2\right )\right )}{16 a^4 d^3 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\left (35 b^2+60 a b c+24 a^2 c^2\right ) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{24 a^3 b d^3 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left (c+d x^2\right )^2 \left (b+a \left (c+d x^2\right )\right )}{6 a^2 d^3 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\left (b \left (35 b^2+60 a b c+24 a^2 c^2\right ) \sqrt{b+a \left (c+d x^2\right )}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b+a x^2}} \, dx,x,\sqrt{c+d x^2}\right )}{16 a^4 d^3 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=\frac{(b+a c)^2 \left (c+d x^2\right )^2}{a^2 b d^3 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left (35 b^2+60 a b c+24 a^2 c^2\right ) \left (b+a \left (c+d x^2\right )\right )}{16 a^4 d^3 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\left (35 b^2+60 a b c+24 a^2 c^2\right ) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{24 a^3 b d^3 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left (c+d x^2\right )^2 \left (b+a \left (c+d x^2\right )\right )}{6 a^2 d^3 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\left (b \left (35 b^2+60 a b c+24 a^2 c^2\right ) \sqrt{b+a \left (c+d x^2\right )}\right ) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\sqrt{c+d x^2}}{\sqrt{b+a \left (c+d x^2\right )}}\right )}{16 a^4 d^3 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ &=\frac{(b+a c)^2 \left (c+d x^2\right )^2}{a^2 b d^3 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left (35 b^2+60 a b c+24 a^2 c^2\right ) \left (b+a \left (c+d x^2\right )\right )}{16 a^4 d^3 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{\left (35 b^2+60 a b c+24 a^2 c^2\right ) \left (c+d x^2\right ) \left (b+a \left (c+d x^2\right )\right )}{24 a^3 b d^3 \sqrt{a+\frac{b}{c+d x^2}}}+\frac{\left (c+d x^2\right )^2 \left (b+a \left (c+d x^2\right )\right )}{6 a^2 d^3 \sqrt{a+\frac{b}{c+d x^2}}}-\frac{b \left (35 b^2+60 a b c+24 a^2 c^2\right ) \sqrt{b+a \left (c+d x^2\right )} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{c+d x^2}}{\sqrt{b+a \left (c+d x^2\right )}}\right )}{16 a^{9/2} d^3 \sqrt{c+d x^2} \sqrt{a+\frac{b}{c+d x^2}}}\\ \end{align*}

Mathematica [C]  time = 11.731, size = 1215, normalized size = 3.92 \[ \frac{b \left (-344 c^2 \, _4F_3\left (\frac{1}{2},2,2,2;1,1,\frac{7}{2};\frac{b}{a d x^2+a c}+1\right ) \left (a+\frac{b}{d x^2+c}\right )^5-192 c^2 \, _5F_4\left (\frac{1}{2},2,2,2,2;1,1,1,\frac{7}{2};\frac{b}{a d x^2+a c}+1\right ) \left (a+\frac{b}{d x^2+c}\right )^5-32 c^2 \, _6F_5\left (\frac{1}{2},2,2,2,2,2;1,1,1,1,\frac{7}{2};\frac{b}{a d x^2+a c}+1\right ) \left (a+\frac{b}{d x^2+c}\right )^5-105 a c^2 \left (a+\frac{b}{d x^2+c}\right )^4+105 a c^2 \sqrt{\frac{b}{a d x^2+a c}+1} \tanh ^{-1}\left (\sqrt{\frac{b}{a d x^2+a c}+1}\right ) \left (a+\frac{b}{d x^2+c}\right )^4+\frac{120 a c^2 \tanh ^{-1}\left (\sqrt{\frac{b}{a d x^2+a c}+1}\right ) \left (a+\frac{b}{d x^2+c}\right )^4}{\sqrt{\frac{b}{a d x^2+a c}+1}}+\frac{60 c (b+a c) \tanh ^{-1}\left (\sqrt{\frac{b}{a d x^2+a c}+1}\right ) \left (a+\frac{b}{d x^2+c}\right )^4}{\left (\frac{b}{a d x^2+a c}+1\right )^{3/2}}+1040 c (b+a c) \, _4F_3\left (\frac{1}{2},2,2,2;1,1,\frac{7}{2};\frac{b}{a d x^2+a c}+1\right ) \left (a+\frac{b}{d x^2+c}\right )^4+448 c (b+a c) \, _5F_4\left (\frac{1}{2},2,2,2,2;1,1,1,\frac{7}{2};\frac{b}{a d x^2+a c}+1\right ) \left (a+\frac{b}{d x^2+c}\right )^4+64 c (b+a c) \, _6F_5\left (\frac{1}{2},2,2,2,2,2;1,1,1,1,\frac{7}{2};\frac{b}{a d x^2+a c}+1\right ) \left (a+\frac{b}{d x^2+c}\right )^4+765 a^2 c^2 \left (a+\frac{b}{d x^2+c}\right )^3+300 a c (b+a c) \left (a+\frac{b}{d x^2+c}\right )^3-300 a c (b+a c) \sqrt{\frac{b}{a d x^2+a c}+1} \tanh ^{-1}\left (\sqrt{\frac{b}{a d x^2+a c}+1}\right ) \left (a+\frac{b}{d x^2+c}\right )^3+\frac{300 (b+a c)^2 \tanh ^{-1}\left (\sqrt{\frac{b}{a d x^2+a c}+1}\right ) \left (a+\frac{b}{d x^2+c}\right )^3}{\left (\frac{b}{a d x^2+a c}+1\right )^{3/2}}-760 (b+a c)^2 \, _4F_3\left (\frac{1}{2},2,2,2;1,1,\frac{7}{2};\frac{b}{a d x^2+a c}+1\right ) \left (a+\frac{b}{d x^2+c}\right )^3-256 (b+a c)^2 \, _5F_4\left (\frac{1}{2},2,2,2,2;1,1,1,\frac{7}{2};\frac{b}{a d x^2+a c}+1\right ) \left (a+\frac{b}{d x^2+c}\right )^3-32 (b+a c)^2 \, _6F_5\left (\frac{1}{2},2,2,2,2,2;1,1,1,1,\frac{7}{2};\frac{b}{a d x^2+a c}+1\right ) \left (a+\frac{b}{d x^2+c}\right )^3+1365 a (b+a c)^2 \left (a+\frac{b}{d x^2+c}\right )^2-3240 a^2 c (b+a c) \left (a+\frac{b}{d x^2+c}\right )^2-765 a^3 c^2 \sqrt{\frac{b}{a d x^2+a c}+1} \tanh ^{-1}\left (\sqrt{\frac{b}{a d x^2+a c}+1}\right ) \left (a+\frac{b}{d x^2+c}\right )^2-1365 a (b+a c)^2 \sqrt{\frac{b}{a d x^2+a c}+1} \tanh ^{-1}\left (\sqrt{\frac{b}{a d x^2+a c}+1}\right ) \left (a+\frac{b}{d x^2+c}\right )^2+2835 a^2 (b+a c)^2 \left (a+\frac{b}{d x^2+c}\right )+3240 a^4 c (b+a c) \left (\frac{b}{a d x^2+a c}+1\right )^{3/2} \tanh ^{-1}\left (\sqrt{\frac{b}{a d x^2+a c}+1}\right )-2835 a^3 (b+a c)^2 \sqrt{\frac{b}{a d x^2+a c}+1} \tanh ^{-1}\left (\sqrt{\frac{b}{a d x^2+a c}+1}\right )\right )}{720 a^5 d^3 \left (a+\frac{b}{d x^2+c}\right )^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(b*(2835*a^2*(b + a*c)^2*(a + b/(c + d*x^2)) - 3240*a^2*c*(b + a*c)*(a + b/(c + d*x^2))^2 + 1365*a*(b + a*c)^2
*(a + b/(c + d*x^2))^2 + 765*a^2*c^2*(a + b/(c + d*x^2))^3 + 300*a*c*(b + a*c)*(a + b/(c + d*x^2))^3 - 105*a*c
^2*(a + b/(c + d*x^2))^4 + (300*(b + a*c)^2*(a + b/(c + d*x^2))^3*ArcTanh[Sqrt[1 + b/(a*c + a*d*x^2)]])/(1 + b
/(a*c + a*d*x^2))^(3/2) + (60*c*(b + a*c)*(a + b/(c + d*x^2))^4*ArcTanh[Sqrt[1 + b/(a*c + a*d*x^2)]])/(1 + b/(
a*c + a*d*x^2))^(3/2) + (120*a*c^2*(a + b/(c + d*x^2))^4*ArcTanh[Sqrt[1 + b/(a*c + a*d*x^2)]])/Sqrt[1 + b/(a*c
+ a*d*x^2)] - 2835*a^3*(b + a*c)^2*Sqrt[1 + b/(a*c + a*d*x^2)]*ArcTanh[Sqrt[1 + b/(a*c + a*d*x^2)]] - 765*a^3
*c^2*(a + b/(c + d*x^2))^2*Sqrt[1 + b/(a*c + a*d*x^2)]*ArcTanh[Sqrt[1 + b/(a*c + a*d*x^2)]] - 1365*a*(b + a*c)
^2*(a + b/(c + d*x^2))^2*Sqrt[1 + b/(a*c + a*d*x^2)]*ArcTanh[Sqrt[1 + b/(a*c + a*d*x^2)]] - 300*a*c*(b + a*c)*
(a + b/(c + d*x^2))^3*Sqrt[1 + b/(a*c + a*d*x^2)]*ArcTanh[Sqrt[1 + b/(a*c + a*d*x^2)]] + 105*a*c^2*(a + b/(c +
d*x^2))^4*Sqrt[1 + b/(a*c + a*d*x^2)]*ArcTanh[Sqrt[1 + b/(a*c + a*d*x^2)]] + 3240*a^4*c*(b + a*c)*(1 + b/(a*c
+ a*d*x^2))^(3/2)*ArcTanh[Sqrt[1 + b/(a*c + a*d*x^2)]] - 760*(b + a*c)^2*(a + b/(c + d*x^2))^3*Hypergeometric
PFQ[{1/2, 2, 2, 2}, {1, 1, 7/2}, 1 + b/(a*c + a*d*x^2)] + 1040*c*(b + a*c)*(a + b/(c + d*x^2))^4*Hypergeometri
cPFQ[{1/2, 2, 2, 2}, {1, 1, 7/2}, 1 + b/(a*c + a*d*x^2)] - 344*c^2*(a + b/(c + d*x^2))^5*HypergeometricPFQ[{1/
2, 2, 2, 2}, {1, 1, 7/2}, 1 + b/(a*c + a*d*x^2)] - 256*(b + a*c)^2*(a + b/(c + d*x^2))^3*HypergeometricPFQ[{1/
2, 2, 2, 2, 2}, {1, 1, 1, 7/2}, 1 + b/(a*c + a*d*x^2)] + 448*c*(b + a*c)*(a + b/(c + d*x^2))^4*HypergeometricP
FQ[{1/2, 2, 2, 2, 2}, {1, 1, 1, 7/2}, 1 + b/(a*c + a*d*x^2)] - 192*c^2*(a + b/(c + d*x^2))^5*HypergeometricPFQ
[{1/2, 2, 2, 2, 2}, {1, 1, 1, 7/2}, 1 + b/(a*c + a*d*x^2)] - 32*(b + a*c)^2*(a + b/(c + d*x^2))^3*Hypergeometr
icPFQ[{1/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 7/2}, 1 + b/(a*c + a*d*x^2)] + 64*c*(b + a*c)*(a + b/(c + d*x^2))^4*H
ypergeometricPFQ[{1/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 7/2}, 1 + b/(a*c + a*d*x^2)] - 32*c^2*(a + b/(c + d*x^2))^
5*HypergeometricPFQ[{1/2, 2, 2, 2, 2, 2}, {1, 1, 1, 1, 7/2}, 1 + b/(a*c + a*d*x^2)]))/(720*a^5*d^3*(a + b/(c +
d*x^2))^(5/2))

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Maple [B]  time = 0.036, size = 1240, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)*(d*x^2+c)/a^4/d^3*(-48*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*
(a*d^2)^(1/2)*x^4*a^3*c*d^2-60*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*d^2)^(1/2)*x^4*a^2*b*d^2-72*
ln(1/2*(2*a*d^2*x^2+2*a*c*d+2*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*d^2)^(1/2)+b*d)/(a*d^2)^(1/2)
)*x^2*a^3*b*c^2*d^2-48*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*d^2)^(1/2)*x^2*a^3*c^2*d-180*ln(1/2*
(2*a*d^2*x^2+2*a*c*d+2*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*d^2)^(1/2)+b*d)/(a*d^2)^(1/2))*x^2*a
^2*b^2*c*d^2+16*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(3/2)*(a*d^2)^(1/2)*x^2*a^2*d-105*ln(1/2*(2*a*d^2*x^
2+2*a*c*d+2*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*d^2)^(1/2)+b*d)/(a*d^2)^(1/2))*x^2*a*b^3*d^2-72
*ln(1/2*(2*a*d^2*x^2+2*a*c*d+2*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*d^2)^(1/2)+b*d)/(a*d^2)^(1/2
))*a^3*b*c^3*d+54*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*d^2)^(1/2)*x^2*a*b^2*d-252*ln(1/2*(2*a*d^
2*x^2+2*a*c*d+2*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*d^2)^(1/2)+b*d)/(a*d^2)^(1/2))*a^2*b^2*c^2*
d+96*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*(a*d^2)^(1/2)*a^2*b*c^2+16*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(3
/2)*(a*d^2)^(1/2)*a^2*c+108*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*d^2)^(1/2)*a^2*b*c^2-285*ln(1/2
*(2*a*d^2*x^2+2*a*c*d+2*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*d^2)^(1/2)+b*d)/(a*d^2)^(1/2))*a*b^
3*c*d+192*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*(a*d^2)^(1/2)*a*b^2*c+16*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)
^(3/2)*(a*d^2)^(1/2)*a*b+222*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*d^2)^(1/2)*a*b^2*c-105*ln(1/2*
(2*a*d^2*x^2+2*a*c*d+2*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a*d^2)^(1/2)+b*d)/(a*d^2)^(1/2))*b^4*d
+96*((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)*(a*d^2)^(1/2)*b^3+114*(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*(a
*d^2)^(1/2)*b^3)/((d*x^2+c)*(a*d*x^2+a*c+b))^(1/2)/(a*d^2)^(1/2)/(a*d*x^2+a*c+b)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.85877, size = 1488, normalized size = 4.8 \begin{align*} \left [\frac{3 \,{\left (24 \, a^{3} b c^{3} + 84 \, a^{2} b^{2} c^{2} + 95 \, a b^{3} c + 35 \, b^{4} +{\left (24 \, a^{3} b c^{2} + 60 \, a^{2} b^{2} c + 35 \, a b^{3}\right )} d x^{2}\right )} \sqrt{a} \log \left (8 \, a^{2} d^{2} x^{4} + 8 \, a^{2} c^{2} + 8 \,{\left (2 \, a^{2} c + a b\right )} d x^{2} + 8 \, a b c + b^{2} - 4 \,{\left (2 \, a d^{2} x^{4} +{\left (4 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + b c\right )} \sqrt{a} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}\right ) + 4 \,{\left (8 \, a^{4} d^{4} x^{8} + 2 \,{\left (4 \, a^{4} c - 7 \, a^{3} b\right )} d^{3} x^{6} + 8 \, a^{4} c^{4} + 118 \, a^{3} b c^{3} +{\left (18 \, a^{3} b c + 35 \, a^{2} b^{2}\right )} d^{2} x^{4} + 215 \, a^{2} b^{2} c^{2} + 105 \, a b^{3} c +{\left (8 \, a^{4} c^{3} + 150 \, a^{3} b c^{2} + 250 \, a^{2} b^{2} c + 105 \, a b^{3}\right )} d x^{2}\right )} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{192 \,{\left (a^{6} d^{4} x^{2} +{\left (a^{6} c + a^{5} b\right )} d^{3}\right )}}, \frac{3 \,{\left (24 \, a^{3} b c^{3} + 84 \, a^{2} b^{2} c^{2} + 95 \, a b^{3} c + 35 \, b^{4} +{\left (24 \, a^{3} b c^{2} + 60 \, a^{2} b^{2} c + 35 \, a b^{3}\right )} d x^{2}\right )} \sqrt{-a} \arctan \left (\frac{{\left (2 \, a d x^{2} + 2 \, a c + b\right )} \sqrt{-a} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{2 \,{\left (a^{2} d x^{2} + a^{2} c + a b\right )}}\right ) + 2 \,{\left (8 \, a^{4} d^{4} x^{8} + 2 \,{\left (4 \, a^{4} c - 7 \, a^{3} b\right )} d^{3} x^{6} + 8 \, a^{4} c^{4} + 118 \, a^{3} b c^{3} +{\left (18 \, a^{3} b c + 35 \, a^{2} b^{2}\right )} d^{2} x^{4} + 215 \, a^{2} b^{2} c^{2} + 105 \, a b^{3} c +{\left (8 \, a^{4} c^{3} + 150 \, a^{3} b c^{2} + 250 \, a^{2} b^{2} c + 105 \, a b^{3}\right )} d x^{2}\right )} \sqrt{\frac{a d x^{2} + a c + b}{d x^{2} + c}}}{96 \,{\left (a^{6} d^{4} x^{2} +{\left (a^{6} c + a^{5} b\right )} d^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/192*(3*(24*a^3*b*c^3 + 84*a^2*b^2*c^2 + 95*a*b^3*c + 35*b^4 + (24*a^3*b*c^2 + 60*a^2*b^2*c + 35*a*b^3)*d*x^
2)*sqrt(a)*log(8*a^2*d^2*x^4 + 8*a^2*c^2 + 8*(2*a^2*c + a*b)*d*x^2 + 8*a*b*c + b^2 - 4*(2*a*d^2*x^4 + (4*a*c +
b)*d*x^2 + 2*a*c^2 + b*c)*sqrt(a)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))) + 4*(8*a^4*d^4*x^8 + 2*(4*a^4*c - 7*
a^3*b)*d^3*x^6 + 8*a^4*c^4 + 118*a^3*b*c^3 + (18*a^3*b*c + 35*a^2*b^2)*d^2*x^4 + 215*a^2*b^2*c^2 + 105*a*b^3*c
+ (8*a^4*c^3 + 150*a^3*b*c^2 + 250*a^2*b^2*c + 105*a*b^3)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/(a^6*
d^4*x^2 + (a^6*c + a^5*b)*d^3), 1/96*(3*(24*a^3*b*c^3 + 84*a^2*b^2*c^2 + 95*a*b^3*c + 35*b^4 + (24*a^3*b*c^2 +
60*a^2*b^2*c + 35*a*b^3)*d*x^2)*sqrt(-a)*arctan(1/2*(2*a*d*x^2 + 2*a*c + b)*sqrt(-a)*sqrt((a*d*x^2 + a*c + b)
/(d*x^2 + c))/(a^2*d*x^2 + a^2*c + a*b)) + 2*(8*a^4*d^4*x^8 + 2*(4*a^4*c - 7*a^3*b)*d^3*x^6 + 8*a^4*c^4 + 118*
a^3*b*c^3 + (18*a^3*b*c + 35*a^2*b^2)*d^2*x^4 + 215*a^2*b^2*c^2 + 105*a*b^3*c + (8*a^4*c^3 + 150*a^3*b*c^2 + 2
50*a^2*b^2*c + 105*a*b^3)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/(a^6*d^4*x^2 + (a^6*c + a^5*b)*d^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\left (\frac{a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**5/((a*c + a*d*x**2 + b)/(c + d*x**2))**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{{\left (a + \frac{b}{d x^{2} + c}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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