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

Optimal. Leaf size=401 \[ -\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} d^{15/4}}+\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} d^{15/4}}+\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{5/4} d^{15/4}}-\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{5/4} d^{15/4}}-\frac{x^{3/2} \left (\frac{3 a^2 d}{c}+42 a b-\frac{77 b^2 c}{d}\right )}{48 c d^2}-\frac{x^{7/2} (b c-a d) (a d+15 b c)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{x^{7/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]

[Out]

-((42*a*b - (77*b^2*c)/d + (3*a^2*d)/c)*x^(3/2))/(48*c*d^2) + ((b*c - a*d)^2*x^(7/2))/(4*c*d^2*(c + d*x^2)^2)
- ((b*c - a*d)*(15*b*c + a*d)*x^(7/2))/(16*c^2*d^2*(c + d*x^2)) + ((77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*ArcTa
n[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*d^(15/4)) - ((77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^
2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*d^(15/4)) - ((77*b^2*c^2 - 42*a*b*c*d -
3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(5/4)*d^(15/4)) + ((77*b^
2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(5/4
)*d^(15/4))

________________________________________________________________________________________

Rubi [A]  time = 0.343386, antiderivative size = 401, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {463, 457, 321, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} d^{15/4}}+\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} d^{15/4}}+\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{5/4} d^{15/4}}-\frac{\left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{32 \sqrt{2} c^{5/4} d^{15/4}}-\frac{x^{3/2} \left (\frac{3 a^2 d}{c}+42 a b-\frac{77 b^2 c}{d}\right )}{48 c d^2}-\frac{x^{7/2} (b c-a d) (a d+15 b c)}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{x^{7/2} (b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((42*a*b - (77*b^2*c)/d + (3*a^2*d)/c)*x^(3/2))/(48*c*d^2) + ((b*c - a*d)^2*x^(7/2))/(4*c*d^2*(c + d*x^2)^2)
- ((b*c - a*d)*(15*b*c + a*d)*x^(7/2))/(16*c^2*d^2*(c + d*x^2)) + ((77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*ArcTa
n[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*d^(15/4)) - ((77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^
2)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(32*Sqrt[2]*c^(5/4)*d^(15/4)) - ((77*b^2*c^2 - 42*a*b*c*d -
3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(5/4)*d^(15/4)) + ((77*b^
2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(64*Sqrt[2]*c^(5/4
)*d^(15/4))

Rule 463

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

Rule 457

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> -Simp[((b*c - a*d
)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*b*e*n*(p + 1)), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b
*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) ||  !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] &&
LeQ[-1, m, -(n*(p + 1))]))

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 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 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 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 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 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]

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]

Rubi steps

\begin{align*} \int \frac{x^{5/2} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx &=\frac{(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{\int \frac{x^{5/2} \left (\frac{1}{2} \left (-8 a^2 d^2+7 (b c-a d)^2\right )-4 b^2 c d x^2\right )}{\left (c+d x^2\right )^2} \, dx}{4 c d^2}\\ &=\frac{(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (15 b c+a d) x^{7/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \int \frac{x^{5/2}}{c+d x^2} \, dx}{32 c^2 d^2}\\ &=\frac{\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) x^{3/2}}{48 c^2 d^3}+\frac{(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (15 b c+a d) x^{7/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac{\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \int \frac{\sqrt{x}}{c+d x^2} \, dx}{32 c d^3}\\ &=\frac{\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) x^{3/2}}{48 c^2 d^3}+\frac{(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (15 b c+a d) x^{7/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac{\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{16 c d^3}\\ &=\frac{\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) x^{3/2}}{48 c^2 d^3}+\frac{(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (15 b c+a d) x^{7/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c}-\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{32 c d^{7/2}}-\frac{\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c}+\sqrt{d} x^2}{c+d x^4} \, dx,x,\sqrt{x}\right )}{32 c d^{7/2}}\\ &=\frac{\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) x^{3/2}}{48 c^2 d^3}+\frac{(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (15 b c+a d) x^{7/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac{\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{c}}{\sqrt{d}}-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt{x}\right )}{64 c d^4}-\frac{\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{c}}{\sqrt{d}}+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}+x^2} \, dx,x,\sqrt{x}\right )}{64 c d^4}-\frac{\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{c}}{\sqrt [4]{d}}+2 x}{-\frac{\sqrt{c}}{\sqrt{d}}-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2} c^{5/4} d^{15/4}}-\frac{\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{c}}{\sqrt [4]{d}}-2 x}{-\frac{\sqrt{c}}{\sqrt{d}}+\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{d}}-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2} c^{5/4} d^{15/4}}\\ &=\frac{\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) x^{3/2}}{48 c^2 d^3}+\frac{(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (15 b c+a d) x^{7/2}}{16 c^2 d^2 \left (c+d x^2\right )}-\frac{\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} d^{15/4}}+\frac{\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} d^{15/4}}-\frac{\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{5/4} d^{15/4}}+\frac{\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{5/4} d^{15/4}}\\ &=\frac{\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) x^{3/2}}{48 c^2 d^3}+\frac{(b c-a d)^2 x^{7/2}}{4 c d^2 \left (c+d x^2\right )^2}-\frac{(b c-a d) (15 b c+a d) x^{7/2}}{16 c^2 d^2 \left (c+d x^2\right )}+\frac{\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{5/4} d^{15/4}}-\frac{\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{32 \sqrt{2} c^{5/4} d^{15/4}}-\frac{\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \log \left (\sqrt{c}-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} d^{15/4}}+\frac{\left (77 b^2 c^2-42 a b c d-3 a^2 d^2\right ) \log \left (\sqrt{c}+\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{d} x\right )}{64 \sqrt{2} c^{5/4} d^{15/4}}\\ \end{align*}

Mathematica [A]  time = 0.235513, size = 363, normalized size = 0.91 \[ \frac{\frac{24 d^{3/4} x^{3/2} \left (3 a^2 d^2-22 a b c d+19 b^2 c^2\right )}{c \left (c+d x^2\right )}-\frac{3 \sqrt{2} \left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{5/4}}+\frac{3 \sqrt{2} \left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt [4]{d} \sqrt{x}+\sqrt{c}+\sqrt{d} x\right )}{c^{5/4}}+\frac{6 \sqrt{2} \left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )}{c^{5/4}}-\frac{6 \sqrt{2} \left (-3 a^2 d^2-42 a b c d+77 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}+1\right )}{c^{5/4}}-\frac{96 d^{3/4} x^{3/2} (b c-a d)^2}{\left (c+d x^2\right )^2}+256 b^2 d^{3/4} x^{3/2}}{384 d^{15/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(256*b^2*d^(3/4)*x^(3/2) - (96*d^(3/4)*(b*c - a*d)^2*x^(3/2))/(c + d*x^2)^2 + (24*d^(3/4)*(19*b^2*c^2 - 22*a*b
*c*d + 3*a^2*d^2)*x^(3/2))/(c*(c + d*x^2)) + (6*Sqrt[2]*(77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*ArcTan[1 - (Sqrt
[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(5/4) - (6*Sqrt[2]*(77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*ArcTan[1 + (Sqrt[2]*
d^(1/4)*Sqrt[x])/c^(1/4)])/c^(5/4) - (3*Sqrt[2]*(77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] - Sqrt[2]*c^
(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(5/4) + (3*Sqrt[2]*(77*b^2*c^2 - 42*a*b*c*d - 3*a^2*d^2)*Log[Sqrt[c] + S
qrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(5/4))/(384*d^(15/4))

________________________________________________________________________________________

Maple [A]  time = 0.018, size = 562, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*b^2/d^3*x^(3/2)+3/16/(d*x^2+c)^2/c*x^(7/2)*a^2-11/8/d/(d*x^2+c)^2*x^(7/2)*a*b+19/16/d^2/(d*x^2+c)^2*c*x^(7
/2)*b^2-1/16/d/(d*x^2+c)^2*x^(3/2)*a^2-7/8/d^2/(d*x^2+c)^2*x^(3/2)*c*a*b+15/16/d^3/(d*x^2+c)^2*x^(3/2)*b^2*c^2
+3/64/d^2/c/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a^2+21/32/d^3/(c/d)^(1/4)*2^(1/2)*arctan
(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*a*b-77/64/d^4*c/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)*b^2+
3/64/d^2/c/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a^2+21/32/d^3/(c/d)^(1/4)*2^(1/2)*arctan(
2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*a*b-77/64/d^4*c/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)*b^2+3
/128/d^2/c/(c/d)^(1/4)*2^(1/2)*ln((x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(
c/d)^(1/2)))*a^2+21/64/d^3/(c/d)^(1/4)*2^(1/2)*ln((x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*x
^(1/2)*2^(1/2)+(c/d)^(1/2)))*a*b-77/128/d^4*c/(c/d)^(1/4)*2^(1/2)*ln((x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2
))/(x+(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))*b^2

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.18075, size = 4574, normalized size = 11.41 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*(12*(c*d^5*x^4 + 2*c^2*d^4*x^2 + c^3*d^3)*(-(35153041*b^8*c^8 - 76697544*a*b^7*c^7*d + 57274140*a^2*b^6
*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 + 539784*a^5*b^3*c^3*d^5 + 86940*a^6*b^2*c^2*d^6
+ 4536*a^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15))^(1/4)*arctan((sqrt((208422380089*b^12*c^12 - 682109607564*a*b^11
*c^11*d + 881427350034*a^2*b^10*c^10*d^2 - 543593843100*a^3*b^9*c^9*d^3 + 136525986135*a^4*b^8*c^8*d^4 + 83346
77736*a^5*b^7*c^7*d^5 - 7849956996*a^6*b^6*c^6*d^6 - 324727704*a^7*b^5*c^5*d^7 + 207241335*a^8*b^4*c^4*d^8 + 3
2148900*a^9*b^3*c^3*d^9 + 2030994*a^10*b^2*c^2*d^10 + 61236*a^11*b*c*d^11 + 729*a^12*d^12)*x - (35153041*b^8*c
^11*d^7 - 76697544*a*b^7*c^10*d^8 + 57274140*a^2*b^6*c^9*d^9 - 13854456*a^3*b^5*c^8*d^10 - 1457946*a^4*b^4*c^7
*d^11 + 539784*a^5*b^3*c^6*d^12 + 86940*a^6*b^2*c^5*d^13 + 4536*a^7*b*c^4*d^14 + 81*a^8*c^3*d^15)*sqrt(-(35153
041*b^8*c^8 - 76697544*a*b^7*c^7*d + 57274140*a^2*b^6*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4
*d^4 + 539784*a^5*b^3*c^3*d^5 + 86940*a^6*b^2*c^2*d^6 + 4536*a^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15)))*c*d^4*(-(3
5153041*b^8*c^8 - 76697544*a*b^7*c^7*d + 57274140*a^2*b^6*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4
*c^4*d^4 + 539784*a^5*b^3*c^3*d^5 + 86940*a^6*b^2*c^2*d^6 + 4536*a^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15))^(1/4) +
(456533*b^6*c^7*d^4 - 747054*a*b^5*c^6*d^5 + 354123*a^2*b^4*c^5*d^6 - 15876*a^3*b^3*c^4*d^7 - 13797*a^4*b^2*c
^3*d^8 - 1134*a^5*b*c^2*d^9 - 27*a^6*c*d^10)*sqrt(x)*(-(35153041*b^8*c^8 - 76697544*a*b^7*c^7*d + 57274140*a^2
*b^6*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 + 539784*a^5*b^3*c^3*d^5 + 86940*a^6*b^2*c^2
*d^6 + 4536*a^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15))^(1/4))/(35153041*b^8*c^8 - 76697544*a*b^7*c^7*d + 57274140*a
^2*b^6*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 + 539784*a^5*b^3*c^3*d^5 + 86940*a^6*b^2*c
^2*d^6 + 4536*a^7*b*c*d^7 + 81*a^8*d^8)) - 3*(c*d^5*x^4 + 2*c^2*d^4*x^2 + c^3*d^3)*(-(35153041*b^8*c^8 - 76697
544*a*b^7*c^7*d + 57274140*a^2*b^6*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 + 539784*a^5*b
^3*c^3*d^5 + 86940*a^6*b^2*c^2*d^6 + 4536*a^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15))^(1/4)*log(c^4*d^11*(-(35153041
*b^8*c^8 - 76697544*a*b^7*c^7*d + 57274140*a^2*b^6*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^
4 + 539784*a^5*b^3*c^3*d^5 + 86940*a^6*b^2*c^2*d^6 + 4536*a^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15))^(3/4) - (45653
3*b^6*c^6 - 747054*a*b^5*c^5*d + 354123*a^2*b^4*c^4*d^2 - 15876*a^3*b^3*c^3*d^3 - 13797*a^4*b^2*c^2*d^4 - 1134
*a^5*b*c*d^5 - 27*a^6*d^6)*sqrt(x)) + 3*(c*d^5*x^4 + 2*c^2*d^4*x^2 + c^3*d^3)*(-(35153041*b^8*c^8 - 76697544*a
*b^7*c^7*d + 57274140*a^2*b^6*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 + 539784*a^5*b^3*c^
3*d^5 + 86940*a^6*b^2*c^2*d^6 + 4536*a^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15))^(1/4)*log(-c^4*d^11*(-(35153041*b^8
*c^8 - 76697544*a*b^7*c^7*d + 57274140*a^2*b^6*c^6*d^2 - 13854456*a^3*b^5*c^5*d^3 - 1457946*a^4*b^4*c^4*d^4 +
539784*a^5*b^3*c^3*d^5 + 86940*a^6*b^2*c^2*d^6 + 4536*a^7*b*c*d^7 + 81*a^8*d^8)/(c^5*d^15))^(3/4) - (456533*b^
6*c^6 - 747054*a*b^5*c^5*d + 354123*a^2*b^4*c^4*d^2 - 15876*a^3*b^3*c^3*d^3 - 13797*a^4*b^2*c^2*d^4 - 1134*a^5
*b*c*d^5 - 27*a^6*d^6)*sqrt(x)) - 4*(32*b^2*c*d^2*x^5 + (121*b^2*c^2*d - 66*a*b*c*d^2 + 9*a^2*d^3)*x^3 + (77*b
^2*c^3 - 42*a*b*c^2*d - 3*a^2*c*d^2)*x)*sqrt(x))/(c*d^5*x^4 + 2*c^2*d^4*x^2 + c^3*d^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.22289, size = 576, normalized size = 1.44 \begin{align*} \frac{2 \, b^{2} x^{\frac{3}{2}}}{3 \, d^{3}} + \frac{19 \, b^{2} c^{2} d x^{\frac{7}{2}} - 22 \, a b c d^{2} x^{\frac{7}{2}} + 3 \, a^{2} d^{3} x^{\frac{7}{2}} + 15 \, b^{2} c^{3} x^{\frac{3}{2}} - 14 \, a b c^{2} d x^{\frac{3}{2}} - a^{2} c d^{2} x^{\frac{3}{2}}}{16 \,{\left (d x^{2} + c\right )}^{2} c d^{3}} - \frac{\sqrt{2}{\left (77 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 42 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{64 \, c^{2} d^{6}} - \frac{\sqrt{2}{\left (77 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 42 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{c}{d}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{c}{d}\right )^{\frac{1}{4}}}\right )}{64 \, c^{2} d^{6}} + \frac{\sqrt{2}{\left (77 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 42 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \log \left (\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{128 \, c^{2} d^{6}} - \frac{\sqrt{2}{\left (77 \, \left (c d^{3}\right )^{\frac{3}{4}} b^{2} c^{2} - 42 \, \left (c d^{3}\right )^{\frac{3}{4}} a b c d - 3 \, \left (c d^{3}\right )^{\frac{3}{4}} a^{2} d^{2}\right )} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{c}{d}\right )^{\frac{1}{4}} + x + \sqrt{\frac{c}{d}}\right )}{128 \, c^{2} d^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*b^2*x^(3/2)/d^3 + 1/16*(19*b^2*c^2*d*x^(7/2) - 22*a*b*c*d^2*x^(7/2) + 3*a^2*d^3*x^(7/2) + 15*b^2*c^3*x^(3/
2) - 14*a*b*c^2*d*x^(3/2) - a^2*c*d^2*x^(3/2))/((d*x^2 + c)^2*c*d^3) - 1/64*sqrt(2)*(77*(c*d^3)^(3/4)*b^2*c^2
- 42*(c*d^3)^(3/4)*a*b*c*d - 3*(c*d^3)^(3/4)*a^2*d^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/
d)^(1/4))/(c^2*d^6) - 1/64*sqrt(2)*(77*(c*d^3)^(3/4)*b^2*c^2 - 42*(c*d^3)^(3/4)*a*b*c*d - 3*(c*d^3)^(3/4)*a^2*
d^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(c^2*d^6) + 1/128*sqrt(2)*(77*(c*d^3)^
(3/4)*b^2*c^2 - 42*(c*d^3)^(3/4)*a*b*c*d - 3*(c*d^3)^(3/4)*a^2*d^2)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt
(c/d))/(c^2*d^6) - 1/128*sqrt(2)*(77*(c*d^3)^(3/4)*b^2*c^2 - 42*(c*d^3)^(3/4)*a*b*c*d - 3*(c*d^3)^(3/4)*a^2*d^
2)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(c^2*d^6)