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

Optimal. Leaf size=374 \[ \frac{d x^{3/2} \left (5 a^2 d^2-11 a b c d+7 b^2 c^2\right )}{2 b^4}+\frac{3 d^2 x^{7/2} (11 b c-5 a d)}{14 b^3}+\frac{3 (b c-5 a d) (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{19/4}}-\frac{3 (b c-5 a d) (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{19/4}}-\frac{3 (b c-5 a d) (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} \sqrt [4]{a} b^{19/4}}+\frac{3 (b c-5 a d) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} \sqrt [4]{a} b^{19/4}}-\frac{x^{3/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{15 d^3 x^{11/2}}{22 b^2} \]

[Out]

(d*(7*b^2*c^2 - 11*a*b*c*d + 5*a^2*d^2)*x^(3/2))/(2*b^4) + (3*d^2*(11*b*c - 5*a*d)*x^(7/2))/(14*b^3) + (15*d^3
*x^(11/2))/(22*b^2) - (x^(3/2)*(c + d*x^2)^3)/(2*b*(a + b*x^2)) - (3*(b*c - 5*a*d)*(b*c - a*d)^2*ArcTan[1 - (S
qrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(1/4)*b^(19/4)) + (3*(b*c - 5*a*d)*(b*c - a*d)^2*ArcTan[1 + (Sq
rt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(1/4)*b^(19/4)) + (3*(b*c - 5*a*d)*(b*c - a*d)^2*Log[Sqrt[a] - S
qrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(1/4)*b^(19/4)) - (3*(b*c - 5*a*d)*(b*c - a*d)^2*Log
[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(1/4)*b^(19/4))

________________________________________________________________________________________

Rubi [A]  time = 0.414985, antiderivative size = 374, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {466, 467, 570, 297, 1162, 617, 204, 1165, 628} \[ \frac{d x^{3/2} \left (5 a^2 d^2-11 a b c d+7 b^2 c^2\right )}{2 b^4}+\frac{3 d^2 x^{7/2} (11 b c-5 a d)}{14 b^3}+\frac{3 (b c-5 a d) (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{19/4}}-\frac{3 (b c-5 a d) (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{19/4}}-\frac{3 (b c-5 a d) (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} \sqrt [4]{a} b^{19/4}}+\frac{3 (b c-5 a d) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} \sqrt [4]{a} b^{19/4}}-\frac{x^{3/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{15 d^3 x^{11/2}}{22 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(7*b^2*c^2 - 11*a*b*c*d + 5*a^2*d^2)*x^(3/2))/(2*b^4) + (3*d^2*(11*b*c - 5*a*d)*x^(7/2))/(14*b^3) + (15*d^3
*x^(11/2))/(22*b^2) - (x^(3/2)*(c + d*x^2)^3)/(2*b*(a + b*x^2)) - (3*(b*c - 5*a*d)*(b*c - a*d)^2*ArcTan[1 - (S
qrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(1/4)*b^(19/4)) + (3*(b*c - 5*a*d)*(b*c - a*d)^2*ArcTan[1 + (Sq
rt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(1/4)*b^(19/4)) + (3*(b*c - 5*a*d)*(b*c - a*d)^2*Log[Sqrt[a] - S
qrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(1/4)*b^(19/4)) - (3*(b*c - 5*a*d)*(b*c - a*d)^2*Log
[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(1/4)*b^(19/4))

Rule 466

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 467

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

Rule 570

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]

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 (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^6 \left (c+d x^4\right )^3}{\left (a+b x^4\right )^2} \, dx,x,\sqrt{x}\right )\\ &=-\frac{x^{3/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (c+d x^4\right )^2 \left (3 c+15 d x^4\right )}{a+b x^4} \, dx,x,\sqrt{x}\right )}{2 b}\\ &=-\frac{x^{3/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{\operatorname{Subst}\left (\int \left (\frac{3 d \left (7 b^2 c^2-11 a b c d+5 a^2 d^2\right ) x^2}{b^3}+\frac{3 d^2 (11 b c-5 a d) x^6}{b^2}+\frac{15 d^3 x^{10}}{b}+\frac{3 \left (b^3 c^3-7 a b^2 c^2 d+11 a^2 b c d^2-5 a^3 d^3\right ) x^2}{b^3 \left (a+b x^4\right )}\right ) \, dx,x,\sqrt{x}\right )}{2 b}\\ &=\frac{d \left (7 b^2 c^2-11 a b c d+5 a^2 d^2\right ) x^{3/2}}{2 b^4}+\frac{3 d^2 (11 b c-5 a d) x^{7/2}}{14 b^3}+\frac{15 d^3 x^{11/2}}{22 b^2}-\frac{x^{3/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{\left (3 (b c-5 a d) (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{2 b^4}\\ &=\frac{d \left (7 b^2 c^2-11 a b c d+5 a^2 d^2\right ) x^{3/2}}{2 b^4}+\frac{3 d^2 (11 b c-5 a d) x^{7/2}}{14 b^3}+\frac{15 d^3 x^{11/2}}{22 b^2}-\frac{x^{3/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac{\left (3 (b c-5 a d) (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 b^{9/2}}+\frac{\left (3 (b c-5 a d) (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 b^{9/2}}\\ &=\frac{d \left (7 b^2 c^2-11 a b c d+5 a^2 d^2\right ) x^{3/2}}{2 b^4}+\frac{3 d^2 (11 b c-5 a d) x^{7/2}}{14 b^3}+\frac{15 d^3 x^{11/2}}{22 b^2}-\frac{x^{3/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{\left (3 (b c-5 a d) (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{x}\right )}{8 b^5}+\frac{\left (3 (b c-5 a d) (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{x}\right )}{8 b^5}+\frac{\left (3 (b c-5 a d) (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{x}\right )}{8 \sqrt{2} \sqrt [4]{a} b^{19/4}}+\frac{\left (3 (b c-5 a d) (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{x}\right )}{8 \sqrt{2} \sqrt [4]{a} b^{19/4}}\\ &=\frac{d \left (7 b^2 c^2-11 a b c d+5 a^2 d^2\right ) x^{3/2}}{2 b^4}+\frac{3 d^2 (11 b c-5 a d) x^{7/2}}{14 b^3}+\frac{15 d^3 x^{11/2}}{22 b^2}-\frac{x^{3/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{3 (b c-5 a d) (b c-a d)^2 \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{19/4}}-\frac{3 (b c-5 a d) (b c-a d)^2 \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{19/4}}+\frac{\left (3 (b c-5 a d) (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} \sqrt [4]{a} b^{19/4}}-\frac{\left (3 (b c-5 a d) (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} \sqrt [4]{a} b^{19/4}}\\ &=\frac{d \left (7 b^2 c^2-11 a b c d+5 a^2 d^2\right ) x^{3/2}}{2 b^4}+\frac{3 d^2 (11 b c-5 a d) x^{7/2}}{14 b^3}+\frac{15 d^3 x^{11/2}}{22 b^2}-\frac{x^{3/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac{3 (b c-5 a d) (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} \sqrt [4]{a} b^{19/4}}+\frac{3 (b c-5 a d) (b c-a d)^2 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} \sqrt [4]{a} b^{19/4}}+\frac{3 (b c-5 a d) (b c-a d)^2 \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{19/4}}-\frac{3 (b c-5 a d) (b c-a d)^2 \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} \sqrt [4]{a} b^{19/4}}\\ \end{align*}

Mathematica [C]  time = 2.03165, size = 377, normalized size = 1.01 \[ \frac{385 \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-\frac{b x^2}{a}\right ) \left (9 a^2 b x^2 \left (50625 c^2 d x^2+16875 c^3+52033 c d^2 x^4+16875 d^3 x^6\right )+a^3 \left (390963 c^2 d x^2+130321 c^3+390963 c d^2 x^4+124561 d^3 x^6\right )+3 a b^2 x^4 \left (41235 c^2 d x^2+14641 c^3+43923 c d^2 x^4+14641 d^3 x^6\right )+b^3 x^6 \left (7203 c^2 d x^2+3553 c^3+7203 c d^2 x^4+2401 d^3 x^6\right )\right )-330 a^2 b x^2 \left (336081 c^2 d x^2+112027 c^3+350865 c d^2 x^4+114907 d^3 x^6\right )-385 a^3 \left (390963 c^2 d x^2+130321 c^3+390963 c d^2 x^4+124561 d^3 x^6\right )-45 a b^2 x^4 \left (299987 c^2 d x^2+122993 c^3+322515 c d^2 x^4+109553 d^3 x^6\right )-32768 b^3 x^6 \left (c+d x^2\right )^3}{887040 a b^4 x^{9/2}}-\frac{128 b x^{11/2} \left (c+d x^2\right )^3 \text{HypergeometricPFQ}\left (\left \{2,2,2,2,\frac{11}{4}\right \},\left \{1,1,1,\frac{27}{4}\right \},-\frac{b x^2}{a}\right )}{72105 a^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-32768*b^3*x^6*(c + d*x^2)^3 - 45*a*b^2*x^4*(122993*c^3 + 299987*c^2*d*x^2 + 322515*c*d^2*x^4 + 109553*d^3*x^
6) - 330*a^2*b*x^2*(112027*c^3 + 336081*c^2*d*x^2 + 350865*c*d^2*x^4 + 114907*d^3*x^6) - 385*a^3*(130321*c^3 +
390963*c^2*d*x^2 + 390963*c*d^2*x^4 + 124561*d^3*x^6) + 385*(b^3*x^6*(3553*c^3 + 7203*c^2*d*x^2 + 7203*c*d^2*
x^4 + 2401*d^3*x^6) + 3*a*b^2*x^4*(14641*c^3 + 41235*c^2*d*x^2 + 43923*c*d^2*x^4 + 14641*d^3*x^6) + 9*a^2*b*x^
2*(16875*c^3 + 50625*c^2*d*x^2 + 52033*c*d^2*x^4 + 16875*d^3*x^6) + a^3*(130321*c^3 + 390963*c^2*d*x^2 + 39096
3*c*d^2*x^4 + 124561*d^3*x^6))*Hypergeometric2F1[3/4, 1, 7/4, -((b*x^2)/a)])/(887040*a*b^4*x^(9/2)) - (128*b*x
^(11/2)*(c + d*x^2)^3*HypergeometricPFQ[{2, 2, 2, 2, 11/4}, {1, 1, 1, 27/4}, -((b*x^2)/a)])/(72105*a^3)

________________________________________________________________________________________

Maple [B]  time = 0.017, size = 748, normalized size = 2. \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)*(d*x^2+c)^3/(b*x^2+a)^2,x)

[Out]

2/11*d^3*x^(11/2)/b^2-4/7*d^3/b^3*x^(7/2)*a+6/7*d^2/b^2*x^(7/2)*c+2*d^3/b^4*x^(3/2)*a^2-4*d^2/b^3*x^(3/2)*a*c+
2*d/b^2*x^(3/2)*c^2+1/2/b^4*x^(3/2)/(b*x^2+a)*a^3*d^3-3/2/b^3*x^(3/2)/(b*x^2+a)*a^2*c*d^2+3/2/b^2*x^(3/2)/(b*x
^2+a)*a*c^2*d-1/2/b*x^(3/2)/(b*x^2+a)*c^3-15/16/b^5/(1/b*a)^(1/4)*2^(1/2)*a^3*d^3*ln((x-(1/b*a)^(1/4)*x^(1/2)*
2^(1/2)+(1/b*a)^(1/2))/(x+(1/b*a)^(1/4)*x^(1/2)*2^(1/2)+(1/b*a)^(1/2)))-15/8/b^5/(1/b*a)^(1/4)*2^(1/2)*a^3*d^3
*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)-15/8/b^5/(1/b*a)^(1/4)*2^(1/2)*a^3*d^3*arctan(2^(1/2)/(1/b*a)^(1/4)*x
^(1/2)-1)+33/16/b^4/(1/b*a)^(1/4)*2^(1/2)*a^2*c*d^2*ln((x-(1/b*a)^(1/4)*x^(1/2)*2^(1/2)+(1/b*a)^(1/2))/(x+(1/b
*a)^(1/4)*x^(1/2)*2^(1/2)+(1/b*a)^(1/2)))+33/8/b^4/(1/b*a)^(1/4)*2^(1/2)*a^2*c*d^2*arctan(2^(1/2)/(1/b*a)^(1/4
)*x^(1/2)+1)+33/8/b^4/(1/b*a)^(1/4)*2^(1/2)*a^2*c*d^2*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)-21/16/b^3/(1/b*a
)^(1/4)*2^(1/2)*a*c^2*d*ln((x-(1/b*a)^(1/4)*x^(1/2)*2^(1/2)+(1/b*a)^(1/2))/(x+(1/b*a)^(1/4)*x^(1/2)*2^(1/2)+(1
/b*a)^(1/2)))-21/8/b^3/(1/b*a)^(1/4)*2^(1/2)*a*c^2*d*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)-21/8/b^3/(1/b*a)^
(1/4)*2^(1/2)*a*c^2*d*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)+3/16/b^2/(1/b*a)^(1/4)*2^(1/2)*c^3*ln((x-(1/b*a)
^(1/4)*x^(1/2)*2^(1/2)+(1/b*a)^(1/2))/(x+(1/b*a)^(1/4)*x^(1/2)*2^(1/2)+(1/b*a)^(1/2)))+3/8/b^2/(1/b*a)^(1/4)*2
^(1/2)*c^3*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)+3/8/b^2/(1/b*a)^(1/4)*2^(1/2)*c^3*arctan(2^(1/2)/(1/b*a)^(1
/4)*x^(1/2)-1)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.65904, size = 6029, normalized size = 16.12 \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)*(d*x^2+c)^3/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

1/616*(924*(b^5*x^2 + a*b^4)*(-(b^12*c^12 - 28*a*b^11*c^11*d + 338*a^2*b^10*c^10*d^2 - 2316*a^3*b^9*c^9*d^3 +
10015*a^4*b^8*c^8*d^4 - 28856*a^5*b^7*c^7*d^5 + 57148*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5*d^7 + 76111*a^8*b^4*
c^4*d^8 - 50220*a^9*b^3*c^3*d^9 + 21650*a^10*b^2*c^2*d^10 - 5500*a^11*b*c*d^11 + 625*a^12*d^12)/(a*b^19))^(1/4
)*arctan((sqrt((b^18*c^18 - 42*a*b^17*c^17*d + 801*a^2*b^16*c^16*d^2 - 9200*a^3*b^15*c^15*d^3 + 71220*a^4*b^14
*c^14*d^4 - 394392*a^5*b^13*c^13*d^5 + 1619684*a^6*b^12*c^12*d^6 - 5050512*a^7*b^11*c^11*d^7 + 12147630*a^8*b^
10*c^10*d^8 - 22765820*a^9*b^9*c^9*d^9 + 33419166*a^10*b^8*c^8*d^10 - 38446992*a^11*b^7*c^7*d^11 + 34503236*a^
12*b^6*c^6*d^12 - 23888280*a^13*b^5*c^5*d^13 + 12508500*a^14*b^4*c^4*d^14 - 4790000*a^15*b^3*c^3*d^15 + 126562
5*a^16*b^2*c^2*d^16 - 206250*a^17*b*c*d^17 + 15625*a^18*d^18)*x - (a*b^21*c^12 - 28*a^2*b^20*c^11*d + 338*a^3*
b^19*c^10*d^2 - 2316*a^4*b^18*c^9*d^3 + 10015*a^5*b^17*c^8*d^4 - 28856*a^6*b^16*c^7*d^5 + 57148*a^7*b^15*c^6*d
^6 - 78968*a^8*b^14*c^5*d^7 + 76111*a^9*b^13*c^4*d^8 - 50220*a^10*b^12*c^3*d^9 + 21650*a^11*b^11*c^2*d^10 - 55
00*a^12*b^10*c*d^11 + 625*a^13*b^9*d^12)*sqrt(-(b^12*c^12 - 28*a*b^11*c^11*d + 338*a^2*b^10*c^10*d^2 - 2316*a^
3*b^9*c^9*d^3 + 10015*a^4*b^8*c^8*d^4 - 28856*a^5*b^7*c^7*d^5 + 57148*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5*d^7
+ 76111*a^8*b^4*c^4*d^8 - 50220*a^9*b^3*c^3*d^9 + 21650*a^10*b^2*c^2*d^10 - 5500*a^11*b*c*d^11 + 625*a^12*d^12
)/(a*b^19)))*b^5*(-(b^12*c^12 - 28*a*b^11*c^11*d + 338*a^2*b^10*c^10*d^2 - 2316*a^3*b^9*c^9*d^3 + 10015*a^4*b^
8*c^8*d^4 - 28856*a^5*b^7*c^7*d^5 + 57148*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5*d^7 + 76111*a^8*b^4*c^4*d^8 - 50
220*a^9*b^3*c^3*d^9 + 21650*a^10*b^2*c^2*d^10 - 5500*a^11*b*c*d^11 + 625*a^12*d^12)/(a*b^19))^(1/4) + (b^14*c^
9 - 21*a*b^13*c^8*d + 180*a^2*b^12*c^7*d^2 - 820*a^3*b^11*c^6*d^3 + 2190*a^4*b^10*c^5*d^4 - 3606*a^5*b^9*c^4*d
^5 + 3716*a^6*b^8*c^3*d^6 - 2340*a^7*b^7*c^2*d^7 + 825*a^8*b^6*c*d^8 - 125*a^9*b^5*d^9)*sqrt(x)*(-(b^12*c^12 -
28*a*b^11*c^11*d + 338*a^2*b^10*c^10*d^2 - 2316*a^3*b^9*c^9*d^3 + 10015*a^4*b^8*c^8*d^4 - 28856*a^5*b^7*c^7*d
^5 + 57148*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5*d^7 + 76111*a^8*b^4*c^4*d^8 - 50220*a^9*b^3*c^3*d^9 + 21650*a^1
0*b^2*c^2*d^10 - 5500*a^11*b*c*d^11 + 625*a^12*d^12)/(a*b^19))^(1/4))/(b^12*c^12 - 28*a*b^11*c^11*d + 338*a^2*
b^10*c^10*d^2 - 2316*a^3*b^9*c^9*d^3 + 10015*a^4*b^8*c^8*d^4 - 28856*a^5*b^7*c^7*d^5 + 57148*a^6*b^6*c^6*d^6 -
78968*a^7*b^5*c^5*d^7 + 76111*a^8*b^4*c^4*d^8 - 50220*a^9*b^3*c^3*d^9 + 21650*a^10*b^2*c^2*d^10 - 5500*a^11*b
*c*d^11 + 625*a^12*d^12)) - 231*(b^5*x^2 + a*b^4)*(-(b^12*c^12 - 28*a*b^11*c^11*d + 338*a^2*b^10*c^10*d^2 - 23
16*a^3*b^9*c^9*d^3 + 10015*a^4*b^8*c^8*d^4 - 28856*a^5*b^7*c^7*d^5 + 57148*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5
*d^7 + 76111*a^8*b^4*c^4*d^8 - 50220*a^9*b^3*c^3*d^9 + 21650*a^10*b^2*c^2*d^10 - 5500*a^11*b*c*d^11 + 625*a^12
*d^12)/(a*b^19))^(1/4)*log(27*a*b^14*(-(b^12*c^12 - 28*a*b^11*c^11*d + 338*a^2*b^10*c^10*d^2 - 2316*a^3*b^9*c^
9*d^3 + 10015*a^4*b^8*c^8*d^4 - 28856*a^5*b^7*c^7*d^5 + 57148*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5*d^7 + 76111*
a^8*b^4*c^4*d^8 - 50220*a^9*b^3*c^3*d^9 + 21650*a^10*b^2*c^2*d^10 - 5500*a^11*b*c*d^11 + 625*a^12*d^12)/(a*b^1
9))^(3/4) - 27*(b^9*c^9 - 21*a*b^8*c^8*d + 180*a^2*b^7*c^7*d^2 - 820*a^3*b^6*c^6*d^3 + 2190*a^4*b^5*c^5*d^4 -
3606*a^5*b^4*c^4*d^5 + 3716*a^6*b^3*c^3*d^6 - 2340*a^7*b^2*c^2*d^7 + 825*a^8*b*c*d^8 - 125*a^9*d^9)*sqrt(x)) +
231*(b^5*x^2 + a*b^4)*(-(b^12*c^12 - 28*a*b^11*c^11*d + 338*a^2*b^10*c^10*d^2 - 2316*a^3*b^9*c^9*d^3 + 10015*
a^4*b^8*c^8*d^4 - 28856*a^5*b^7*c^7*d^5 + 57148*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5*d^7 + 76111*a^8*b^4*c^4*d^
8 - 50220*a^9*b^3*c^3*d^9 + 21650*a^10*b^2*c^2*d^10 - 5500*a^11*b*c*d^11 + 625*a^12*d^12)/(a*b^19))^(1/4)*log(
-27*a*b^14*(-(b^12*c^12 - 28*a*b^11*c^11*d + 338*a^2*b^10*c^10*d^2 - 2316*a^3*b^9*c^9*d^3 + 10015*a^4*b^8*c^8*
d^4 - 28856*a^5*b^7*c^7*d^5 + 57148*a^6*b^6*c^6*d^6 - 78968*a^7*b^5*c^5*d^7 + 76111*a^8*b^4*c^4*d^8 - 50220*a^
9*b^3*c^3*d^9 + 21650*a^10*b^2*c^2*d^10 - 5500*a^11*b*c*d^11 + 625*a^12*d^12)/(a*b^19))^(3/4) - 27*(b^9*c^9 -
21*a*b^8*c^8*d + 180*a^2*b^7*c^7*d^2 - 820*a^3*b^6*c^6*d^3 + 2190*a^4*b^5*c^5*d^4 - 3606*a^5*b^4*c^4*d^5 + 371
6*a^6*b^3*c^3*d^6 - 2340*a^7*b^2*c^2*d^7 + 825*a^8*b*c*d^8 - 125*a^9*d^9)*sqrt(x)) + 4*(28*b^3*d^3*x^7 + 12*(1
1*b^3*c*d^2 - 5*a*b^2*d^3)*x^5 + 44*(7*b^3*c^2*d - 11*a*b^2*c*d^2 + 5*a^2*b*d^3)*x^3 - 77*(b^3*c^3 - 7*a*b^2*c
^2*d + 11*a^2*b*c*d^2 - 5*a^3*d^3)*x)*sqrt(x))/(b^5*x^2 + a*b^4)

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

[Out]

Timed out

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Giac [A]  time = 1.27006, size = 745, normalized size = 1.99 \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)*(d*x^2+c)^3/(b*x^2+a)^2,x, algorithm="giac")

[Out]

-1/2*(b^3*c^3*x^(3/2) - 3*a*b^2*c^2*d*x^(3/2) + 3*a^2*b*c*d^2*x^(3/2) - a^3*d^3*x^(3/2))/((b*x^2 + a)*b^4) + 3
/8*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 7*(a*b^3)^(3/4)*a*b^2*c^2*d + 11*(a*b^3)^(3/4)*a^2*b*c*d^2 - 5*(a*b^3)^(3/
4)*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a*b^7) + 3/8*sqrt(2)*((a*b^3)^(
3/4)*b^3*c^3 - 7*(a*b^3)^(3/4)*a*b^2*c^2*d + 11*(a*b^3)^(3/4)*a^2*b*c*d^2 - 5*(a*b^3)^(3/4)*a^3*d^3)*arctan(-1
/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a*b^7) - 3/16*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 7*(a
*b^3)^(3/4)*a*b^2*c^2*d + 11*(a*b^3)^(3/4)*a^2*b*c*d^2 - 5*(a*b^3)^(3/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*(a/b)^(1
/4) + x + sqrt(a/b))/(a*b^7) + 3/16*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 7*(a*b^3)^(3/4)*a*b^2*c^2*d + 11*(a*b^3)^
(3/4)*a^2*b*c*d^2 - 5*(a*b^3)^(3/4)*a^3*d^3)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a*b^7) + 2/77*
(7*b^20*d^3*x^(11/2) + 33*b^20*c*d^2*x^(7/2) - 22*a*b^19*d^3*x^(7/2) + 77*b^20*c^2*d*x^(3/2) - 154*a*b^19*c*d^
2*x^(3/2) + 77*a^2*b^18*d^3*x^(3/2))/b^22