3.452 \(\int \frac{x^{7/2} (c+d x^2)^3}{(a+b x^2)^2} \, dx\)

Optimal. Leaf size=409 \[ \frac{d x^{5/2} \left (17 a^2 d^2-39 a b c d+27 b^2 c^2\right )}{10 b^4}+\frac{d^2 x^{9/2} (39 b c-17 a d)}{18 b^3}+\frac{\sqrt{x} (5 b c-17 a d) (b c-a d)^2}{2 b^5}+\frac{\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{21/4}}-\frac{\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{21/4}}+\frac{\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} b^{21/4}}-\frac{\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} b^{21/4}}-\frac{x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{17 d^3 x^{13/2}}{26 b^2} \]

[Out]

((5*b*c - 17*a*d)*(b*c - a*d)^2*Sqrt[x])/(2*b^5) + (d*(27*b^2*c^2 - 39*a*b*c*d + 17*a^2*d^2)*x^(5/2))/(10*b^4)
+ (d^2*(39*b*c - 17*a*d)*x^(9/2))/(18*b^3) + (17*d^3*x^(13/2))/(26*b^2) - (x^(5/2)*(c + d*x^2)^3)/(2*b*(a + b
*x^2)) + (a^(1/4)*(5*b*c - 17*a*d)*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*b^(
21/4)) - (a^(1/4)*(5*b*c - 17*a*d)*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*b^(
21/4)) + (a^(1/4)*(5*b*c - 17*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]*b^(21/4)) - (a^(1/4)*(5*b*c - 17*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]*b^(21/4))

________________________________________________________________________________________

Rubi [A]  time = 0.46727, antiderivative size = 409, 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, 211, 1165, 628, 1162, 617, 204} \[ \frac{d x^{5/2} \left (17 a^2 d^2-39 a b c d+27 b^2 c^2\right )}{10 b^4}+\frac{d^2 x^{9/2} (39 b c-17 a d)}{18 b^3}+\frac{\sqrt{x} (5 b c-17 a d) (b c-a d)^2}{2 b^5}+\frac{\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{21/4}}-\frac{\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{21/4}}+\frac{\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} b^{21/4}}-\frac{\sqrt [4]{a} (5 b c-17 a d) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} b^{21/4}}-\frac{x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{17 d^3 x^{13/2}}{26 b^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^(7/2)*(c + d*x^2)^3)/(a + b*x^2)^2,x]

[Out]

((5*b*c - 17*a*d)*(b*c - a*d)^2*Sqrt[x])/(2*b^5) + (d*(27*b^2*c^2 - 39*a*b*c*d + 17*a^2*d^2)*x^(5/2))/(10*b^4)
+ (d^2*(39*b*c - 17*a*d)*x^(9/2))/(18*b^3) + (17*d^3*x^(13/2))/(26*b^2) - (x^(5/2)*(c + d*x^2)^3)/(2*b*(a + b
*x^2)) + (a^(1/4)*(5*b*c - 17*a*d)*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*b^(
21/4)) - (a^(1/4)*(5*b*c - 17*a*d)*(b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*b^(
21/4)) + (a^(1/4)*(5*b*c - 17*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]*b^(21/4)) - (a^(1/4)*(5*b*c - 17*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]*b^(21/4))

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 211

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 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]

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])

Rubi steps

\begin{align*} \int \frac{x^{7/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^8 \left (c+d x^4\right )^3}{\left (a+b x^4\right )^2} \, dx,x,\sqrt{x}\right )\\ &=-\frac{x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{x^4 \left (c+d x^4\right )^2 \left (5 c+17 d x^4\right )}{a+b x^4} \, dx,x,\sqrt{x}\right )}{2 b}\\ &=-\frac{x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{\operatorname{Subst}\left (\int \left (\frac{(5 b c-17 a d) (b c-a d)^2}{b^4}+\frac{d \left (27 b^2 c^2-39 a b c d+17 a^2 d^2\right ) x^4}{b^3}+\frac{d^2 (39 b c-17 a d) x^8}{b^2}+\frac{17 d^3 x^{12}}{b}+\frac{-5 a b^3 c^3+27 a^2 b^2 c^2 d-39 a^3 b c d^2+17 a^4 d^3}{b^4 \left (a+b x^4\right )}\right ) \, dx,x,\sqrt{x}\right )}{2 b}\\ &=\frac{(5 b c-17 a d) (b c-a d)^2 \sqrt{x}}{2 b^5}+\frac{d \left (27 b^2 c^2-39 a b c d+17 a^2 d^2\right ) x^{5/2}}{10 b^4}+\frac{d^2 (39 b c-17 a d) x^{9/2}}{18 b^3}+\frac{17 d^3 x^{13/2}}{26 b^2}-\frac{x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac{\left (a (5 b c-17 a d) (b c-a d)^2\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^4} \, dx,x,\sqrt{x}\right )}{2 b^5}\\ &=\frac{(5 b c-17 a d) (b c-a d)^2 \sqrt{x}}{2 b^5}+\frac{d \left (27 b^2 c^2-39 a b c d+17 a^2 d^2\right ) x^{5/2}}{10 b^4}+\frac{d^2 (39 b c-17 a d) x^{9/2}}{18 b^3}+\frac{17 d^3 x^{13/2}}{26 b^2}-\frac{x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac{\left (\sqrt{a} (5 b c-17 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^5}-\frac{\left (\sqrt{a} (5 b c-17 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^5}\\ &=\frac{(5 b c-17 a d) (b c-a d)^2 \sqrt{x}}{2 b^5}+\frac{d \left (27 b^2 c^2-39 a b c d+17 a^2 d^2\right ) x^{5/2}}{10 b^4}+\frac{d^2 (39 b c-17 a d) x^{9/2}}{18 b^3}+\frac{17 d^3 x^{13/2}}{26 b^2}-\frac{x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac{\left (\sqrt{a} (5 b c-17 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^{11/2}}-\frac{\left (\sqrt{a} (5 b c-17 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^{11/2}}+\frac{\left (\sqrt [4]{a} (5 b c-17 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} b^{21/4}}+\frac{\left (\sqrt [4]{a} (5 b c-17 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} b^{21/4}}\\ &=\frac{(5 b c-17 a d) (b c-a d)^2 \sqrt{x}}{2 b^5}+\frac{d \left (27 b^2 c^2-39 a b c d+17 a^2 d^2\right ) x^{5/2}}{10 b^4}+\frac{d^2 (39 b c-17 a d) x^{9/2}}{18 b^3}+\frac{17 d^3 x^{13/2}}{26 b^2}-\frac{x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{\sqrt [4]{a} (5 b c-17 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} b^{21/4}}-\frac{\sqrt [4]{a} (5 b c-17 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} b^{21/4}}-\frac{\left (\sqrt [4]{a} (5 b c-17 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} b^{21/4}}+\frac{\left (\sqrt [4]{a} (5 b c-17 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} b^{21/4}}\\ &=\frac{(5 b c-17 a d) (b c-a d)^2 \sqrt{x}}{2 b^5}+\frac{d \left (27 b^2 c^2-39 a b c d+17 a^2 d^2\right ) x^{5/2}}{10 b^4}+\frac{d^2 (39 b c-17 a d) x^{9/2}}{18 b^3}+\frac{17 d^3 x^{13/2}}{26 b^2}-\frac{x^{5/2} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac{\sqrt [4]{a} (5 b c-17 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} b^{21/4}}-\frac{\sqrt [4]{a} (5 b c-17 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} b^{21/4}}+\frac{\sqrt [4]{a} (5 b c-17 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} b^{21/4}}-\frac{\sqrt [4]{a} (5 b c-17 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} b^{21/4}}\\ \end{align*}

Mathematica [C]  time = 2.57421, size = 419, normalized size = 1.02 \[ \frac{35 a^2 \left (-9945 a \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};-\frac{b x^2}{a}\right ) \left (a^2 b x^2 \left (250563 c^2 d x^2+83521 c^3+255555 c d^2 x^4+83521 d^3 x^6\right )+a^3 \left (194481 c^2 d x^2+64827 c^3+194481 c d^2 x^4+62651 d^3 x^6\right )+a b^2 x^4 \left (82227 c^2 d x^2+28561 c^3+85683 c d^2 x^4+28561 d^3 x^6\right )+b^3 x^6 \left (6561 c^2 d x^2+2827 c^3+6561 c d^2 x^4+2187 d^3 x^6\right )\right )+221 a^2 b^2 x^4 \left (2417553 c^2 d x^2+857691 c^3+2528145 c d^2 x^4+846811 d^3 x^6\right )+3978 a^3 b x^2 \left (529167 c^2 d x^2+176389 c^3+541647 c d^2 x^4+177477 d^3 x^6\right )+9945 a^4 \left (194481 c^2 d x^2+64827 c^3+194481 c d^2 x^4+62651 d^3 x^6\right )+8704 a b^3 x^6 \left (3423 c^2 d x^2+1609 c^3+3267 c d^2 x^4+1069 d^3 x^6\right )-32768 b^4 x^8 \left (c+d x^2\right )^3\right )-98304 b^6 x^{12} \left (c+d x^2\right )^3 \text{HypergeometricPFQ}\left (\left \{2,2,2,2,\frac{13}{4}\right \},\left \{1,1,1,\frac{29}{4}\right \},-\frac{b x^2}{a}\right )}{89107200 a^3 b^5 x^{11/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^(7/2)*(c + d*x^2)^3)/(a + b*x^2)^2,x]

[Out]

(35*a^2*(-32768*b^4*x^8*(c + d*x^2)^3 + 8704*a*b^3*x^6*(1609*c^3 + 3423*c^2*d*x^2 + 3267*c*d^2*x^4 + 1069*d^3*
x^6) + 9945*a^4*(64827*c^3 + 194481*c^2*d*x^2 + 194481*c*d^2*x^4 + 62651*d^3*x^6) + 3978*a^3*b*x^2*(176389*c^3
+ 529167*c^2*d*x^2 + 541647*c*d^2*x^4 + 177477*d^3*x^6) + 221*a^2*b^2*x^4*(857691*c^3 + 2417553*c^2*d*x^2 + 2
528145*c*d^2*x^4 + 846811*d^3*x^6) - 9945*a*(b^3*x^6*(2827*c^3 + 6561*c^2*d*x^2 + 6561*c*d^2*x^4 + 2187*d^3*x^
6) + a*b^2*x^4*(28561*c^3 + 82227*c^2*d*x^2 + 85683*c*d^2*x^4 + 28561*d^3*x^6) + a^3*(64827*c^3 + 194481*c^2*d
*x^2 + 194481*c*d^2*x^4 + 62651*d^3*x^6) + a^2*b*x^2*(83521*c^3 + 250563*c^2*d*x^2 + 255555*c*d^2*x^4 + 83521*
d^3*x^6))*Hypergeometric2F1[1/4, 1, 5/4, -((b*x^2)/a)]) - 98304*b^6*x^12*(c + d*x^2)^3*HypergeometricPFQ[{2, 2
, 2, 2, 13/4}, {1, 1, 1, 29/4}, -((b*x^2)/a)])/(89107200*a^3*b^5*x^(11/2))

________________________________________________________________________________________

Maple [B]  time = 0.018, size = 804, 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^(7/2)*(d*x^2+c)^3/(b*x^2+a)^2,x)

[Out]

-3/2*a^2/b^3*x^(1/2)/(b*x^2+a)*c^2*d+17/8*a^3/b^5*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1
)*d^3+17/8*a^3/b^5*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)*d^3+17/16*a^3/b^5*(1/b*a)^(1/
4)*2^(1/2)*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)))
*d^3+3/2*a^3/b^4*x^(1/2)/(b*x^2+a)*c*d^2-39/8*a^2/b^4*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/
2)-1)*c*d^2+27/8*a/b^3*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)*c^2*d-39/16*a^2/b^4*(1/b*
a)^(1/4)*2^(1/2)*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)))*c*d^2+27/16*a/b^3*(1/b*a)^(1/4)*2^(1/2)*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)))*c^2*d-39/8*a^2/b^4*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(
1/2)+1)*c*d^2+27/8*a/b^3*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)*c^2*d-12/5/b^3*x^(5/2)*
a*c*d^2+18/b^4*a^2*c*d^2*x^(1/2)+2/b^2*c^3*x^(1/2)+2/13*d^3*x^(13/2)/b^2-12/b^3*a*c^2*d*x^(1/2)-1/2*a^4/b^5*x^
(1/2)/(b*x^2+a)*d^3+1/2*a/b^2*x^(1/2)/(b*x^2+a)*c^3-5/8/b^2*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)
*x^(1/2)+1)*c^3-5/8/b^2*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)*c^3-5/16/b^2*(1/b*a)^(1/
4)*2^(1/2)*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)))
*c^3+6/5/b^4*x^(5/2)*a^2*d^3+6/5/b^2*x^(5/2)*c^2*d-8/b^5*a^3*d^3*x^(1/2)-4/9/b^3*x^(9/2)*a*d^3+2/3/b^2*x^(9/2)
*c*d^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^(7/2)*(d*x^2+c)^3/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.42914, size = 5115, normalized size = 12.51 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(d*x^2+c)^3/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

1/4680*(2340*(b^6*x^2 + a*b^5)*(-(625*a*b^12*c^12 - 13500*a^2*b^11*c^11*d + 128850*a^3*b^10*c^10*d^2 - 718060*
a^4*b^9*c^9*d^3 + 2603151*a^5*b^8*c^8*d^4 - 6477048*a^6*b^7*c^7*d^5 + 11369148*a^7*b^6*c^6*d^6 - 14225976*a^8*
b^5*c^5*d^7 + 12631455*a^9*b^4*c^4*d^8 - 7783756*a^10*b^3*c^3*d^9 + 3168018*a^11*b^2*c^2*d^10 - 766428*a^12*b*
c*d^11 + 83521*a^13*d^12)/b^21)^(1/4)*arctan((sqrt(b^10*sqrt(-(625*a*b^12*c^12 - 13500*a^2*b^11*c^11*d + 12885
0*a^3*b^10*c^10*d^2 - 718060*a^4*b^9*c^9*d^3 + 2603151*a^5*b^8*c^8*d^4 - 6477048*a^6*b^7*c^7*d^5 + 11369148*a^
7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631455*a^9*b^4*c^4*d^8 - 7783756*a^10*b^3*c^3*d^9 + 3168018*a^11*
b^2*c^2*d^10 - 766428*a^12*b*c*d^11 + 83521*a^13*d^12)/b^21) + (25*b^6*c^6 - 270*a*b^5*c^5*d + 1119*a^2*b^4*c^
4*d^2 - 2276*a^3*b^3*c^3*d^3 + 2439*a^4*b^2*c^2*d^4 - 1326*a^5*b*c*d^5 + 289*a^6*d^6)*x)*b^16*(-(625*a*b^12*c^
12 - 13500*a^2*b^11*c^11*d + 128850*a^3*b^10*c^10*d^2 - 718060*a^4*b^9*c^9*d^3 + 2603151*a^5*b^8*c^8*d^4 - 647
7048*a^6*b^7*c^7*d^5 + 11369148*a^7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631455*a^9*b^4*c^4*d^8 - 778375
6*a^10*b^3*c^3*d^9 + 3168018*a^11*b^2*c^2*d^10 - 766428*a^12*b*c*d^11 + 83521*a^13*d^12)/b^21)^(3/4) + (5*b^19
*c^3 - 27*a*b^18*c^2*d + 39*a^2*b^17*c*d^2 - 17*a^3*b^16*d^3)*sqrt(x)*(-(625*a*b^12*c^12 - 13500*a^2*b^11*c^11
*d + 128850*a^3*b^10*c^10*d^2 - 718060*a^4*b^9*c^9*d^3 + 2603151*a^5*b^8*c^8*d^4 - 6477048*a^6*b^7*c^7*d^5 + 1
1369148*a^7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631455*a^9*b^4*c^4*d^8 - 7783756*a^10*b^3*c^3*d^9 + 316
8018*a^11*b^2*c^2*d^10 - 766428*a^12*b*c*d^11 + 83521*a^13*d^12)/b^21)^(3/4))/(625*a*b^12*c^12 - 13500*a^2*b^1
1*c^11*d + 128850*a^3*b^10*c^10*d^2 - 718060*a^4*b^9*c^9*d^3 + 2603151*a^5*b^8*c^8*d^4 - 6477048*a^6*b^7*c^7*d
^5 + 11369148*a^7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631455*a^9*b^4*c^4*d^8 - 7783756*a^10*b^3*c^3*d^9
+ 3168018*a^11*b^2*c^2*d^10 - 766428*a^12*b*c*d^11 + 83521*a^13*d^12)) + 585*(b^6*x^2 + a*b^5)*(-(625*a*b^12*
c^12 - 13500*a^2*b^11*c^11*d + 128850*a^3*b^10*c^10*d^2 - 718060*a^4*b^9*c^9*d^3 + 2603151*a^5*b^8*c^8*d^4 - 6
477048*a^6*b^7*c^7*d^5 + 11369148*a^7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631455*a^9*b^4*c^4*d^8 - 7783
756*a^10*b^3*c^3*d^9 + 3168018*a^11*b^2*c^2*d^10 - 766428*a^12*b*c*d^11 + 83521*a^13*d^12)/b^21)^(1/4)*log(b^5
*(-(625*a*b^12*c^12 - 13500*a^2*b^11*c^11*d + 128850*a^3*b^10*c^10*d^2 - 718060*a^4*b^9*c^9*d^3 + 2603151*a^5*
b^8*c^8*d^4 - 6477048*a^6*b^7*c^7*d^5 + 11369148*a^7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631455*a^9*b^4
*c^4*d^8 - 7783756*a^10*b^3*c^3*d^9 + 3168018*a^11*b^2*c^2*d^10 - 766428*a^12*b*c*d^11 + 83521*a^13*d^12)/b^21
)^(1/4) - (5*b^3*c^3 - 27*a*b^2*c^2*d + 39*a^2*b*c*d^2 - 17*a^3*d^3)*sqrt(x)) - 585*(b^6*x^2 + a*b^5)*(-(625*a
*b^12*c^12 - 13500*a^2*b^11*c^11*d + 128850*a^3*b^10*c^10*d^2 - 718060*a^4*b^9*c^9*d^3 + 2603151*a^5*b^8*c^8*d
^4 - 6477048*a^6*b^7*c^7*d^5 + 11369148*a^7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631455*a^9*b^4*c^4*d^8
- 7783756*a^10*b^3*c^3*d^9 + 3168018*a^11*b^2*c^2*d^10 - 766428*a^12*b*c*d^11 + 83521*a^13*d^12)/b^21)^(1/4)*l
og(-b^5*(-(625*a*b^12*c^12 - 13500*a^2*b^11*c^11*d + 128850*a^3*b^10*c^10*d^2 - 718060*a^4*b^9*c^9*d^3 + 26031
51*a^5*b^8*c^8*d^4 - 6477048*a^6*b^7*c^7*d^5 + 11369148*a^7*b^6*c^6*d^6 - 14225976*a^8*b^5*c^5*d^7 + 12631455*
a^9*b^4*c^4*d^8 - 7783756*a^10*b^3*c^3*d^9 + 3168018*a^11*b^2*c^2*d^10 - 766428*a^12*b*c*d^11 + 83521*a^13*d^1
2)/b^21)^(1/4) - (5*b^3*c^3 - 27*a*b^2*c^2*d + 39*a^2*b*c*d^2 - 17*a^3*d^3)*sqrt(x)) + 4*(180*b^4*d^3*x^8 + 29
25*a*b^3*c^3 - 15795*a^2*b^2*c^2*d + 22815*a^3*b*c*d^2 - 9945*a^4*d^3 + 20*(39*b^4*c*d^2 - 17*a*b^3*d^3)*x^6 +
52*(27*b^4*c^2*d - 39*a*b^3*c*d^2 + 17*a^2*b^2*d^3)*x^4 + 468*(5*b^4*c^3 - 27*a*b^3*c^2*d + 39*a^2*b^2*c*d^2
- 17*a^3*b*d^3)*x^2)*sqrt(x))/(b^6*x^2 + a*b^5)

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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**(7/2)*(d*x**2+c)**3/(b*x**2+a)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.22219, size = 810, normalized size = 1.98 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(d*x^2+c)^3/(b*x^2+a)^2,x, algorithm="giac")

[Out]

-1/8*sqrt(2)*(5*(a*b^3)^(1/4)*b^3*c^3 - 27*(a*b^3)^(1/4)*a*b^2*c^2*d + 39*(a*b^3)^(1/4)*a^2*b*c*d^2 - 17*(a*b^
3)^(1/4)*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/b^6 - 1/8*sqrt(2)*(5*(a*b^
3)^(1/4)*b^3*c^3 - 27*(a*b^3)^(1/4)*a*b^2*c^2*d + 39*(a*b^3)^(1/4)*a^2*b*c*d^2 - 17*(a*b^3)^(1/4)*a^3*d^3)*arc
tan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/b^6 - 1/16*sqrt(2)*(5*(a*b^3)^(1/4)*b^3*c^3 -
27*(a*b^3)^(1/4)*a*b^2*c^2*d + 39*(a*b^3)^(1/4)*a^2*b*c*d^2 - 17*(a*b^3)^(1/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*(a
/b)^(1/4) + x + sqrt(a/b))/b^6 + 1/16*sqrt(2)*(5*(a*b^3)^(1/4)*b^3*c^3 - 27*(a*b^3)^(1/4)*a*b^2*c^2*d + 39*(a*
b^3)^(1/4)*a^2*b*c*d^2 - 17*(a*b^3)^(1/4)*a^3*d^3)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^6 + 1/2
*(a*b^3*c^3*sqrt(x) - 3*a^2*b^2*c^2*d*sqrt(x) + 3*a^3*b*c*d^2*sqrt(x) - a^4*d^3*sqrt(x))/((b*x^2 + a)*b^5) + 2
/585*(45*b^24*d^3*x^(13/2) + 195*b^24*c*d^2*x^(9/2) - 130*a*b^23*d^3*x^(9/2) + 351*b^24*c^2*d*x^(5/2) - 702*a*
b^23*c*d^2*x^(5/2) + 351*a^2*b^22*d^3*x^(5/2) + 585*b^24*c^3*sqrt(x) - 3510*a*b^23*c^2*d*sqrt(x) + 5265*a^2*b^
22*c*d^2*sqrt(x) - 2340*a^3*b^21*d^3*sqrt(x))/b^26