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

Optimal. Leaf size=386 \[ -\frac {(b c-13 a d) (b c-a d)^2 \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} b^{17/4}}+\frac {(b c-13 a d) (b c-a d)^2 \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{8 \sqrt {2} a^{3/4} b^{17/4}}-\frac {(b c-13 a d) (b c-a d)^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} b^{17/4}}+\frac {(b c-13 a d) (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt {2} a^{3/4} b^{17/4}}+\frac {d \sqrt {x} \left (585 a^2 d^2-1098 a b c d+497 b^2 c^2\right )}{90 b^4}+\frac {d \sqrt {x} \left (c+d x^2\right ) (113 b c-117 a d)}{90 b^3}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{18 b^2} \]

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Rubi [A]  time = 0.53, antiderivative size = 386, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {466, 467, 528, 388, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {d \sqrt {x} \left (585 a^2 d^2-1098 a b c d+497 b^2 c^2\right )}{90 b^4}-\frac {(b c-13 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} a^{3/4} b^{17/4}}+\frac {(b c-13 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} a^{3/4} b^{17/4}}-\frac {(b c-13 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} a^{3/4} b^{17/4}}+\frac {(b c-13 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} a^{3/4} b^{17/4}}+\frac {d \sqrt {x} \left (c+d x^2\right ) (113 b c-117 a d)}{90 b^3}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{18 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d*(497*b^2*c^2 - 1098*a*b*c*d + 585*a^2*d^2)*Sqrt[x])/(90*b^4) + (d*(113*b*c - 117*a*d)*Sqrt[x]*(c + d*x^2))/
(90*b^3) + (13*d*Sqrt[x]*(c + d*x^2)^2)/(18*b^2) - (Sqrt[x]*(c + d*x^2)^3)/(2*b*(a + b*x^2)) - ((b*c - 13*a*d)
*(b*c - a*d)^2*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(3/4)*b^(17/4)) + ((b*c - 13*a*d)*(
b*c - a*d)^2*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(3/4)*b^(17/4)) - ((b*c - 13*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^(3/4)*b^(17/4)) + ((b*c -
13*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^(3/4)*b^(17/4))

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 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 388

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

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 528

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

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 {x^{3/2} \left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^4 \left (c+d x^4\right )^3}{\left (a+b x^4\right )^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\sqrt {x} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {\left (c+d x^4\right )^2 \left (c+13 d x^4\right )}{a+b x^4} \, dx,x,\sqrt {x}\right )}{2 b}\\ &=\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{18 b^2}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {\left (c+d x^4\right ) \left (c (9 b c-13 a d)+d (113 b c-117 a d) x^4\right )}{a+b x^4} \, dx,x,\sqrt {x}\right )}{18 b^2}\\ &=\frac {d (113 b c-117 a d) \sqrt {x} \left (c+d x^2\right )}{90 b^3}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{18 b^2}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {c \left (45 b^2 c^2-178 a b c d+117 a^2 d^2\right )+d \left (497 b^2 c^2-1098 a b c d+585 a^2 d^2\right ) x^4}{a+b x^4} \, dx,x,\sqrt {x}\right )}{90 b^3}\\ &=\frac {d \left (497 b^2 c^2-1098 a b c d+585 a^2 d^2\right ) \sqrt {x}}{90 b^4}+\frac {d (113 b c-117 a d) \sqrt {x} \left (c+d x^2\right )}{90 b^3}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{18 b^2}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\left ((b c-13 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^4}\\ &=\frac {d \left (497 b^2 c^2-1098 a b c d+585 a^2 d^2\right ) \sqrt {x}}{90 b^4}+\frac {d (113 b c-117 a d) \sqrt {x} \left (c+d x^2\right )}{90 b^3}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{18 b^2}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\left ((b c-13 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 \sqrt {a} b^4}+\frac {\left ((b c-13 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 \sqrt {a} b^4}\\ &=\frac {d \left (497 b^2 c^2-1098 a b c d+585 a^2 d^2\right ) \sqrt {x}}{90 b^4}+\frac {d (113 b c-117 a d) \sqrt {x} \left (c+d x^2\right )}{90 b^3}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{18 b^2}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}+\frac {\left ((b c-13 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 \sqrt {a} b^{9/2}}+\frac {\left ((b c-13 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 \sqrt {a} b^{9/2}}-\frac {\left ((b c-13 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} a^{3/4} b^{17/4}}-\frac {\left ((b c-13 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} a^{3/4} b^{17/4}}\\ &=\frac {d \left (497 b^2 c^2-1098 a b c d+585 a^2 d^2\right ) \sqrt {x}}{90 b^4}+\frac {d (113 b c-117 a d) \sqrt {x} \left (c+d x^2\right )}{90 b^3}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{18 b^2}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac {(b c-13 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} a^{3/4} b^{17/4}}+\frac {(b c-13 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} a^{3/4} b^{17/4}}+\frac {\left ((b c-13 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} a^{3/4} b^{17/4}}-\frac {\left ((b c-13 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} a^{3/4} b^{17/4}}\\ &=\frac {d \left (497 b^2 c^2-1098 a b c d+585 a^2 d^2\right ) \sqrt {x}}{90 b^4}+\frac {d (113 b c-117 a d) \sqrt {x} \left (c+d x^2\right )}{90 b^3}+\frac {13 d \sqrt {x} \left (c+d x^2\right )^2}{18 b^2}-\frac {\sqrt {x} \left (c+d x^2\right )^3}{2 b \left (a+b x^2\right )}-\frac {(b c-13 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} a^{3/4} b^{17/4}}+\frac {(b c-13 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} a^{3/4} b^{17/4}}-\frac {(b c-13 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} a^{3/4} b^{17/4}}+\frac {(b c-13 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} a^{3/4} b^{17/4}}\\ \end {align*}

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Mathematica [C]  time = 2.69, size = 377, normalized size = 0.98 \begin {gather*} \frac {-585 a^3 \left (83521 c^3+250563 c^2 d x^2+250563 c d^2 x^4+78529 d^3 x^6\right )-234 a^2 b x^2 \left (172447 c^3+517341 c^2 d x^2+543261 c d^2 x^4+174943 d^3 x^6\right )+585 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};-\frac {b x^2}{a}\right ) \left (a^3 \left (83521 c^3+250563 c^2 d x^2+250563 c d^2 x^4+78529 d^3 x^6\right )+3 a^2 b x^2 \left (28561 c^3+85683 c^2 d x^2+89139 c d^2 x^4+28561 d^3 x^6\right )+9 a b^2 x^4 \left (2187 c^3+5921 c^2 d x^2+6561 c d^2 x^4+2187 d^3 x^6\right )+b^3 x^6 \left (1009 c^3+1875 c^2 d x^2+1875 c d^2 x^4+625 d^3 x^6\right )\right )-13 a b^2 x^4 \left (532193 c^3+1337379 c^2 d x^2+1503267 c d^2 x^4+507233 d^3 x^6\right )-98304 b^3 x^6 \left (c+d x^2\right )^3}{449280 a b^4 x^{11/2}}-\frac {128 b x^{9/2} \left (c+d x^2\right )^3 \, _5F_4\left (2,2,2,2,\frac {9}{4};1,1,1,\frac {25}{4};-\frac {b x^2}{a}\right )}{41769 a^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-98304*b^3*x^6*(c + d*x^2)^3 - 585*a^3*(83521*c^3 + 250563*c^2*d*x^2 + 250563*c*d^2*x^4 + 78529*d^3*x^6) - 23
4*a^2*b*x^2*(172447*c^3 + 517341*c^2*d*x^2 + 543261*c*d^2*x^4 + 174943*d^3*x^6) - 13*a*b^2*x^4*(532193*c^3 + 1
337379*c^2*d*x^2 + 1503267*c*d^2*x^4 + 507233*d^3*x^6) + 585*(b^3*x^6*(1009*c^3 + 1875*c^2*d*x^2 + 1875*c*d^2*
x^4 + 625*d^3*x^6) + 9*a*b^2*x^4*(2187*c^3 + 5921*c^2*d*x^2 + 6561*c*d^2*x^4 + 2187*d^3*x^6) + 3*a^2*b*x^2*(28
561*c^3 + 85683*c^2*d*x^2 + 89139*c*d^2*x^4 + 28561*d^3*x^6) + a^3*(83521*c^3 + 250563*c^2*d*x^2 + 250563*c*d^
2*x^4 + 78529*d^3*x^6))*Hypergeometric2F1[1/4, 1, 5/4, -((b*x^2)/a)])/(449280*a*b^4*x^(11/2)) - (128*b*x^(9/2)
*(c + d*x^2)^3*HypergeometricPFQ[{2, 2, 2, 2, 9/4}, {1, 1, 1, 25/4}, -((b*x^2)/a)])/(41769*a^3)

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IntegrateAlgebraic [A]  time = 0.54, size = 280, normalized size = 0.73 \begin {gather*} \frac {(13 a d-b c) (a d-b c)^2 \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{4 \sqrt {2} a^{3/4} b^{17/4}}-\frac {(13 a d-b c) (a d-b c)^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{4 \sqrt {2} a^{3/4} b^{17/4}}+\frac {\sqrt {x} \left (585 a^3 d^3-1215 a^2 b c d^2+468 a^2 b d^3 x^2+675 a b^2 c^2 d-972 a b^2 c d^2 x^2-52 a b^2 d^3 x^4-45 b^3 c^3+540 b^3 c^2 d x^2+108 b^3 c d^2 x^4+20 b^3 d^3 x^6\right )}{90 b^4 \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*(-45*b^3*c^3 + 675*a*b^2*c^2*d - 1215*a^2*b*c*d^2 + 585*a^3*d^3 + 540*b^3*c^2*d*x^2 - 972*a*b^2*c*d^2
*x^2 + 468*a^2*b*d^3*x^2 + 108*b^3*c*d^2*x^4 - 52*a*b^2*d^3*x^4 + 20*b^3*d^3*x^6))/(90*b^4*(a + b*x^2)) + ((-(
b*c) + a*d)^2*(-(b*c) + 13*a*d)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])])/(4*Sqrt[2]*a^
(3/4)*b^(17/4)) - ((-(b*c) + a*d)^2*(-(b*c) + 13*a*d)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqr
t[b]*x)])/(4*Sqrt[2]*a^(3/4)*b^(17/4))

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fricas [B]  time = 1.06, size = 1961, normalized size = 5.08

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/360*(180*(b^5*x^2 + a*b^4)*(-(b^12*c^12 - 60*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 18412*a^3*b^9*c^9*d^3
+ 130239*a^4*b^8*c^8*d^4 - 535032*a^5*b^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 + 25202
07*a^8*b^4*c^4*d^8 - 1853644*a^9*b^3*c^3*d^9 + 871026*a^10*b^2*c^2*d^10 - 237276*a^11*b*c*d^11 + 28561*a^12*d^
12)/(a^3*b^17))^(1/4)*arctan((sqrt(a^2*b^8*sqrt(-(b^12*c^12 - 60*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 1841
2*a^3*b^9*c^9*d^3 + 130239*a^4*b^8*c^8*d^4 - 535032*a^5*b^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6 - 2272824*a^7*b^
5*c^5*d^7 + 2520207*a^8*b^4*c^4*d^8 - 1853644*a^9*b^3*c^3*d^9 + 871026*a^10*b^2*c^2*d^10 - 237276*a^11*b*c*d^1
1 + 28561*a^12*d^12)/(a^3*b^17)) + (b^6*c^6 - 30*a*b^5*c^5*d + 279*a^2*b^4*c^4*d^2 - 836*a^3*b^3*c^3*d^3 + 111
9*a^4*b^2*c^2*d^4 - 702*a^5*b*c*d^5 + 169*a^6*d^6)*x)*a^2*b^13*(-(b^12*c^12 - 60*a*b^11*c^11*d + 1458*a^2*b^10
*c^10*d^2 - 18412*a^3*b^9*c^9*d^3 + 130239*a^4*b^8*c^8*d^4 - 535032*a^5*b^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6
- 2272824*a^7*b^5*c^5*d^7 + 2520207*a^8*b^4*c^4*d^8 - 1853644*a^9*b^3*c^3*d^9 + 871026*a^10*b^2*c^2*d^10 - 237
276*a^11*b*c*d^11 + 28561*a^12*d^12)/(a^3*b^17))^(3/4) + (a^2*b^16*c^3 - 15*a^3*b^15*c^2*d + 27*a^4*b^14*c*d^2
- 13*a^5*b^13*d^3)*sqrt(x)*(-(b^12*c^12 - 60*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 18412*a^3*b^9*c^9*d^3 +
130239*a^4*b^8*c^8*d^4 - 535032*a^5*b^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 + 2520207
*a^8*b^4*c^4*d^8 - 1853644*a^9*b^3*c^3*d^9 + 871026*a^10*b^2*c^2*d^10 - 237276*a^11*b*c*d^11 + 28561*a^12*d^12
)/(a^3*b^17))^(3/4))/(b^12*c^12 - 60*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 18412*a^3*b^9*c^9*d^3 + 130239*a
^4*b^8*c^8*d^4 - 535032*a^5*b^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 + 2520207*a^8*b^4*
c^4*d^8 - 1853644*a^9*b^3*c^3*d^9 + 871026*a^10*b^2*c^2*d^10 - 237276*a^11*b*c*d^11 + 28561*a^12*d^12)) + 45*(
b^5*x^2 + a*b^4)*(-(b^12*c^12 - 60*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 18412*a^3*b^9*c^9*d^3 + 130239*a^4
*b^8*c^8*d^4 - 535032*a^5*b^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 + 2520207*a^8*b^4*c^
4*d^8 - 1853644*a^9*b^3*c^3*d^9 + 871026*a^10*b^2*c^2*d^10 - 237276*a^11*b*c*d^11 + 28561*a^12*d^12)/(a^3*b^17
))^(1/4)*log(a*b^4*(-(b^12*c^12 - 60*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 18412*a^3*b^9*c^9*d^3 + 130239*a
^4*b^8*c^8*d^4 - 535032*a^5*b^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 + 2520207*a^8*b^4*
c^4*d^8 - 1853644*a^9*b^3*c^3*d^9 + 871026*a^10*b^2*c^2*d^10 - 237276*a^11*b*c*d^11 + 28561*a^12*d^12)/(a^3*b^
17))^(1/4) - (b^3*c^3 - 15*a*b^2*c^2*d + 27*a^2*b*c*d^2 - 13*a^3*d^3)*sqrt(x)) - 45*(b^5*x^2 + a*b^4)*(-(b^12*
c^12 - 60*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 18412*a^3*b^9*c^9*d^3 + 130239*a^4*b^8*c^8*d^4 - 535032*a^5
*b^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 + 2520207*a^8*b^4*c^4*d^8 - 1853644*a^9*b^3*c
^3*d^9 + 871026*a^10*b^2*c^2*d^10 - 237276*a^11*b*c*d^11 + 28561*a^12*d^12)/(a^3*b^17))^(1/4)*log(-a*b^4*(-(b^
12*c^12 - 60*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 18412*a^3*b^9*c^9*d^3 + 130239*a^4*b^8*c^8*d^4 - 535032*
a^5*b^7*c^7*d^5 + 1365756*a^6*b^6*c^6*d^6 - 2272824*a^7*b^5*c^5*d^7 + 2520207*a^8*b^4*c^4*d^8 - 1853644*a^9*b^
3*c^3*d^9 + 871026*a^10*b^2*c^2*d^10 - 237276*a^11*b*c*d^11 + 28561*a^12*d^12)/(a^3*b^17))^(1/4) - (b^3*c^3 -
15*a*b^2*c^2*d + 27*a^2*b*c*d^2 - 13*a^3*d^3)*sqrt(x)) - 4*(20*b^3*d^3*x^6 - 45*b^3*c^3 + 675*a*b^2*c^2*d - 12
15*a^2*b*c*d^2 + 585*a^3*d^3 + 4*(27*b^3*c*d^2 - 13*a*b^2*d^3)*x^4 + 36*(15*b^3*c^2*d - 27*a*b^2*c*d^2 + 13*a^
2*b*d^3)*x^2)*sqrt(x))/(b^5*x^2 + a*b^4)

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giac [A]  time = 0.50, size = 552, normalized size = 1.43 \begin {gather*} \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 15 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 13 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a b^{5}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 15 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 13 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, a b^{5}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 15 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 13 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a b^{5}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{3} c^{3} - 15 \, \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} c^{2} d + 27 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{2} b c d^{2} - 13 \, \left (a b^{3}\right )^{\frac {1}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, a b^{5}} - \frac {b^{3} c^{3} \sqrt {x} - 3 \, a b^{2} c^{2} d \sqrt {x} + 3 \, a^{2} b c d^{2} \sqrt {x} - a^{3} d^{3} \sqrt {x}}{2 \, {\left (b x^{2} + a\right )} b^{4}} + \frac {2 \, {\left (5 \, b^{16} d^{3} x^{\frac {9}{2}} + 27 \, b^{16} c d^{2} x^{\frac {5}{2}} - 18 \, a b^{15} d^{3} x^{\frac {5}{2}} + 135 \, b^{16} c^{2} d \sqrt {x} - 270 \, a b^{15} c d^{2} \sqrt {x} + 135 \, a^{2} b^{14} d^{3} \sqrt {x}\right )}}{45 \, b^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 15*(a*b^3)^(1/4)*a*b^2*c^2*d + 27*(a*b^3)^(1/4)*a^2*b*c*d^2 - 13*(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))/(a*b^5) + 1/8*sqrt(2)*((a*b^3
)^(1/4)*b^3*c^3 - 15*(a*b^3)^(1/4)*a*b^2*c^2*d + 27*(a*b^3)^(1/4)*a^2*b*c*d^2 - 13*(a*b^3)^(1/4)*a^3*d^3)*arct
an(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a*b^5) + 1/16*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 -
15*(a*b^3)^(1/4)*a*b^2*c^2*d + 27*(a*b^3)^(1/4)*a^2*b*c*d^2 - 13*(a*b^3)^(1/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*(
a/b)^(1/4) + x + sqrt(a/b))/(a*b^5) - 1/16*sqrt(2)*((a*b^3)^(1/4)*b^3*c^3 - 15*(a*b^3)^(1/4)*a*b^2*c^2*d + 27*
(a*b^3)^(1/4)*a^2*b*c*d^2 - 13*(a*b^3)^(1/4)*a^3*d^3)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a*b^5
) - 1/2*(b^3*c^3*sqrt(x) - 3*a*b^2*c^2*d*sqrt(x) + 3*a^2*b*c*d^2*sqrt(x) - a^3*d^3*sqrt(x))/((b*x^2 + a)*b^4)
+ 2/45*(5*b^16*d^3*x^(9/2) + 27*b^16*c*d^2*x^(5/2) - 18*a*b^15*d^3*x^(5/2) + 135*b^16*c^2*d*sqrt(x) - 270*a*b^
15*c*d^2*sqrt(x) + 135*a^2*b^14*d^3*sqrt(x))/b^18

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maple [B]  time = 0.02, size = 748, normalized size = 1.94 \begin {gather*} \frac {2 d^{3} x^{\frac {9}{2}}}{9 b^{2}}-\frac {4 a \,d^{3} x^{\frac {5}{2}}}{5 b^{3}}+\frac {6 c \,d^{2} x^{\frac {5}{2}}}{5 b^{2}}+\frac {a^{3} d^{3} \sqrt {x}}{2 \left (b \,x^{2}+a \right ) b^{4}}-\frac {3 a^{2} c \,d^{2} \sqrt {x}}{2 \left (b \,x^{2}+a \right ) b^{3}}+\frac {3 a \,c^{2} d \sqrt {x}}{2 \left (b \,x^{2}+a \right ) b^{2}}-\frac {c^{3} \sqrt {x}}{2 \left (b \,x^{2}+a \right ) b}-\frac {13 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 b^{4}}-\frac {13 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} d^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 b^{4}}-\frac {13 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a^{2} d^{3} \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 b^{4}}+\frac {27 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a c \,d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 b^{3}}+\frac {27 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a c \,d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 b^{3}}+\frac {27 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, a c \,d^{2} \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 b^{3}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 a b}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 a b}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c^{3} \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 a b}-\frac {15 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c^{2} d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{8 b^{2}}-\frac {15 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c^{2} d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{8 b^{2}}-\frac {15 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, c^{2} d \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{16 b^{2}}+\frac {6 a^{2} d^{3} \sqrt {x}}{b^{4}}-\frac {12 a c \,d^{2} \sqrt {x}}{b^{3}}+\frac {6 c^{2} d \sqrt {x}}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*d^3/b^2*x^(9/2)-4/5*d^3/b^3*x^(5/2)*a+6/5*d^2/b^2*x^(5/2)*c+6*d^3/b^4*a^2*x^(1/2)-12*d^2/b^3*a*c*x^(1/2)+6
*d/b^2*c^2*x^(1/2)+1/2/b^4*x^(1/2)/(b*x^2+a)*a^3*d^3-3/2/b^3*x^(1/2)/(b*x^2+a)*a^2*c*d^2+3/2/b^2*x^(1/2)/(b*x^
2+a)*a*c^2*d-1/2/b*x^(1/2)/(b*x^2+a)*c^3-13/16/b^4*(a/b)^(1/4)*a^2*2^(1/2)*ln((x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(
a/b)^(1/2))/(x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2)))*d^3+27/16/b^3*(a/b)^(1/4)*a*2^(1/2)*ln((x+(a/b)^(1/4)
*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2)))*c*d^2-15/16/b^2*(a/b)^(1/4)*2^(1/2)
*ln((x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2)))*c^2*d+1/16/b*(a/b
)^(1/4)/a*2^(1/2)*ln((x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2)))*
c^3-13/8/b^4*(a/b)^(1/4)*a^2*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*d^3+27/8/b^3*(a/b)^(1/4)*a*2^(1/2)*
arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*c*d^2-15/8/b^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)
*c^2*d+1/8/b*(a/b)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)*c^3-13/8/b^4*(a/b)^(1/4)*a^2*2^(1/2)*
arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*d^3+27/8/b^3*(a/b)^(1/4)*a*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)
*c*d^2-15/8/b^2*(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*c^2*d+1/8/b*(a/b)^(1/4)/a*2^(1/2)*ar
ctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)*c^3

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maxima [A]  time = 2.35, size = 445, normalized size = 1.15 \begin {gather*} -\frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {x}}{2 \, {\left (b^{5} x^{2} + a b^{4}\right )}} + \frac {2 \, {\left (5 \, b^{2} d^{3} x^{\frac {9}{2}} + 9 \, {\left (3 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{\frac {5}{2}} + 135 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \sqrt {x}\right )}}{45 \, b^{4}} + \frac {\frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 15 \, a b^{2} c^{2} d + 27 \, a^{2} b c d^{2} - 13 \, a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (b^{3} c^{3} - 15 \, a b^{2} c^{2} d + 27 \, a^{2} b c d^{2} - 13 \, a^{3} d^{3}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (b^{3} c^{3} - 15 \, a b^{2} c^{2} d + 27 \, a^{2} b c d^{2} - 13 \, a^{3} d^{3}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (b^{3} c^{3} - 15 \, a b^{2} c^{2} d + 27 \, a^{2} b c d^{2} - 13 \, a^{3} d^{3}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{16 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(x)/(b^5*x^2 + a*b^4) + 2/45*(5*b^2*d^3*x^(9/2) +
9*(3*b^2*c*d^2 - 2*a*b*d^3)*x^(5/2) + 135*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*sqrt(x))/b^4 + 1/16*(2*sqrt(2)*
(b^3*c^3 - 15*a*b^2*c^2*d + 27*a^2*b*c*d^2 - 13*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(
b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(b^3*c^3 - 15*a*b^2*c^2*d + 27*
a^2*b*c*d^2 - 13*a^3*d^3)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(
b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(b^3*c^3 - 15*a*b^2*c^2*d + 27*a^2*b*c*d^2 - 13*a^3*d^3)*log(sq
rt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(b^3*c^3 - 15*a*b^2*c^2*d + 2
7*a^2*b*c*d^2 - 13*a^3*d^3)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))/b^4

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mupad [B]  time = 0.45, size = 1691, normalized size = 4.38

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(1/2)*((6*c^2*d)/b^2 + (2*a*((4*a*d^3)/b^3 - (6*c*d^2)/b^2))/b - (2*a^2*d^3)/b^4) - x^(5/2)*((4*a*d^3)/(5*b^
3) - (6*c*d^2)/(5*b^2)) + (2*d^3*x^(9/2))/(9*b^2) + (x^(1/2)*((a^3*d^3)/2 - (b^3*c^3)/2 + (3*a*b^2*c^2*d)/2 -
(3*a^2*b*c*d^2)/2))/(a*b^4 + b^5*x^2) + (atan(((((x^(1/2)*(169*a^6*d^6 + b^6*c^6 + 279*a^2*b^4*c^4*d^2 - 836*a
^3*b^3*c^3*d^3 + 1119*a^4*b^2*c^2*d^4 - 30*a*b^5*c^5*d - 702*a^5*b*c*d^5))/b^5 + ((a*d - b*c)^2*(13*a*d - b*c)
*(13*a^4*d^3 - a*b^3*c^3 + 15*a^2*b^2*c^2*d - 27*a^3*b*c*d^2))/((-a)^(3/4)*b^(21/4)))*(a*d - b*c)^2*(13*a*d -
b*c)*1i)/(8*(-a)^(3/4)*b^(17/4)) + (((x^(1/2)*(169*a^6*d^6 + b^6*c^6 + 279*a^2*b^4*c^4*d^2 - 836*a^3*b^3*c^3*d
^3 + 1119*a^4*b^2*c^2*d^4 - 30*a*b^5*c^5*d - 702*a^5*b*c*d^5))/b^5 - ((a*d - b*c)^2*(13*a*d - b*c)*(13*a^4*d^3
- a*b^3*c^3 + 15*a^2*b^2*c^2*d - 27*a^3*b*c*d^2))/((-a)^(3/4)*b^(21/4)))*(a*d - b*c)^2*(13*a*d - b*c)*1i)/(8*
(-a)^(3/4)*b^(17/4)))/((((x^(1/2)*(169*a^6*d^6 + b^6*c^6 + 279*a^2*b^4*c^4*d^2 - 836*a^3*b^3*c^3*d^3 + 1119*a^
4*b^2*c^2*d^4 - 30*a*b^5*c^5*d - 702*a^5*b*c*d^5))/b^5 + ((a*d - b*c)^2*(13*a*d - b*c)*(13*a^4*d^3 - a*b^3*c^3
+ 15*a^2*b^2*c^2*d - 27*a^3*b*c*d^2))/((-a)^(3/4)*b^(21/4)))*(a*d - b*c)^2*(13*a*d - b*c))/(8*(-a)^(3/4)*b^(1
7/4)) - (((x^(1/2)*(169*a^6*d^6 + b^6*c^6 + 279*a^2*b^4*c^4*d^2 - 836*a^3*b^3*c^3*d^3 + 1119*a^4*b^2*c^2*d^4 -
30*a*b^5*c^5*d - 702*a^5*b*c*d^5))/b^5 - ((a*d - b*c)^2*(13*a*d - b*c)*(13*a^4*d^3 - a*b^3*c^3 + 15*a^2*b^2*c
^2*d - 27*a^3*b*c*d^2))/((-a)^(3/4)*b^(21/4)))*(a*d - b*c)^2*(13*a*d - b*c))/(8*(-a)^(3/4)*b^(17/4))))*(a*d -
b*c)^2*(13*a*d - b*c)*1i)/(4*(-a)^(3/4)*b^(17/4)) - (atan(((((x^(1/2)*(169*a^6*d^6 + b^6*c^6 + 279*a^2*b^4*c^4
*d^2 - 836*a^3*b^3*c^3*d^3 + 1119*a^4*b^2*c^2*d^4 - 30*a*b^5*c^5*d - 702*a^5*b*c*d^5))/b^5 - ((a*d - b*c)^2*(1
3*a*d - b*c)*(13*a^4*d^3 - a*b^3*c^3 + 15*a^2*b^2*c^2*d - 27*a^3*b*c*d^2)*1i)/((-a)^(3/4)*b^(21/4)))*(a*d - b*
c)^2*(13*a*d - b*c))/(8*(-a)^(3/4)*b^(17/4)) + (((x^(1/2)*(169*a^6*d^6 + b^6*c^6 + 279*a^2*b^4*c^4*d^2 - 836*a
^3*b^3*c^3*d^3 + 1119*a^4*b^2*c^2*d^4 - 30*a*b^5*c^5*d - 702*a^5*b*c*d^5))/b^5 + ((a*d - b*c)^2*(13*a*d - b*c)
*(13*a^4*d^3 - a*b^3*c^3 + 15*a^2*b^2*c^2*d - 27*a^3*b*c*d^2)*1i)/((-a)^(3/4)*b^(21/4)))*(a*d - b*c)^2*(13*a*d
- b*c))/(8*(-a)^(3/4)*b^(17/4)))/((((x^(1/2)*(169*a^6*d^6 + b^6*c^6 + 279*a^2*b^4*c^4*d^2 - 836*a^3*b^3*c^3*d
^3 + 1119*a^4*b^2*c^2*d^4 - 30*a*b^5*c^5*d - 702*a^5*b*c*d^5))/b^5 - ((a*d - b*c)^2*(13*a*d - b*c)*(13*a^4*d^3
- a*b^3*c^3 + 15*a^2*b^2*c^2*d - 27*a^3*b*c*d^2)*1i)/((-a)^(3/4)*b^(21/4)))*(a*d - b*c)^2*(13*a*d - b*c)*1i)/
(8*(-a)^(3/4)*b^(17/4)) - (((x^(1/2)*(169*a^6*d^6 + b^6*c^6 + 279*a^2*b^4*c^4*d^2 - 836*a^3*b^3*c^3*d^3 + 1119
*a^4*b^2*c^2*d^4 - 30*a*b^5*c^5*d - 702*a^5*b*c*d^5))/b^5 + ((a*d - b*c)^2*(13*a*d - b*c)*(13*a^4*d^3 - a*b^3*
c^3 + 15*a^2*b^2*c^2*d - 27*a^3*b*c*d^2)*1i)/((-a)^(3/4)*b^(21/4)))*(a*d - b*c)^2*(13*a*d - b*c)*1i)/(8*(-a)^(
3/4)*b^(17/4))))*(a*d - b*c)^2*(13*a*d - b*c))/(4*(-a)^(3/4)*b^(17/4))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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