3.5.49 \(\int \frac {1}{x^{3/2} (a+b x^2) (c+d x^2)} \, dx\)

Optimal. Leaf size=476 \[ -\frac {b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)}+\frac {b^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)}+\frac {b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)}-\frac {b^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{5/4} (b c-a d)}+\frac {d^{5/4} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} c^{5/4} (b c-a d)}-\frac {d^{5/4} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} c^{5/4} (b c-a d)}-\frac {d^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{5/4} (b c-a d)}+\frac {d^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} c^{5/4} (b c-a d)}-\frac {2}{a c \sqrt {x}} \]

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Rubi [A]  time = 0.54, antiderivative size = 476, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {466, 480, 584, 297, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)}+\frac {b^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)}+\frac {b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)}-\frac {b^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{5/4} (b c-a d)}+\frac {d^{5/4} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} c^{5/4} (b c-a d)}-\frac {d^{5/4} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} c^{5/4} (b c-a d)}-\frac {d^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{5/4} (b c-a d)}+\frac {d^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} c^{5/4} (b c-a d)}-\frac {2}{a c \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*(a + b*x^2)*(c + d*x^2)),x]

[Out]

-2/(a*c*Sqrt[x]) + (b^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(5/4)*(b*c - a*d)) - (b^
(5/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(5/4)*(b*c - a*d)) - (d^(5/4)*ArcTan[1 - (Sqrt
[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(Sqrt[2]*c^(5/4)*(b*c - a*d)) + (d^(5/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c
^(1/4)])/(Sqrt[2]*c^(5/4)*(b*c - a*d)) - (b^(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/
(2*Sqrt[2]*a^(5/4)*(b*c - a*d)) + (b^(5/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt
[2]*a^(5/4)*(b*c - a*d)) + (d^(5/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*c^(
5/4)*(b*c - a*d)) - (d^(5/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*c^(5/4)*(b
*c - a*d))

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 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 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 480

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

Rule 584

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> Int[ExpandIntegrand[((g*x)^m*(a + b*x^n)^p*(e + f*x^n))/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, g, m, p}, x] && IGtQ[n, 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 {1}{x^{3/2} \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )\\ &=-\frac {2}{a c \sqrt {x}}+\frac {2 \operatorname {Subst}\left (\int \frac {x^2 \left (-b c-a d-b d x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{a c}\\ &=-\frac {2}{a c \sqrt {x}}+\frac {2 \operatorname {Subst}\left (\int \left (-\frac {b^2 c x^2}{(b c-a d) \left (a+b x^4\right )}-\frac {a d^2 x^2}{(-b c+a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{a c}\\ &=-\frac {2}{a c \sqrt {x}}-\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a (b c-a d)}+\frac {\left (2 d^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{c (b c-a d)}\\ &=-\frac {2}{a c \sqrt {x}}+\frac {b^{3/2} \operatorname {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a (b c-a d)}-\frac {b^{3/2} \operatorname {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{a (b c-a d)}-\frac {d^{3/2} \operatorname {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{c (b c-a d)}+\frac {d^{3/2} \operatorname {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{c (b c-a d)}\\ &=-\frac {2}{a c \sqrt {x}}-\frac {b \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 )}{2 a (b c-a d)}-\frac {b \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 )}{2 a (b c-a d)}-\frac {b^{5/4} \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 )}{2 \sqrt {2} a^{5/4} (b c-a d)}-\frac {b^{5/4} \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 )}{2 \sqrt {2} a^{5/4} (b c-a d)}+\frac {d \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 )}{2 c (b c-a d)}+\frac {d \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 )}{2 c (b c-a d)}+\frac {d^{5/4} \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 )}{2 \sqrt {2} c^{5/4} (b c-a d)}+\frac {d^{5/4} \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 )}{2 \sqrt {2} c^{5/4} (b c-a d)}\\ &=-\frac {2}{a c \sqrt {x}}-\frac {b^{5/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)}+\frac {b^{5/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)}+\frac {d^{5/4} \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{5/4} (b c-a d)}-\frac {d^{5/4} \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{5/4} (b c-a d)}-\frac {b^{5/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)}+\frac {b^{5/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)}+\frac {d^{5/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{5/4} (b c-a d)}-\frac {d^{5/4} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{5/4} (b c-a d)}\\ &=-\frac {2}{a c \sqrt {x}}+\frac {b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)}-\frac {b^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} (b c-a d)}-\frac {d^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{5/4} (b c-a d)}+\frac {d^{5/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{5/4} (b c-a d)}-\frac {b^{5/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)}+\frac {b^{5/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} a^{5/4} (b c-a d)}+\frac {d^{5/4} \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{5/4} (b c-a d)}-\frac {d^{5/4} \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} c^{5/4} (b c-a d)}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 409, normalized size = 0.86 \begin {gather*} \frac {\frac {\sqrt {2} b^{5/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{5/4}}-\frac {\sqrt {2} b^{5/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{5/4}}-\frac {2 \sqrt {2} b^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{a^{5/4}}+\frac {2 \sqrt {2} b^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{a^{5/4}}+\frac {8 b}{a \sqrt {x}}-\frac {\sqrt {2} d^{5/4} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{5/4}}+\frac {\sqrt {2} d^{5/4} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{5/4}}+\frac {2 \sqrt {2} d^{5/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{c^{5/4}}-\frac {2 \sqrt {2} d^{5/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{c^{5/4}}-\frac {8 d}{c \sqrt {x}}}{4 a d-4 b c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(3/2)*(a + b*x^2)*(c + d*x^2)),x]

[Out]

((8*b)/(a*Sqrt[x]) - (8*d)/(c*Sqrt[x]) - (2*Sqrt[2]*b^(5/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(
5/4) + (2*Sqrt[2]*b^(5/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(5/4) + (2*Sqrt[2]*d^(5/4)*ArcTan[1
- (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(5/4) - (2*Sqrt[2]*d^(5/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/
4)])/c^(5/4) + (Sqrt[2]*b^(5/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(5/4) - (Sqrt[2]
*b^(5/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(5/4) - (Sqrt[2]*d^(5/4)*Log[Sqrt[c] -
Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(5/4) + (Sqrt[2]*d^(5/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)
*Sqrt[x] + Sqrt[d]*x])/c^(5/4))/(-4*b*c + 4*a*d)

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IntegrateAlgebraic [A]  time = 0.62, size = 280, normalized size = 0.59 \begin {gather*} -\frac {b^{5/4} \tan ^{-1}\left (\frac {\frac {\sqrt [4]{a}}{\sqrt {2} \sqrt [4]{b}}-\frac {\sqrt [4]{b} x}{\sqrt {2} \sqrt [4]{a}}}{\sqrt {x}}\right )}{\sqrt {2} a^{5/4} (a d-b c)}-\frac {b^{5/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt {2} a^{5/4} (a d-b c)}-\frac {d^{5/4} \tan ^{-1}\left (\frac {\frac {\sqrt [4]{c}}{\sqrt {2} \sqrt [4]{d}}-\frac {\sqrt [4]{d} x}{\sqrt {2} \sqrt [4]{c}}}{\sqrt {x}}\right )}{\sqrt {2} c^{5/4} (b c-a d)}-\frac {d^{5/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{\sqrt {2} c^{5/4} (b c-a d)}-\frac {2}{a c \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^(3/2)*(a + b*x^2)*(c + d*x^2)),x]

[Out]

-2/(a*c*Sqrt[x]) - (b^(5/4)*ArcTan[(a^(1/4)/(Sqrt[2]*b^(1/4)) - (b^(1/4)*x)/(Sqrt[2]*a^(1/4)))/Sqrt[x]])/(Sqrt
[2]*a^(5/4)*(-(b*c) + a*d)) - (d^(5/4)*ArcTan[(c^(1/4)/(Sqrt[2]*d^(1/4)) - (d^(1/4)*x)/(Sqrt[2]*c^(1/4)))/Sqrt
[x]])/(Sqrt[2]*c^(5/4)*(b*c - a*d)) - (b^(5/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)
])/(Sqrt[2]*a^(5/4)*(-(b*c) + a*d)) - (d^(5/4)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)
])/(Sqrt[2]*c^(5/4)*(b*c - a*d))

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fricas [B]  time = 2.49, size = 1421, normalized size = 2.99

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(b*x^2+a)/(d*x^2+c),x, algorithm="fricas")

[Out]

-1/2*(4*(-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4))^(1/4)*a*c*x*arcta
n(-(sqrt(b^8*x - (a^3*b^7*c^2 - 2*a^4*b^6*c*d + a^5*b^5*d^2)*sqrt(-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*
b^2*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4)))*(-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*
d^3 + a^9*d^4))^(1/4)*(a*b*c - a^2*d) - (a*b^5*c - a^2*b^4*d)*(-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2
*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4))^(1/4)*sqrt(x))/b^5) - 4*(-d^5/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^
2 - 4*a^3*b*c^6*d^3 + a^4*c^5*d^4))^(1/4)*a*c*x*arctan(-(sqrt(d^8*x - (b^2*c^5*d^5 - 2*a*b*c^4*d^6 + a^2*c^3*d
^7)*sqrt(-d^5/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c^6*d^3 + a^4*c^5*d^4)))*(-d^5/(b^4*c^9 -
4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c^6*d^3 + a^4*c^5*d^4))^(1/4)*(b*c^2 - a*c*d) - (b*c^2*d^4 - a*c*
d^5)*(-d^5/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c^6*d^3 + a^4*c^5*d^4))^(1/4)*sqrt(x))/d^5)
+ (-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4))^(1/4)*a*c*x*log(b^4*sqr
t(x) + (a^4*b^3*c^3 - 3*a^5*b^2*c^2*d + 3*a^6*b*c*d^2 - a^7*d^3)*(-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*
b^2*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4))^(3/4)) - (-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*
a^8*b*c*d^3 + a^9*d^4))^(1/4)*a*c*x*log(b^4*sqrt(x) - (a^4*b^3*c^3 - 3*a^5*b^2*c^2*d + 3*a^6*b*c*d^2 - a^7*d^3
)*(-b^5/(a^5*b^4*c^4 - 4*a^6*b^3*c^3*d + 6*a^7*b^2*c^2*d^2 - 4*a^8*b*c*d^3 + a^9*d^4))^(3/4)) - (-d^5/(b^4*c^9
- 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c^6*d^3 + a^4*c^5*d^4))^(1/4)*a*c*x*log(d^4*sqrt(x) + (b^3*c^7
- 3*a*b^2*c^6*d + 3*a^2*b*c^5*d^2 - a^3*c^4*d^3)*(-d^5/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*
c^6*d^3 + a^4*c^5*d^4))^(3/4)) + (-d^5/(b^4*c^9 - 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c^6*d^3 + a^4*c^
5*d^4))^(1/4)*a*c*x*log(d^4*sqrt(x) - (b^3*c^7 - 3*a*b^2*c^6*d + 3*a^2*b*c^5*d^2 - a^3*c^4*d^3)*(-d^5/(b^4*c^9
- 4*a*b^3*c^8*d + 6*a^2*b^2*c^7*d^2 - 4*a^3*b*c^6*d^3 + a^4*c^5*d^4))^(3/4)) + 4*sqrt(x))/(a*c*x)

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giac [A]  time = 0.86, size = 492, normalized size = 1.03 \begin {gather*} -\frac {\left (a b^{3}\right )^{\frac {3}{4}} \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 )}{\sqrt {2} a^{2} b^{2} c - \sqrt {2} a^{3} b d} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} \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 )}{\sqrt {2} a^{2} b^{2} c - \sqrt {2} a^{3} b d} + \frac {\left (c d^{3}\right )^{\frac {3}{4}} \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 )}{\sqrt {2} b c^{3} d - \sqrt {2} a c^{2} d^{2}} + \frac {\left (c d^{3}\right )^{\frac {3}{4}} \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 )}{\sqrt {2} b c^{3} d - \sqrt {2} a c^{2} d^{2}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a^{2} b^{2} c - \sqrt {2} a^{3} b d\right )}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} a^{2} b^{2} c - \sqrt {2} a^{3} b d\right )}} - \frac {\left (c d^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{2 \, {\left (\sqrt {2} b c^{3} d - \sqrt {2} a c^{2} d^{2}\right )}} + \frac {\left (c d^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {c}{d}\right )^{\frac {1}{4}} + x + \sqrt {\frac {c}{d}}\right )}{2 \, {\left (\sqrt {2} b c^{3} d - \sqrt {2} a c^{2} d^{2}\right )}} - \frac {2}{a c \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(b*x^2+a)/(d*x^2+c),x, algorithm="giac")

[Out]

-(a*b^3)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^2*b^2*c - sqrt(2)*
a^3*b*d) - (a*b^3)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*a^2*b^2*c
- sqrt(2)*a^3*b*d) + (c*d^3)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)
*b*c^3*d - sqrt(2)*a*c^2*d^2) + (c*d^3)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4
))/(sqrt(2)*b*c^3*d - sqrt(2)*a*c^2*d^2) + 1/2*(a*b^3)^(3/4)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/
(sqrt(2)*a^2*b^2*c - sqrt(2)*a^3*b*d) - 1/2*(a*b^3)^(3/4)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(s
qrt(2)*a^2*b^2*c - sqrt(2)*a^3*b*d) - 1/2*(c*d^3)^(3/4)*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt
(2)*b*c^3*d - sqrt(2)*a*c^2*d^2) + 1/2*(c*d^3)^(3/4)*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2
)*b*c^3*d - sqrt(2)*a*c^2*d^2) - 2/(a*c*sqrt(x))

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maple [A]  time = 0.02, size = 339, normalized size = 0.71 \begin {gather*} \frac {\sqrt {2}\, b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} a}+\frac {\sqrt {2}\, b \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} a}+\frac {\sqrt {2}\, b \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 )}{4 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} a}-\frac {\sqrt {2}\, d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{2 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} c}-\frac {\sqrt {2}\, d \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{2 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} c}-\frac {\sqrt {2}\, d \ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{4 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} c}-\frac {2}{a c \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(3/2)/(b*x^2+a)/(d*x^2+c),x)

[Out]

1/4*b/a/(a*d-b*c)/(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)))+1/2*b/a/(a*d-b*c)/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+1/2*b/a/(a*d-b
*c)/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)-1/4*d/c/(a*d-b*c)/(c/d)^(1/4)*2^(1/2)*ln((x-(c/d
)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2))/(x+(c/d)^(1/4)*2^(1/2)*x^(1/2)+(c/d)^(1/2)))-1/2*d/c/(a*d-b*c)/(c/d)^(1/4
)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)-1/2*d/c/(a*d-b*c)/(c/d)^(1/4)*2^(1/2)*arctan(2^(1/2)/(c/d)^(1/
4)*x^(1/2)-1)-2/a/c/x^(1/2)

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maxima [A]  time = 2.54, size = 390, normalized size = 0.82 \begin {gather*} -\frac {b^{2} {\left (\frac {2 \, \sqrt {2} \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 {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \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 {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, {\left (a b c - a^{2} d\right )}} + \frac {d^{2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{4 \, {\left (b c^{2} - a c d\right )}} - \frac {2}{a c \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(b*x^2+a)/(d*x^2+c),x, algorithm="maxima")

[Out]

-1/4*b^2*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(s
qrt(sqrt(a)*sqrt(b))*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sq
rt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqrt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x
+ sqrt(a))/(a^(1/4)*b^(3/4)) + sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b
^(3/4)))/(a*b*c - a^2*d) + 1/4*d^2*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/4) + 2*sqrt(d)*sqrt(x))
/sqrt(sqrt(c)*sqrt(d)))/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*c^(1/4)*d^(1/
4) - 2*sqrt(d)*sqrt(x))/sqrt(sqrt(c)*sqrt(d)))/(sqrt(sqrt(c)*sqrt(d))*sqrt(d)) - sqrt(2)*log(sqrt(2)*c^(1/4)*d
^(1/4)*sqrt(x) + sqrt(d)*x + sqrt(c))/(c^(1/4)*d^(3/4)) + sqrt(2)*log(-sqrt(2)*c^(1/4)*d^(1/4)*sqrt(x) + sqrt(
d)*x + sqrt(c))/(c^(1/4)*d^(3/4)))/(b*c^2 - a*c*d) - 2/(a*c*sqrt(x))

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mupad [B]  time = 1.64, size = 6038, normalized size = 12.68

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(3/2)*(a + b*x^2)*(c + d*x^2)),x)

[Out]

atan((a^6*b^8*c^9*x^(1/2)*(-b^5/(16*a^9*d^4 + 16*a^5*b^4*c^4 - 64*a^6*b^3*c^3*d + 96*a^7*b^2*c^2*d^2 - 64*a^8*
b*c*d^3))^(5/4)*32i + a^6*b^4*d^5*x^(1/2)*(-b^5/(16*a^9*d^4 + 16*a^5*b^4*c^4 - 64*a^6*b^3*c^3*d + 96*a^7*b^2*c
^2*d^2 - 64*a^8*b*c*d^3))^(1/4)*2i + a^14*c*d^8*x^(1/2)*(-b^5/(16*a^9*d^4 + 16*a^5*b^4*c^4 - 64*a^6*b^3*c^3*d
+ 96*a^7*b^2*c^2*d^2 - 64*a^8*b*c*d^3))^(5/4)*32i + a^8*b^6*c^7*d^2*x^(1/2)*(-b^5/(16*a^9*d^4 + 16*a^5*b^4*c^4
- 64*a^6*b^3*c^3*d + 96*a^7*b^2*c^2*d^2 - 64*a^8*b*c*d^3))^(5/4)*192i - a^9*b^5*c^6*d^3*x^(1/2)*(-b^5/(16*a^9
*d^4 + 16*a^5*b^4*c^4 - 64*a^6*b^3*c^3*d + 96*a^7*b^2*c^2*d^2 - 64*a^8*b*c*d^3))^(5/4)*128i + a^10*b^4*c^5*d^4
*x^(1/2)*(-b^5/(16*a^9*d^4 + 16*a^5*b^4*c^4 - 64*a^6*b^3*c^3*d + 96*a^7*b^2*c^2*d^2 - 64*a^8*b*c*d^3))^(5/4)*6
4i - a^11*b^3*c^4*d^5*x^(1/2)*(-b^5/(16*a^9*d^4 + 16*a^5*b^4*c^4 - 64*a^6*b^3*c^3*d + 96*a^7*b^2*c^2*d^2 - 64*
a^8*b*c*d^3))^(5/4)*128i + a^12*b^2*c^3*d^6*x^(1/2)*(-b^5/(16*a^9*d^4 + 16*a^5*b^4*c^4 - 64*a^6*b^3*c^3*d + 96
*a^7*b^2*c^2*d^2 - 64*a^8*b*c*d^3))^(5/4)*192i + a^5*b^5*c*d^4*x^(1/2)*(-b^5/(16*a^9*d^4 + 16*a^5*b^4*c^4 - 64
*a^6*b^3*c^3*d + 96*a^7*b^2*c^2*d^2 - 64*a^8*b*c*d^3))^(1/4)*2i - a^7*b^7*c^8*d*x^(1/2)*(-b^5/(16*a^9*d^4 + 16
*a^5*b^4*c^4 - 64*a^6*b^3*c^3*d + 96*a^7*b^2*c^2*d^2 - 64*a^8*b*c*d^3))^(5/4)*128i - a^13*b*c^2*d^7*x^(1/2)*(-
b^5/(16*a^9*d^4 + 16*a^5*b^4*c^4 - 64*a^6*b^3*c^3*d + 96*a^7*b^2*c^2*d^2 - 64*a^8*b*c*d^3))^(5/4)*128i)/(b^9*c
^4 + a^4*b^5*d^4 + a^3*b^6*c*d^3 + a^2*b^7*c^2*d^2 + a*b^8*c^3*d))*(-b^5/(16*a^9*d^4 + 16*a^5*b^4*c^4 - 64*a^6
*b^3*c^3*d + 96*a^7*b^2*c^2*d^2 - 64*a^8*b*c*d^3))^(1/4)*2i + atan((a^9*c^6*d^8*x^(1/2)*(-d^5/(16*b^4*c^9 + 16
*a^4*c^5*d^4 - 64*a^3*b*c^6*d^3 + 96*a^2*b^2*c^7*d^2 - 64*a*b^3*c^8*d))^(5/4)*32i + b^5*c^6*d^4*x^(1/2)*(-d^5/
(16*b^4*c^9 + 16*a^4*c^5*d^4 - 64*a^3*b*c^6*d^3 + 96*a^2*b^2*c^7*d^2 - 64*a*b^3*c^8*d))^(1/4)*2i + a*b^8*c^14*
x^(1/2)*(-d^5/(16*b^4*c^9 + 16*a^4*c^5*d^4 - 64*a^3*b*c^6*d^3 + 96*a^2*b^2*c^7*d^2 - 64*a*b^3*c^8*d))^(5/4)*32
i + a^3*b^6*c^12*d^2*x^(1/2)*(-d^5/(16*b^4*c^9 + 16*a^4*c^5*d^4 - 64*a^3*b*c^6*d^3 + 96*a^2*b^2*c^7*d^2 - 64*a
*b^3*c^8*d))^(5/4)*192i - a^4*b^5*c^11*d^3*x^(1/2)*(-d^5/(16*b^4*c^9 + 16*a^4*c^5*d^4 - 64*a^3*b*c^6*d^3 + 96*
a^2*b^2*c^7*d^2 - 64*a*b^3*c^8*d))^(5/4)*128i + a^5*b^4*c^10*d^4*x^(1/2)*(-d^5/(16*b^4*c^9 + 16*a^4*c^5*d^4 -
64*a^3*b*c^6*d^3 + 96*a^2*b^2*c^7*d^2 - 64*a*b^3*c^8*d))^(5/4)*64i - a^6*b^3*c^9*d^5*x^(1/2)*(-d^5/(16*b^4*c^9
+ 16*a^4*c^5*d^4 - 64*a^3*b*c^6*d^3 + 96*a^2*b^2*c^7*d^2 - 64*a*b^3*c^8*d))^(5/4)*128i + a^7*b^2*c^8*d^6*x^(1
/2)*(-d^5/(16*b^4*c^9 + 16*a^4*c^5*d^4 - 64*a^3*b*c^6*d^3 + 96*a^2*b^2*c^7*d^2 - 64*a*b^3*c^8*d))^(5/4)*192i +
a*b^4*c^5*d^5*x^(1/2)*(-d^5/(16*b^4*c^9 + 16*a^4*c^5*d^4 - 64*a^3*b*c^6*d^3 + 96*a^2*b^2*c^7*d^2 - 64*a*b^3*c
^8*d))^(1/4)*2i - a^2*b^7*c^13*d*x^(1/2)*(-d^5/(16*b^4*c^9 + 16*a^4*c^5*d^4 - 64*a^3*b*c^6*d^3 + 96*a^2*b^2*c^
7*d^2 - 64*a*b^3*c^8*d))^(5/4)*128i - a^8*b*c^7*d^7*x^(1/2)*(-d^5/(16*b^4*c^9 + 16*a^4*c^5*d^4 - 64*a^3*b*c^6*
d^3 + 96*a^2*b^2*c^7*d^2 - 64*a*b^3*c^8*d))^(5/4)*128i)/(a^4*d^9 + b^4*c^4*d^5 + a*b^3*c^3*d^6 + a^2*b^2*c^2*d
^7 + a^3*b*c*d^8))*(-d^5/(16*b^4*c^9 + 16*a^4*c^5*d^4 - 64*a^3*b*c^6*d^3 + 96*a^2*b^2*c^7*d^2 - 64*a*b^3*c^8*d
))^(1/4)*2i + 2*atan(((-b^5/(16*a^9*d^4 + 16*a^5*b^4*c^4 - 64*a^6*b^3*c^3*d + 96*a^7*b^2*c^2*d^2 - 64*a^8*b*c*
d^3))^(1/4)*(x^(1/2)*(256*a^11*b^9*c^12*d^8 + 256*a^12*b^8*c^11*d^9) - (-b^5/(16*a^9*d^4 + 16*a^5*b^4*c^4 - 64
*a^6*b^3*c^3*d + 96*a^7*b^2*c^2*d^2 - 64*a^8*b*c*d^3))^(3/4)*(x^(1/2)*(-b^5/(16*a^9*d^4 + 16*a^5*b^4*c^4 - 64*
a^6*b^3*c^3*d + 96*a^7*b^2*c^2*d^2 - 64*a^8*b*c*d^3))^(1/4)*(4096*a^12*b^12*c^20*d^4 - 16384*a^13*b^11*c^19*d^
5 + 24576*a^14*b^10*c^18*d^6 - 16384*a^15*b^9*c^17*d^7 + 8192*a^16*b^8*c^16*d^8 - 16384*a^17*b^7*c^15*d^9 + 24
576*a^18*b^6*c^14*d^10 - 16384*a^19*b^5*c^13*d^11 + 4096*a^20*b^4*c^12*d^12)*1i - 2048*a^11*b^12*c^19*d^4 + 61
44*a^12*b^11*c^18*d^5 - 6144*a^13*b^10*c^17*d^6 + 2048*a^14*b^9*c^16*d^7 + 2048*a^16*b^7*c^14*d^9 - 6144*a^17*
b^6*c^13*d^10 + 6144*a^18*b^5*c^12*d^11 - 2048*a^19*b^4*c^11*d^12)*1i) + (-b^5/(16*a^9*d^4 + 16*a^5*b^4*c^4 -
64*a^6*b^3*c^3*d + 96*a^7*b^2*c^2*d^2 - 64*a^8*b*c*d^3))^(1/4)*(x^(1/2)*(256*a^11*b^9*c^12*d^8 + 256*a^12*b^8*
c^11*d^9) - (-b^5/(16*a^9*d^4 + 16*a^5*b^4*c^4 - 64*a^6*b^3*c^3*d + 96*a^7*b^2*c^2*d^2 - 64*a^8*b*c*d^3))^(3/4
)*(x^(1/2)*(-b^5/(16*a^9*d^4 + 16*a^5*b^4*c^4 - 64*a^6*b^3*c^3*d + 96*a^7*b^2*c^2*d^2 - 64*a^8*b*c*d^3))^(1/4)
*(4096*a^12*b^12*c^20*d^4 - 16384*a^13*b^11*c^19*d^5 + 24576*a^14*b^10*c^18*d^6 - 16384*a^15*b^9*c^17*d^7 + 81
92*a^16*b^8*c^16*d^8 - 16384*a^17*b^7*c^15*d^9 + 24576*a^18*b^6*c^14*d^10 - 16384*a^19*b^5*c^13*d^11 + 4096*a^
20*b^4*c^12*d^12)*1i + 2048*a^11*b^12*c^19*d^4 - 6144*a^12*b^11*c^18*d^5 + 6144*a^13*b^10*c^17*d^6 - 2048*a^14
*b^9*c^16*d^7 - 2048*a^16*b^7*c^14*d^9 + 6144*a^17*b^6*c^13*d^10 - 6144*a^18*b^5*c^12*d^11 + 2048*a^19*b^4*c^1
1*d^12)*1i))/((-b^5/(16*a^9*d^4 + 16*a^5*b^4*c^4 - 64*a^6*b^3*c^3*d + 96*a^7*b^2*c^2*d^2 - 64*a^8*b*c*d^3))^(1
/4)*(x^(1/2)*(256*a^11*b^9*c^12*d^8 + 256*a^12*b^8*c^11*d^9) - (-b^5/(16*a^9*d^4 + 16*a^5*b^4*c^4 - 64*a^6*b^3
*c^3*d + 96*a^7*b^2*c^2*d^2 - 64*a^8*b*c*d^3))^(3/4)*(x^(1/2)*(-b^5/(16*a^9*d^4 + 16*a^5*b^4*c^4 - 64*a^6*b^3*
c^3*d + 96*a^7*b^2*c^2*d^2 - 64*a^8*b*c*d^3))^(1/4)*(4096*a^12*b^12*c^20*d^4 - 16384*a^13*b^11*c^19*d^5 + 2457
6*a^14*b^10*c^18*d^6 - 16384*a^15*b^9*c^17*d^7 + 8192*a^16*b^8*c^16*d^8 - 16384*a^17*b^7*c^15*d^9 + 24576*a^18
*b^6*c^14*d^10 - 16384*a^19*b^5*c^13*d^11 + 4096*a^20*b^4*c^12*d^12)*1i - 2048*a^11*b^12*c^19*d^4 + 6144*a^12*
b^11*c^18*d^5 - 6144*a^13*b^10*c^17*d^6 + 2048*a^14*b^9*c^16*d^7 + 2048*a^16*b^7*c^14*d^9 - 6144*a^17*b^6*c^13
*d^10 + 6144*a^18*b^5*c^12*d^11 - 2048*a^19*b^4*c^11*d^12)*1i)*1i - (-b^5/(16*a^9*d^4 + 16*a^5*b^4*c^4 - 64*a^
6*b^3*c^3*d + 96*a^7*b^2*c^2*d^2 - 64*a^8*b*c*d^3))^(1/4)*(x^(1/2)*(256*a^11*b^9*c^12*d^8 + 256*a^12*b^8*c^11*
d^9) - (-b^5/(16*a^9*d^4 + 16*a^5*b^4*c^4 - 64*a^6*b^3*c^3*d + 96*a^7*b^2*c^2*d^2 - 64*a^8*b*c*d^3))^(3/4)*(x^
(1/2)*(-b^5/(16*a^9*d^4 + 16*a^5*b^4*c^4 - 64*a^6*b^3*c^3*d + 96*a^7*b^2*c^2*d^2 - 64*a^8*b*c*d^3))^(1/4)*(409
6*a^12*b^12*c^20*d^4 - 16384*a^13*b^11*c^19*d^5 + 24576*a^14*b^10*c^18*d^6 - 16384*a^15*b^9*c^17*d^7 + 8192*a^
16*b^8*c^16*d^8 - 16384*a^17*b^7*c^15*d^9 + 24576*a^18*b^6*c^14*d^10 - 16384*a^19*b^5*c^13*d^11 + 4096*a^20*b^
4*c^12*d^12)*1i + 2048*a^11*b^12*c^19*d^4 - 6144*a^12*b^11*c^18*d^5 + 6144*a^13*b^10*c^17*d^6 - 2048*a^14*b^9*
c^16*d^7 - 2048*a^16*b^7*c^14*d^9 + 6144*a^17*b^6*c^13*d^10 - 6144*a^18*b^5*c^12*d^11 + 2048*a^19*b^4*c^11*d^1
2)*1i)*1i))*(-b^5/(16*a^9*d^4 + 16*a^5*b^4*c^4 - 64*a^6*b^3*c^3*d + 96*a^7*b^2*c^2*d^2 - 64*a^8*b*c*d^3))^(1/4
) + 2*atan(((-d^5/(16*b^4*c^9 + 16*a^4*c^5*d^4 - 64*a^3*b*c^6*d^3 + 96*a^2*b^2*c^7*d^2 - 64*a*b^3*c^8*d))^(1/4
)*(x^(1/2)*(256*a^11*b^9*c^12*d^8 + 256*a^12*b^8*c^11*d^9) - (-d^5/(16*b^4*c^9 + 16*a^4*c^5*d^4 - 64*a^3*b*c^6
*d^3 + 96*a^2*b^2*c^7*d^2 - 64*a*b^3*c^8*d))^(3/4)*(x^(1/2)*(-d^5/(16*b^4*c^9 + 16*a^4*c^5*d^4 - 64*a^3*b*c^6*
d^3 + 96*a^2*b^2*c^7*d^2 - 64*a*b^3*c^8*d))^(1/4)*(4096*a^12*b^12*c^20*d^4 - 16384*a^13*b^11*c^19*d^5 + 24576*
a^14*b^10*c^18*d^6 - 16384*a^15*b^9*c^17*d^7 + 8192*a^16*b^8*c^16*d^8 - 16384*a^17*b^7*c^15*d^9 + 24576*a^18*b
^6*c^14*d^10 - 16384*a^19*b^5*c^13*d^11 + 4096*a^20*b^4*c^12*d^12)*1i - 2048*a^11*b^12*c^19*d^4 + 6144*a^12*b^
11*c^18*d^5 - 6144*a^13*b^10*c^17*d^6 + 2048*a^14*b^9*c^16*d^7 + 2048*a^16*b^7*c^14*d^9 - 6144*a^17*b^6*c^13*d
^10 + 6144*a^18*b^5*c^12*d^11 - 2048*a^19*b^4*c^11*d^12)*1i) + (-d^5/(16*b^4*c^9 + 16*a^4*c^5*d^4 - 64*a^3*b*c
^6*d^3 + 96*a^2*b^2*c^7*d^2 - 64*a*b^3*c^8*d))^(1/4)*(x^(1/2)*(256*a^11*b^9*c^12*d^8 + 256*a^12*b^8*c^11*d^9)
- (-d^5/(16*b^4*c^9 + 16*a^4*c^5*d^4 - 64*a^3*b*c^6*d^3 + 96*a^2*b^2*c^7*d^2 - 64*a*b^3*c^8*d))^(3/4)*(x^(1/2)
*(-d^5/(16*b^4*c^9 + 16*a^4*c^5*d^4 - 64*a^3*b*c^6*d^3 + 96*a^2*b^2*c^7*d^2 - 64*a*b^3*c^8*d))^(1/4)*(4096*a^1
2*b^12*c^20*d^4 - 16384*a^13*b^11*c^19*d^5 + 24576*a^14*b^10*c^18*d^6 - 16384*a^15*b^9*c^17*d^7 + 8192*a^16*b^
8*c^16*d^8 - 16384*a^17*b^7*c^15*d^9 + 24576*a^18*b^6*c^14*d^10 - 16384*a^19*b^5*c^13*d^11 + 4096*a^20*b^4*c^1
2*d^12)*1i + 2048*a^11*b^12*c^19*d^4 - 6144*a^12*b^11*c^18*d^5 + 6144*a^13*b^10*c^17*d^6 - 2048*a^14*b^9*c^16*
d^7 - 2048*a^16*b^7*c^14*d^9 + 6144*a^17*b^6*c^13*d^10 - 6144*a^18*b^5*c^12*d^11 + 2048*a^19*b^4*c^11*d^12)*1i
))/((-d^5/(16*b^4*c^9 + 16*a^4*c^5*d^4 - 64*a^3*b*c^6*d^3 + 96*a^2*b^2*c^7*d^2 - 64*a*b^3*c^8*d))^(1/4)*(x^(1/
2)*(256*a^11*b^9*c^12*d^8 + 256*a^12*b^8*c^11*d^9) - (-d^5/(16*b^4*c^9 + 16*a^4*c^5*d^4 - 64*a^3*b*c^6*d^3 + 9
6*a^2*b^2*c^7*d^2 - 64*a*b^3*c^8*d))^(3/4)*(x^(1/2)*(-d^5/(16*b^4*c^9 + 16*a^4*c^5*d^4 - 64*a^3*b*c^6*d^3 + 96
*a^2*b^2*c^7*d^2 - 64*a*b^3*c^8*d))^(1/4)*(4096*a^12*b^12*c^20*d^4 - 16384*a^13*b^11*c^19*d^5 + 24576*a^14*b^1
0*c^18*d^6 - 16384*a^15*b^9*c^17*d^7 + 8192*a^16*b^8*c^16*d^8 - 16384*a^17*b^7*c^15*d^9 + 24576*a^18*b^6*c^14*
d^10 - 16384*a^19*b^5*c^13*d^11 + 4096*a^20*b^4*c^12*d^12)*1i - 2048*a^11*b^12*c^19*d^4 + 6144*a^12*b^11*c^18*
d^5 - 6144*a^13*b^10*c^17*d^6 + 2048*a^14*b^9*c^16*d^7 + 2048*a^16*b^7*c^14*d^9 - 6144*a^17*b^6*c^13*d^10 + 61
44*a^18*b^5*c^12*d^11 - 2048*a^19*b^4*c^11*d^12)*1i)*1i - (-d^5/(16*b^4*c^9 + 16*a^4*c^5*d^4 - 64*a^3*b*c^6*d^
3 + 96*a^2*b^2*c^7*d^2 - 64*a*b^3*c^8*d))^(1/4)*(x^(1/2)*(256*a^11*b^9*c^12*d^8 + 256*a^12*b^8*c^11*d^9) - (-d
^5/(16*b^4*c^9 + 16*a^4*c^5*d^4 - 64*a^3*b*c^6*d^3 + 96*a^2*b^2*c^7*d^2 - 64*a*b^3*c^8*d))^(3/4)*(x^(1/2)*(-d^
5/(16*b^4*c^9 + 16*a^4*c^5*d^4 - 64*a^3*b*c^6*d^3 + 96*a^2*b^2*c^7*d^2 - 64*a*b^3*c^8*d))^(1/4)*(4096*a^12*b^1
2*c^20*d^4 - 16384*a^13*b^11*c^19*d^5 + 24576*a^14*b^10*c^18*d^6 - 16384*a^15*b^9*c^17*d^7 + 8192*a^16*b^8*c^1
6*d^8 - 16384*a^17*b^7*c^15*d^9 + 24576*a^18*b^6*c^14*d^10 - 16384*a^19*b^5*c^13*d^11 + 4096*a^20*b^4*c^12*d^1
2)*1i + 2048*a^11*b^12*c^19*d^4 - 6144*a^12*b^11*c^18*d^5 + 6144*a^13*b^10*c^17*d^6 - 2048*a^14*b^9*c^16*d^7 -
2048*a^16*b^7*c^14*d^9 + 6144*a^17*b^6*c^13*d^10 - 6144*a^18*b^5*c^12*d^11 + 2048*a^19*b^4*c^11*d^12)*1i)*1i)
)*(-d^5/(16*b^4*c^9 + 16*a^4*c^5*d^4 - 64*a^3*b*c^6*d^3 + 96*a^2*b^2*c^7*d^2 - 64*a*b^3*c^8*d))^(1/4) - 2/(a*c
*x^(1/2))

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

[Out]

Timed out

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