3.469 \(\int \frac {1}{x^{7/2} (a+b x^2) (c+d x^2)} \, dx\)

Optimal. Leaf size=498 \[ \frac {b^{9/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)}-\frac {b^{9/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)}-\frac {b^{9/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)}+\frac {b^{9/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{9/4} (b c-a d)}+\frac {2 (a d+b c)}{a^2 c^2 \sqrt {x}}-\frac {d^{9/4} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} c^{9/4} (b c-a d)}+\frac {d^{9/4} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} c^{9/4} (b c-a d)}+\frac {d^{9/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{9/4} (b c-a d)}-\frac {d^{9/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} c^{9/4} (b c-a d)}-\frac {2}{5 a c x^{5/2}} \]

[Out]

-2/5/a/c/x^(5/2)-1/2*b^(9/4)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(9/4)/(-a*d+b*c)*2^(1/2)+1/2*b^(9/4)*
arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(9/4)/(-a*d+b*c)*2^(1/2)+1/2*d^(9/4)*arctan(1-d^(1/4)*2^(1/2)*x^(1
/2)/c^(1/4))/c^(9/4)/(-a*d+b*c)*2^(1/2)-1/2*d^(9/4)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/c^(9/4)/(-a*d+b*
c)*2^(1/2)+1/4*b^(9/4)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(9/4)/(-a*d+b*c)*2^(1/2)-1/4*b^
(9/4)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(9/4)/(-a*d+b*c)*2^(1/2)-1/4*d^(9/4)*ln(c^(1/2)+
x*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/c^(9/4)/(-a*d+b*c)*2^(1/2)+1/4*d^(9/4)*ln(c^(1/2)+x*d^(1/2)+c^(1/4)
*d^(1/4)*2^(1/2)*x^(1/2))/c^(9/4)/(-a*d+b*c)*2^(1/2)+2*(a*d+b*c)/a^2/c^2/x^(1/2)

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Rubi [A]  time = 0.69, antiderivative size = 498, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {466, 480, 583, 584, 297, 1162, 617, 204, 1165, 628} \[ \frac {b^{9/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)}-\frac {b^{9/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} a^{9/4} (b c-a d)}-\frac {b^{9/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{9/4} (b c-a d)}+\frac {b^{9/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} a^{9/4} (b c-a d)}+\frac {2 (a d+b c)}{a^2 c^2 \sqrt {x}}-\frac {d^{9/4} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} c^{9/4} (b c-a d)}+\frac {d^{9/4} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} c^{9/4} (b c-a d)}+\frac {d^{9/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} c^{9/4} (b c-a d)}-\frac {d^{9/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} c^{9/4} (b c-a d)}-\frac {2}{5 a c x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)),x]

[Out]

-2/(5*a*c*x^(5/2)) + (2*(b*c + a*d))/(a^2*c^2*Sqrt[x]) - (b^(9/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)
])/(Sqrt[2]*a^(9/4)*(b*c - a*d)) + (b^(9/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(9/4)*(b
*c - a*d)) + (d^(9/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(Sqrt[2]*c^(9/4)*(b*c - a*d)) - (d^(9/4)*
ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(Sqrt[2]*c^(9/4)*(b*c - a*d)) + (b^(9/4)*Log[Sqrt[a] - Sqrt[2]*
a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(9/4)*(b*c - a*d)) - (b^(9/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)
*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(9/4)*(b*c - a*d)) - (d^(9/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4
)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*c^(9/4)*(b*c - a*d)) + (d^(9/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[
x] + Sqrt[d]*x])/(2*Sqrt[2]*c^(9/4)*(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 583

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

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

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Mathematica [A]  time = 0.28, size = 437, normalized size = 0.88 \[ \frac {-\frac {5 \sqrt {2} b^{9/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{9/4}}+\frac {5 \sqrt {2} b^{9/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{a^{9/4}}+\frac {10 \sqrt {2} b^{9/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{a^{9/4}}-\frac {10 \sqrt {2} b^{9/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{a^{9/4}}-\frac {40 b^2}{a^2 \sqrt {x}}+\frac {8 b}{a x^{5/2}}+\frac {5 \sqrt {2} d^{9/4} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{9/4}}-\frac {5 \sqrt {2} d^{9/4} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{c^{9/4}}-\frac {10 \sqrt {2} d^{9/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{c^{9/4}}+\frac {10 \sqrt {2} d^{9/4} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{c^{9/4}}+\frac {40 d^2}{c^2 \sqrt {x}}-\frac {8 d}{c x^{5/2}}}{20 a d-20 b c} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)),x]

[Out]

((8*b)/(a*x^(5/2)) - (8*d)/(c*x^(5/2)) - (40*b^2)/(a^2*Sqrt[x]) + (40*d^2)/(c^2*Sqrt[x]) + (10*Sqrt[2]*b^(9/4)
*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/a^(9/4) - (10*Sqrt[2]*b^(9/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt
[x])/a^(1/4)])/a^(9/4) - (10*Sqrt[2]*d^(9/4)*ArcTan[1 - (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(9/4) + (10*Sqrt
[2]*d^(9/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/c^(9/4) - (5*Sqrt[2]*b^(9/4)*Log[Sqrt[a] - Sqrt[2]*
a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/a^(9/4) + (5*Sqrt[2]*b^(9/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[
x] + Sqrt[b]*x])/a^(9/4) + (5*Sqrt[2]*d^(9/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(9
/4) - (5*Sqrt[2]*d^(9/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/c^(9/4))/(-20*b*c + 20*a*
d)

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fricas [B]  time = 9.86, size = 1484, normalized size = 2.98 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(b*x^2+a)/(d*x^2+c),x, algorithm="fricas")

[Out]

1/10*(20*(-b^9/(a^9*b^4*c^4 - 4*a^10*b^3*c^3*d + 6*a^11*b^2*c^2*d^2 - 4*a^12*b*c*d^3 + a^13*d^4))^(1/4)*a^2*c^
2*x^3*arctan(-(sqrt(b^14*x - (a^5*b^11*c^2 - 2*a^6*b^10*c*d + a^7*b^9*d^2)*sqrt(-b^9/(a^9*b^4*c^4 - 4*a^10*b^3
*c^3*d + 6*a^11*b^2*c^2*d^2 - 4*a^12*b*c*d^3 + a^13*d^4)))*(-b^9/(a^9*b^4*c^4 - 4*a^10*b^3*c^3*d + 6*a^11*b^2*
c^2*d^2 - 4*a^12*b*c*d^3 + a^13*d^4))^(1/4)*(a^2*b*c - a^3*d) - (a^2*b^8*c - a^3*b^7*d)*(-b^9/(a^9*b^4*c^4 - 4
*a^10*b^3*c^3*d + 6*a^11*b^2*c^2*d^2 - 4*a^12*b*c*d^3 + a^13*d^4))^(1/4)*sqrt(x))/b^9) - 20*(-d^9/(b^4*c^13 -
4*a*b^3*c^12*d + 6*a^2*b^2*c^11*d^2 - 4*a^3*b*c^10*d^3 + a^4*c^9*d^4))^(1/4)*a^2*c^2*x^3*arctan(-(sqrt(d^14*x
- (b^2*c^7*d^9 - 2*a*b*c^6*d^10 + a^2*c^5*d^11)*sqrt(-d^9/(b^4*c^13 - 4*a*b^3*c^12*d + 6*a^2*b^2*c^11*d^2 - 4*
a^3*b*c^10*d^3 + a^4*c^9*d^4)))*(-d^9/(b^4*c^13 - 4*a*b^3*c^12*d + 6*a^2*b^2*c^11*d^2 - 4*a^3*b*c^10*d^3 + a^4
*c^9*d^4))^(1/4)*(b*c^3 - a*c^2*d) - (b*c^3*d^7 - a*c^2*d^8)*(-d^9/(b^4*c^13 - 4*a*b^3*c^12*d + 6*a^2*b^2*c^11
*d^2 - 4*a^3*b*c^10*d^3 + a^4*c^9*d^4))^(1/4)*sqrt(x))/d^9) + 5*(-b^9/(a^9*b^4*c^4 - 4*a^10*b^3*c^3*d + 6*a^11
*b^2*c^2*d^2 - 4*a^12*b*c*d^3 + a^13*d^4))^(1/4)*a^2*c^2*x^3*log(b^7*sqrt(x) + (a^7*b^3*c^3 - 3*a^8*b^2*c^2*d
+ 3*a^9*b*c*d^2 - a^10*d^3)*(-b^9/(a^9*b^4*c^4 - 4*a^10*b^3*c^3*d + 6*a^11*b^2*c^2*d^2 - 4*a^12*b*c*d^3 + a^13
*d^4))^(3/4)) - 5*(-b^9/(a^9*b^4*c^4 - 4*a^10*b^3*c^3*d + 6*a^11*b^2*c^2*d^2 - 4*a^12*b*c*d^3 + a^13*d^4))^(1/
4)*a^2*c^2*x^3*log(b^7*sqrt(x) - (a^7*b^3*c^3 - 3*a^8*b^2*c^2*d + 3*a^9*b*c*d^2 - a^10*d^3)*(-b^9/(a^9*b^4*c^4
- 4*a^10*b^3*c^3*d + 6*a^11*b^2*c^2*d^2 - 4*a^12*b*c*d^3 + a^13*d^4))^(3/4)) - 5*(-d^9/(b^4*c^13 - 4*a*b^3*c^
12*d + 6*a^2*b^2*c^11*d^2 - 4*a^3*b*c^10*d^3 + a^4*c^9*d^4))^(1/4)*a^2*c^2*x^3*log(d^7*sqrt(x) + (b^3*c^10 - 3
*a*b^2*c^9*d + 3*a^2*b*c^8*d^2 - a^3*c^7*d^3)*(-d^9/(b^4*c^13 - 4*a*b^3*c^12*d + 6*a^2*b^2*c^11*d^2 - 4*a^3*b*
c^10*d^3 + a^4*c^9*d^4))^(3/4)) + 5*(-d^9/(b^4*c^13 - 4*a*b^3*c^12*d + 6*a^2*b^2*c^11*d^2 - 4*a^3*b*c^10*d^3 +
a^4*c^9*d^4))^(1/4)*a^2*c^2*x^3*log(d^7*sqrt(x) - (b^3*c^10 - 3*a*b^2*c^9*d + 3*a^2*b*c^8*d^2 - a^3*c^7*d^3)*
(-d^9/(b^4*c^13 - 4*a*b^3*c^12*d + 6*a^2*b^2*c^11*d^2 - 4*a^3*b*c^10*d^3 + a^4*c^9*d^4))^(3/4)) + 4*(5*(b*c +
a*d)*x^2 - a*c)*sqrt(x))/(a^2*c^2*x^3)

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giac [A]  time = 0.74, size = 487, normalized size = 0.98 \[ \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^{3} b c - \sqrt {2} a^{4} 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^{3} b c - \sqrt {2} a^{4} 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^{4} - \sqrt {2} a c^{3} 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^{4} - \sqrt {2} a c^{3} d} - \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^{3} b c - \sqrt {2} a^{4} 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^{3} b c - \sqrt {2} a^{4} 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^{4} - \sqrt {2} a c^{3} 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^{4} - \sqrt {2} a c^{3} d\right )}} + \frac {2 \, {\left (5 \, b c x^{2} + 5 \, a d x^{2} - a c\right )}}{5 \, a^{2} c^{2} x^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(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^3*b*c - sqrt(2)*a^4
*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^3*b*c - sqrt
(2)*a^4*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^4 -
sqrt(2)*a*c^3*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^4 - sqrt(2)*a*c^3*d) - 1/2*(a*b^3)^(3/4)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^3*b*c
- sqrt(2)*a^4*d) + 1/2*(a*b^3)^(3/4)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*a^3*b*c - sqrt
(2)*a^4*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^4 - sqrt(2)*a*c^3
*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^4 - sqrt(2)*a*c^3*d) +
2/5*(5*b*c*x^2 + 5*a*d*x^2 - a*c)/(a^2*c^2*x^(5/2))

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maple [A]  time = 0.02, size = 375, normalized size = 0.75 \[ -\frac {\sqrt {2}\, b^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2}}-\frac {\sqrt {2}\, b^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2}}-\frac {\sqrt {2}\, b^{2} \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )}{4 \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2}}+\frac {\sqrt {2}\, d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{2 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} c^{2}}+\frac {\sqrt {2}\, d^{2} \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{2 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} c^{2}}+\frac {\sqrt {2}\, d^{2} \ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {c}{d}}}\right )}{4 \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}} c^{2}}+\frac {2 d}{a \,c^{2} \sqrt {x}}+\frac {2 b}{a^{2} c \sqrt {x}}-\frac {2}{5 a c \,x^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(7/2)/(b*x^2+a)/(d*x^2+c),x)

[Out]

-1/4*b^2/a^2/(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^2/a^2/(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
^2/a^2/(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^2/c^2/(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^2/c^2
/(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^2/c^2/(a*d-b*c)/(c/d)^(1/4)*2^(1/2)
*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)-1)-2/5/a/c/x^(5/2)+2/a/c^2/x^(1/2)*d+2/a^2/c/x^(1/2)*b

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maxima [A]  time = 2.55, size = 411, normalized size = 0.83 \[ \frac {b^{3} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{4 \, {\left (a^{2} b c - a^{3} d\right )}} - \frac {d^{3} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} + 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} - 2 \, \sqrt {d} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {c} \sqrt {d}}}\right )}{\sqrt {\sqrt {c} \sqrt {d}} \sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {x} + \sqrt {d} x + \sqrt {c}\right )}{c^{\frac {1}{4}} d^{\frac {3}{4}}}\right )}}{4 \, {\left (b c^{3} - a c^{2} d\right )}} + \frac {2 \, {\left (5 \, {\left (b c + a d\right )} x^{2} - a c\right )}}{5 \, a^{2} c^{2} x^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(7/2)/(b*x^2+a)/(d*x^2+c),x, algorithm="maxima")

[Out]

1/4*b^3*(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)))/(sq
rt(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))/sqr
t(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^2*b*c - a^3*d) - 1/4*d^3*(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^3 - a*c^2*d) + 2/5*(5*(b*c + a*d)*x^2 - a*c)/(a^2*c^2*x^(5/2))

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mupad [B]  time = 2.73, size = 4643, normalized size = 9.32 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(7/2)*(a + b*x^2)*(c + d*x^2)),x)

[Out]

- 2*atan((32*a^11*b^10*c^13*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*
d^2 - 64*a^12*b*c*d^3))^(5/4) + 2*a^11*b^6*d^9*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d
+ 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(1/4) + 32*a^21*c^3*d^10*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^
4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(5/4) + 2*a^8*b^9*c^3*d^6*x^(1/2)*(-b^9/(16*a^
13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(1/4) + 192*a^13*b^8*c^1
1*d^2*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3)
)^(5/4) - 128*a^14*b^7*c^10*d^3*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*
c^2*d^2 - 64*a^12*b*c*d^3))^(5/4) + 32*a^15*b^6*c^9*d^4*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*
b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(5/4) + 32*a^17*b^4*c^7*d^6*x^(1/2)*(-b^9/(16*a^13*d^4 + 1
6*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(5/4) - 128*a^18*b^3*c^6*d^7*x^(1/
2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(5/4) + 1
92*a^19*b^2*c^5*d^8*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64
*a^12*b*c*d^3))^(5/4) - 128*a^12*b^9*c^12*d*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d +
96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(5/4) - 128*a^20*b*c^4*d^9*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4
- 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(5/4))/(b^16*c^8 + a^8*b^8*d^8 + a^7*b^9*c*d^7
+ a^2*b^14*c^6*d^2 + a^3*b^13*c^5*d^3 + a^4*b^12*c^4*d^4 + a^5*b^11*c^3*d^5 + a^6*b^10*c^2*d^6 + a*b^15*c^7*d)
)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(1/4) - at
an((a^11*b^10*c^13*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*
a^12*b*c*d^3))^(5/4)*32i + a^11*b^6*d^9*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d + 96*a
^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(1/4)*2i + a^21*c^3*d^10*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*
a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(5/4)*32i + a^8*b^9*c^3*d^6*x^(1/2)*(-b^9/(16*a^13*d^
4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(1/4)*2i + a^13*b^8*c^11*d^2*
x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(5/4
)*192i - a^14*b^7*c^10*d^3*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d
^2 - 64*a^12*b*c*d^3))^(5/4)*128i + a^15*b^6*c^9*d^4*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3
*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(5/4)*32i + a^17*b^4*c^7*d^6*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*
a^9*b^4*c^4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(5/4)*32i - a^18*b^3*c^6*d^7*x^(1/2)
*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(5/4)*128i
+ a^19*b^2*c^5*d^8*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*
a^12*b*c*d^3))^(5/4)*192i - a^12*b^9*c^12*d*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d +
96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(5/4)*128i - a^20*b*c^4*d^9*x^(1/2)*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^
4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(5/4)*128i)/(b^16*c^8 + a^8*b^8*d^8 + a^7*b^9*
c*d^7 + a^2*b^14*c^6*d^2 + a^3*b^13*c^5*d^3 + a^4*b^12*c^4*d^4 + a^5*b^11*c^3*d^5 + a^6*b^10*c^2*d^6 + a*b^15*
c^7*d))*(-b^9/(16*a^13*d^4 + 16*a^9*b^4*c^4 - 64*a^10*b^3*c^3*d + 96*a^11*b^2*c^2*d^2 - 64*a^12*b*c*d^3))^(1/4
)*2i - (2/(5*a*c) - (2*x^2*(a*d + b*c))/(a^2*c^2))/x^(5/2) - 2*atan((32*a^3*b^10*c^21*x^(1/2)*(-d^9/(16*b^4*c^
13 + 16*a^4*c^9*d^4 - 64*a^3*b*c^10*d^3 + 96*a^2*b^2*c^11*d^2 - 64*a*b^3*c^12*d))^(5/4) + 32*a^13*c^11*d^10*x^
(1/2)*(-d^9/(16*b^4*c^13 + 16*a^4*c^9*d^4 - 64*a^3*b*c^10*d^3 + 96*a^2*b^2*c^11*d^2 - 64*a*b^3*c^12*d))^(5/4)
+ 2*b^9*c^11*d^6*x^(1/2)*(-d^9/(16*b^4*c^13 + 16*a^4*c^9*d^4 - 64*a^3*b*c^10*d^3 + 96*a^2*b^2*c^11*d^2 - 64*a*
b^3*c^12*d))^(1/4) + 2*a^3*b^6*c^8*d^9*x^(1/2)*(-d^9/(16*b^4*c^13 + 16*a^4*c^9*d^4 - 64*a^3*b*c^10*d^3 + 96*a^
2*b^2*c^11*d^2 - 64*a*b^3*c^12*d))^(1/4) + 192*a^5*b^8*c^19*d^2*x^(1/2)*(-d^9/(16*b^4*c^13 + 16*a^4*c^9*d^4 -
64*a^3*b*c^10*d^3 + 96*a^2*b^2*c^11*d^2 - 64*a*b^3*c^12*d))^(5/4) - 128*a^6*b^7*c^18*d^3*x^(1/2)*(-d^9/(16*b^4
*c^13 + 16*a^4*c^9*d^4 - 64*a^3*b*c^10*d^3 + 96*a^2*b^2*c^11*d^2 - 64*a*b^3*c^12*d))^(5/4) + 32*a^7*b^6*c^17*d
^4*x^(1/2)*(-d^9/(16*b^4*c^13 + 16*a^4*c^9*d^4 - 64*a^3*b*c^10*d^3 + 96*a^2*b^2*c^11*d^2 - 64*a*b^3*c^12*d))^(
5/4) + 32*a^9*b^4*c^15*d^6*x^(1/2)*(-d^9/(16*b^4*c^13 + 16*a^4*c^9*d^4 - 64*a^3*b*c^10*d^3 + 96*a^2*b^2*c^11*d
^2 - 64*a*b^3*c^12*d))^(5/4) - 128*a^10*b^3*c^14*d^7*x^(1/2)*(-d^9/(16*b^4*c^13 + 16*a^4*c^9*d^4 - 64*a^3*b*c^
10*d^3 + 96*a^2*b^2*c^11*d^2 - 64*a*b^3*c^12*d))^(5/4) + 192*a^11*b^2*c^13*d^8*x^(1/2)*(-d^9/(16*b^4*c^13 + 16
*a^4*c^9*d^4 - 64*a^3*b*c^10*d^3 + 96*a^2*b^2*c^11*d^2 - 64*a*b^3*c^12*d))^(5/4) - 128*a^4*b^9*c^20*d*x^(1/2)*
(-d^9/(16*b^4*c^13 + 16*a^4*c^9*d^4 - 64*a^3*b*c^10*d^3 + 96*a^2*b^2*c^11*d^2 - 64*a*b^3*c^12*d))^(5/4) - 128*
a^12*b*c^12*d^9*x^(1/2)*(-d^9/(16*b^4*c^13 + 16*a^4*c^9*d^4 - 64*a^3*b*c^10*d^3 + 96*a^2*b^2*c^11*d^2 - 64*a*b
^3*c^12*d))^(5/4))/(a^8*d^16 + b^8*c^8*d^8 + a*b^7*c^7*d^9 + a^2*b^6*c^6*d^10 + a^3*b^5*c^5*d^11 + a^4*b^4*c^4
*d^12 + a^5*b^3*c^3*d^13 + a^6*b^2*c^2*d^14 + a^7*b*c*d^15))*(-d^9/(16*b^4*c^13 + 16*a^4*c^9*d^4 - 64*a^3*b*c^
10*d^3 + 96*a^2*b^2*c^11*d^2 - 64*a*b^3*c^12*d))^(1/4) - atan((a^3*b^10*c^21*x^(1/2)*(-d^9/(16*b^4*c^13 + 16*a
^4*c^9*d^4 - 64*a^3*b*c^10*d^3 + 96*a^2*b^2*c^11*d^2 - 64*a*b^3*c^12*d))^(5/4)*32i + a^13*c^11*d^10*x^(1/2)*(-
d^9/(16*b^4*c^13 + 16*a^4*c^9*d^4 - 64*a^3*b*c^10*d^3 + 96*a^2*b^2*c^11*d^2 - 64*a*b^3*c^12*d))^(5/4)*32i + b^
9*c^11*d^6*x^(1/2)*(-d^9/(16*b^4*c^13 + 16*a^4*c^9*d^4 - 64*a^3*b*c^10*d^3 + 96*a^2*b^2*c^11*d^2 - 64*a*b^3*c^
12*d))^(1/4)*2i + a^3*b^6*c^8*d^9*x^(1/2)*(-d^9/(16*b^4*c^13 + 16*a^4*c^9*d^4 - 64*a^3*b*c^10*d^3 + 96*a^2*b^2
*c^11*d^2 - 64*a*b^3*c^12*d))^(1/4)*2i + a^5*b^8*c^19*d^2*x^(1/2)*(-d^9/(16*b^4*c^13 + 16*a^4*c^9*d^4 - 64*a^3
*b*c^10*d^3 + 96*a^2*b^2*c^11*d^2 - 64*a*b^3*c^12*d))^(5/4)*192i - a^6*b^7*c^18*d^3*x^(1/2)*(-d^9/(16*b^4*c^13
+ 16*a^4*c^9*d^4 - 64*a^3*b*c^10*d^3 + 96*a^2*b^2*c^11*d^2 - 64*a*b^3*c^12*d))^(5/4)*128i + a^7*b^6*c^17*d^4*
x^(1/2)*(-d^9/(16*b^4*c^13 + 16*a^4*c^9*d^4 - 64*a^3*b*c^10*d^3 + 96*a^2*b^2*c^11*d^2 - 64*a*b^3*c^12*d))^(5/4
)*32i + a^9*b^4*c^15*d^6*x^(1/2)*(-d^9/(16*b^4*c^13 + 16*a^4*c^9*d^4 - 64*a^3*b*c^10*d^3 + 96*a^2*b^2*c^11*d^2
- 64*a*b^3*c^12*d))^(5/4)*32i - a^10*b^3*c^14*d^7*x^(1/2)*(-d^9/(16*b^4*c^13 + 16*a^4*c^9*d^4 - 64*a^3*b*c^10
*d^3 + 96*a^2*b^2*c^11*d^2 - 64*a*b^3*c^12*d))^(5/4)*128i + a^11*b^2*c^13*d^8*x^(1/2)*(-d^9/(16*b^4*c^13 + 16*
a^4*c^9*d^4 - 64*a^3*b*c^10*d^3 + 96*a^2*b^2*c^11*d^2 - 64*a*b^3*c^12*d))^(5/4)*192i - a^4*b^9*c^20*d*x^(1/2)*
(-d^9/(16*b^4*c^13 + 16*a^4*c^9*d^4 - 64*a^3*b*c^10*d^3 + 96*a^2*b^2*c^11*d^2 - 64*a*b^3*c^12*d))^(5/4)*128i -
a^12*b*c^12*d^9*x^(1/2)*(-d^9/(16*b^4*c^13 + 16*a^4*c^9*d^4 - 64*a^3*b*c^10*d^3 + 96*a^2*b^2*c^11*d^2 - 64*a*
b^3*c^12*d))^(5/4)*128i)/(a^8*d^16 + b^8*c^8*d^8 + a*b^7*c^7*d^9 + a^2*b^6*c^6*d^10 + a^3*b^5*c^5*d^11 + a^4*b
^4*c^4*d^12 + a^5*b^3*c^3*d^13 + a^6*b^2*c^2*d^14 + a^7*b*c*d^15))*(-d^9/(16*b^4*c^13 + 16*a^4*c^9*d^4 - 64*a^
3*b*c^10*d^3 + 96*a^2*b^2*c^11*d^2 - 64*a*b^3*c^12*d))^(1/4)*2i

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(7/2)/(b*x**2+a)/(d*x**2+c),x)

[Out]

Timed out

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