3.5.61 \(\int \frac {x^{9/2}}{(a+b x^2) (c+d x^2)} \, dx\) [461]

Optimal. Leaf size=478 \[ \frac {2 x^{3/2}}{3 b d}-\frac {a^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{7/4} (b c-a d)}+\frac {a^{7/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{7/4} (b c-a d)}+\frac {c^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{7/4} (b c-a d)}-\frac {c^{7/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{7/4} (b c-a d)}+\frac {a^{7/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{7/4} (b c-a d)}-\frac {a^{7/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{7/4} (b c-a d)}-\frac {c^{7/4} \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{7/4} (b c-a d)}+\frac {c^{7/4} \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{7/4} (b c-a d)} \]

[Out]

2/3*x^(3/2)/b/d-1/2*a^(7/4)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/b^(7/4)/(-a*d+b*c)*2^(1/2)+1/2*a^(7/4)*a
rctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/b^(7/4)/(-a*d+b*c)*2^(1/2)+1/2*c^(7/4)*arctan(1-d^(1/4)*2^(1/2)*x^(1/
2)/c^(1/4))/d^(7/4)/(-a*d+b*c)*2^(1/2)-1/2*c^(7/4)*arctan(1+d^(1/4)*2^(1/2)*x^(1/2)/c^(1/4))/d^(7/4)/(-a*d+b*c
)*2^(1/2)+1/4*a^(7/4)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/b^(7/4)/(-a*d+b*c)*2^(1/2)-1/4*a^(
7/4)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/b^(7/4)/(-a*d+b*c)*2^(1/2)-1/4*c^(7/4)*ln(c^(1/2)+x
*d^(1/2)-c^(1/4)*d^(1/4)*2^(1/2)*x^(1/2))/d^(7/4)/(-a*d+b*c)*2^(1/2)+1/4*c^(7/4)*ln(c^(1/2)+x*d^(1/2)+c^(1/4)*
d^(1/4)*2^(1/2)*x^(1/2))/d^(7/4)/(-a*d+b*c)*2^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.38, antiderivative size = 478, 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 = {477, 490, 598, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {a^{7/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{7/4} (b c-a d)}+\frac {a^{7/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} b^{7/4} (b c-a d)}+\frac {a^{7/4} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{7/4} (b c-a d)}-\frac {a^{7/4} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} b^{7/4} (b c-a d)}+\frac {c^{7/4} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{7/4} (b c-a d)}-\frac {c^{7/4} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}+1\right )}{\sqrt {2} d^{7/4} (b c-a d)}-\frac {c^{7/4} \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{7/4} (b c-a d)}+\frac {c^{7/4} \log \left (\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {c}+\sqrt {d} x\right )}{2 \sqrt {2} d^{7/4} (b c-a d)}+\frac {2 x^{3/2}}{3 b d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(9/2)/((a + b*x^2)*(c + d*x^2)),x]

[Out]

(2*x^(3/2))/(3*b*d) - (a^(7/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(7/4)*(b*c - a*d)) +
(a^(7/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*b^(7/4)*(b*c - a*d)) + (c^(7/4)*ArcTan[1 - (S
qrt[2]*d^(1/4)*Sqrt[x])/c^(1/4)])/(Sqrt[2]*d^(7/4)*(b*c - a*d)) - (c^(7/4)*ArcTan[1 + (Sqrt[2]*d^(1/4)*Sqrt[x]
)/c^(1/4)])/(Sqrt[2]*d^(7/4)*(b*c - a*d)) + (a^(7/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x
])/(2*Sqrt[2]*b^(7/4)*(b*c - a*d)) - (a^(7/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*S
qrt[2]*b^(7/4)*(b*c - a*d)) - (c^(7/4)*Log[Sqrt[c] - Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*
d^(7/4)*(b*c - a*d)) + (c^(7/4)*Log[Sqrt[c] + Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x] + Sqrt[d]*x])/(2*Sqrt[2]*d^(7/4)
*(b*c - a*d))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 303

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 477

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 490

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

Rule 598

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 631

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 642

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 1176

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 1179

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^{9/2}}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx &=2 \text {Subst}\left (\int \frac {x^{10}}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 x^{3/2}}{3 b d}-\frac {2 \text {Subst}\left (\int \frac {x^2 \left (3 a c+3 (b c+a d) x^4\right )}{\left (a+b x^4\right ) \left (c+d x^4\right )} \, dx,x,\sqrt {x}\right )}{3 b d}\\ &=\frac {2 x^{3/2}}{3 b d}-\frac {2 \text {Subst}\left (\int \left (\frac {3 a^2 d x^2}{(-b c+a d) \left (a+b x^4\right )}+\frac {3 b c^2 x^2}{(b c-a d) \left (c+d x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{3 b d}\\ &=\frac {2 x^{3/2}}{3 b d}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b (b c-a d)}-\frac {\left (2 c^2\right ) \text {Subst}\left (\int \frac {x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{d (b c-a d)}\\ &=\frac {2 x^{3/2}}{3 b d}-\frac {a^2 \text {Subst}\left (\int \frac {\sqrt {a}-\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^{3/2} (b c-a d)}+\frac {a^2 \text {Subst}\left (\int \frac {\sqrt {a}+\sqrt {b} x^2}{a+b x^4} \, dx,x,\sqrt {x}\right )}{b^{3/2} (b c-a d)}+\frac {c^2 \text {Subst}\left (\int \frac {\sqrt {c}-\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{d^{3/2} (b c-a d)}-\frac {c^2 \text {Subst}\left (\int \frac {\sqrt {c}+\sqrt {d} x^2}{c+d x^4} \, dx,x,\sqrt {x}\right )}{d^{3/2} (b c-a d)}\\ &=\frac {2 x^{3/2}}{3 b d}+\frac {a^2 \text {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 b^2 (b c-a d)}+\frac {a^2 \text {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 b^2 (b c-a d)}+\frac {a^{7/4} \text {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} b^{7/4} (b c-a d)}+\frac {a^{7/4} \text {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} b^{7/4} (b c-a d)}-\frac {c^2 \text {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 d^2 (b c-a d)}-\frac {c^2 \text {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 d^2 (b c-a d)}-\frac {c^{7/4} \text {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} d^{7/4} (b c-a d)}-\frac {c^{7/4} \text {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} d^{7/4} (b c-a d)}\\ &=\frac {2 x^{3/2}}{3 b d}+\frac {a^{7/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{7/4} (b c-a d)}-\frac {a^{7/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{7/4} (b c-a d)}-\frac {c^{7/4} \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{7/4} (b c-a d)}+\frac {c^{7/4} \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{7/4} (b c-a d)}+\frac {a^{7/4} \text {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} b^{7/4} (b c-a d)}-\frac {a^{7/4} \text {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} b^{7/4} (b c-a d)}-\frac {c^{7/4} \text {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} d^{7/4} (b c-a d)}+\frac {c^{7/4} \text {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} d^{7/4} (b c-a d)}\\ &=\frac {2 x^{3/2}}{3 b d}-\frac {a^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{7/4} (b c-a d)}+\frac {a^{7/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} b^{7/4} (b c-a d)}+\frac {c^{7/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{7/4} (b c-a d)}-\frac {c^{7/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{d} \sqrt {x}}{\sqrt [4]{c}}\right )}{\sqrt {2} d^{7/4} (b c-a d)}+\frac {a^{7/4} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{7/4} (b c-a d)}-\frac {a^{7/4} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{2 \sqrt {2} b^{7/4} (b c-a d)}-\frac {c^{7/4} \log \left (\sqrt {c}-\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{7/4} (b c-a d)}+\frac {c^{7/4} \log \left (\sqrt {c}+\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}+\sqrt {d} x\right )}{2 \sqrt {2} d^{7/4} (b c-a d)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.51, size = 249, normalized size = 0.52 \begin {gather*} \frac {-\frac {4 a x^{3/2}}{b}+\frac {4 c x^{3/2}}{d}-\frac {3 \sqrt {2} a^{7/4} \tan ^{-1}\left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{b^{7/4}}+\frac {3 \sqrt {2} c^{7/4} \tan ^{-1}\left (\frac {\sqrt {c}-\sqrt {d} x}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}\right )}{d^{7/4}}-\frac {3 \sqrt {2} a^{7/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{b^{7/4}}+\frac {3 \sqrt {2} c^{7/4} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{d} \sqrt {x}}{\sqrt {c}+\sqrt {d} x}\right )}{d^{7/4}}}{6 b c-6 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(9/2)/((a + b*x^2)*(c + d*x^2)),x]

[Out]

((-4*a*x^(3/2))/b + (4*c*x^(3/2))/d - (3*Sqrt[2]*a^(7/4)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)
*Sqrt[x])])/b^(7/4) + (3*Sqrt[2]*c^(7/4)*ArcTan[(Sqrt[c] - Sqrt[d]*x)/(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])])/d^(7
/4) - (3*Sqrt[2]*a^(7/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/b^(7/4) + (3*Sqrt[2
]*c^(7/4)*ArcTanh[(Sqrt[2]*c^(1/4)*d^(1/4)*Sqrt[x])/(Sqrt[c] + Sqrt[d]*x)])/d^(7/4))/(6*b*c - 6*a*d)

________________________________________________________________________________________

Maple [A]
time = 0.11, size = 249, normalized size = 0.52

method	result	size
derivativedivides	\(\frac {2 x^{\frac {3}{2}}}{3 b d}-\frac {a^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{2} \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {c^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d^{2} \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}}}\)	\(249\)
default	\(\frac {2 x^{\frac {3}{2}}}{3 b d}-\frac {a^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{4 b^{2} \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {c^{2} \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )\right )}{4 d^{2} \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}}}\)	\(249\)
risch	\(\frac {2 x^{\frac {3}{2}}}{3 b d}-\frac {a^{2} \sqrt {2}\, \ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )}{4 b^{2} \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {a^{2} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )}{2 b^{2} \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {a^{2} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )}{2 b^{2} \left (a d -b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {c^{2} \sqrt {2}\, \ln \left (\frac {x -\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}{x +\left (\frac {c}{d}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {c}{d}}}\right )}{4 d^{2} \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {c^{2} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}+1\right )}{2 d^{2} \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}}}+\frac {c^{2} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {c}{d}\right )^{\frac {1}{4}}}-1\right )}{2 d^{2} \left (a d -b c \right ) \left (\frac {c}{d}\right )^{\frac {1}{4}}}\)	\(351\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(9/2)/(b*x^2+a)/(d*x^2+c),x,method=_RETURNVERBOSE)

[Out]

2/3*x^(3/2)/b/d-1/4*a^2/b^2/(a*d-b*c)/(a/b)^(1/4)*2^(1/2)*(ln((x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(
a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*
x^(1/2)-1))+1/4*c^2/d^2/(a*d-b*c)/(c/d)^(1/4)*2^(1/2)*(ln((x-(c/d)^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2))/(x+(c/d)
^(1/4)*x^(1/2)*2^(1/2)+(c/d)^(1/2)))+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(c/d)^(1/4)*x^(1
/2)-1))

________________________________________________________________________________________

Maxima [A]
time = 0.54, size = 390, normalized size = 0.82 \begin {gather*} \frac {a^{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 (b^{2} c - a b d\right )}} - \frac {c^{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 d - a d^{2}\right )}} + \frac {2 \, x^{\frac {3}{2}}}{3 \, b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(9/2)/(b*x^2+a)/(d*x^2+c),x, algorithm="maxima")

[Out]

1/4*a^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)))/(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)))/(b^2*c - a*b*d) - 1/4*c^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*d - a*d^2) + 2/3*x^(3/2)/(b*d)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1422 vs. \(2 (340) = 680\).
time = 1.81, size = 1422, normalized size = 2.97 \begin {gather*} \frac {12 \, \left (-\frac {a^{7}}{b^{11} c^{4} - 4 \, a b^{10} c^{3} d + 6 \, a^{2} b^{9} c^{2} d^{2} - 4 \, a^{3} b^{8} c d^{3} + a^{4} b^{7} d^{4}}\right )^{\frac {1}{4}} b d \arctan \left (-\frac {\sqrt {a^{10} x - {\left (a^{7} b^{5} c^{2} - 2 \, a^{8} b^{4} c d + a^{9} b^{3} d^{2}\right )} \sqrt {-\frac {a^{7}}{b^{11} c^{4} - 4 \, a b^{10} c^{3} d + 6 \, a^{2} b^{9} c^{2} d^{2} - 4 \, a^{3} b^{8} c d^{3} + a^{4} b^{7} d^{4}}}} \left (-\frac {a^{7}}{b^{11} c^{4} - 4 \, a b^{10} c^{3} d + 6 \, a^{2} b^{9} c^{2} d^{2} - 4 \, a^{3} b^{8} c d^{3} + a^{4} b^{7} d^{4}}\right )^{\frac {1}{4}} {\left (b^{3} c - a b^{2} d\right )} - {\left (a^{5} b^{3} c - a^{6} b^{2} d\right )} \left (-\frac {a^{7}}{b^{11} c^{4} - 4 \, a b^{10} c^{3} d + 6 \, a^{2} b^{9} c^{2} d^{2} - 4 \, a^{3} b^{8} c d^{3} + a^{4} b^{7} d^{4}}\right )^{\frac {1}{4}} \sqrt {x}}{a^{7}}\right ) - 12 \, \left (-\frac {c^{7}}{b^{4} c^{4} d^{7} - 4 \, a b^{3} c^{3} d^{8} + 6 \, a^{2} b^{2} c^{2} d^{9} - 4 \, a^{3} b c d^{10} + a^{4} d^{11}}\right )^{\frac {1}{4}} b d \arctan \left (-\frac {\sqrt {c^{10} x - {\left (b^{2} c^{9} d^{3} - 2 \, a b c^{8} d^{4} + a^{2} c^{7} d^{5}\right )} \sqrt {-\frac {c^{7}}{b^{4} c^{4} d^{7} - 4 \, a b^{3} c^{3} d^{8} + 6 \, a^{2} b^{2} c^{2} d^{9} - 4 \, a^{3} b c d^{10} + a^{4} d^{11}}}} \left (-\frac {c^{7}}{b^{4} c^{4} d^{7} - 4 \, a b^{3} c^{3} d^{8} + 6 \, a^{2} b^{2} c^{2} d^{9} - 4 \, a^{3} b c d^{10} + a^{4} d^{11}}\right )^{\frac {1}{4}} {\left (b c d^{2} - a d^{3}\right )} - {\left (b c^{6} d^{2} - a c^{5} d^{3}\right )} \left (-\frac {c^{7}}{b^{4} c^{4} d^{7} - 4 \, a b^{3} c^{3} d^{8} + 6 \, a^{2} b^{2} c^{2} d^{9} - 4 \, a^{3} b c d^{10} + a^{4} d^{11}}\right )^{\frac {1}{4}} \sqrt {x}}{c^{7}}\right ) + 3 \, \left (-\frac {a^{7}}{b^{11} c^{4} - 4 \, a b^{10} c^{3} d + 6 \, a^{2} b^{9} c^{2} d^{2} - 4 \, a^{3} b^{8} c d^{3} + a^{4} b^{7} d^{4}}\right )^{\frac {1}{4}} b d \log \left (a^{5} \sqrt {x} + {\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} \left (-\frac {a^{7}}{b^{11} c^{4} - 4 \, a b^{10} c^{3} d + 6 \, a^{2} b^{9} c^{2} d^{2} - 4 \, a^{3} b^{8} c d^{3} + a^{4} b^{7} d^{4}}\right )^{\frac {3}{4}}\right ) - 3 \, \left (-\frac {a^{7}}{b^{11} c^{4} - 4 \, a b^{10} c^{3} d + 6 \, a^{2} b^{9} c^{2} d^{2} - 4 \, a^{3} b^{8} c d^{3} + a^{4} b^{7} d^{4}}\right )^{\frac {1}{4}} b d \log \left (a^{5} \sqrt {x} - {\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} \left (-\frac {a^{7}}{b^{11} c^{4} - 4 \, a b^{10} c^{3} d + 6 \, a^{2} b^{9} c^{2} d^{2} - 4 \, a^{3} b^{8} c d^{3} + a^{4} b^{7} d^{4}}\right )^{\frac {3}{4}}\right ) - 3 \, \left (-\frac {c^{7}}{b^{4} c^{4} d^{7} - 4 \, a b^{3} c^{3} d^{8} + 6 \, a^{2} b^{2} c^{2} d^{9} - 4 \, a^{3} b c d^{10} + a^{4} d^{11}}\right )^{\frac {1}{4}} b d \log \left (c^{5} \sqrt {x} + {\left (b^{3} c^{3} d^{5} - 3 \, a b^{2} c^{2} d^{6} + 3 \, a^{2} b c d^{7} - a^{3} d^{8}\right )} \left (-\frac {c^{7}}{b^{4} c^{4} d^{7} - 4 \, a b^{3} c^{3} d^{8} + 6 \, a^{2} b^{2} c^{2} d^{9} - 4 \, a^{3} b c d^{10} + a^{4} d^{11}}\right )^{\frac {3}{4}}\right ) + 3 \, \left (-\frac {c^{7}}{b^{4} c^{4} d^{7} - 4 \, a b^{3} c^{3} d^{8} + 6 \, a^{2} b^{2} c^{2} d^{9} - 4 \, a^{3} b c d^{10} + a^{4} d^{11}}\right )^{\frac {1}{4}} b d \log \left (c^{5} \sqrt {x} - {\left (b^{3} c^{3} d^{5} - 3 \, a b^{2} c^{2} d^{6} + 3 \, a^{2} b c d^{7} - a^{3} d^{8}\right )} \left (-\frac {c^{7}}{b^{4} c^{4} d^{7} - 4 \, a b^{3} c^{3} d^{8} + 6 \, a^{2} b^{2} c^{2} d^{9} - 4 \, a^{3} b c d^{10} + a^{4} d^{11}}\right )^{\frac {3}{4}}\right ) + 4 \, x^{\frac {3}{2}}}{6 \, b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(9/2)/(b*x^2+a)/(d*x^2+c),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)] Timed out
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**(9/2)/(b*x**2+a)/(d*x**2+c),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 1.77, size = 476, normalized size = 1.00 \begin {gather*} \frac {\left (a b^{3}\right )^{\frac {3}{4}} a \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} b^{5} c - \sqrt {2} a b^{4} d} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} a \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} b^{5} c - \sqrt {2} a b^{4} d} - \frac {\left (c d^{3}\right )^{\frac {3}{4}} c \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 d^{4} - \sqrt {2} a d^{5}} - \frac {\left (c d^{3}\right )^{\frac {3}{4}} c \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 d^{4} - \sqrt {2} a d^{5}} - \frac {\left (a b^{3}\right )^{\frac {3}{4}} a \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} b^{5} c - \sqrt {2} a b^{4} d\right )}} + \frac {\left (a b^{3}\right )^{\frac {3}{4}} a \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{2 \, {\left (\sqrt {2} b^{5} c - \sqrt {2} a b^{4} d\right )}} + \frac {\left (c d^{3}\right )^{\frac {3}{4}} c \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 d^{4} - \sqrt {2} a d^{5}\right )}} - \frac {\left (c d^{3}\right )^{\frac {3}{4}} c \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 d^{4} - \sqrt {2} a d^{5}\right )}} + \frac {2 \, x^{\frac {3}{2}}}{3 \, b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(9/2)/(b*x^2+a)/(d*x^2+c),x, algorithm="giac")

[Out]

(a*b^3)^(3/4)*a*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*b^5*c - sqrt(2)*a*b
^4*d) + (a*b^3)^(3/4)*a*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(sqrt(2)*b^5*c - sq
rt(2)*a*b^4*d) - (c*d^3)^(3/4)*c*arctan(1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) + 2*sqrt(x))/(c/d)^(1/4))/(sqrt(2)*b*
c*d^4 - sqrt(2)*a*d^5) - (c*d^3)^(3/4)*c*arctan(-1/2*sqrt(2)*(sqrt(2)*(c/d)^(1/4) - 2*sqrt(x))/(c/d)^(1/4))/(s
qrt(2)*b*c*d^4 - sqrt(2)*a*d^5) - 1/2*(a*b^3)^(3/4)*a*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2
)*b^5*c - sqrt(2)*a*b^4*d) + 1/2*(a*b^3)^(3/4)*a*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(sqrt(2)*b^
5*c - sqrt(2)*a*b^4*d) + 1/2*(c*d^3)^(3/4)*c*log(sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b*c*d^4
- sqrt(2)*a*d^5) - 1/2*(c*d^3)^(3/4)*c*log(-sqrt(2)*sqrt(x)*(c/d)^(1/4) + x + sqrt(c/d))/(sqrt(2)*b*c*d^4 - s
qrt(2)*a*d^5) + 2/3*x^(3/2)/(b*d)

________________________________________________________________________________________

Mupad [B]
time = 1.08, size = 2500, normalized size = 5.23 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(9/2)/((a + b*x^2)*(c + d*x^2)),x)

[Out]

atan(((-c^7/(16*a^4*d^11 + 16*b^4*c^4*d^7 - 64*a*b^3*c^3*d^8 + 96*a^2*b^2*c^2*d^9 - 64*a^3*b*c*d^10))^(1/4)*((
-c^7/(16*a^4*d^11 + 16*b^4*c^4*d^7 - 64*a*b^3*c^3*d^8 + 96*a^2*b^2*c^2*d^9 - 64*a^3*b*c*d^10))^(3/4)*((128*(16
*a^3*b^10*c^10*d^3 - 48*a^4*b^9*c^9*d^4 + 48*a^5*b^8*c^8*d^5 - 16*a^6*b^7*c^7*d^6 - 16*a^7*b^6*c^6*d^7 + 48*a^
8*b^5*c^5*d^8 - 48*a^9*b^4*c^4*d^9 + 16*a^10*b^3*c^3*d^10))/(b^3*d^3) - (256*x^(1/2)*(-c^7/(16*a^4*d^11 + 16*b
^4*c^4*d^7 - 64*a*b^3*c^3*d^8 + 96*a^2*b^2*c^2*d^9 - 64*a^3*b*c*d^10))^(1/4)*(16*a^3*b^11*c^9*d^5 - 64*a^4*b^1
0*c^8*d^6 + 112*a^5*b^9*c^7*d^7 - 128*a^6*b^8*c^6*d^8 + 112*a^7*b^7*c^5*d^9 - 64*a^8*b^6*c^4*d^10 + 16*a^9*b^5
*c^3*d^11))/(b^3*d^3)) - (256*x^(1/2)*(a^5*b^5*c^10 + a^10*c^5*d^5))/(b^3*d^3))*1i - (-c^7/(16*a^4*d^11 + 16*b
^4*c^4*d^7 - 64*a*b^3*c^3*d^8 + 96*a^2*b^2*c^2*d^9 - 64*a^3*b*c*d^10))^(1/4)*((-c^7/(16*a^4*d^11 + 16*b^4*c^4*
d^7 - 64*a*b^3*c^3*d^8 + 96*a^2*b^2*c^2*d^9 - 64*a^3*b*c*d^10))^(3/4)*((128*(16*a^3*b^10*c^10*d^3 - 48*a^4*b^9
*c^9*d^4 + 48*a^5*b^8*c^8*d^5 - 16*a^6*b^7*c^7*d^6 - 16*a^7*b^6*c^6*d^7 + 48*a^8*b^5*c^5*d^8 - 48*a^9*b^4*c^4*
d^9 + 16*a^10*b^3*c^3*d^10))/(b^3*d^3) + (256*x^(1/2)*(-c^7/(16*a^4*d^11 + 16*b^4*c^4*d^7 - 64*a*b^3*c^3*d^8 +
96*a^2*b^2*c^2*d^9 - 64*a^3*b*c*d^10))^(1/4)*(16*a^3*b^11*c^9*d^5 - 64*a^4*b^10*c^8*d^6 + 112*a^5*b^9*c^7*d^7
- 128*a^6*b^8*c^6*d^8 + 112*a^7*b^7*c^5*d^9 - 64*a^8*b^6*c^4*d^10 + 16*a^9*b^5*c^3*d^11))/(b^3*d^3)) + (256*x
^(1/2)*(a^5*b^5*c^10 + a^10*c^5*d^5))/(b^3*d^3))*1i)/((-c^7/(16*a^4*d^11 + 16*b^4*c^4*d^7 - 64*a*b^3*c^3*d^8 +
96*a^2*b^2*c^2*d^9 - 64*a^3*b*c*d^10))^(1/4)*((-c^7/(16*a^4*d^11 + 16*b^4*c^4*d^7 - 64*a*b^3*c^3*d^8 + 96*a^2
*b^2*c^2*d^9 - 64*a^3*b*c*d^10))^(3/4)*((128*(16*a^3*b^10*c^10*d^3 - 48*a^4*b^9*c^9*d^4 + 48*a^5*b^8*c^8*d^5 -
16*a^6*b^7*c^7*d^6 - 16*a^7*b^6*c^6*d^7 + 48*a^8*b^5*c^5*d^8 - 48*a^9*b^4*c^4*d^9 + 16*a^10*b^3*c^3*d^10))/(b
^3*d^3) - (256*x^(1/2)*(-c^7/(16*a^4*d^11 + 16*b^4*c^4*d^7 - 64*a*b^3*c^3*d^8 + 96*a^2*b^2*c^2*d^9 - 64*a^3*b*
c*d^10))^(1/4)*(16*a^3*b^11*c^9*d^5 - 64*a^4*b^10*c^8*d^6 + 112*a^5*b^9*c^7*d^7 - 128*a^6*b^8*c^6*d^8 + 112*a^
7*b^7*c^5*d^9 - 64*a^8*b^6*c^4*d^10 + 16*a^9*b^5*c^3*d^11))/(b^3*d^3)) - (256*x^(1/2)*(a^5*b^5*c^10 + a^10*c^5
*d^5))/(b^3*d^3)) + (-c^7/(16*a^4*d^11 + 16*b^4*c^4*d^7 - 64*a*b^3*c^3*d^8 + 96*a^2*b^2*c^2*d^9 - 64*a^3*b*c*d
^10))^(1/4)*((-c^7/(16*a^4*d^11 + 16*b^4*c^4*d^7 - 64*a*b^3*c^3*d^8 + 96*a^2*b^2*c^2*d^9 - 64*a^3*b*c*d^10))^(
3/4)*((128*(16*a^3*b^10*c^10*d^3 - 48*a^4*b^9*c^9*d^4 + 48*a^5*b^8*c^8*d^5 - 16*a^6*b^7*c^7*d^6 - 16*a^7*b^6*c
^6*d^7 + 48*a^8*b^5*c^5*d^8 - 48*a^9*b^4*c^4*d^9 + 16*a^10*b^3*c^3*d^10))/(b^3*d^3) + (256*x^(1/2)*(-c^7/(16*a
^4*d^11 + 16*b^4*c^4*d^7 - 64*a*b^3*c^3*d^8 + 96*a^2*b^2*c^2*d^9 - 64*a^3*b*c*d^10))^(1/4)*(16*a^3*b^11*c^9*d^
5 - 64*a^4*b^10*c^8*d^6 + 112*a^5*b^9*c^7*d^7 - 128*a^6*b^8*c^6*d^8 + 112*a^7*b^7*c^5*d^9 - 64*a^8*b^6*c^4*d^1
0 + 16*a^9*b^5*c^3*d^11))/(b^3*d^3)) + (256*x^(1/2)*(a^5*b^5*c^10 + a^10*c^5*d^5))/(b^3*d^3)) - (256*(a^7*b^2*
c^9 + a^9*c^7*d^2 + a^8*b*c^8*d))/(b^3*d^3)))*(-c^7/(16*a^4*d^11 + 16*b^4*c^4*d^7 - 64*a*b^3*c^3*d^8 + 96*a^2*
b^2*c^2*d^9 - 64*a^3*b*c*d^10))^(1/4)*2i + 2*atan(((-c^7/(16*a^4*d^11 + 16*b^4*c^4*d^7 - 64*a*b^3*c^3*d^8 + 96
*a^2*b^2*c^2*d^9 - 64*a^3*b*c*d^10))^(1/4)*((-c^7/(16*a^4*d^11 + 16*b^4*c^4*d^7 - 64*a*b^3*c^3*d^8 + 96*a^2*b^
2*c^2*d^9 - 64*a^3*b*c*d^10))^(3/4)*((128*(16*a^3*b^10*c^10*d^3 - 48*a^4*b^9*c^9*d^4 + 48*a^5*b^8*c^8*d^5 - 16
*a^6*b^7*c^7*d^6 - 16*a^7*b^6*c^6*d^7 + 48*a^8*b^5*c^5*d^8 - 48*a^9*b^4*c^4*d^9 + 16*a^10*b^3*c^3*d^10))/(b^3*
d^3) - (x^(1/2)*(-c^7/(16*a^4*d^11 + 16*b^4*c^4*d^7 - 64*a*b^3*c^3*d^8 + 96*a^2*b^2*c^2*d^9 - 64*a^3*b*c*d^10)
)^(1/4)*(16*a^3*b^11*c^9*d^5 - 64*a^4*b^10*c^8*d^6 + 112*a^5*b^9*c^7*d^7 - 128*a^6*b^8*c^6*d^8 + 112*a^7*b^7*c
^5*d^9 - 64*a^8*b^6*c^4*d^10 + 16*a^9*b^5*c^3*d^11)*256i)/(b^3*d^3))*1i + (256*x^(1/2)*(a^5*b^5*c^10 + a^10*c^
5*d^5))/(b^3*d^3)) - (-c^7/(16*a^4*d^11 + 16*b^4*c^4*d^7 - 64*a*b^3*c^3*d^8 + 96*a^2*b^2*c^2*d^9 - 64*a^3*b*c*
d^10))^(1/4)*((-c^7/(16*a^4*d^11 + 16*b^4*c^4*d^7 - 64*a*b^3*c^3*d^8 + 96*a^2*b^2*c^2*d^9 - 64*a^3*b*c*d^10))^
(3/4)*((128*(16*a^3*b^10*c^10*d^3 - 48*a^4*b^9*c^9*d^4 + 48*a^5*b^8*c^8*d^5 - 16*a^6*b^7*c^7*d^6 - 16*a^7*b^6*
c^6*d^7 + 48*a^8*b^5*c^5*d^8 - 48*a^9*b^4*c^4*d^9 + 16*a^10*b^3*c^3*d^10))/(b^3*d^3) + (x^(1/2)*(-c^7/(16*a^4*
d^11 + 16*b^4*c^4*d^7 - 64*a*b^3*c^3*d^8 + 96*a^2*b^2*c^2*d^9 - 64*a^3*b*c*d^10))^(1/4)*(16*a^3*b^11*c^9*d^5 -
64*a^4*b^10*c^8*d^6 + 112*a^5*b^9*c^7*d^7 - 128*a^6*b^8*c^6*d^8 + 112*a^7*b^7*c^5*d^9 - 64*a^8*b^6*c^4*d^10 +
16*a^9*b^5*c^3*d^11)*256i)/(b^3*d^3))*1i - (256*x^(1/2)*(a^5*b^5*c^10 + a^10*c^5*d^5))/(b^3*d^3)))/((-c^7/(16
*a^4*d^11 + 16*b^4*c^4*d^7 - 64*a*b^3*c^3*d^8 + 96*a^2*b^2*c^2*d^9 - 64*a^3*b*c*d^10))^(1/4)*((-c^7/(16*a^4*d^
11 + 16*b^4*c^4*d^7 - 64*a*b^3*c^3*d^8 + 96*a^2*b^2*c^2*d^9 - 64*a^3*b*c*d^10))^(3/4)*((128*(16*a^3*b^10*c^10*
d^3 - 48*a^4*b^9*c^9*d^4 + 48*a^5*b^8*c^8*d^5 - 16*a^6*b^7*c^7*d^6 - 16*a^7*b^6*c^6*d^7 + 48*a^8*b^5*c^5*d^8 -
48*a^9*b^4*c^4*d^9 + 16*a^10*b^3*c^3*d^10))/(b...

________________________________________________________________________________________