3.6.48 \(\int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx\) [548]

Optimal. Leaf size=172 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{5/2} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{5/2} d}-\frac {2 a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {4 a^2 \left (a^2+4 b^2\right )}{3 b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}} \]

[Out]

arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/(a-I*b)^(5/2)/d+arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/(a
+I*b)^(5/2)/d-4/3*a^2*(a^2+4*b^2)/b^2/(a^2+b^2)^2/d/(a+b*tan(d*x+c))^(1/2)-2/3*a^2*tan(d*x+c)/b/(a^2+b^2)/d/(a
+b*tan(d*x+c))^(3/2)

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Rubi [A]
time = 0.26, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3646, 3709, 3620, 3618, 65, 214} \begin {gather*} -\frac {4 a^2 \left (a^2+4 b^2\right )}{3 b^2 d \left (a^2+b^2\right )^2 \sqrt {a+b \tan (c+d x)}}-\frac {2 a^2 \tan (c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d (a-i b)^{5/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d (a+i b)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^3/(a + b*Tan[c + d*x])^(5/2),x]

[Out]

ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/((a - I*b)^(5/2)*d) + ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a
+ I*b]]/((a + I*b)^(5/2)*d) - (2*a^2*Tan[c + d*x])/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) - (4*a^2*(a^
2 + 4*b^2))/(3*b^2*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rule 3618

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(
d/f), Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]

Rule 3620

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]
)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
&& NeQ[c^2 + d^2, 0] &&  !IntegerQ[m]

Rule 3646

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

Rule 3709

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2)
)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e +
f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2
+ b^2, 0]

Rubi steps

\begin {align*} \int \frac {\tan ^3(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx &=-\frac {2 a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac {2 \int \frac {a^2-\frac {3}{2} a b \tan (c+d x)+\frac {1}{2} \left (2 a^2+3 b^2\right ) \tan ^2(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx}{3 b \left (a^2+b^2\right )}\\ &=-\frac {2 a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {4 a^2 \left (a^2+4 b^2\right )}{3 b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {2 \int \frac {-3 a b^2-\frac {3}{2} b \left (a^2-b^2\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{3 b \left (a^2+b^2\right )^2}\\ &=-\frac {2 a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {4 a^2 \left (a^2+4 b^2\right )}{3 b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}+\frac {i \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (a-i b)^2}-\frac {i \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{2 (a+i b)^2}\\ &=-\frac {2 a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {4 a^2 \left (a^2+4 b^2\right )}{3 b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}-\frac {\text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 (a-i b)^2 d}-\frac {\text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 (a+i b)^2 d}\\ &=-\frac {2 a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {4 a^2 \left (a^2+4 b^2\right )}{3 b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}-\frac {i \text {Subst}\left (\int \frac {1}{-1-\frac {i a}{b}+\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{(a-i b)^2 b d}+\frac {i \text {Subst}\left (\int \frac {1}{-1+\frac {i a}{b}-\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{(a+i b)^2 b d}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{5/2} d}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{5/2} d}-\frac {2 a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac {4 a^2 \left (a^2+4 b^2\right )}{3 b^2 \left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 1.52, size = 220, normalized size = 1.28 \begin {gather*} \frac {-\frac {4 a}{b}-\frac {a \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {a+b \tan (c+d x)}{a-i b}\right )}{i a+b}+\frac {a \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {a+b \tan (c+d x)}{a+i b}\right )}{i a-b}-6 \tan (c+d x)+\frac {3 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {a+b \tan (c+d x)}{a-i b}\right ) (a+b \tan (c+d x))}{i a+b}+\frac {3 i \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {a+b \tan (c+d x)}{a+i b}\right ) (a+b \tan (c+d x))}{a+i b}}{3 b d (a+b \tan (c+d x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^3/(a + b*Tan[c + d*x])^(5/2),x]

[Out]

((-4*a)/b - (a*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a - I*b)])/(I*a + b) + (a*Hypergeometric
2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a + I*b)])/(I*a - b) - 6*Tan[c + d*x] + (3*Hypergeometric2F1[-1/2, 1,
1/2, (a + b*Tan[c + d*x])/(a - I*b)]*(a + b*Tan[c + d*x]))/(I*a + b) + ((3*I)*Hypergeometric2F1[-1/2, 1, 1/2,
(a + b*Tan[c + d*x])/(a + I*b)]*(a + b*Tan[c + d*x]))/(a + I*b))/(3*b*d*(a + b*Tan[c + d*x])^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(967\) vs. \(2(148)=296\).
time = 0.13, size = 968, normalized size = 5.63

method	result	size
derivativedivides	\(\frac {\frac {2 b^{2} \left (\frac {\frac {\left (-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{3}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a \,b^{2}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{4}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2} b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{4}\right ) \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (2 a^{5}-4 a^{3} b^{2}-6 a \,b^{4}-\frac {\left (-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{3}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a \,b^{2}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{4}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2} b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{4}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{4 \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {-\frac {\left (-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{3}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a \,b^{2}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{4}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2} b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{4}\right ) \ln \left (-b \tan \left (d x +c \right )-a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (-2 a^{5}+4 a^{3} b^{2}+6 a \,b^{4}+\frac {\left (-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{3}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a \,b^{2}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{4}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2} b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{4}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {-2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{4 \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {2 a^{2} \left (a^{2}+3 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{2} \sqrt {a +b \tan \left (d x +c \right )}}+\frac {2 a^{3}}{3 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}}{b^{2} d}\)	\(968\)
default	\(\frac {\frac {2 b^{2} \left (\frac {\frac {\left (-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{3}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a \,b^{2}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{4}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2} b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{4}\right ) \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (2 a^{5}-4 a^{3} b^{2}-6 a \,b^{4}-\frac {\left (-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{3}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a \,b^{2}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{4}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2} b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{4}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{4 \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}+\frac {-\frac {\left (-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{3}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a \,b^{2}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{4}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2} b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{4}\right ) \ln \left (-b \tan \left (d x +c \right )-a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-\sqrt {a^{2}+b^{2}}\right )}{2}+\frac {2 \left (-2 a^{5}+4 a^{3} b^{2}+6 a \,b^{4}+\frac {\left (-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a^{3}-2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a \,b^{2}+3 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{4}+2 \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2} b^{2}-\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, b^{4}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{2}\right ) \arctan \left (\frac {-2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}}{4 \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {2 a^{2} \left (a^{2}+3 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{2} \sqrt {a +b \tan \left (d x +c \right )}}+\frac {2 a^{3}}{3 \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}}{b^{2} d}\)	\(968\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^3/(a+b*tan(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/d/b^2*(b^2/(a^2+b^2)^2*(1/4/(a^2+b^2)^(3/2)*(1/2*(-2*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^3-2*(2*
(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a*b^2+3*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^4+2*(2*(a^2+b^2)^(1/2)+2*a)
^(1/2)*a^2*b^2-(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*b^4)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+
2*a)^(1/2)+(a^2+b^2)^(1/2))+2*(2*a^5-4*a^3*b^2-6*a*b^4-1/2*(-2*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a
^3-2*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a*b^2+3*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^4+2*(2*(a^2+b^2)^(1
/2)+2*a)^(1/2)*a^2*b^2-(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*b^4)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*
a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)))+1/4/(
a^2+b^2)^(3/2)*(-1/2*(-2*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^3-2*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^
2+b^2)^(1/2)*a*b^2+3*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^4+2*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^2*b^2-(2*(a^2+b^2)^(1
/2)+2*a)^(1/2)*b^4)*ln(-b*tan(d*x+c)-a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-(a^2+b^2)^(1/2))+2
*(-2*a^5+4*a^3*b^2+6*a*b^4+1/2*(-2*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)*a^3-2*(2*(a^2+b^2)^(1/2)+2*a)
^(1/2)*(a^2+b^2)^(1/2)*a*b^2+3*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^4+2*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^2*b^2-(2*(a
^2+b^2)^(1/2)+2*a)^(1/2)*b^4)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((-2*(a+b*tan
(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))))-a^2*(a^2+3*b^2)/(a^2+b^2)^2/(a+
b*tan(d*x+c))^(1/2)+1/3*a^3/(a^2+b^2)/(a+b*tan(d*x+c))^(3/2))

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Maxima [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(tan(d*x+c)^3/(a+b*tan(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 9834 vs. \(2 (144) = 288\).
time = 1.61, size = 9834, normalized size = 57.17 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^3/(a+b*tan(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

-1/12*(12*sqrt(2)*((a^18*b^2 + a^16*b^4 - 20*a^14*b^6 - 84*a^12*b^8 - 154*a^10*b^10 - 154*a^8*b^12 - 84*a^6*b^
14 - 20*a^4*b^16 + a^2*b^18 + b^20)*d^5*cos(d*x + c)^4 + 2*(3*a^16*b^4 + 20*a^14*b^6 + 56*a^12*b^8 + 84*a^10*b
^10 + 70*a^8*b^12 + 28*a^6*b^14 - 4*a^2*b^18 - b^20)*d^5*cos(d*x + c)^2 + (a^14*b^6 + 7*a^12*b^8 + 21*a^10*b^1
0 + 35*a^8*b^12 + 35*a^6*b^14 + 21*a^4*b^16 + 7*a^2*b^18 + b^20)*d^5 + 4*((a^17*b^3 + 6*a^15*b^5 + 14*a^13*b^7
+ 14*a^11*b^9 - 14*a^7*b^13 - 14*a^5*b^15 - 6*a^3*b^17 - a*b^19)*d^5*cos(d*x + c)^3 + (a^15*b^5 + 7*a^13*b^7
+ 21*a^11*b^9 + 35*a^9*b^11 + 35*a^7*b^13 + 21*a^5*b^15 + 7*a^3*b^17 + a*b^19)*d^5*cos(d*x + c))*sin(d*x + c))
*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a
^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b
^6 + 5*a^2*b^8 + b^10)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((25*a^8*b^2 -
100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b
^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*(1/((a^10 + 5*a^8*b
^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))^(3/4)*arctan(((5*a^20 + 30*a^18*b^2 + 61*a^16*b^4 + 8*a
^14*b^6 - 182*a^12*b^8 - 364*a^10*b^10 - 350*a^8*b^12 - 184*a^6*b^14 - 47*a^4*b^16 - 2*a^2*b^18 + b^20)*d^4*sq
rt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*
b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*sqr
t(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)) + (5*a^15 + 15*a^13*b^2 + a^11*b^4
- 45*a^9*b^6 - 65*a^7*b^8 - 35*a^5*b^10 - 5*a^3*b^12 + a*b^14)*d^2*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^
6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*
a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)) - sqrt(2)*((a^23 + 7*a^21*b^2 + 15*a^19*b^4
- 15*a^17*b^6 - 150*a^15*b^8 - 378*a^13*b^10 - 546*a^11*b^12 - 510*a^9*b^14 - 315*a^7*b^16 - 125*a^5*b^18 - 29
*a^3*b^20 - 3*a*b^22)*d^7*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b
^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 1
0*a^2*b^18 + b^20)*d^4))*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)) + (a^18
+ 7*a^16*b^2 + 20*a^14*b^4 + 28*a^12*b^6 + 14*a^10*b^8 - 14*a^8*b^10 - 28*a^6*b^12 - 20*a^4*b^14 - 7*a^2*b^16
- b^18)*d^5*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*
b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 +
b^20)*d^4)))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^15 - 5*a^13*b^2 - 35*a^1
1*b^4 - 65*a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^
4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((
(25*a^14 - 25*a^12*b^2 - 115*a^10*b^4 + 35*a^8*b^6 + 171*a^6*b^8 + 53*a^4*b^10 - 17*a^2*b^12 + b^14)*d^2*sqrt(
1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))*cos(d*x + c) + sqrt(2)*((25*a^16 - 50
*a^14*b^2 - 90*a^12*b^4 + 150*a^10*b^6 + 136*a^8*b^8 - 118*a^6*b^10 - 70*a^4*b^12 + 18*a^2*b^14 - b^16)*d^3*sq
rt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))*cos(d*x + c) + (25*a^11 - 175*a^9*
b^2 + 410*a^7*b^4 - 350*a^5*b^6 + 61*a^3*b^8 - 3*a*b^10)*d*cos(d*x + c))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 +
10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^
3*b^12 + 5*a*b^14)*d^2*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)))/(25*a^8*
b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/
((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))^(1/4) + (25*a^9 - 100*a^7*b^2 + 110*a^5
*b^4 - 20*a^3*b^6 + a*b^8)*cos(d*x + c) + (25*a^8*b - 100*a^6*b^3 + 110*a^4*b^5 - 20*a^2*b^7 + b^9)*sin(d*x +
c))/cos(d*x + c))*(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))^(3/4) + sqrt(2)*((
5*a^27 + 25*a^25*b^2 + 6*a^23*b^4 - 218*a^21*b^6 - 585*a^19*b^8 - 405*a^17*b^10 + 900*a^15*b^12 + 2532*a^13*b^
14 + 2979*a^11*b^16 + 2015*a^9*b^18 + 790*a^7*b^20 + 150*a^5*b^22 + a^3*b^24 - 3*a*b^26)*d^7*sqrt((25*a^8*b^2
- 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*
b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*sqrt(1/((a^10 + 5*
a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)) + (5*a^22 + 25*a^20*b^2 + 31*a^18*b^4 - 53*a^16*b^
6 - 190*a^14*b^8 - 182*a^12*b^10 + 14*a^10*b^12...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tan ^{3}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**3/(a+b*tan(d*x+c))**(5/2),x)

[Out]

Integral(tan(c + d*x)**3/(a + b*tan(c + d*x))**(5/2), x)

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Giac [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(tan(d*x+c)^3/(a+b*tan(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [B]
time = 9.45, size = 2500, normalized size = 14.53 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^3/(a + b*tan(c + d*x))^(5/2),x)

[Out]

atan(((1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*((
1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*(96*a*b^2
0*d^4 + 736*a^3*b^18*d^4 + 2432*a^5*b^16*d^4 + 4480*a^7*b^14*d^4 + 4928*a^9*b^12*d^4 + 3136*a^11*b^10*d^4 + 89
6*a^13*b^8*d^4 - 128*a^15*b^6*d^4 - 160*a^17*b^4*d^4 - 32*a^19*b^2*d^4 + (1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*
d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^22*d^5 + 6
40*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440*a^1
3*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b^2*d^5)) + (a + b*tan(c + d*x
))^(1/2)*(320*a^4*b^14*d^3 - 16*b^18*d^3 + 1024*a^6*b^12*d^3 + 1440*a^8*b^10*d^3 + 1024*a^10*b^8*d^3 + 320*a^1
2*b^6*d^3 - 16*a^16*b^2*d^3))*1i - (1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2
- a^3*b^2*d^2*10i)))^(1/2)*((1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3*b
^2*d^2*10i)))^(1/2)*(96*a*b^20*d^4 + 736*a^3*b^18*d^4 + 2432*a^5*b^16*d^4 + 4480*a^7*b^14*d^4 + 4928*a^9*b^12*
d^4 + 3136*a^11*b^10*d^4 + 896*a^13*b^8*d^4 - 128*a^15*b^6*d^4 - 160*a^17*b^4*d^4 - 32*a^19*b^2*d^4 - (1i/(4*(
a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*(a + b*tan(c + d
*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 16
128*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b
^2*d^5)) - (a + b*tan(c + d*x))^(1/2)*(320*a^4*b^14*d^3 - 16*b^18*d^3 + 1024*a^6*b^12*d^3 + 1440*a^8*b^10*d^3
+ 1024*a^10*b^8*d^3 + 320*a^12*b^6*d^3 - 16*a^16*b^2*d^3))*1i)/((1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i -
5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*((1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b
*d^2 + 10*a^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*(96*a*b^20*d^4 + 736*a^3*b^18*d^4 + 2432*a^5*b^16*d^4 + 4480*
a^7*b^14*d^4 + 4928*a^9*b^12*d^4 + 3136*a^11*b^10*d^4 + 896*a^13*b^8*d^4 - 128*a^15*b^6*d^4 - 160*a^17*b^4*d^4
- 32*a^19*b^2*d^4 + (1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3*b^2*d^2*
10i)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 7680*a^7*b^16*
d^5 + 13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680*a^15*b^8*d^5 + 2880*a^17*b^6*d^5 +
640*a^19*b^4*d^5 + 64*a^21*b^2*d^5)) + (a + b*tan(c + d*x))^(1/2)*(320*a^4*b^14*d^3 - 16*b^18*d^3 + 1024*a^6*
b^12*d^3 + 1440*a^8*b^10*d^3 + 1024*a^10*b^8*d^3 + 320*a^12*b^6*d^3 - 16*a^16*b^2*d^3)) + (1i/(4*(a^5*d^2*1i -
b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*((1i/(4*(a^5*d^2*1i - b^5*d^
2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*(96*a*b^20*d^4 + 736*a^3*b^18*d^4 +
2432*a^5*b^16*d^4 + 4480*a^7*b^14*d^4 + 4928*a^9*b^12*d^4 + 3136*a^11*b^10*d^4 + 896*a^13*b^8*d^4 - 128*a^15*
b^6*d^4 - 160*a^17*b^4*d^4 - 32*a^19*b^2*d^4 - (1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*
a^2*b^3*d^2 - a^3*b^2*d^2*10i)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5
*b^18*d^5 + 7680*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680*a^15*b^8
*d^5 + 2880*a^17*b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b^2*d^5)) - (a + b*tan(c + d*x))^(1/2)*(320*a^4*b^14*d^3
- 16*b^18*d^3 + 1024*a^6*b^12*d^3 + 1440*a^8*b^10*d^3 + 1024*a^10*b^8*d^3 + 320*a^12*b^6*d^3 - 16*a^16*b^2*d^
3)) - 16*b^16*d^2 - 80*a^2*b^14*d^2 - 144*a^4*b^12*d^2 - 80*a^6*b^10*d^2 + 80*a^8*b^8*d^2 + 144*a^10*b^6*d^2 +
80*a^12*b^4*d^2 + 16*a^14*b^2*d^2))*(1i/(4*(a^5*d^2*1i - b^5*d^2 + a*b^4*d^2*5i - 5*a^4*b*d^2 + 10*a^2*b^3*d^
2 - a^3*b^2*d^2*10i)))^(1/2)*2i - atan(((((1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2*
10i - 10*a^3*b^2*d^2))^(1/2)*(48*a*b^20*d^4 - ((1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3
*d^2*10i - 10*a^3*b^2*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^18
*d^5 + 7680*a^7*b^16*d^5 + 13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680*a^15*b^8*d^5
+ 2880*a^17*b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b^2*d^5))/4 + 368*a^3*b^18*d^4 + 1216*a^5*b^16*d^4 + 2240*a^7
*b^14*d^4 + 2464*a^9*b^12*d^4 + 1568*a^11*b^10*d^4 + 448*a^13*b^8*d^4 - 64*a^15*b^6*d^4 - 80*a^17*b^4*d^4 - 16
*a^19*b^2*d^4))/2 - ((a + b*tan(c + d*x))^(1/2)*(320*a^4*b^14*d^3 - 16*b^18*d^3 + 1024*a^6*b^12*d^3 + 1440*a^8
*b^10*d^3 + 1024*a^10*b^8*d^3 + 320*a^12*b^6*d^3 - 16*a^16*b^2*d^3))/2)*(1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2
- a^4*b*d^2*5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d^2))^(1/2)*1i - (((1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4
*b*d^2*5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d^2))^(1/2)*(48*a*b^20*d^4 + ((1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2
- a^4*b*d^2*5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d...

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