3.222 \(\int e^{\frac{3}{2} i \tan ^{-1}(a+b x)} x \, dx\)

Optimal. Leaf size=410 \[ \frac{\sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{2 b^2}+\frac{(3-4 i a) \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}{4 b^2}-\frac{3 (3-4 i a) \log \left (\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}-\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{8 \sqrt{2} b^2}+\frac{3 (3-4 i a) \log \left (\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}+\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{8 \sqrt{2} b^2}-\frac{3 (3-4 i a) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{4 \sqrt{2} b^2}+\frac{3 (3-4 i a) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{4 \sqrt{2} b^2} \]

[Out]

((3 - (4*I)*a)*(1 - I*a - I*b*x)^(1/4)*(1 + I*a + I*b*x)^(3/4))/(4*b^2) + ((1 - I*a - I*b*x)^(1/4)*(1 + I*a +
I*b*x)^(7/4))/(2*b^2) - (3*(3 - (4*I)*a)*ArcTan[1 - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)]
)/(4*Sqrt[2]*b^2) + (3*(3 - (4*I)*a)*ArcTan[1 + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(4
*Sqrt[2]*b^2) - (3*(3 - (4*I)*a)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] - (Sqrt[2]*(1 - I*a - I*b
*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(8*Sqrt[2]*b^2) + (3*(3 - (4*I)*a)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 +
I*a + I*b*x] + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(8*Sqrt[2]*b^2)

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Rubi [A]  time = 0.292121, antiderivative size = 410, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.688, Rules used = {5095, 80, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ \frac{\sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{2 b^2}+\frac{(3-4 i a) \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}{4 b^2}-\frac{3 (3-4 i a) \log \left (\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}-\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{8 \sqrt{2} b^2}+\frac{3 (3-4 i a) \log \left (\frac{\sqrt{-i a-i b x+1}}{\sqrt{i a+i b x+1}}+\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{8 \sqrt{2} b^2}-\frac{3 (3-4 i a) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{4 \sqrt{2} b^2}+\frac{3 (3-4 i a) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{4 \sqrt{2} b^2} \]

Antiderivative was successfully verified.

[In]

Int[E^(((3*I)/2)*ArcTan[a + b*x])*x,x]

[Out]

((3 - (4*I)*a)*(1 - I*a - I*b*x)^(1/4)*(1 + I*a + I*b*x)^(3/4))/(4*b^2) + ((1 - I*a - I*b*x)^(1/4)*(1 + I*a +
I*b*x)^(7/4))/(2*b^2) - (3*(3 - (4*I)*a)*ArcTan[1 - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)]
)/(4*Sqrt[2]*b^2) + (3*(3 - (4*I)*a)*ArcTan[1 + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(4
*Sqrt[2]*b^2) - (3*(3 - (4*I)*a)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 + I*a + I*b*x] - (Sqrt[2]*(1 - I*a - I*b
*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(8*Sqrt[2]*b^2) + (3*(3 - (4*I)*a)*Log[1 + Sqrt[1 - I*a - I*b*x]/Sqrt[1 +
I*a + I*b*x] + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)])/(8*Sqrt[2]*b^2)

Rule 5095

Int[E^(ArcTan[(c_.)*((a_) + (b_.)*(x_))]*(n_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[((d + e*x)^m*(1 -
I*a*c - I*b*c*x)^((I*n)/2))/(1 + I*a*c + I*b*c*x)^((I*n)/2), x] /; FreeQ[{a, b, c, d, e, m, n}, x]

Rule 80

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

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 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 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]

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 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 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])

Rubi steps

\begin{align*} \int e^{\frac{3}{2} i \tan ^{-1}(a+b x)} x \, dx &=\int \frac{x (1+i a+i b x)^{3/4}}{(1-i a-i b x)^{3/4}} \, dx\\ &=\frac{\sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{2 b^2}-\frac{(3 i+4 a) \int \frac{(1+i a+i b x)^{3/4}}{(1-i a-i b x)^{3/4}} \, dx}{4 b}\\ &=\frac{(3-4 i a) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{4 b^2}+\frac{\sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{2 b^2}-\frac{(3 (3 i+4 a)) \int \frac{1}{(1-i a-i b x)^{3/4} \sqrt [4]{1+i a+i b x}} \, dx}{8 b}\\ &=\frac{(3-4 i a) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{4 b^2}+\frac{\sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{2 b^2}+\frac{(3 (3-4 i a)) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-i a-i b x}\right )}{2 b^2}\\ &=\frac{(3-4 i a) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{4 b^2}+\frac{\sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{2 b^2}+\frac{(3 (3-4 i a)) \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{2 b^2}\\ &=\frac{(3-4 i a) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{4 b^2}+\frac{\sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{2 b^2}+\frac{(3 (3-4 i a)) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{4 b^2}+\frac{(3 (3-4 i a)) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{4 b^2}\\ &=\frac{(3-4 i a) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{4 b^2}+\frac{\sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{2 b^2}+\frac{(3 (3-4 i a)) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 b^2}+\frac{(3 (3-4 i a)) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 b^2}-\frac{(3 (3-4 i a)) \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt{2} b^2}-\frac{(3 (3-4 i a)) \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt{2} b^2}\\ &=\frac{(3-4 i a) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{4 b^2}+\frac{\sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{2 b^2}-\frac{3 (3-4 i a) \log \left (1+\frac{\sqrt{1-i a-i b x}}{\sqrt{1+i a+i b x}}-\frac{\sqrt{2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt{2} b^2}+\frac{3 (3-4 i a) \log \left (1+\frac{\sqrt{1-i a-i b x}}{\sqrt{1+i a+i b x}}+\frac{\sqrt{2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt{2} b^2}+\frac{(3 (3-4 i a)) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{4 \sqrt{2} b^2}-\frac{(3 (3-4 i a)) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{4 \sqrt{2} b^2}\\ &=\frac{(3-4 i a) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{4 b^2}+\frac{\sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{2 b^2}-\frac{3 (3-4 i a) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{4 \sqrt{2} b^2}+\frac{3 (3-4 i a) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{4 \sqrt{2} b^2}-\frac{3 (3-4 i a) \log \left (1+\frac{\sqrt{1-i a-i b x}}{\sqrt{1+i a+i b x}}-\frac{\sqrt{2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt{2} b^2}+\frac{3 (3-4 i a) \log \left (1+\frac{\sqrt{1-i a-i b x}}{\sqrt{1+i a+i b x}}+\frac{\sqrt{2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt{2} b^2}\\ \end{align*}

Mathematica [C]  time = 0.0419517, size = 79, normalized size = 0.19 \[ \frac{\sqrt [4]{-i (a+b x+i)} \left (2\ 2^{3/4} (3-4 i a) \, _2F_1\left (-\frac{3}{4},\frac{1}{4};\frac{5}{4};-\frac{1}{2} i (a+b x+i)\right )+(i a+i b x+1)^{7/4}\right )}{2 b^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(((3*I)/2)*ArcTan[a + b*x])*x,x]

[Out]

(((-I)*(I + a + b*x))^(1/4)*((1 + I*a + I*b*x)^(7/4) + 2*2^(3/4)*(3 - (4*I)*a)*Hypergeometric2F1[-3/4, 1/4, 5/
4, (-I/2)*(I + a + b*x)]))/(2*b^2)

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Maple [F]  time = 0.22, size = 0, normalized size = 0. \begin{align*} \int \left ({(1+i \left ( bx+a \right ) ){\frac{1}{\sqrt{1+ \left ( bx+a \right ) ^{2}}}}} \right ) ^{{\frac{3}{2}}}x\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (\frac{i \, b x + i \, a + 1}{\sqrt{{\left (b x + a\right )}^{2} + 1}}\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*((I*b*x + I*a + 1)/sqrt((b*x + a)^2 + 1))^(3/2), x)

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Fricas [A]  time = 2.316, size = 1123, normalized size = 2.74 \begin{align*} -\frac{b^{2} \sqrt{\frac{144 i \, a^{2} - 216 \, a - 81 i}{b^{4}}} \log \left (\frac{b^{2} \sqrt{\frac{144 i \, a^{2} - 216 \, a - 81 i}{b^{4}}} +{\left (12 \, a + 9 i\right )} \sqrt{\frac{i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{12 \, a + 9 i}\right ) - b^{2} \sqrt{\frac{144 i \, a^{2} - 216 \, a - 81 i}{b^{4}}} \log \left (-\frac{b^{2} \sqrt{\frac{144 i \, a^{2} - 216 \, a - 81 i}{b^{4}}} -{\left (12 \, a + 9 i\right )} \sqrt{\frac{i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{12 \, a + 9 i}\right ) - b^{2} \sqrt{\frac{-144 i \, a^{2} + 216 \, a + 81 i}{b^{4}}} \log \left (\frac{b^{2} \sqrt{\frac{-144 i \, a^{2} + 216 \, a + 81 i}{b^{4}}} +{\left (12 \, a + 9 i\right )} \sqrt{\frac{i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{12 \, a + 9 i}\right ) + b^{2} \sqrt{\frac{-144 i \, a^{2} + 216 \, a + 81 i}{b^{4}}} \log \left (-\frac{b^{2} \sqrt{\frac{-144 i \, a^{2} + 216 \, a + 81 i}{b^{4}}} -{\left (12 \, a + 9 i\right )} \sqrt{\frac{i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{12 \, a + 9 i}\right ) - 2 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{\left (2 i \, b x - 2 i \, a + 5\right )} \sqrt{\frac{i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{8 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(b^2*sqrt((144*I*a^2 - 216*a - 81*I)/b^4)*log((b^2*sqrt((144*I*a^2 - 216*a - 81*I)/b^4) + (12*a + 9*I)*sq
rt(I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/(b*x + a + I)))/(12*a + 9*I)) - b^2*sqrt((144*I*a^2 - 216*a - 81*I)/b^4
)*log(-(b^2*sqrt((144*I*a^2 - 216*a - 81*I)/b^4) - (12*a + 9*I)*sqrt(I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/(b*x
+ a + I)))/(12*a + 9*I)) - b^2*sqrt((-144*I*a^2 + 216*a + 81*I)/b^4)*log((b^2*sqrt((-144*I*a^2 + 216*a + 81*I)
/b^4) + (12*a + 9*I)*sqrt(I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/(b*x + a + I)))/(12*a + 9*I)) + b^2*sqrt((-144*I
*a^2 + 216*a + 81*I)/b^4)*log(-(b^2*sqrt((-144*I*a^2 + 216*a + 81*I)/b^4) - (12*a + 9*I)*sqrt(I*sqrt(b^2*x^2 +
2*a*b*x + a^2 + 1)/(b*x + a + I)))/(12*a + 9*I)) - 2*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(2*I*b*x - 2*I*a + 5)*
sqrt(I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/(b*x + a + I)))/b^2

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1+I*(b*x+a))/(1+(b*x+a)**2)**(1/2))**(3/2)*x,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (\frac{i \, b x + i \, a + 1}{\sqrt{{\left (b x + a\right )}^{2} + 1}}\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*((I*b*x + I*a + 1)/sqrt((b*x + a)^2 + 1))^(3/2), x)