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

Optimal. Leaf size=249 \[ -\frac{i (-i a-i b x+1)^{3/2} \sqrt{i a+i b x+1} \left (-22 i a^2+2 (11+10 i a) b x-54 a+29 i\right )}{8 b^4}+\frac{3 \left (-8 i a^3-36 a^2+44 i a+17\right ) \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{8 b^4}+\frac{3 \left (8 a^3-36 i a^2-44 a+17 i\right ) \sinh ^{-1}(a+b x)}{8 b^4}-\frac{9 x^2 (-i a-i b x+1)^{3/2} \sqrt{i a+i b x+1}}{4 b^2}+\frac{2 i x^3 (-i a-i b x+1)^{3/2}}{b \sqrt{i a+i b x+1}} \]

[Out]

((2*I)*x^3*(1 - I*a - I*b*x)^(3/2))/(b*Sqrt[1 + I*a + I*b*x]) + (3*(17 + (44*I)*a - 36*a^2 - (8*I)*a^3)*Sqrt[1
- I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(8*b^4) - (9*x^2*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x])/(4*b^2)
- ((I/8)*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x]*(29*I - 54*a - (22*I)*a^2 + 2*(11 + (10*I)*a)*b*x))/b^
4 + (3*(17*I - 44*a - (36*I)*a^2 + 8*a^3)*ArcSinh[a + b*x])/(8*b^4)

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Rubi [A]  time = 0.245515, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5095, 97, 153, 147, 50, 53, 619, 215} \[ -\frac{i (-i a-i b x+1)^{3/2} \sqrt{i a+i b x+1} \left (-22 i a^2+2 (11+10 i a) b x-54 a+29 i\right )}{8 b^4}+\frac{3 \left (-8 i a^3-36 a^2+44 i a+17\right ) \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{8 b^4}+\frac{3 \left (8 a^3-36 i a^2-44 a+17 i\right ) \sinh ^{-1}(a+b x)}{8 b^4}-\frac{9 x^2 (-i a-i b x+1)^{3/2} \sqrt{i a+i b x+1}}{4 b^2}+\frac{2 i x^3 (-i a-i b x+1)^{3/2}}{b \sqrt{i a+i b x+1}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/E^((3*I)*ArcTan[a + b*x]),x]

[Out]

((2*I)*x^3*(1 - I*a - I*b*x)^(3/2))/(b*Sqrt[1 + I*a + I*b*x]) + (3*(17 + (44*I)*a - 36*a^2 - (8*I)*a^3)*Sqrt[1
- I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(8*b^4) - (9*x^2*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x])/(4*b^2)
- ((I/8)*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x]*(29*I - 54*a - (22*I)*a^2 + 2*(11 + (10*I)*a)*b*x))/b^
4 + (3*(17*I - 44*a - (36*I)*a^2 + 8*a^3)*ArcSinh[a + b*x])/(8*b^4)

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 97

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

Rule 153

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 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 53

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Int[1/Sqrt[a*c - b*(a - c)*x - b^2*x^2]
, x] /; FreeQ[{a, b, c, d}, x] && EqQ[b + d, 0] && GtQ[a + c, 0]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 215

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

Rubi steps

\begin{align*} \int e^{-3 i \tan ^{-1}(a+b x)} x^3 \, dx &=\int \frac{x^3 (1-i a-i b x)^{3/2}}{(1+i a+i b x)^{3/2}} \, dx\\ &=\frac{2 i x^3 (1-i a-i b x)^{3/2}}{b \sqrt{1+i a+i b x}}-\frac{(2 i) \int \frac{x^2 \sqrt{1-i a-i b x} \left (3 (1-i a)-\frac{9 i b x}{2}\right )}{\sqrt{1+i a+i b x}} \, dx}{b}\\ &=\frac{2 i x^3 (1-i a-i b x)^{3/2}}{b \sqrt{1+i a+i b x}}-\frac{9 x^2 (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{4 b^2}-\frac{i \int \frac{x \sqrt{1-i a-i b x} \left (9 i \left (1+a^2\right ) b+\frac{3}{2} (11+10 i a) b^2 x\right )}{\sqrt{1+i a+i b x}} \, dx}{2 b^3}\\ &=\frac{2 i x^3 (1-i a-i b x)^{3/2}}{b \sqrt{1+i a+i b x}}-\frac{9 x^2 (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{4 b^2}-\frac{i (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x} \left (29 i-54 a-22 i a^2+2 (11+10 i a) b x\right )}{8 b^4}+\frac{\left (3 \left (17 i-44 a-36 i a^2+8 a^3\right )\right ) \int \frac{\sqrt{1-i a-i b x}}{\sqrt{1+i a+i b x}} \, dx}{8 b^3}\\ &=\frac{2 i x^3 (1-i a-i b x)^{3/2}}{b \sqrt{1+i a+i b x}}+\frac{3 \left (17+44 i a-36 a^2-8 i a^3\right ) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{8 b^4}-\frac{9 x^2 (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{4 b^2}-\frac{i (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x} \left (29 i-54 a-22 i a^2+2 (11+10 i a) b x\right )}{8 b^4}+\frac{\left (3 \left (17 i-44 a-36 i a^2+8 a^3\right )\right ) \int \frac{1}{\sqrt{1-i a-i b x} \sqrt{1+i a+i b x}} \, dx}{8 b^3}\\ &=\frac{2 i x^3 (1-i a-i b x)^{3/2}}{b \sqrt{1+i a+i b x}}+\frac{3 \left (17+44 i a-36 a^2-8 i a^3\right ) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{8 b^4}-\frac{9 x^2 (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{4 b^2}-\frac{i (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x} \left (29 i-54 a-22 i a^2+2 (11+10 i a) b x\right )}{8 b^4}+\frac{\left (3 \left (17 i-44 a-36 i a^2+8 a^3\right )\right ) \int \frac{1}{\sqrt{(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{8 b^3}\\ &=\frac{2 i x^3 (1-i a-i b x)^{3/2}}{b \sqrt{1+i a+i b x}}+\frac{3 \left (17+44 i a-36 a^2-8 i a^3\right ) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{8 b^4}-\frac{9 x^2 (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{4 b^2}-\frac{i (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x} \left (29 i-54 a-22 i a^2+2 (11+10 i a) b x\right )}{8 b^4}+\frac{\left (3 \left (17 i-44 a-36 i a^2+8 a^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{4 b^2}}} \, dx,x,2 a b+2 b^2 x\right )}{16 b^5}\\ &=\frac{2 i x^3 (1-i a-i b x)^{3/2}}{b \sqrt{1+i a+i b x}}+\frac{3 \left (17+44 i a-36 a^2-8 i a^3\right ) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{8 b^4}-\frac{9 x^2 (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{4 b^2}-\frac{i (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x} \left (29 i-54 a-22 i a^2+2 (11+10 i a) b x\right )}{8 b^4}+\frac{3 \left (17 i-44 a-36 i a^2+8 a^3\right ) \sinh ^{-1}(a+b x)}{8 b^4}\\ \end{align*}

Mathematica [A]  time = 0.3541, size = 244, normalized size = 0.98 \[ \frac{a^2 \left (-12 b^2 x^2+265 i b x+4\right )+a^4 (-76-2 i b x)-5 a^3 (20 b x-31 i)-2 i a^5+a \left (2 i b^4 x^4+4 b^3 x^3+53 i b^2 x^2+212 b x+157 i\right )+2 i b^5 x^5-8 b^4 x^4-17 i b^3 x^3+40 b^2 x^2-51 i b x+80}{8 b^4 \sqrt{a^2+2 a b x+b^2 x^2+1}}+\frac{3 \sqrt [4]{-1} \left (8 a^3-36 i a^2-44 a+17 i\right ) \sqrt{-i b} \sinh ^{-1}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right )}{4 b^{9/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(80 - (2*I)*a^5 - (51*I)*b*x + 40*b^2*x^2 - (17*I)*b^3*x^3 - 8*b^4*x^4 + (2*I)*b^5*x^5 + a^4*(-76 - (2*I)*b*x)
- 5*a^3*(-31*I + 20*b*x) + a^2*(4 + (265*I)*b*x - 12*b^2*x^2) + a*(157*I + 212*b*x + (53*I)*b^2*x^2 + 4*b^3*x
^3 + (2*I)*b^4*x^4))/(8*b^4*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]) + (3*(-1)^(1/4)*(17*I - 44*a - (36*I)*a^2 + 8*a
^3)*Sqrt[(-I)*b]*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/(4*b^(9/2))

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Maple [B]  time = 0.102, size = 1529, normalized size = 6.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/2*I/b^3*ln((I*b+(x-(I-a)/b)*b^2)/(b^2)^(1/2)+((x-(I-a)/b)^2*b^2+2*I*(x-(I-a)/b)*b)^(1/2))/(b^2)^(1/2)*a^2-
3*I/b^7/(x-I/b+a/b)^3*((x-(I-a)/b)^2*b^2+2*I*(x-(I-a)/b)*b)^(5/2)*a^2+2*I/b^6/(x-I/b+a/b)^2*((x-(I-a)/b)^2*b^2
+2*I*(x-(I-a)/b)*b)^(5/2)*a^3-12*I/b^6/(x-I/b+a/b)^2*((x-(I-a)/b)^2*b^2+2*I*(x-(I-a)/b)*b)^(5/2)*a-5/b^6/(x-I/
b+a/b)^2*((x-(I-a)/b)^2*b^2+2*I*(x-(I-a)/b)*b)^(5/2)-9/b^4*((x-(I-a)/b)^2*b^2+2*I*(x-(I-a)/b)*b)^(3/2)*a^2+3/b
^4*((x-(I-a)/b)^2*b^2+2*I*(x-(I-a)/b)*b)^(1/2)*a^4+3/b^3*ln((I*b+(x-(I-a)/b)*b^2)/(b^2)^(1/2)+((x-(I-a)/b)^2*b
^2+2*I*(x-(I-a)/b)*b)^(1/2))/(b^2)^(1/2)*a^3-33/2/b^3*ln((I*b+(x-(I-a)/b)*b^2)/(b^2)^(1/2)+((x-(I-a)/b)^2*b^2+
2*I*(x-(I-a)/b)*b)^(1/2))/(b^2)^(1/2)*a-33/2/b^4*((x-(I-a)/b)^2*b^2+2*I*(x-(I-a)/b)*b)^(1/2)*a^2-27/2*I/b^3*((
x-(I-a)/b)^2*b^2+2*I*(x-(I-a)/b)*b)^(1/2)*x*a^2+3/8*I/b^4*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a+3/8*I/b^3*ln((b^2*x+
a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2)+1/b^7/(x-I/b+a/b)^3*((x-(I-a)/b)^2*b^2+2*I*(x-(I-a
)/b)*b)^(5/2)*a^3-3/b^7/(x-I/b+a/b)^3*((x-(I-a)/b)^2*b^2+2*I*(x-(I-a)/b)*b)^(5/2)*a+I/b^7/(x-I/b+a/b)^3*((x-(I
-a)/b)^2*b^2+2*I*(x-(I-a)/b)*b)^(5/2)+9/b^6/(x-I/b+a/b)^2*((x-(I-a)/b)^2*b^2+2*I*(x-(I-a)/b)*b)^(5/2)*a^2+3/b^
3*((x-(I-a)/b)^2*b^2+2*I*(x-(I-a)/b)*b)^(1/2)*x*a^3-33/2/b^3*((x-(I-a)/b)^2*b^2+2*I*(x-(I-a)/b)*b)^(1/2)*x*a-2
*I/b^4*((x-(I-a)/b)^2*b^2+2*I*(x-(I-a)/b)*b)^(3/2)*a^3+1/4*I/b^3*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)*x+1/4*I/b^4*(b^
2*x^2+2*a*b*x+a^2+1)^(3/2)*a+3/8*I/b^3*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*x-27/2*I/b^4*((x-(I-a)/b)^2*b^2+2*I*(x-(I
-a)/b)*b)^(1/2)*a^3+11*I/b^4*((x-(I-a)/b)^2*b^2+2*I*(x-(I-a)/b)*b)^(3/2)*a+6*I/b^3*((x-(I-a)/b)^2*b^2+2*I*(x-(
I-a)/b)*b)^(1/2)*x+6*I/b^4*((x-(I-a)/b)^2*b^2+2*I*(x-(I-a)/b)*b)^(1/2)*a+6*I/b^3*ln((I*b+(x-(I-a)/b)*b^2)/(b^2
)^(1/2)+((x-(I-a)/b)^2*b^2+2*I*(x-(I-a)/b)*b)^(1/2))/(b^2)^(1/2)+4/b^4*((x-(I-a)/b)^2*b^2+2*I*(x-(I-a)/b)*b)^(
3/2)

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Maxima [B]  time = 1.658, size = 1322, normalized size = 5.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*a^3/(b^6*x^2 + 2*a*b^5*x + a^2*b^4 - 2*I*b^5*x - 2*I*a*b^4 - b^4) - 3*(
b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*a^2/(b^6*x^2 + 2*a*b^5*x + a^2*b^4 - 2*I*b^5*x - 2*I*a*b^4 - b^4) - 3*(b^2*
x^2 + 2*a*b*x + a^2 + 1)^(3/2)*a^2/(2*I*b^5*x + 2*I*a*b^4 + 2*b^4) - 6*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^3
/(I*b^5*x + I*a*b^4 + b^4) + 3*I*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*a/(b^6*x^2 + 2*a*b^5*x + a^2*b^4 - 2*I*b^
5*x - 2*I*a*b^4 - b^4) + 6*I*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*a/(2*I*b^5*x + 2*I*a*b^4 + 2*b^4) - 18*sqrt(b
^2*x^2 + 2*a*b*x + a^2 + 1)*a^2/(I*b^5*x + I*a*b^4 + b^4) + (b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)/(b^6*x^2 + 2*a
*b^5*x + a^2*b^4 - 2*I*b^5*x - 2*I*a*b^4 - b^4) + 3*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)/(2*I*b^5*x + 2*I*a*b^4
+ 2*b^4) + 18*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a/(I*b^5*x + I*a*b^4 + b^4) + 3*a^3*arcsinh(b*x + a)/b^4 +
6*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/(I*b^5*x + I*a*b^4 + b^4) + 1/4*I*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*x/b^
3 + 3/2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 4*I*b*x + 4*I*a + 3)*a*x/b^3 - 27/2*I*a^2*arcsinh(b*x + a)/b^4 - 3/4*I
*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*a/b^4 - 9/2*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^2/b^4 + 3/2*sqrt(-b^2*x^2
- 2*a*b*x - a^2 + 4*I*b*x + 4*I*a + 3)*a^2/b^4 + 3/8*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*x/b^3 - 3/2*I*sqrt(-
b^2*x^2 - 2*a*b*x - a^2 + 4*I*b*x + 4*I*a + 3)*x/b^3 - 3/2*a*arcsin(I*b*x + I*a + 2)/b^4 - 18*a*arcsinh(b*x +
a)/b^4 - (b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)/b^4 + 75/8*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a/b^4 - 9/2*I*sqrt
(-b^2*x^2 - 2*a*b*x - a^2 + 4*I*b*x + 4*I*a + 3)*a/b^4 + 3/2*I*arcsin(I*b*x + I*a + 2)/b^4 + 63/8*I*arcsinh(b*
x + a)/b^4 + 9/2*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/b^4 - 3*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 4*I*b*x + 4*I*a + 3
)/b^4

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Fricas [A]  time = 2.36326, size = 635, normalized size = 2.55 \begin{align*} \frac{-15 i \, a^{5} - 495 \, a^{4} + 1664 i \, a^{3} +{\left (-15 i \, a^{4} - 480 \, a^{3} + 1184 i \, a^{2} + 968 \, a - 256 i\right )} b x + 2152 \, a^{2} -{\left (192 \, a^{4} - 1056 i \, a^{3} +{\left (192 \, a^{3} - 864 i \, a^{2} - 1056 \, a + 408 i\right )} b x - 1920 \, a^{2} + 1464 i \, a + 408\right )} \log \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) +{\left (16 i \, b^{4} x^{4} - 48 \, b^{3} x^{3} +{\left (80 \, a - 88 i\right )} b^{2} x^{2} - 16 i \, a^{4} - 624 \, a^{3} - 8 \,{\left (22 \, a^{2} - 54 i \, a - 29\right )} b x + 1864 i \, a^{2} + 1896 \, a - 640 i\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 1224 i \, a - 256}{64 \, b^{5} x +{\left (64 \, a - 64 i\right )} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-15*I*a^5 - 495*a^4 + 1664*I*a^3 + (-15*I*a^4 - 480*a^3 + 1184*I*a^2 + 968*a - 256*I)*b*x + 2152*a^2 - (192*a
^4 - 1056*I*a^3 + (192*a^3 - 864*I*a^2 - 1056*a + 408*I)*b*x - 1920*a^2 + 1464*I*a + 408)*log(-b*x - a + sqrt(
b^2*x^2 + 2*a*b*x + a^2 + 1)) + (16*I*b^4*x^4 - 48*b^3*x^3 + (80*a - 88*I)*b^2*x^2 - 16*I*a^4 - 624*a^3 - 8*(2
2*a^2 - 54*I*a - 29)*b*x + 1864*I*a^2 + 1896*a - 640*I)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) - 1224*I*a - 256)/(6
4*b^5*x + (64*a - 64*I)*b^4)

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

[Out]

Timed out

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Giac [A]  time = 1.1616, size = 459, normalized size = 1.84 \begin{align*} -\frac{1}{8} \, \sqrt{{\left (b x + a\right )}^{2} + 1}{\left ({\left (2 \, x{\left (\frac{x}{b i} - \frac{a b^{11} - 4 \, b^{11} i}{b^{13} i}\right )} + \frac{2 \, a^{2} b^{10} - 20 \, a b^{10} i - 19 \, b^{10}}{b^{13} i}\right )} x - \frac{2 \, a^{3} b^{9} - 44 \, a^{2} b^{9} i - 93 \, a b^{9} + 48 \, b^{9} i}{b^{13} i}\right )} - \frac{{\left (8 \, a^{3} - 36 \, a^{2} i - 44 \, a + 17 \, i\right )} \log \left (3 \,{\left (x{\left | b \right |} - \sqrt{{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b + a^{3} b - 2 \,{\left (x{\left | b \right |} - \sqrt{{\left (b x + a\right )}^{2} + 1}\right )}^{2} b i - 2 \, a^{2} b i +{\left (x{\left | b \right |} - \sqrt{{\left (b x + a\right )}^{2} + 1}\right )}^{3}{\left | b \right |} + 3 \,{\left (x{\left | b \right |} - \sqrt{{\left (b x + a\right )}^{2} + 1}\right )} a^{2}{\left | b \right |} - 4 \,{\left (x{\left | b \right |} - \sqrt{{\left (b x + a\right )}^{2} + 1}\right )} a i{\left | b \right |} - a b -{\left (x{\left | b \right |} - \sqrt{{\left (b x + a\right )}^{2} + 1}\right )}{\left | b \right |}\right )}{8 \, b^{3}{\left | b \right |}} - \frac{{\left (8 \, a^{3}{\left | b \right |} - 36 \, a^{2} i{\left | b \right |} - 44 \, a{\left | b \right |} + 17 \, i{\left | b \right |}\right )} \log \left (96 \, b^{4}\right )}{4 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*sqrt((b*x + a)^2 + 1)*((2*x*(x/(b*i) - (a*b^11 - 4*b^11*i)/(b^13*i)) + (2*a^2*b^10 - 20*a*b^10*i - 19*b^1
0)/(b^13*i))*x - (2*a^3*b^9 - 44*a^2*b^9*i - 93*a*b^9 + 48*b^9*i)/(b^13*i)) - 1/8*(8*a^3 - 36*a^2*i - 44*a + 1
7*i)*log(3*(x*abs(b) - sqrt((b*x + a)^2 + 1))^2*a*b + a^3*b - 2*(x*abs(b) - sqrt((b*x + a)^2 + 1))^2*b*i - 2*a
^2*b*i + (x*abs(b) - sqrt((b*x + a)^2 + 1))^3*abs(b) + 3*(x*abs(b) - sqrt((b*x + a)^2 + 1))*a^2*abs(b) - 4*(x*
abs(b) - sqrt((b*x + a)^2 + 1))*a*i*abs(b) - a*b - (x*abs(b) - sqrt((b*x + a)^2 + 1))*abs(b))/(b^3*abs(b)) - 1
/4*(8*a^3*abs(b) - 36*a^2*i*abs(b) - 44*a*abs(b) + 17*i*abs(b))*log(96*b^4)/b^5