3.189 \(\int e^{-i \tan ^{-1}(a+b x)} x^4 \, dx\)

Optimal. Leaf size=276 \[ -\frac{(-i a-i b x+1)^{3/2} \sqrt{i a+i b x+1} \left (2 \left (-36 a^2+14 i a+13\right ) b x+96 a^3-86 i a^2-114 a+19 i\right )}{120 b^5}-\frac{\left (8 i a^4+16 a^3-24 i a^2-12 a+3 i\right ) \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{8 b^5}+\frac{\left (8 a^4-16 i a^3-24 a^2+12 i a+3\right ) \sinh ^{-1}(a+b x)}{8 b^5}+\frac{x^3 (-i a-i b x+1)^{3/2} \sqrt{i a+i b x+1}}{5 b^2}+\frac{(-8 a+i) x^2 (-i a-i b x+1)^{3/2} \sqrt{i a+i b x+1}}{20 b^3} \]

[Out]

-((3*I - 12*a - (24*I)*a^2 + 16*a^3 + (8*I)*a^4)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(8*b^5) + ((I -
8*a)*x^2*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x])/(20*b^3) + (x^3*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a +
I*b*x])/(5*b^2) - ((1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x]*(19*I - 114*a - (86*I)*a^2 + 96*a^3 + 2*(13
+ (14*I)*a - 36*a^2)*b*x))/(120*b^5) + ((3 + (12*I)*a - 24*a^2 - (16*I)*a^3 + 8*a^4)*ArcSinh[a + b*x])/(8*b^5)

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Rubi [A]  time = 0.224759, antiderivative size = 276, 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, 100, 153, 147, 50, 53, 619, 215} \[ -\frac{(-i a-i b x+1)^{3/2} \sqrt{i a+i b x+1} \left (2 \left (-36 a^2+14 i a+13\right ) b x+96 a^3-86 i a^2-114 a+19 i\right )}{120 b^5}-\frac{\left (8 i a^4+16 a^3-24 i a^2-12 a+3 i\right ) \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{8 b^5}+\frac{\left (8 a^4-16 i a^3-24 a^2+12 i a+3\right ) \sinh ^{-1}(a+b x)}{8 b^5}+\frac{x^3 (-i a-i b x+1)^{3/2} \sqrt{i a+i b x+1}}{5 b^2}+\frac{(-8 a+i) x^2 (-i a-i b x+1)^{3/2} \sqrt{i a+i b x+1}}{20 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^4/E^(I*ArcTan[a + b*x]),x]

[Out]

-((3*I - 12*a - (24*I)*a^2 + 16*a^3 + (8*I)*a^4)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(8*b^5) + ((I -
8*a)*x^2*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x])/(20*b^3) + (x^3*(1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a +
I*b*x])/(5*b^2) - ((1 - I*a - I*b*x)^(3/2)*Sqrt[1 + I*a + I*b*x]*(19*I - 114*a - (86*I)*a^2 + 96*a^3 + 2*(13
+ (14*I)*a - 36*a^2)*b*x))/(120*b^5) + ((3 + (12*I)*a - 24*a^2 - (16*I)*a^3 + 8*a^4)*ArcSinh[a + b*x])/(8*b^5)

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 100

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

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^{-i \tan ^{-1}(a+b x)} x^4 \, dx &=\int \frac{x^4 \sqrt{1-i a-i b x}}{\sqrt{1+i a+i b x}} \, dx\\ &=\frac{x^3 (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{5 b^2}+\frac{\int \frac{x^2 \sqrt{1-i a-i b x} \left (-3 \left (1+a^2\right )+(i-8 a) b x\right )}{\sqrt{1+i a+i b x}} \, dx}{5 b^2}\\ &=\frac{(i-8 a) x^2 (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{20 b^3}+\frac{x^3 (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{5 b^2}+\frac{\int \frac{x \sqrt{1-i a-i b x} \left (2 (i-8 a) (i-a) (i+a) b-\left (13+14 i a-36 a^2\right ) b^2 x\right )}{\sqrt{1+i a+i b x}} \, dx}{20 b^4}\\ &=\frac{(i-8 a) x^2 (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{20 b^3}+\frac{x^3 (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{5 b^2}-\frac{(1-i a-i b x)^{3/2} \sqrt{1+i a+i b x} \left (19 i-114 a-86 i a^2+96 a^3+2 \left (13+14 i a-36 a^2\right ) b x\right )}{120 b^5}+\frac{\left (3+12 i a-24 a^2-16 i a^3+8 a^4\right ) \int \frac{\sqrt{1-i a-i b x}}{\sqrt{1+i a+i b x}} \, dx}{8 b^4}\\ &=-\frac{\left (3 i-12 a-24 i a^2+16 a^3+8 i a^4\right ) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{8 b^5}+\frac{(i-8 a) x^2 (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{20 b^3}+\frac{x^3 (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{5 b^2}-\frac{(1-i a-i b x)^{3/2} \sqrt{1+i a+i b x} \left (19 i-114 a-86 i a^2+96 a^3+2 \left (13+14 i a-36 a^2\right ) b x\right )}{120 b^5}+\frac{\left (3+12 i a-24 a^2-16 i a^3+8 a^4\right ) \int \frac{1}{\sqrt{1-i a-i b x} \sqrt{1+i a+i b x}} \, dx}{8 b^4}\\ &=-\frac{\left (3 i-12 a-24 i a^2+16 a^3+8 i a^4\right ) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{8 b^5}+\frac{(i-8 a) x^2 (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{20 b^3}+\frac{x^3 (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{5 b^2}-\frac{(1-i a-i b x)^{3/2} \sqrt{1+i a+i b x} \left (19 i-114 a-86 i a^2+96 a^3+2 \left (13+14 i a-36 a^2\right ) b x\right )}{120 b^5}+\frac{\left (3+12 i a-24 a^2-16 i a^3+8 a^4\right ) \int \frac{1}{\sqrt{(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{8 b^4}\\ &=-\frac{\left (3 i-12 a-24 i a^2+16 a^3+8 i a^4\right ) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{8 b^5}+\frac{(i-8 a) x^2 (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{20 b^3}+\frac{x^3 (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{5 b^2}-\frac{(1-i a-i b x)^{3/2} \sqrt{1+i a+i b x} \left (19 i-114 a-86 i a^2+96 a^3+2 \left (13+14 i a-36 a^2\right ) b x\right )}{120 b^5}+\frac{\left (3+12 i a-24 a^2-16 i a^3+8 a^4\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^6}\\ &=-\frac{\left (3 i-12 a-24 i a^2+16 a^3+8 i a^4\right ) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{8 b^5}+\frac{(i-8 a) x^2 (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{20 b^3}+\frac{x^3 (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{5 b^2}-\frac{(1-i a-i b x)^{3/2} \sqrt{1+i a+i b x} \left (19 i-114 a-86 i a^2+96 a^3+2 \left (13+14 i a-36 a^2\right ) b x\right )}{120 b^5}+\frac{\left (3+12 i a-24 a^2-16 i a^3+8 a^4\right ) \sinh ^{-1}(a+b x)}{8 b^5}\\ \end{align*}

Mathematica [A]  time = 0.758265, size = 248, normalized size = 0.9 \[ \frac{i \sqrt{i a+i b x+1} \left (a^2 \left (-84 b^2 x^2-346 i b x+57\right )+2 a^3 (72 b x-41 i)+24 i a^5+226 a^4+a \left (64 b^3 x^3+154 i b^2 x^2-346 b x-211 i\right )+24 i b^5 x^5-54 b^4 x^4-62 i b^3 x^3+77 b^2 x^2+109 i b x-64\right )}{120 b^5 \sqrt{-i (a+b x+i)}}+\frac{\sqrt [4]{-1} \left (-8 i a^4-16 a^3+24 i a^2+12 a-3 i\right ) \sinh ^{-1}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right )}{4 \sqrt{-i b} b^{9/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4/E^(I*ArcTan[a + b*x]),x]

[Out]

((I/120)*Sqrt[1 + I*a + I*b*x]*(-64 + 226*a^4 + (24*I)*a^5 + (109*I)*b*x + 77*b^2*x^2 - (62*I)*b^3*x^3 - 54*b^
4*x^4 + (24*I)*b^5*x^5 + 2*a^3*(-41*I + 72*b*x) + a^2*(57 - (346*I)*b*x - 84*b^2*x^2) + a*(-211*I - 346*b*x +
(154*I)*b^2*x^2 + 64*b^3*x^3)))/(b^5*Sqrt[(-I)*(I + a + b*x)]) + ((-1)^(1/4)*(-3*I + 12*a + (24*I)*a^2 - 16*a^
3 - (8*I)*a^4)*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/(4*Sqrt[(-I)*b]*b^(9/2))

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Maple [B]  time = 0.223, size = 1208, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(1+I*(b*x+a))*(1+(b*x+a)^2)^(1/2),x)

[Out]

7/15*I/b^5*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)-4/b^5*((x-(I-a)/b)^2*b^2+2*I*(x-(I-a)/b)*b)^(1/2)*a^3+4/b^5*((x-(I-a)
/b)^2*b^2+2*I*(x-(I-a)/b)*b)^(1/2)*a+1/b^4*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)-I/b^5*((x-(I-a)/b)^2*b^2+2*I*(x-(I-a)/b)*b)^(1/2)+3*a^3/b^5*(b^2*x^2+2*a*b*x+a^2+1)
^(1/2)-5/8/b^4*x*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-5/8/b^4*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)-5/2*I/b^4*a*x*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+2*I/b^4*a^3*x*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-5/2*I/b
^4*a*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)-4*I/b^4*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+4*I/b^4*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+3/5*I/b^4*a*x*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)-5
/8*a/b^5*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/4/b^4*x*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)-13/12/b^5*a*(b^2*x^2+2*a*b*x+a^
2+1)^(3/2)+2*I/b^4*a^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^4*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^4-6/b^4*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-I/b^5*((x-(I-a)/b)^2*b^2+2*I*
(x-(I-a)/b)*b)^(1/2)*a^4+6*I/b^5*((x-(I-a)/b)^2*b^2+2*I*(x-(I-a)/b)*b)^(1/2)*a^2-6/5*I/b^5*a^2*(b^2*x^2+2*a*b*
x+a^2+1)^(3/2)-5/2*I/b^5*a^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+2*I/b^5*a^4*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+3*a^2/b^4
*x*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-1/5*I/b^3*x^2*(b^2*x^2+2*a*b*x+a^2+1)^(3/2)+3*a^2/b^4*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)

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Maxima [B]  time = 1.63244, size = 616, normalized size = 2.23 \begin{align*} \frac{2 i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{3} x}{b^{4}} - \frac{i \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac{3}{2}} x^{2}}{5 \, b^{3}} + \frac{a^{4} \operatorname{arsinh}\left (b x + a\right )}{b^{5}} + \frac{i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{4}}{b^{5}} + \frac{3 i \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac{3}{2}} a x}{5 \, b^{4}} + \frac{3 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2} x}{b^{4}} - \frac{2 i \, a^{3} \operatorname{arsinh}\left (b x + a\right )}{b^{5}} - \frac{6 i \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac{3}{2}} a^{2}}{5 \, b^{5}} - \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{3}}{b^{5}} + \frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac{3}{2}} x}{4 \, b^{4}} - \frac{5 i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a x}{2 \, b^{4}} - \frac{3 \, a^{2} \operatorname{arsinh}\left (b x + a\right )}{b^{5}} - \frac{13 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac{3}{2}} a}{12 \, b^{5}} + \frac{7 i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2}}{2 \, b^{5}} - \frac{5 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x}{8 \, b^{4}} + \frac{3 i \, a \operatorname{arsinh}\left (b x + a\right )}{2 \, b^{5}} + \frac{7 i \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac{3}{2}}}{15 \, b^{5}} + \frac{27 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{8 \, b^{5}} + \frac{3 \, \operatorname{arsinh}\left (b x + a\right )}{8 \, b^{5}} - \frac{i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(1+I*(b*x+a))*(1+(b*x+a)^2)^(1/2),x, algorithm="maxima")

[Out]

2*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^3*x/b^4 - 1/5*I*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*x^2/b^3 + a^4*arcs
inh(b*x + a)/b^5 + I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^4/b^5 + 3/5*I*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*a*x
/b^4 + 3*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^2*x/b^4 - 2*I*a^3*arcsinh(b*x + a)/b^5 - 6/5*I*(b^2*x^2 + 2*a*b*x
+ a^2 + 1)^(3/2)*a^2/b^5 - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^3/b^5 + 1/4*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2
)*x/b^4 - 5/2*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a*x/b^4 - 3*a^2*arcsinh(b*x + a)/b^5 - 13/12*(b^2*x^2 + 2*a*
b*x + a^2 + 1)^(3/2)*a/b^5 + 7/2*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^2/b^5 - 5/8*sqrt(b^2*x^2 + 2*a*b*x + a^
2 + 1)*x/b^4 + 3/2*I*a*arcsinh(b*x + a)/b^5 + 7/15*I*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)/b^5 + 27/8*sqrt(b^2*x
^2 + 2*a*b*x + a^2 + 1)*a/b^5 + 3/8*arcsinh(b*x + a)/b^5 - I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/b^5

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Fricas [A]  time = 2.36044, size = 524, normalized size = 1.9 \begin{align*} \frac{-186 i \, a^{5} - 1345 \, a^{4} + 1730 i \, a^{3} + 1320 \, a^{2} -{\left (960 \, a^{4} - 1920 i \, a^{3} - 2880 \, a^{2} + 1440 i \, a + 360\right )} \log \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) +{\left (-192 i \, b^{4} x^{4} - 48 \,{\left (-4 i \, a - 5\right )} b^{3} x^{3} +{\left (-192 i \, a^{2} - 560 \, a + 256 i\right )} b^{2} x^{2} - 192 i \, a^{4} - 2000 \, a^{3} +{\left (192 i \, a^{3} + 1040 \, a^{2} - 928 i \, a - 360\right )} b x + 2656 i \, a^{2} + 2200 \, a - 512 i\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 300 i \, a}{960 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(1+I*(b*x+a))*(1+(b*x+a)^2)^(1/2),x, algorithm="fricas")

[Out]

1/960*(-186*I*a^5 - 1345*a^4 + 1730*I*a^3 + 1320*a^2 - (960*a^4 - 1920*I*a^3 - 2880*a^2 + 1440*I*a + 360)*log(
-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) + (-192*I*b^4*x^4 - 48*(-4*I*a - 5)*b^3*x^3 + (-192*I*a^2 - 560*
a + 256*I)*b^2*x^2 - 192*I*a^4 - 2000*a^3 + (192*I*a^3 + 1040*a^2 - 928*I*a - 360)*b*x + 2656*I*a^2 + 2200*a -
512*I)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) - 300*I*a)/b^5

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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**4/(1+I*(b*x+a))*(1+(b*x+a)**2)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.12711, size = 289, normalized size = 1.05 \begin{align*} -\frac{1}{120} \, \sqrt{{\left (b x + a\right )}^{2} + 1}{\left ({\left (2 \,{\left (3 \,{\left (\frac{4 \, i x}{b} - \frac{4 \, a b^{7} i + 5 \, b^{7}}{b^{9}}\right )} x + \frac{12 \, a^{2} b^{6} i + 35 \, a b^{6} - 16 \, b^{6} i}{b^{9}}\right )} x - \frac{24 \, a^{3} b^{5} i + 130 \, a^{2} b^{5} - 116 \, a b^{5} i - 45 \, b^{5}}{b^{9}}\right )} x + \frac{24 \, a^{4} b^{4} i + 250 \, a^{3} b^{4} - 332 \, a^{2} b^{4} i - 275 \, a b^{4} + 64 \, b^{4} i}{b^{9}}\right )} - \frac{{\left (8 \, a^{4} - 16 \, a^{3} i - 24 \, a^{2} + 12 \, a i + 3\right )} \log \left (-a b -{\left (x{\left | b \right |} - \sqrt{{\left (b x + a\right )}^{2} + 1}\right )}{\left | b \right |}\right )}{8 \, b^{4}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(1+I*(b*x+a))*(1+(b*x+a)^2)^(1/2),x, algorithm="giac")

[Out]

-1/120*sqrt((b*x + a)^2 + 1)*((2*(3*(4*i*x/b - (4*a*b^7*i + 5*b^7)/b^9)*x + (12*a^2*b^6*i + 35*a*b^6 - 16*b^6*
i)/b^9)*x - (24*a^3*b^5*i + 130*a^2*b^5 - 116*a*b^5*i - 45*b^5)/b^9)*x + (24*a^4*b^4*i + 250*a^3*b^4 - 332*a^2
*b^4*i - 275*a*b^4 + 64*b^4*i)/b^9) - 1/8*(8*a^4 - 16*a^3*i - 24*a^2 + 12*a*i + 3)*log(-a*b - (x*abs(b) - sqrt
((b*x + a)^2 + 1))*abs(b))/(b^4*abs(b))