3.2.81 \(\int e^{3 i \text {ArcTan}(a+b x)} x^3 \, dx\) [181]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.17, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5203, 99, 158, 152, 52, 55, 633, 221} \begin {gather*} -\frac {i \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2} \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 \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{4 b^2}-\frac {2 i x^3 (i a+i b x+1)^{3/2}}{b \sqrt {-i a-i b x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^((3*I)*ArcTan[a + b*x])*x^3,x]

[Out]

(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) - ((2*I)*x^3*(1 +
I*a + I*b*x)^(3/2))/(b*Sqrt[1 - I*a - I*b*x]) - (9*x^2*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2))/(4*b^2)
- ((I/8)*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2)*(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 52

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 55

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 99

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 152

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 158

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 221

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]

Rule 633

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 5203

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]

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 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{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 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}-\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \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 {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 {2 i x^3 (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}-\frac {9 x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}-\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \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 {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 {2 i x^3 (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}-\frac {9 x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}-\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \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 {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 {2 i x^3 (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}-\frac {9 x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}-\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \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 ) \text {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 {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 {2 i x^3 (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}-\frac {9 x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}-\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \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 = 0.19, size = 201, normalized size = 0.81 \begin {gather*} \frac {\sqrt {1+i a+i b x} \left (80+78 i a^3+2 a^4-29 i b x+11 b^2 x^2+6 i b^3 x^3-2 b^4 x^4+a^2 (-233+22 i b x)-i a \left (237-54 i b x+10 b^2 x^2\right )\right )}{8 b^4 \sqrt {-i (i+a+b x)}}+\frac {3 \sqrt [4]{-1} \left (-17 i-44 a+36 i a^2+8 a^3\right ) \sqrt {-i b} \sinh ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (i+a+b x)}}{\sqrt {-i b}}\right )}{4 b^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 + I*a + I*b*x]*(80 + (78*I)*a^3 + 2*a^4 - (29*I)*b*x + 11*b^2*x^2 + (6*I)*b^3*x^3 - 2*b^4*x^4 + a^2*(-
233 + (22*I)*b*x) - I*a*(237 - (54*I)*b*x + 10*b^2*x^2)))/(8*b^4*Sqrt[(-I)*(I + a + b*x)]) + (3*(-1)^(1/4)*(-1
7*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))

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2971 vs. \(2 (200 ) = 400\).
time = 0.18, size = 2972, normalized size = 11.94

method	result	size
risch	\(\frac {i \left (-2 b^{3} x^{3}+2 a \,b^{2} x^{2}+8 i b^{2} x^{2}-2 a^{2} b x -20 i a b x +2 a^{3}+44 i a^{2}+19 b x -93 a -48 i\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{8 b^{4}}+\frac {27 i \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right ) a^{2}}{2 b^{3} \sqrt {b^{2}}}+\frac {3 \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right ) a^{3}}{b^{3} \sqrt {b^{2}}}-\frac {51 i \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{3} \sqrt {b^{2}}}-\frac {33 \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right ) a}{2 b^{3} \sqrt {b^{2}}}-\frac {4 \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}\, a^{3}}{b^{5} \left (x +\frac {i}{b}+\frac {a}{b}\right )}+\frac {12 \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}\, a}{b^{5} \left (x +\frac {i}{b}+\frac {a}{b}\right )}-\frac {12 i \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}\, a^{2}}{b^{5} \left (x +\frac {i}{b}+\frac {a}{b}\right )}+\frac {4 i \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}}{b^{5} \left (x +\frac {i}{b}+\frac {a}{b}\right )}\)	\(501\)
default	\(\text {Expression too large to display}\)	\(2972\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*b^3*(1/4*x^5/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-9/4*a/b*(1/3*x^4/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-7/3*a/b*(
1/2*x^3/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-5/2*a/b*(x^2/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-3*a/b*(-x/b^2/(b^2*x^
2+2*a*b*x+a^2+1)^(1/2)-a/b*(-1/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^
2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2))+1/b^2*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))
-2*(a^2+1)/b^2*(-1/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+
2*a*b*x+a^2+1)^(1/2)))-3/2*(a^2+1)/b^2*(-x/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-a/b*(-1/b^2/(b^2*x^2+2*a*b*x+a^2+
1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2))+1/b^2*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)))-4/3*(a^2+1)/b^2*(x^2/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)
-3*a/b*(-x/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-a/b*(-1/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(
4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2))+1/b^2*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))-2*(a^2+1)/b^2*(-1/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1
)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2))))-5/4*(a^2+1)/b^2*(1/2*x^3/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-5/2*a
/b*(x^2/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-3*a/b*(-x/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-a/b*(-1/b^2/(b^2*x^2+2*a
*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2))+1/b^2*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))-2*(a^2+1)/b^2*(-1/b^2/(b^2*x^2+2*a*b*x+a^2+1)^
(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)))-3/2*(a^2+1)/b^2*(-x/b^2/
(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-a/b*(-1/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4
*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2))+1/b^2*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))))-3*(1+I*a)*b^2*(1/3*x^4/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-7/3*a/b*(1/2*x^3/b^2/(b^2*x^2+2*a*b*x+a^2+1
)^(1/2)-5/2*a/b*(x^2/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-3*a/b*(-x/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-a/b*(-1/b^2
/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2))+
1/b^2*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))-2*(a^2+1)/b^2*(-1/b^2/(b^2*x^2+2*
a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)))-3/2*(a^2+1)
/b^2*(-x/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-a/b*(-1/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*
b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2))+1/b^2*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)))-4/3*(a^2+1)/b^2*(x^2/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-3*a/b*(-x/b^2/(b^2*x^2+2*a*b*x+a^
2+1)^(1/2)-a/b*(-1/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+
2*a*b*x+a^2+1)^(1/2))+1/b^2*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))-2*(a^2+1)/b
^2*(-1/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+
1)^(1/2))))+3*I*(1+I*a)^2*b*(1/2*x^3/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-5/2*a/b*(x^2/b^2/(b^2*x^2+2*a*b*x+a^2+1
)^(1/2)-3*a/b*(-x/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-a/b*(-1/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2
*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2))+1/b^2*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))-2*(a^2+1)/b^2*(-1/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2
*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)))-3/2*(a^2+1)/b^2*(-x/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-a/b*
(-1/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^
(1/2))+1/b^2*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+I*a)^3*(x^2/b^2/(b^2*x
^2+2*a*b*x+a^2+1)^(1/2)-3*a/b*(-x/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-a/b*(-1/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-
2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2))+1/b^2*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))-2*(a^2+1)/b^2*(-1/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2
*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)))

________________________________________________________________________________________

Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2295 vs. \(2 (175) = 350\).
time = 0.29, size = 2295, normalized size = 9.22 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*I*b*x^5/sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) + 315/4*I*a^6*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b
*x + a^2 + 1)*b) + 3/4*I*a*x^4/sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) - 945/8*I*(a^2 + 1)*a^4*x/((a^2*b^2 - (a^2 +
1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b) - 21/8*I*a^2*x^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b) + 105/8*I*
(a^2 + 1)*a^5/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - 105*(I*a*b^2 + b^2)*a^5*x/((
a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^3) + 45*(I*a^2*b + 2*a*b - I*b)*a^4*x/((a^2*b^2 -
(a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) + 169/4*I*(a^2 + 1)^2*a^2*x/((a^2*b^2 - (a^2 + 1)*b^2)*
sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b) + 6*(-I*a^3 - 3*a^2 + 3*I*a + 1)*a^3*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^
2*x^2 + 2*a*b*x + a^2 + 1)*b) + 105/8*I*a^3*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - 5/8*(-I*a^2 - I)*x^3
/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b) - (I*a*b^2 + b^2)*x^4/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - 14*I*(a
^2 + 1)^2*a^3/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) + 265/2*(I*a*b^2 + b^2)*(a^2 +
1)*a^3*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^3) - 93/2*(I*a^2*b + 2*a*b - I*b)*(a^
2 + 1)*a^2*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - 15/8*I*(a^2 + 1)^3*x/((a^2*b^
2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b) - 5*(-I*a^3 - 3*a^2 + 3*I*a + 1)*(a^2 + 1)*a*x/((a^2*b
^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b) - 49/8*I*(a^2 + 1)*a*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^
2 + 1)*b^2) + 7/2*(I*a*b^2 + b^2)*a*x^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^3) - 3/2*(I*a^2*b + 2*a*b - I*b)*
x^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - 315/8*I*a^4*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1
)*b^2))/b^4 - 35/2*(I*a*b^2 + b^2)*(a^2 + 1)*a^4/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*
b^4) + 15/2*(I*a^2*b + 2*a*b - I*b)*(a^2 + 1)*a^3/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)
*b^3) + 15/8*I*(a^2 + 1)^3*a/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) + (-I*a^3 - 3*a
^2 + 3*I*a + 1)*(a^2 + 1)*a^2/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - 61/2*(I*a*b^
2 + b^2)*(a^2 + 1)^2*a*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^3) + 9/2*(I*a^2*b + 2*
a*b - I*b)*(a^2 + 1)^2*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - 35/2*(I*a*b^2 + b
^2)*a^2*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^4) + 15/2*(I*a^2*b + 2*a*b - I*b)*a*x^2/(sqrt(b^2*x^2 + 2*a*b
*x + a^2 + 1)*b^3) + (-I*a^3 - 3*a^2 + 3*I*a + 1)*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) + 105/4*I*(a^2 +
1)*a^2*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^4 + 29/2*(I*a*b^2 + b^2)*(a^2 + 1)^2*a^2
/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^4) + 105/4*I*(a^2 + 1)*a^3/(sqrt(b^2*x^2 + 2*a
*b*x + a^2 + 1)*b^4) - 9/2*(I*a^2*b + 2*a*b - I*b)*(a^2 + 1)^2*a/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a
*b*x + a^2 + 1)*b^3) + 4*(I*a*b^2 + b^2)*(a^2 + 1)*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^4) + 105/2*(I*a*b^
2 + b^2)*a^3*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^6 - 45/2*(I*a^2*b + 2*a*b - I*b)*a^
2*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^5 - 15/8*I*(a^2 + 1)^2*arcsinh(2*(b^2*x + a*b)
/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^4 - 3*(-I*a^3 - 3*a^2 + 3*I*a + 1)*a*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^
2*b^2 + 4*(a^2 + 1)*b^2))/b^4 - 49/4*I*(a^2 + 1)^2*a/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^4) - 45/2*(I*a*b^2 +
b^2)*(a^2 + 1)*a*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^6 + 9/2*(I*a^2*b + 2*a*b - I*b
)*(a^2 + 1)*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^5 - 35*(I*a*b^2 + b^2)*(a^2 + 1)*a^2
/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^6) + 15*(I*a^2*b + 2*a*b - I*b)*(a^2 + 1)*a/(sqrt(b^2*x^2 + 2*a*b*x + a^
2 + 1)*b^5) + 2*(-I*a^3 - 3*a^2 + 3*I*a + 1)*(a^2 + 1)/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^4) + 8*(I*a*b^2 +
b^2)*(a^2 + 1)^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^6)

________________________________________________________________________________________

Fricas [A]
time = 2.82, size = 216, normalized size = 0.87 \begin {gather*} \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} - 24 \, {\left (8 \, a^{4} + 44 i \, a^{3} + {\left (8 \, a^{3} + 36 i \, a^{2} - 44 \, a - 17 i\right )} b x - 80 \, a^{2} - 61 i \, a + 17\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 8 \, {\left (2 i \, b^{4} x^{4} + 6 \, b^{3} x^{3} - {\left (10 \, a + 11 i\right )} b^{2} x^{2} - 2 i \, a^{4} + 78 \, a^{3} + {\left (22 \, a^{2} + 54 i \, a - 29\right )} b x + 233 i \, a^{2} - 237 \, a - 80 i\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 1224 i \, a - 256}{64 \, {\left (b^{5} x + {\left (a + i\right )} b^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(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 - 24*
(8*a^4 + 44*I*a^3 + (8*a^3 + 36*I*a^2 - 44*a - 17*I)*b*x - 80*a^2 - 61*I*a + 17)*log(-b*x - a + sqrt(b^2*x^2 +
2*a*b*x + a^2 + 1)) - 8*(2*I*b^4*x^4 + 6*b^3*x^3 - (10*a + 11*I)*b^2*x^2 - 2*I*a^4 + 78*a^3 + (22*a^2 + 54*I*
a - 29)*b*x + 233*I*a^2 - 237*a - 80*I)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) + 1224*I*a - 256)/(b^5*x + (a + I)*b
^4)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - i \left (\int \frac {i x^{3}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 a x^{3}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {a^{3} x^{3}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 b x^{4}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {b^{3} x^{6}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i a^{2} x^{3}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \left (- \frac {3 i b^{2} x^{5}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {3 a b^{2} x^{5}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} b x^{4}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {6 i a b x^{4}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I*(Integral(I*x**3/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)
+ b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*a*x
**3/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**2*sqrt
(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(a**3*x**3/(a**2*sqrt(a
**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**2*sqrt(a**2 + 2*a*b*x
+ b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*b*x**4/(a**2*sqrt(a**2 + 2*a*b*x +
b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)
+ sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(b**3*x**6/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) +
2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2
*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*I*a**2*x**3/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sq
rt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b*
*2*x**2 + 1)), x) + Integral(-3*I*b**2*x**5/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2
*a*b*x + b**2*x**2 + 1) + b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1
)), x) + Integral(3*a*b**2*x**5/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**
2*x**2 + 1) + b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Int
egral(3*a**2*b*x**4/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)
+ b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-6*I*a
*b*x**4/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**2*
sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x))

________________________________________________________________________________________

Giac [A]
time = 0.46, size = 285, normalized size = 1.14 \begin {gather*} -\frac {1}{8} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left ({\left (2 \, x {\left (\frac {i \, x}{b} - \frac {i \, a b^{11} - 4 \, b^{11}}{b^{13}}\right )} - \frac {-2 i \, a^{2} b^{10} + 20 \, a b^{10} + 19 i \, b^{10}}{b^{13}}\right )} x - \frac {2 i \, a^{3} b^{9} - 44 \, a^{2} b^{9} - 93 i \, a b^{9} + 48 \, b^{9}}{b^{13}}\right )} - \frac {{\left (8 \, a^{3} + 36 i \, a^{2} - 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 + {\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 |} + 2 i \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b + 2 i \, a^{2} b + 4 \, {\left (i \, x {\left | b \right |} - i \, \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a {\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 |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3\,{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3}{{\left ({\left (a+b\,x\right )}^2+1\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________