3.2.88 \(\int \frac {e^{3 i \text {ArcTan}(a+b x)}}{x^4} \, dx\) [188]

Optimal. Leaf size=338 \[ \frac {\left (52+51 i a-2 a^2\right ) b^3 \sqrt {1+i a+i b x}}{6 (i-a) (i+a)^4 \sqrt {1-i a-i b x}}-\frac {(i-a) \sqrt {1+i a+i b x}}{3 (i+a) x^3 \sqrt {1-i a-i b x}}+\frac {7 i b \sqrt {1+i a+i b x}}{6 (i+a)^2 x^2 \sqrt {1-i a-i b x}}+\frac {(19+16 i a) b^2 \sqrt {1+i a+i b x}}{6 (i-a) (i+a)^3 x \sqrt {1-i a-i b x}}-\frac {\left (11 i-18 a-6 i a^2\right ) b^3 \tanh ^{-1}\left (\frac {\sqrt {i+a} \sqrt {1+i a+i b x}}{\sqrt {i-a} \sqrt {1-i a-i b x}}\right )}{(i-a)^{3/2} (i+a)^{9/2}} \]

[Out]

-(11*I-18*a-6*I*a^2)*b^3*arctanh((I+a)^(1/2)*(1+I*a+I*b*x)^(1/2)/(I-a)^(1/2)/(1-I*a-I*b*x)^(1/2))/(I-a)^(3/2)/
(I+a)^(9/2)+1/6*(52+51*I*a-2*a^2)*b^3*(1+I*a+I*b*x)^(1/2)/(I-a)/(I+a)^4/(1-I*a-I*b*x)^(1/2)-1/3*(I-a)*(1+I*a+I
*b*x)^(1/2)/(I+a)/x^3/(1-I*a-I*b*x)^(1/2)+7/6*I*b*(1+I*a+I*b*x)^(1/2)/(I+a)^2/x^2/(1-I*a-I*b*x)^(1/2)+1/6*(19+
16*I*a)*b^2*(1+I*a+I*b*x)^(1/2)/(I-a)/(I+a)^3/x/(1-I*a-I*b*x)^(1/2)

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Rubi [A]
time = 0.24, antiderivative size = 338, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5203, 100, 156, 157, 12, 95, 214} \begin {gather*} \frac {\left (-2 a^2+51 i a+52\right ) b^3 \sqrt {i a+i b x+1}}{6 (-a+i) (a+i)^4 \sqrt {-i a-i b x+1}}-\frac {\left (-6 i a^2-18 a+11 i\right ) b^3 \tanh ^{-1}\left (\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-a+i} \sqrt {-i a-i b x+1}}\right )}{(-a+i)^{3/2} (a+i)^{9/2}}+\frac {(19+16 i a) b^2 \sqrt {i a+i b x+1}}{6 (-a+i) (a+i)^3 x \sqrt {-i a-i b x+1}}-\frac {(-a+i) \sqrt {i a+i b x+1}}{3 (a+i) x^3 \sqrt {-i a-i b x+1}}+\frac {7 i b \sqrt {i a+i b x+1}}{6 (a+i)^2 x^2 \sqrt {-i a-i b x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^((3*I)*ArcTan[a + b*x])/x^4,x]

[Out]

((52 + (51*I)*a - 2*a^2)*b^3*Sqrt[1 + I*a + I*b*x])/(6*(I - a)*(I + a)^4*Sqrt[1 - I*a - I*b*x]) - ((I - a)*Sqr
t[1 + I*a + I*b*x])/(3*(I + a)*x^3*Sqrt[1 - I*a - I*b*x]) + (((7*I)/6)*b*Sqrt[1 + I*a + I*b*x])/((I + a)^2*x^2
*Sqrt[1 - I*a - I*b*x]) + ((19 + (16*I)*a)*b^2*Sqrt[1 + I*a + I*b*x])/(6*(I - a)*(I + a)^3*x*Sqrt[1 - I*a - I*
b*x]) - ((11*I - 18*a - (6*I)*a^2)*b^3*ArcTanh[(Sqrt[I + a]*Sqrt[1 + I*a + I*b*x])/(Sqrt[I - a]*Sqrt[1 - I*a -
I*b*x])])/((I - a)^(3/2)*(I + a)^(9/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 100

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

Rule 156

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

Rule 157

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

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 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 \frac {e^{3 i \tan ^{-1}(a+b x)}}{x^4} \, dx &=\int \frac {(1+i a+i b x)^{3/2}}{x^4 (1-i a-i b x)^{3/2}} \, dx\\ &=-\frac {(i-a) \sqrt {1+i a+i b x}}{3 (i+a) x^3 \sqrt {1-i a-i b x}}-\frac {\int \frac {-7 (i-a) b+6 b^2 x}{x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}} \, dx}{3 (1-i a)}\\ &=-\frac {(i-a) \sqrt {1+i a+i b x}}{3 (i+a) x^3 \sqrt {1-i a-i b x}}+\frac {7 i b \sqrt {1+i a+i b x}}{6 (i+a)^2 x^2 \sqrt {1-i a-i b x}}+\frac {\int \frac {-\left (19+35 i a-16 a^2\right ) b^2-14 (i-a) b^3 x}{x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}} \, dx}{6 (1-i a) \left (1+a^2\right )}\\ &=-\frac {(i-a) \sqrt {1+i a+i b x}}{3 (i+a) x^3 \sqrt {1-i a-i b x}}+\frac {7 i b \sqrt {1+i a+i b x}}{6 (i+a)^2 x^2 \sqrt {1-i a-i b x}}+\frac {(19 i-16 a) b^2 \sqrt {1+i a+i b x}}{6 (i-a) (1-i a)^3 x \sqrt {1-i a-i b x}}-\frac {\int \frac {3 (i-a) \left (11+18 i a-6 a^2\right ) b^3-\left (19+35 i a-16 a^2\right ) b^4 x}{x (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}} \, dx}{6 (1-i a) \left (1+a^2\right )^2}\\ &=\frac {\left (52+51 i a-2 a^2\right ) b^3 \sqrt {1+i a+i b x}}{6 (i-a) (i+a)^4 \sqrt {1-i a-i b x}}-\frac {(i-a) \sqrt {1+i a+i b x}}{3 (i+a) x^3 \sqrt {1-i a-i b x}}+\frac {7 i b \sqrt {1+i a+i b x}}{6 (i+a)^2 x^2 \sqrt {1-i a-i b x}}+\frac {(19 i-16 a) b^2 \sqrt {1+i a+i b x}}{6 (i-a) (1-i a)^3 x \sqrt {1-i a-i b x}}+\frac {i \int \frac {3 \left (11+29 i a-24 a^2-6 i a^3\right ) b^4}{x \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{6 (i-a)^2 (i+a)^4 b}\\ &=\frac {\left (52+51 i a-2 a^2\right ) b^3 \sqrt {1+i a+i b x}}{6 (i-a) (i+a)^4 \sqrt {1-i a-i b x}}-\frac {(i-a) \sqrt {1+i a+i b x}}{3 (i+a) x^3 \sqrt {1-i a-i b x}}+\frac {7 i b \sqrt {1+i a+i b x}}{6 (i+a)^2 x^2 \sqrt {1-i a-i b x}}+\frac {(19 i-16 a) b^2 \sqrt {1+i a+i b x}}{6 (i-a) (1-i a)^3 x \sqrt {1-i a-i b x}}+\frac {\left (\left (11+18 i a-6 a^2\right ) b^3\right ) \int \frac {1}{x \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{2 (i-a) (i+a)^4}\\ &=\frac {\left (52+51 i a-2 a^2\right ) b^3 \sqrt {1+i a+i b x}}{6 (i-a) (i+a)^4 \sqrt {1-i a-i b x}}-\frac {(i-a) \sqrt {1+i a+i b x}}{3 (i+a) x^3 \sqrt {1-i a-i b x}}+\frac {7 i b \sqrt {1+i a+i b x}}{6 (i+a)^2 x^2 \sqrt {1-i a-i b x}}+\frac {(19 i-16 a) b^2 \sqrt {1+i a+i b x}}{6 (i-a) (1-i a)^3 x \sqrt {1-i a-i b x}}+\frac {\left (\left (11+18 i a-6 a^2\right ) b^3\right ) \text {Subst}\left (\int \frac {1}{-1-i a-(-1+i a) x^2} \, dx,x,\frac {\sqrt {1+i a+i b x}}{\sqrt {1-i a-i b x}}\right )}{(i-a) (i+a)^4}\\ &=\frac {\left (52+51 i a-2 a^2\right ) b^3 \sqrt {1+i a+i b x}}{6 (i-a) (i+a)^4 \sqrt {1-i a-i b x}}-\frac {(i-a) \sqrt {1+i a+i b x}}{3 (i+a) x^3 \sqrt {1-i a-i b x}}+\frac {7 i b \sqrt {1+i a+i b x}}{6 (i+a)^2 x^2 \sqrt {1-i a-i b x}}+\frac {(19 i-16 a) b^2 \sqrt {1+i a+i b x}}{6 (i-a) (1-i a)^3 x \sqrt {1-i a-i b x}}+\frac {\left (18 a-i \left (11-6 a^2\right )\right ) b^3 \tanh ^{-1}\left (\frac {\sqrt {i+a} \sqrt {1+i a+i b x}}{\sqrt {i-a} \sqrt {1-i a-i b x}}\right )}{(i-a)^{3/2} (i+a)^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 282, normalized size = 0.83 \begin {gather*} -\frac {2 (-1+i a)^{3/2} (1+i a) (i+a)^2 (1+i a+i b x)^{5/2}+(3 i-4 a) (-1+i a)^{5/2} b x (1+i a+i b x)^{5/2}-i \left (-11-18 i a+6 a^2\right ) b^2 x^2 \left (i \sqrt {-1+i a} \sqrt {1+i a+i b x} \left (1+a^2-5 i b x+a b x\right )-6 \sqrt {-1-i a} b x \sqrt {-i (i+a+b x)} \tanh ^{-1}\left (\frac {\sqrt {-1-i a} \sqrt {-i (i+a+b x)}}{\sqrt {-1+i a} \sqrt {1+i a+i b x}}\right )\right )}{6 (-1+i a)^{5/2} \left (1+a^2\right )^2 x^3 \sqrt {-i (i+a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^((3*I)*ArcTan[a + b*x])/x^4,x]

[Out]

-1/6*(2*(-1 + I*a)^(3/2)*(1 + I*a)*(I + a)^2*(1 + I*a + I*b*x)^(5/2) + (3*I - 4*a)*(-1 + I*a)^(5/2)*b*x*(1 + I
*a + I*b*x)^(5/2) - I*(-11 - (18*I)*a + 6*a^2)*b^2*x^2*(I*Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x]*(1 + a^2 - (5*I
)*b*x + a*b*x) - 6*Sqrt[-1 - I*a]*b*x*Sqrt[(-I)*(I + a + b*x)]*ArcTanh[(Sqrt[-1 - I*a]*Sqrt[(-I)*(I + a + b*x)
])/(Sqrt[-1 + I*a]*Sqrt[1 + I*a + I*b*x])]))/((-1 + I*a)^(5/2)*(1 + a^2)^2*x^3*Sqrt[(-I)*(I + a + b*x)])

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1648 vs. \(2 (266 ) = 532\).
time = 0.26, size = 1649, normalized size = 4.88

method	result	size
risch	\(\frac {i \left (2 a^{2} b^{4} x^{4}-27 i a \,b^{4} x^{4}+2 a^{3} b^{3} x^{3}-45 i a^{2} b^{3} x^{3}-9 i a^{3} b^{2} x^{2}-28 x^{4} b^{4}+2 a^{5} b x +9 i a^{4} b x -58 a \,b^{3} x^{3}+9 i b^{3} x^{3}+2 a^{6}-26 a^{2} b^{2} x^{2}-9 i a \,b^{2} x^{2}+4 a^{3} b x +18 i x \,a^{2} b +6 a^{4}-26 b^{2} x^{2}+2 a b x +9 i b x +6 a^{2}+2\right )}{6 x^{3} \left (a -i\right ) \left (i+a \right )^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {6 i b^{3} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right ) a^{2}}{\left (a -i\right ) \left (a^{4}+4 i a^{3}-6 a^{2}-4 i a +1\right ) \left (i+a \right ) \sqrt {a^{2}+1}}-\frac {3 b^{3} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right ) a^{3}}{\left (a -i\right ) \left (a^{4}+4 i a^{3}-6 a^{2}-4 i a +1\right ) \left (i+a \right ) \sqrt {a^{2}+1}}+\frac {11 i b^{3} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{2 \left (a -i\right ) \left (a^{4}+4 i a^{3}-6 a^{2}-4 i a +1\right ) \left (i+a \right ) \sqrt {a^{2}+1}}-\frac {7 b^{3} \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right ) a}{2 \left (a -i\right ) \left (a^{4}+4 i a^{3}-6 a^{2}-4 i a +1\right ) \left (i+a \right ) \sqrt {a^{2}+1}}+\frac {4 b^{2} \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}\, a^{2}}{\left (a -i\right ) \left (a^{4}+4 i a^{3}-6 a^{2}-4 i a +1\right ) \left (i+a \right ) \left (x +\frac {i}{b}+\frac {a}{b}\right )}+\frac {4 b^{2} \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}}{\left (a -i\right ) \left (a^{4}+4 i a^{3}-6 a^{2}-4 i a +1\right ) \left (i+a \right ) \left (x +\frac {i}{b}+\frac {a}{b}\right )}\)	\(757\)
default	\(\text {Expression too large to display}\)	\(1649\)

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^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2313 vs. \(2 (223) = 446\).
time = 0.28, size = 2313, normalized size = 6.84 \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^4,x, algorithm="maxima")

[Out]

-35/2*(-I*a^3 - 3*a^2 + 3*I*a + 1)*a^4*b^6*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2
+ 1)^4) - 35/2*(-I*a^3 - 3*a^2 + 3*I*a + 1)*a^5*b^5/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 +
1)*(a^2 + 1)^4) - 45/2*(I*a^2*b + 2*a*b - I*b)*a^3*b^5*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x +
a^2 + 1)*(a^2 + 1)^3) - I*a*b^6*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)) + 11
5/6*(-I*a^3 - 3*a^2 + 3*I*a + 1)*a^2*b^6*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 +
1)^3) - 45/2*(I*a^2*b + 2*a*b - I*b)*a^4*b^4/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^
2 + 1)^3) - I*a^2*b^5/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)) + 115/6*(-I*a^3
- 3*a^2 + 3*I*a + 1)*a^3*b^5/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^3) + 9*(I*
a*b^2 + b^2)*a^2*b^4*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^2) + 39/2*(I*a^2
*b + 2*a*b - I*b)*a*b^5*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^2) - 8/3*(-I*
a^3 - 3*a^2 + 3*I*a + 1)*b^6*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^2) + 9*(
I*a*b^2 + b^2)*a^3*b^3/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^2) + 39/2*(I*a^2
*b + 2*a*b - I*b)*a^2*b^4/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^2) - 8/3*(-I*
a^3 - 3*a^2 + 3*I*a + 1)*a*b^5/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^2) - 6*(
I*a*b^2 + b^2)*b^4*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)) + 35/2*(-I*a^3 -
3*a^2 + 3*I*a + 1)*a^3*b^3*arcsinh(2*a*b*x/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2*a^2/(sqrt(-4*a^2*b^
2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)))/(a^2 + 1)^(9/2) - 6*(I*a*b^2 + b
^2)*a*b^3/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)) - 35/2*(-I*a^3 - 3*a^2 + 3*I
*a + 1)*a^3*b^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^4) + 45/2*(I*a^2*b + 2*a*b - I*b)*a^2*b^2*arcsinh
(2*a*b*x/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2*a^2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2/(
sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)))/(a^2 + 1)^(7/2) + I*b^3*arcsinh(2*a*b*x/(sqrt(-4*a^2*b^2 + 4*(a^2
+ 1)*b^2)*abs(x)) + 2*a^2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*
abs(x)))/(a^2 + 1)^(3/2) - 15/2*(-I*a^3 - 3*a^2 + 3*I*a + 1)*a*b^3*arcsinh(2*a*b*x/(sqrt(-4*a^2*b^2 + 4*(a^2 +
1)*b^2)*abs(x)) + 2*a^2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*a
bs(x)))/(a^2 + 1)^(7/2) - 45/2*(I*a^2*b + 2*a*b - I*b)*a^2*b^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^3)
- I*b^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)) + 15/2*(-I*a^3 - 3*a^2 + 3*I*a + 1)*a*b^3/(sqrt(b^2*x^2
+ 2*a*b*x + a^2 + 1)*(a^2 + 1)^3) - 9*(I*a*b^2 + b^2)*a*b*arcsinh(2*a*b*x/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)
*abs(x)) + 2*a^2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)))/
(a^2 + 1)^(5/2) - 9/2*(I*a^2*b + 2*a*b - I*b)*b^2*arcsinh(2*a*b*x/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x))
+ 2*a^2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)) + 2/(sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2)*abs(x)))/(a^2 + 1)
^(5/2) + 9*(I*a*b^2 + b^2)*a*b/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^2) + 9/2*(I*a^2*b + 2*a*b - I*b)*b
^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^2) - 35/6*(-I*a^3 - 3*a^2 + 3*I*a + 1)*a^2*b^2/(sqrt(b^2*x^2 +
2*a*b*x + a^2 + 1)*(a^2 + 1)^3*x) - 15/2*(I*a^2*b + 2*a*b - I*b)*a*b/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2
+ 1)^2*x) + 4/3*(-I*a^3 - 3*a^2 + 3*I*a + 1)*b^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^2*x) + 7/6*(-I*a
^3 - 3*a^2 + 3*I*a + 1)*a*b/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^2*x^2) + 3*(I*a*b^2 + b^2)/(sqrt(b^2*
x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)*x) + 3/2*(I*a^2*b + 2*a*b - I*b)/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 +
1)*x^2) - 1/3*(-I*a^3 - 3*a^2 + 3*I*a + 1)/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)*x^3)

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 839 vs. \(2 (223) = 446\).
time = 4.74, size = 839, normalized size = 2.48 \begin {gather*} \frac {{\left (2 i \, a^{2} + 51 \, a - 52 i\right )} b^{4} x^{4} + {\left (2 i \, a^{3} + 49 \, a^{2} - i \, a + 52\right )} b^{3} x^{3} + 3 \, \sqrt {\frac {{\left (36 \, a^{4} - 216 i \, a^{3} - 456 \, a^{2} + 396 i \, a + 121\right )} b^{6}}{a^{12} + 6 i \, a^{11} - 12 \, a^{10} - 2 i \, a^{9} - 27 \, a^{8} - 36 i \, a^{7} - 36 i \, a^{5} + 27 \, a^{4} - 2 i \, a^{3} + 12 \, a^{2} + 6 i \, a - 1}} {\left ({\left (a^{5} + 3 i \, a^{4} - 2 \, a^{3} + 2 i \, a^{2} - 3 \, a - i\right )} b x^{4} + {\left (a^{6} + 4 i \, a^{5} - 5 \, a^{4} - 5 \, a^{2} - 4 i \, a + 1\right )} x^{3}\right )} \log \left (-\frac {{\left (6 \, a^{2} - 18 i \, a - 11\right )} b^{4} x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (6 \, a^{2} - 18 i \, a - 11\right )} b^{3} + {\left (a^{7} + 3 i \, a^{6} - a^{5} + 5 i \, a^{4} - 5 \, a^{3} + i \, a^{2} - 3 \, a - i\right )} \sqrt {\frac {{\left (36 \, a^{4} - 216 i \, a^{3} - 456 \, a^{2} + 396 i \, a + 121\right )} b^{6}}{a^{12} + 6 i \, a^{11} - 12 \, a^{10} - 2 i \, a^{9} - 27 \, a^{8} - 36 i \, a^{7} - 36 i \, a^{5} + 27 \, a^{4} - 2 i \, a^{3} + 12 \, a^{2} + 6 i \, a - 1}}}{{\left (6 \, a^{2} - 18 i \, a - 11\right )} b^{3}}\right ) - 3 \, \sqrt {\frac {{\left (36 \, a^{4} - 216 i \, a^{3} - 456 \, a^{2} + 396 i \, a + 121\right )} b^{6}}{a^{12} + 6 i \, a^{11} - 12 \, a^{10} - 2 i \, a^{9} - 27 \, a^{8} - 36 i \, a^{7} - 36 i \, a^{5} + 27 \, a^{4} - 2 i \, a^{3} + 12 \, a^{2} + 6 i \, a - 1}} {\left ({\left (a^{5} + 3 i \, a^{4} - 2 \, a^{3} + 2 i \, a^{2} - 3 \, a - i\right )} b x^{4} + {\left (a^{6} + 4 i \, a^{5} - 5 \, a^{4} - 5 \, a^{2} - 4 i \, a + 1\right )} x^{3}\right )} \log \left (-\frac {{\left (6 \, a^{2} - 18 i \, a - 11\right )} b^{4} x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (6 \, a^{2} - 18 i \, a - 11\right )} b^{3} - {\left (a^{7} + 3 i \, a^{6} - a^{5} + 5 i \, a^{4} - 5 \, a^{3} + i \, a^{2} - 3 \, a - i\right )} \sqrt {\frac {{\left (36 \, a^{4} - 216 i \, a^{3} - 456 \, a^{2} + 396 i \, a + 121\right )} b^{6}}{a^{12} + 6 i \, a^{11} - 12 \, a^{10} - 2 i \, a^{9} - 27 \, a^{8} - 36 i \, a^{7} - 36 i \, a^{5} + 27 \, a^{4} - 2 i \, a^{3} + 12 \, a^{2} + 6 i \, a - 1}}}{{\left (6 \, a^{2} - 18 i \, a - 11\right )} b^{3}}\right ) + {\left ({\left (2 i \, a^{2} + 51 \, a - 52 i\right )} b^{3} x^{3} + 2 i \, a^{5} + {\left (16 \, a^{2} - 3 i \, a + 19\right )} b^{2} x^{2} - 2 \, a^{4} + 4 i \, a^{3} - 7 \, {\left (a^{3} + i \, a^{2} + a + i\right )} b x - 4 \, a^{2} + 2 i \, a - 2\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{6 \, {\left ({\left (a^{5} + 3 i \, a^{4} - 2 \, a^{3} + 2 i \, a^{2} - 3 \, a - i\right )} b x^{4} + {\left (a^{6} + 4 i \, a^{5} - 5 \, a^{4} - 5 \, a^{2} - 4 i \, a + 1\right )} x^{3}\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^4,x, algorithm="fricas")

[Out]

1/6*((2*I*a^2 + 51*a - 52*I)*b^4*x^4 + (2*I*a^3 + 49*a^2 - I*a + 52)*b^3*x^3 + 3*sqrt((36*a^4 - 216*I*a^3 - 45
6*a^2 + 396*I*a + 121)*b^6/(a^12 + 6*I*a^11 - 12*a^10 - 2*I*a^9 - 27*a^8 - 36*I*a^7 - 36*I*a^5 + 27*a^4 - 2*I*
a^3 + 12*a^2 + 6*I*a - 1))*((a^5 + 3*I*a^4 - 2*a^3 + 2*I*a^2 - 3*a - I)*b*x^4 + (a^6 + 4*I*a^5 - 5*a^4 - 5*a^2
- 4*I*a + 1)*x^3)*log(-((6*a^2 - 18*I*a - 11)*b^4*x - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(6*a^2 - 18*I*a - 11)
*b^3 + (a^7 + 3*I*a^6 - a^5 + 5*I*a^4 - 5*a^3 + I*a^2 - 3*a - I)*sqrt((36*a^4 - 216*I*a^3 - 456*a^2 + 396*I*a
+ 121)*b^6/(a^12 + 6*I*a^11 - 12*a^10 - 2*I*a^9 - 27*a^8 - 36*I*a^7 - 36*I*a^5 + 27*a^4 - 2*I*a^3 + 12*a^2 + 6
*I*a - 1)))/((6*a^2 - 18*I*a - 11)*b^3)) - 3*sqrt((36*a^4 - 216*I*a^3 - 456*a^2 + 396*I*a + 121)*b^6/(a^12 + 6
*I*a^11 - 12*a^10 - 2*I*a^9 - 27*a^8 - 36*I*a^7 - 36*I*a^5 + 27*a^4 - 2*I*a^3 + 12*a^2 + 6*I*a - 1))*((a^5 + 3
*I*a^4 - 2*a^3 + 2*I*a^2 - 3*a - I)*b*x^4 + (a^6 + 4*I*a^5 - 5*a^4 - 5*a^2 - 4*I*a + 1)*x^3)*log(-((6*a^2 - 18
*I*a - 11)*b^4*x - sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(6*a^2 - 18*I*a - 11)*b^3 - (a^7 + 3*I*a^6 - a^5 + 5*I*a^
4 - 5*a^3 + I*a^2 - 3*a - I)*sqrt((36*a^4 - 216*I*a^3 - 456*a^2 + 396*I*a + 121)*b^6/(a^12 + 6*I*a^11 - 12*a^1
0 - 2*I*a^9 - 27*a^8 - 36*I*a^7 - 36*I*a^5 + 27*a^4 - 2*I*a^3 + 12*a^2 + 6*I*a - 1)))/((6*a^2 - 18*I*a - 11)*b
^3)) + ((2*I*a^2 + 51*a - 52*I)*b^3*x^3 + 2*I*a^5 + (16*a^2 - 3*I*a + 19)*b^2*x^2 - 2*a^4 + 4*I*a^3 - 7*(a^3 +
I*a^2 + a + I)*b*x - 4*a^2 + 2*I*a - 2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))/((a^5 + 3*I*a^4 - 2*a^3 + 2*I*a^2
- 3*a - I)*b*x^4 + (a^6 + 4*I*a^5 - 5*a^4 - 5*a^2 - 4*I*a + 1)*x^3)

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

[Out]

-I*(Integral(I/(a**2*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2 +
1) + b**2*x**6*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integra
l(-3*a/(a**2*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**
2*x**6*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(a**3/(
a**2*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**6*s
qrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*I*a**2/(a**
2*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**6*sqrt
(a**2 + 2*a*b*x + b**2*x**2 + 1) + x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*b*x/(a**2*x**4
*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**6*sqrt(a**2
+ 2*a*b*x + b**2*x**2 + 1) + x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(b**3*x**3/(a**2*x**4*sq
rt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**6*sqrt(a**2 + 2
*a*b*x + b**2*x**2 + 1) + x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*I*b**2*x**2/(a**2*x**4*
sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**6*sqrt(a**2 +
2*a*b*x + b**2*x**2 + 1) + x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(3*a*b**2*x**2/(a**2*x**4
*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**6*sqrt(a**2
+ 2*a*b*x + b**2*x**2 + 1) + x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(3*a**2*b*x/(a**2*x**4*s
qrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**6*sqrt(a**2 +
2*a*b*x + b**2*x**2 + 1) + x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-6*I*a*b*x/(a**2*x**4*sqr
t(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x**5*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**6*sqrt(a**2 + 2*
a*b*x + b**2*x**2 + 1) + x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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^4,x, algorithm="giac")

[Out]

undef

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3}{x^4\,{\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((a*1i + b*x*1i + 1)^3/(x^4*((a + b*x)^2 + 1)^(3/2)),x)

[Out]

int((a*1i + b*x*1i + 1)^3/(x^4*((a + b*x)^2 + 1)^(3/2)), x)

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