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

Optimal. Leaf size=176 \[ -\frac {(i a+i b x+1)^{3/2}}{(1-i a) x \sqrt {-i a-i b x+1}}-\frac {6 i b \sqrt {i a+i b x+1}}{(a+i)^2 \sqrt {-i a-i b x+1}}+\frac {6 i \sqrt {-a+i} b \tanh ^{-1}\left (\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-a+i} \sqrt {-i a-i b x+1}}\right )}{(a+i)^{5/2}} \]

[Out]

6*I*b*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)^(1/2)/(I+a)^(5/2)-(1+I*a+
I*b*x)^(3/2)/(1-I*a)/x/(1-I*a-I*b*x)^(1/2)-6*I*b*(1+I*a+I*b*x)^(1/2)/(I+a)^2/(1-I*a-I*b*x)^(1/2)

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Rubi [A]  time = 0.08, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5095, 94, 93, 208} \[ -\frac {(i a+i b x+1)^{3/2}}{(1-i a) x \sqrt {-i a-i b x+1}}-\frac {6 i b \sqrt {i a+i b x+1}}{(a+i)^2 \sqrt {-i a-i b x+1}}+\frac {6 i \sqrt {-a+i} b \tanh ^{-1}\left (\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-a+i} \sqrt {-i a-i b x+1}}\right )}{(a+i)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[E^((3*I)*ArcTan[a + b*x])/x^2,x]

[Out]

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

Rule 93

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 94

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 + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

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]

Rubi steps

\begin {align*} \int \frac {e^{3 i \tan ^{-1}(a+b x)}}{x^2} \, dx &=\int \frac {(1+i a+i b x)^{3/2}}{x^2 (1-i a-i b x)^{3/2}} \, dx\\ &=-\frac {(1+i a+i b x)^{3/2}}{(1-i a) x \sqrt {1-i a-i b x}}-\frac {(3 b) \int \frac {\sqrt {1+i a+i b x}}{x (1-i a-i b x)^{3/2}} \, dx}{i+a}\\ &=-\frac {6 i b \sqrt {1+i a+i b x}}{(i+a)^2 \sqrt {1-i a-i b x}}-\frac {(1+i a+i b x)^{3/2}}{(1-i a) x \sqrt {1-i a-i b x}}-\frac {(3 (i-a) b) \int \frac {1}{x \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{(i+a)^2}\\ &=-\frac {6 i b \sqrt {1+i a+i b x}}{(i+a)^2 \sqrt {1-i a-i b x}}-\frac {(1+i a+i b x)^{3/2}}{(1-i a) x \sqrt {1-i a-i b x}}-\frac {(6 (i-a) b) \operatorname {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)^2}\\ &=-\frac {6 i b \sqrt {1+i a+i b x}}{(i+a)^2 \sqrt {1-i a-i b x}}-\frac {(1+i a+i b x)^{3/2}}{(1-i a) x \sqrt {1-i a-i b x}}+\frac {6 i \sqrt {i-a} b \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)^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 145, normalized size = 0.82 \[ \frac {\frac {\sqrt {i a+i b x+1} \left (a^2+a b x-5 i b x+1\right )}{x \sqrt {-i (a+b x+i)}}+\frac {6 i \sqrt {-1-i a} b \tanh ^{-1}\left (\frac {\sqrt {-1-i a} \sqrt {-i (a+b x+i)}}{\sqrt {-1+i a} \sqrt {i a+i b x+1}}\right )}{\sqrt {-1+i a}}}{(a+i)^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [B]  time = 0.47, size = 404, normalized size = 2.30 \[ -\frac {2 \, {\left (-i \, a - 5\right )} b^{2} x^{2} - {\left (2 i \, a^{2} + 8 \, a + 10 i\right )} b x - {\left ({\left (a^{2} + 2 i \, a - 1\right )} b x^{2} + {\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} x\right )} \sqrt {\frac {{\left (36 \, a - 36 i\right )} b^{2}}{a^{5} + 5 i \, a^{4} - 10 \, a^{3} - 10 i \, a^{2} + 5 \, a + i}} \log \left (-\frac {6 \, b^{2} x + {\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} \sqrt {\frac {{\left (36 \, a - 36 i\right )} b^{2}}{a^{5} + 5 i \, a^{4} - 10 \, a^{3} - 10 i \, a^{2} + 5 \, a + i}} - 6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b}{6 \, b}\right ) + {\left ({\left (a^{2} + 2 i \, a - 1\right )} b x^{2} + {\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} x\right )} \sqrt {\frac {{\left (36 \, a - 36 i\right )} b^{2}}{a^{5} + 5 i \, a^{4} - 10 \, a^{3} - 10 i \, a^{2} + 5 \, a + i}} \log \left (-\frac {6 \, b^{2} x - {\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} \sqrt {\frac {{\left (36 \, a - 36 i\right )} b^{2}}{a^{5} + 5 i \, a^{4} - 10 \, a^{3} - 10 i \, a^{2} + 5 \, a + i}} - 6 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b}{6 \, b}\right ) + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (2 \, {\left (-i \, a - 5\right )} b x - 2 i \, a^{2} - 2 i\right )}}{2 \, {\left (a^{2} + 2 i \, a - 1\right )} b x^{2} + {\left (2 \, a^{3} + 6 i \, a^{2} - 6 \, a - 2 i\right )} x} \]

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^2,x, algorithm="fricas")

[Out]

-(2*(-I*a - 5)*b^2*x^2 - (2*I*a^2 + 8*a + 10*I)*b*x - ((a^2 + 2*I*a - 1)*b*x^2 + (a^3 + 3*I*a^2 - 3*a - I)*x)*
sqrt((36*a - 36*I)*b^2/(a^5 + 5*I*a^4 - 10*a^3 - 10*I*a^2 + 5*a + I))*log(-1/6*(6*b^2*x + (a^3 + 3*I*a^2 - 3*a
- I)*sqrt((36*a - 36*I)*b^2/(a^5 + 5*I*a^4 - 10*a^3 - 10*I*a^2 + 5*a + I)) - 6*sqrt(b^2*x^2 + 2*a*b*x + a^2 +
1)*b)/b) + ((a^2 + 2*I*a - 1)*b*x^2 + (a^3 + 3*I*a^2 - 3*a - I)*x)*sqrt((36*a - 36*I)*b^2/(a^5 + 5*I*a^4 - 10
*a^3 - 10*I*a^2 + 5*a + I))*log(-1/6*(6*b^2*x - (a^3 + 3*I*a^2 - 3*a - I)*sqrt((36*a - 36*I)*b^2/(a^5 + 5*I*a^
4 - 10*a^3 - 10*I*a^2 + 5*a + I)) - 6*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b)/b) + sqrt(b^2*x^2 + 2*a*b*x + a^2 +
1)*(2*(-I*a - 5)*b*x - 2*I*a^2 - 2*I))/(2*(a^2 + 2*I*a - 1)*b*x^2 + (2*a^3 + 6*I*a^2 - 6*a - 2*I)*x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {undef} \]

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^2,x, algorithm="giac")

[Out]

undef

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maple [B]  time = 0.17, size = 1358, normalized size = 7.72 \[ \text {result too large to display} \]

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^2,x)

[Out]

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

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maxima [B]  time = 0.35, size = 993, normalized size = 5.64 \[ \text {result too large to display} \]

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^2,x, algorithm="maxima")

[Out]

-I*a*b^4*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 3*(-I*a^3 - 3*a^2 + 3*I*a + 1)*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) - I*a^2*b^3/((a^2*b^2 - (a^2 +
1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 3*(-I*a^3 - 3*a^2 + 3*I*a + 1)*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) + (-3*I*a^2*b - 6*a*b + 3*I*b)*a*b^3*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*(-I*a^3 - 3*a^2 + 3*I*a + 1)*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)) + (-3*I*a^2*b - 6*a*b + 3*I*b)*a^2*b^2/((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*(-I*a^3 - 3*a^2 + 3*I*a + 1)*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)) - (-3*I*a*b^2 - 3*b^2)*b^2*x/((a^2*b^2 - (a^2 + 1)*b^2)*s
qrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - (-3*I*a*b^2 - 3*b^2)*a*b/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x
+ a^2 + 1)) + 3*(-I*a^3 - 3*a^2 + 3*I*a + 1)*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) + I*b/sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) - 3*(-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) - (-3*I*a^2*b - 6*a*b + 3*I*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)^(3/2) + (-3*I*a^2*b - 6*a*b + 3*I*b)/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)) - (-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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3}{x^2\,{\left ({\left (a+b\,x\right )}^2+1\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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**2,x)

[Out]

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

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