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

Optimal. Leaf size=163 \[ -\frac {(1-i a) (i a+i b x+1)^{5/2}}{b^2 \sqrt {-i a-i b x+1}}-\frac {(3-2 i a) \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{2 b^2}-\frac {3 (3-2 i a) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{2 b^2}+\frac {3 (2 a+3 i) \sinh ^{-1}(a+b x)}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5095, 78, 50, 53, 619, 215} \[ -\frac {(1-i a) (i a+i b x+1)^{5/2}}{b^2 \sqrt {-i a-i b x+1}}-\frac {(3-2 i a) \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{2 b^2}-\frac {3 (3-2 i a) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{2 b^2}+\frac {3 (2 a+3 i) \sinh ^{-1}(a+b x)}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[E^((3*I)*ArcTan[a + b*x])*x,x]

[Out]

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

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 78

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

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]

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 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 e^{3 i \tan ^{-1}(a+b x)} x \, dx &=\int \frac {x (1+i a+i b x)^{3/2}}{(1-i a-i b x)^{3/2}} \, dx\\ &=-\frac {(1-i a) (1+i a+i b x)^{5/2}}{b^2 \sqrt {1-i a-i b x}}+\frac {(3 i+2 a) \int \frac {(1+i a+i b x)^{3/2}}{\sqrt {1-i a-i b x}} \, dx}{b}\\ &=-\frac {(3-2 i a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 b^2}-\frac {(1-i a) (1+i a+i b x)^{5/2}}{b^2 \sqrt {1-i a-i b x}}+\frac {(3 (3 i+2 a)) \int \frac {\sqrt {1+i a+i b x}}{\sqrt {1-i a-i b x}} \, dx}{2 b}\\ &=-\frac {3 (3-2 i a) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^2}-\frac {(3-2 i a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 b^2}-\frac {(1-i a) (1+i a+i b x)^{5/2}}{b^2 \sqrt {1-i a-i b x}}+\frac {(3 (3 i+2 a)) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{2 b}\\ &=-\frac {3 (3-2 i a) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^2}-\frac {(3-2 i a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 b^2}-\frac {(1-i a) (1+i a+i b x)^{5/2}}{b^2 \sqrt {1-i a-i b x}}+\frac {(3 (3 i+2 a)) \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{2 b}\\ &=-\frac {3 (3-2 i a) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^2}-\frac {(3-2 i a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 b^2}-\frac {(1-i a) (1+i a+i b x)^{5/2}}{b^2 \sqrt {1-i a-i b x}}+\frac {(3 (3 i+2 a)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{4 b^2}}} \, dx,x,2 a b+2 b^2 x\right )}{4 b^3}\\ &=-\frac {3 (3-2 i a) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^2}-\frac {(3-2 i a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 b^2}-\frac {(1-i a) (1+i a+i b x)^{5/2}}{b^2 \sqrt {1-i a-i b x}}+\frac {3 (3 i+2 a) \sinh ^{-1}(a+b x)}{2 b^2}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 132, normalized size = 0.81 \[ \frac {\sqrt {i a+i b x+1} \left (a^2+15 i a-b^2 x^2+5 i b x-14\right )}{2 b^2 \sqrt {-i (a+b x+i)}}+\frac {3 \sqrt [4]{-1} (2 a+3 i) \sqrt {-i b} \sinh ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (a+b x+i)}}{\sqrt {-i b}}\right )}{b^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[1 + I*a + I*b*x]*(-14 + (15*I)*a + a^2 + (5*I)*b*x - b^2*x^2))/(2*b^2*Sqrt[(-I)*(I + a + b*x)]) + (3*(-1
)^(1/4)*(3*I + 2*a)*Sqrt[(-I)*b]*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/b^(5/2)

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fricas [A]  time = 0.51, size = 137, normalized size = 0.84 \[ \frac {3 i \, a^{3} + {\left (3 i \, a^{2} - 44 \, a - 32 i\right )} b x - 47 \, a^{2} - {\left ({\left (24 \, a + 36 i\right )} b x + 24 \, a^{2} + 60 i \, a - 36\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (-4 i \, b^{2} x^{2} + 4 i \, a^{2} - 20 \, b x - 60 \, a - 56 i\right )} - 76 i \, a + 32}{8 \, b^{3} x + {\left (8 \, a + 8 i\right )} b^{2}} \]

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,x, algorithm="fricas")

[Out]

(3*I*a^3 + (3*I*a^2 - 44*a - 32*I)*b*x - 47*a^2 - ((24*a + 36*I)*b*x + 24*a^2 + 60*I*a - 36)*log(-b*x - a + sq
rt(b^2*x^2 + 2*a*b*x + a^2 + 1)) + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(-4*I*b^2*x^2 + 4*I*a^2 - 20*b*x - 60*a -
56*I) - 76*I*a + 32)/(8*b^3*x + (8*a + 8*I)*b^2)

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giac [A]  time = 0.19, size = 214, normalized size = 1.31 \[ -\frac {1}{2} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left (\frac {i x}{b} - \frac {a b^{2} i - 6 \, b^{2}}{b^{4}}\right )} - \frac {{\left (2 \, a + 3 \, i\right )} \log \left (3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b + a^{3} b + 2 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b i + 2 \, a^{2} b i + {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{3} {\left | b \right |} + 3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a^{2} {\left | b \right |} + 4 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a i {\left | b \right |} - a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{2 \, b {\left | b \right |}} \]

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,x, algorithm="giac")

[Out]

-1/2*sqrt((b*x + a)^2 + 1)*(i*x/b - (a*b^2*i - 6*b^2)/b^4) - 1/2*(2*a + 3*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*ab
s(b) - a*b - (x*abs(b) - sqrt((b*x + a)^2 + 1))*abs(b))/(b*abs(b))

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maple [B]  time = 0.18, size = 358, normalized size = 2.20 \[ -\frac {10 a x}{b \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {9 i \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 b \sqrt {b^{2}}}+\frac {i a^{3}}{2 b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {i b \,x^{3}}{2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {7}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {3 a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b \sqrt {b^{2}}}+\frac {i a^{2} x}{2 b \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {3 x^{2}}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {i a \,x^{2}}{2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {i a}{2 b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {9 i x}{2 b \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {7 a^{2}}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}} \]

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,x)

[Out]

-10*a/b/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*x+9/2*I/b*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)+1/2*I/b^2*a^3/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-1/2*I*b*x^3/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-7/b^2/(b^2*x^2+2
*a*b*x+a^2+1)^(1/2)+3*a/b*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)+1/2*I/b*a^2*x/
(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-3*x^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-1/2*I*a*x^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/
2*I/b^2*a/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-9/2*I/b*x/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-7*a^2/b^2/(b^2*x^2+2*a*b*x+a^2
+1)^(1/2)

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maxima [B]  time = 0.37, size = 1116, normalized size = 6.85 \[ \text {result too large to display} \]

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,x, algorithm="maxima")

[Out]

15*I*a^4*b*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 31/2*I*(a^2 + 1)*a^2*b*x/((a^2*b^
2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 1/2*I*b*x^3/sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) + 5/2*I*
(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)) - 2*(-3*I*a^2*b - 6*a*b + 3*I*b)*a
^2*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) + 6*(-3*I*a*b^2 - 3*b^2)*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) + 3/2*I*(a^2 + 1)^2*b*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^
2*x^2 + 2*a*b*x + a^2 + 1)) + (-I*a^3 - 3*a^2 + 3*I*a + 1)*a*b*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a
*b*x + a^2 + 1)) + 5/2*I*a*x^2/sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) - 3/2*I*(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)) + (-I*a^3 - 3*a^2 + 3*I*a + 1)*a^2/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*
x^2 + 2*a*b*x + a^2 + 1)) + (-3*I*a^2*b - 6*a*b + 3*I*b)*(a^2 + 1)*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 +
2*a*b*x + a^2 + 1)) - 5*(-3*I*a*b^2 - 3*b^2)*(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) - 15/2*I*a^2*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^2 + (-3*I*a*b^2 - 3*b
^2)*(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) - (-3*I*a^2*b - 6*a*b + 3*
I*b)*(a^2 + 1)*a/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b) + (-3*I*a*b^2 - 3*b^2)*x^2/(s
qrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - 1/2*(-3*I*a^2 - 3*I)*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2
+ 1)*b^2))/b^2 + 5*I*(a^2 + 1)*a/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - 3*(-3*I*a*b^2 - 3*b^2)*a*arcsinh(2*
(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^4 + (-3*I*a^2*b - 6*a*b + 3*I*b)*arcsinh(2*(b^2*x + a*b)/s
qrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^3 + (I*a^3 + 3*a^2 - 3*I*a - 1)/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) +
2*(-3*I*a*b^2 - 3*b^2)*(a^2 + 1)/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((x*(a*1i + b*x*1i + 1)^3)/((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 x}{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}{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}{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^{2}}{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^{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 {3 i a^{2} x}{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^{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 {3 a b^{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}}\, dx + \int \frac {3 a^{2} b x^{2}}{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^{2}}{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 ) \]

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,x)

[Out]

-I*(Integral(I*x/(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/(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/(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**2/(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**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(-3*I*a**2*x/(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*b**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)), x) + Int
egral(3*a*b**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)), x) + Integral(3*a**2
*b*x**2/(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**2/(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))

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