3.180 \(\int e^{3 i \tan ^{-1}(a+b x)} x^4 \, dx\)

Optimal. Leaf size=324 \[ -\frac {i \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2} \left (112 i a^3+2 \left (-52 i a^2+118 a+61 i\right ) b x-422 a^2-458 i a+163\right )}{40 b^5}-\frac {3 \left (8 i a^4-48 a^3-88 i a^2+68 a+19 i\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{8 b^5}-\frac {3 \left (8 a^4+48 i a^3-88 a^2-68 i a+19\right ) \sinh ^{-1}(a+b x)}{8 b^5}+\frac {3 (16 a+17 i) x^2 \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{20 b^3}-\frac {11 x^3 \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{5 b^2}-\frac {2 i x^4 (i a+i b x+1)^{3/2}}{b \sqrt {-i a-i b x+1}} \]

[Out]

-3/8*(19-68*I*a-88*a^2+48*I*a^3+8*a^4)*arcsinh(b*x+a)/b^5-2*I*x^4*(1+I*a+I*b*x)^(3/2)/b/(1-I*a-I*b*x)^(1/2)+3/
20*(17*I+16*a)*x^2*(1+I*a+I*b*x)^(3/2)*(1-I*a-I*b*x)^(1/2)/b^3-11/5*x^3*(1+I*a+I*b*x)^(3/2)*(1-I*a-I*b*x)^(1/2
)/b^2-1/40*I*(1+I*a+I*b*x)^(3/2)*(163-458*I*a-422*a^2+112*I*a^3+2*(61*I+118*a-52*I*a^2)*b*x)*(1-I*a-I*b*x)^(1/
2)/b^5-3/8*(19*I+68*a-88*I*a^2-48*a^3+8*I*a^4)*(1-I*a-I*b*x)^(1/2)*(1+I*a+I*b*x)^(1/2)/b^5

________________________________________________________________________________________

Rubi [A]  time = 0.27, antiderivative size = 324, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5095, 97, 153, 147, 50, 53, 619, 215} \[ -\frac {i \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2} \left (2 \left (-52 i a^2+118 a+61 i\right ) b x+112 i a^3-422 a^2-458 i a+163\right )}{40 b^5}-\frac {3 \left (8 i a^4-48 a^3-88 i a^2+68 a+19 i\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{8 b^5}-\frac {3 \left (8 a^4+48 i a^3-88 a^2-68 i a+19\right ) \sinh ^{-1}(a+b x)}{8 b^5}-\frac {11 x^3 \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{5 b^2}+\frac {3 (16 a+17 i) x^2 \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{20 b^3}-\frac {2 i x^4 (i a+i b x+1)^{3/2}}{b \sqrt {-i a-i b x+1}} \]

Antiderivative was successfully verified.

[In]

Int[E^((3*I)*ArcTan[a + b*x])*x^4,x]

[Out]

(-3*(19*I + 68*a - (88*I)*a^2 - 48*a^3 + (8*I)*a^4)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(8*b^5) - ((2
*I)*x^4*(1 + I*a + I*b*x)^(3/2))/(b*Sqrt[1 - I*a - I*b*x]) + (3*(17*I + 16*a)*x^2*Sqrt[1 - I*a - I*b*x]*(1 + I
*a + I*b*x)^(3/2))/(20*b^3) - (11*x^3*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2))/(5*b^2) - ((I/40)*Sqrt[1
- I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2)*(163 - (458*I)*a - 422*a^2 + (112*I)*a^3 + 2*(61*I + 118*a - (52*I)*a^2
)*b*x))/b^5 - (3*(19 - (68*I)*a - 88*a^2 + (48*I)*a^3 + 8*a^4)*ArcSinh[a + b*x])/(8*b^5)

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 97

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 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 153

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

Rule 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^4 \, dx &=\int \frac {x^4 (1+i a+i b x)^{3/2}}{(1-i a-i b x)^{3/2}} \, dx\\ &=-\frac {2 i x^4 (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}+\frac {(2 i) \int \frac {x^3 \sqrt {1+i a+i b x} \left (4 (1+i a)+\frac {11 i b x}{2}\right )}{\sqrt {1-i a-i b x}} \, dx}{b}\\ &=-\frac {2 i x^4 (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}-\frac {11 x^3 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}+\frac {(2 i) \int \frac {x^2 \sqrt {1+i a+i b x} \left (-\frac {33}{2} (i-a) (1-i a) b+\frac {3}{2} (17-16 i a) b^2 x\right )}{\sqrt {1-i a-i b x}} \, dx}{5 b^3}\\ &=-\frac {2 i x^4 (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}+\frac {3 (17 i+16 a) x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}-\frac {11 x^3 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}+\frac {i \int \frac {x \sqrt {1+i a+i b x} \left (3 (1+i a) (i+a) (17 i+16 a) b^2-\frac {3}{2} \left (118 a+i \left (61-52 a^2\right )\right ) b^3 x\right )}{\sqrt {1-i a-i b x}} \, dx}{10 b^5}\\ &=-\frac {2 i x^4 (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}+\frac {3 (17 i+16 a) x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}-\frac {11 x^3 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}-\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (163-458 i a-422 a^2+112 i a^3+2 \left (61 i+118 a-52 i a^2\right ) b x\right )}{40 b^5}-\frac {\left (3 \left (19-68 i a-88 a^2+48 i a^3+8 a^4\right )\right ) \int \frac {\sqrt {1+i a+i b x}}{\sqrt {1-i a-i b x}} \, dx}{8 b^4}\\ &=-\frac {3 \left (19 i+68 a-88 i a^2-48 a^3+8 i a^4\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^5}-\frac {2 i x^4 (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}+\frac {3 (17 i+16 a) x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}-\frac {11 x^3 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}-\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (163-458 i a-422 a^2+112 i a^3+2 \left (61 i+118 a-52 i a^2\right ) b x\right )}{40 b^5}-\frac {\left (3 \left (19-68 i a-88 a^2+48 i a^3+8 a^4\right )\right ) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{8 b^4}\\ &=-\frac {3 \left (19 i+68 a-88 i a^2-48 a^3+8 i a^4\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^5}-\frac {2 i x^4 (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}+\frac {3 (17 i+16 a) x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}-\frac {11 x^3 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}-\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (163-458 i a-422 a^2+112 i a^3+2 \left (61 i+118 a-52 i a^2\right ) b x\right )}{40 b^5}-\frac {\left (3 \left (19-68 i a-88 a^2+48 i a^3+8 a^4\right )\right ) \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{8 b^4}\\ &=-\frac {3 \left (19 i+68 a-88 i a^2-48 a^3+8 i a^4\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^5}-\frac {2 i x^4 (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}+\frac {3 (17 i+16 a) x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}-\frac {11 x^3 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}-\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (163-458 i a-422 a^2+112 i a^3+2 \left (61 i+118 a-52 i a^2\right ) b x\right )}{40 b^5}-\frac {\left (3 \left (19-68 i a-88 a^2+48 i a^3+8 a^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{4 b^2}}} \, dx,x,2 a b+2 b^2 x\right )}{16 b^6}\\ &=-\frac {3 \left (19 i+68 a-88 i a^2-48 a^3+8 i a^4\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^5}-\frac {2 i x^4 (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}+\frac {3 (17 i+16 a) x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}-\frac {11 x^3 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}-\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (163-458 i a-422 a^2+112 i a^3+2 \left (61 i+118 a-52 i a^2\right ) b x\right )}{40 b^5}-\frac {3 \left (19-68 i a-88 a^2+48 i a^3+8 a^4\right ) \sinh ^{-1}(a+b x)}{8 b^5}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.48, size = 249, normalized size = 0.77 \[ \frac {3 (-1)^{3/4} \left (8 a^4+48 i a^3-88 a^2-68 i a+19\right ) \sinh ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (a+b x+i)}}{\sqrt {-i b}}\right )}{4 \sqrt {-i b} b^{9/2}}-\frac {\sqrt {i a+i b x+1} \left (8 a^5+418 i a^4+14 i a^3 (8 b x+121 i)-i a^2 \left (52 b^2 x^2-422 i b x+2599\right )+a \left (32 i b^3 x^3+118 b^2 x^2-458 i b x+1763\right )+8 b^5 x^5-22 i b^4 x^4-34 b^3 x^3+61 i b^2 x^2+163 b x+448 i\right )}{40 b^5 \sqrt {-i (a+b x+i)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/40*(Sqrt[1 + I*a + I*b*x]*(448*I + (418*I)*a^4 + 8*a^5 + 163*b*x + (61*I)*b^2*x^2 - 34*b^3*x^3 - (22*I)*b^4
*x^4 + 8*b^5*x^5 + (14*I)*a^3*(121*I + 8*b*x) - I*a^2*(2599 - (422*I)*b*x + 52*b^2*x^2) + a*(1763 - (458*I)*b*
x + 118*b^2*x^2 + (32*I)*b^3*x^3)))/(b^5*Sqrt[(-I)*(I + a + b*x)]) + (3*(-1)^(3/4)*(19 - (68*I)*a - 88*a^2 + (
48*I)*a^3 + 8*a^4)*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/(4*Sqrt[(-I)*b]*b^(9/
2))

________________________________________________________________________________________

fricas [A]  time = 0.50, size = 263, normalized size = 0.81 \[ \frac {-62 i \, a^{6} + 2687 \, a^{5} + 11575 i \, a^{4} - 20350 \, a^{3} + {\left (-62 i \, a^{5} + 2625 \, a^{4} + 8950 i \, a^{3} - 11400 \, a^{2} - 6340 i \, a + 1280\right )} b x - 17740 i \, a^{2} + {\left (960 \, a^{5} + 6720 i \, a^{4} - 16320 \, a^{3} + {\left (960 \, a^{4} + 5760 i \, a^{3} - 10560 \, a^{2} - 8160 i \, a + 2280\right )} b x - 18720 i \, a^{2} + 10440 \, a + 2280 i\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + {\left (-64 i \, b^{5} x^{5} - 176 \, b^{4} x^{4} + {\left (256 \, a + 272 i\right )} b^{3} x^{3} - 64 i \, a^{5} - 8 \, {\left (52 \, a^{2} + 118 i \, a - 61\right )} b^{2} x^{2} + 3344 \, a^{4} + 13552 i \, a^{3} + {\left (896 \, a^{3} + 3376 i \, a^{2} - 3664 \, a - 1304 i\right )} b x - 20792 \, a^{2} - 14104 i \, a + 3584\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 7620 \, a + 1280 i}{320 \, b^{6} x + {\left (320 \, a + 320 i\right )} b^{5}} \]

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]

(-62*I*a^6 + 2687*a^5 + 11575*I*a^4 - 20350*a^3 + (-62*I*a^5 + 2625*a^4 + 8950*I*a^3 - 11400*a^2 - 6340*I*a +
1280)*b*x - 17740*I*a^2 + (960*a^5 + 6720*I*a^4 - 16320*a^3 + (960*a^4 + 5760*I*a^3 - 10560*a^2 - 8160*I*a + 2
280)*b*x - 18720*I*a^2 + 10440*a + 2280*I)*log(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) + (-64*I*b^5*x^5
- 176*b^4*x^4 + (256*a + 272*I)*b^3*x^3 - 64*I*a^5 - 8*(52*a^2 + 118*I*a - 61)*b^2*x^2 + 3344*a^4 + 13552*I*a^
3 + (896*a^3 + 3376*I*a^2 - 3664*a - 1304*I)*b*x - 20792*a^2 - 14104*I*a + 3584)*sqrt(b^2*x^2 + 2*a*b*x + a^2
+ 1) + 7620*a + 1280*I)/(320*b^6*x + (320*a + 320*I)*b^5)

________________________________________________________________________________________

giac [A]  time = 0.21, size = 345, normalized size = 1.06 \[ -\frac {1}{40} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left ({\left (2 \, {\left ({\left (\frac {4 \, i x}{b} - \frac {4 \, a b^{17} i - 15 \, b^{17}}{b^{19}}\right )} x + \frac {4 \, a^{2} b^{16} i - 35 \, a b^{16} - 32 \, b^{16} i}{b^{19}}\right )} x - \frac {8 \, a^{3} b^{15} i - 130 \, a^{2} b^{15} - 252 \, a b^{15} i + 125 \, b^{15}}{b^{19}}\right )} x + \frac {8 \, a^{4} b^{14} i - 250 \, a^{3} b^{14} - 804 \, a^{2} b^{14} i + 835 \, a b^{14} + 288 \, b^{14} i}{b^{19}}\right )} + \frac {{\left (8 \, a^{4} + 48 \, a^{3} i - 88 \, a^{2} - 68 \, a i + 19\right )} \log \left (3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b + a^{3} b + 2 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b i + 2 \, a^{2} b i + {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{3} {\left | b \right |} + 3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a^{2} {\left | b \right |} + 4 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a i {\left | b \right |} - a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{8 \, b^{4} {\left | b \right |}} \]

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^4,x, algorithm="giac")

[Out]

-1/40*sqrt((b*x + a)^2 + 1)*((2*((4*i*x/b - (4*a*b^17*i - 15*b^17)/b^19)*x + (4*a^2*b^16*i - 35*a*b^16 - 32*b^
16*i)/b^19)*x - (8*a^3*b^15*i - 130*a^2*b^15 - 252*a*b^15*i + 125*b^15)/b^19)*x + (8*a^4*b^14*i - 250*a^3*b^14
- 804*a^2*b^14*i + 835*a*b^14 + 288*b^14*i)/b^19) + 1/8*(8*a^4 + 48*a^3*i - 88*a^2 - 68*a*i + 19)*log(3*(x*ab
s(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*abs(b) - a*b - (x*abs(b) - sqrt((b*x + a)^2 + 1))*abs(b))/(b^4*abs(b))

________________________________________________________________________________________

maple [B]  time = 0.20, size = 933, normalized size = 2.88 \[ \frac {33 a^{2} \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^{4} \sqrt {b^{2}}}+\frac {89 i a^{2} x^{2}}{10 b^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {501 i a^{3} x}{10 b^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {7 i x^{4}}{5 b \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {28 i x^{2}}{5 b^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {i b \,x^{6}}{5 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {319 i a^{4}}{10 b^{5} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {i a^{6}}{5 b^{5} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {209 i a^{2}}{10 b^{5} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {i a \,x^{5}}{5 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {x^{3} a^{2}}{2 b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {3 a^{3} x^{2}}{2 b^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {3 a^{4} \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^{4} \sqrt {b^{2}}}+\frac {a \,x^{4}}{4 b \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {29 i a \,x^{3}}{10 b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {367 i a x}{10 b^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {i a^{5} x}{5 b^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {3 x^{5}}{4 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {18 i a^{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 )}{b^{4} \sqrt {b^{2}}}-\frac {103 a \,x^{2}}{8 b^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {527 a^{2} x}{8 b^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {53 a^{4} x}{4 b^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {51 i 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 )}{2 b^{4} \sqrt {b^{2}}}-\frac {56 i}{5 b^{5} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {57 x}{8 b^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {57 \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^{4} \sqrt {b^{2}}}-\frac {181 a^{3}}{8 b^{5} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {41 a^{5}}{4 b^{5} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {263 a}{8 b^{5} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {19 x^{3}}{8 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^4,x)

[Out]

33*a^2/b^4*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2)+89/10*I/b^3*a^2*x^2/(b^2*x^2+
2*a*b*x+a^2+1)^(1/2)+501/10*I/b^4*a^3*x/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-29/10*I/b^2*a*x^3/(b^2*x^2+2*a*b*x+a^2+1
)^(1/2)-367/10*I/b^4*a*x/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-28/5*I/b^3*x^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-1/2*x^3/b^
2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a^2+3/2*a^3/b^3*x^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-3*a^4/b^4*ln((b^2*x+a*b)/(b^
2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2)+1/4/b*a*x^4/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-3/4*x^5/(b^2*x^2
+2*a*b*x+a^2+1)^(1/2)-103/8*a/b^3*x^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-1/5*I*b*x^6/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-
527/8*a^2/b^4*x/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+53/4*a^4/b^4/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*x+319/10*I/b^5*a^4/(b
^2*x^2+2*a*b*x+a^2+1)^(1/2)-1/5*I/b^5*a^6/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+209/10*I/b^5*a^2/(b^2*x^2+2*a*b*x+a^2+
1)^(1/2)-1/5*I*a*x^5/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+7/5*I/b*x^4/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-56/5*I/b^5/(b^2*x
^2+2*a*b*x+a^2+1)^(1/2)+57/8/b^4*x/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-57/8/b^4*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+
2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2)-181/8*a^3/b^5/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+41/4*a^5/b^5/(b^2*x^2+2*a*b*x+a^
2+1)^(1/2)-263/8*a/b^5/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+19/8*x^3/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-1/5*I/b^4*a^5*
x/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-18*I/b^4*a^3*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(
1/2)+51/2*I/b^4*a*ln((b^2*x+a*b)/(b^2)^(1/2)+(b^2*x^2+2*a*b*x+a^2+1)^(1/2))/(b^2)^(1/2)

________________________________________________________________________________________

maxima [B]  time = 0.41, size = 3117, normalized size = 9.62 \[ \text {result too large to display} \]

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^4,x, algorithm="maxima")

[Out]

-1/5*I*b*x^6/sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) + 11/20*I*a*x^5/sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) - 693/4*I*a^7
*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - 33/20*I*a^2*x^4/(sqrt(b^2*x^2 + 2*a*b*x
+ a^2 + 1)*b) + 2415/8*I*(a^2 + 1)*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^2) +
231/40*I*a^3*x^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - 1/5*(-2*I*a^2 - 2*I)*x^4/(sqrt(b^2*x^2 + 2*a*b*x +
a^2 + 1)*b) + 1/4*(-3*I*a*b^2 - 3*b^2)*x^5/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - 231/8*I*(a^2 + 1)*a^6/((a
^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^3) - 315/4*(-3*I*a*b^2 - 3*b^2)*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^4) + 35*(-3*I*a^2*b - 6*a*b + 3*I*b)*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) - 2919/20*I*(a^2 + 1)^2*a^3*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqr
t(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - 15*(-I*a^3 - 3*a^2 + 3*I*a + 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^2) - 231/8*I*a^4*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^3) - 111/40*I*(a^2 + 1)
*a*x^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - 3/4*(-3*I*a*b^2 - 3*b^2)*a*x^4/(sqrt(b^2*x^2 + 2*a*b*x + a^2
+ 1)*b^3) + 1/3*(-3*I*a^2*b - 6*a*b + 3*I*b)*x^4/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) + 189/5*I*(a^2 + 1)^2
*a^4/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^3) + 945/8*(-3*I*a*b^2 - 3*b^2)*(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^4) - 265/6*(-3*I*a^2*b - 6*a*b + 3*I*b)*(
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) + 653/40*I*(a^2 + 1)^3*a*x/((
a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) + 31/2*(-I*a^3 - 3*a^2 + 3*I*a + 1)*(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) + 819/40*I*(a^2 + 1)*a^2*x^2/(sqrt(b^2
*x^2 + 2*a*b*x + a^2 + 1)*b^3) + 21/8*(-3*I*a*b^2 - 3*b^2)*a^2*x^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^4) - 7
/6*(-3*I*a^2*b - 6*a*b + 3*I*b)*a*x^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^3) - 1/2*(I*a^3 + 3*a^2 - 3*I*a - 1
)*x^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) + 693/8*I*a^5*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 +
1)*b^2))/b^5 - 105/8*(-3*I*a*b^2 - 3*b^2)*(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^5) + 35/6*(-3*I*a^2*b - 6*a*b + 3*I*b)*(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) - 397/40*I*(a^2 + 1)^3*a^2/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^3
) - 5/2*(-I*a^3 - 3*a^2 + 3*I*a + 1)*(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) - 169/4*(-3*I*a*b^2 - 3*b^2)*(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^4) + 61/6*(-3*I*a^2*b - 6*a*b + 3*I*b)*(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) - 3/2*(-I*a^3 - 3*a^2 + 3*I*a + 1)*(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) - 105/8*(-3*I*a*b^2 - 3*b^2)*a^3*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^5) + 35/6*(-3
*I*a^2*b - 6*a*b + 3*I*b)*a^2*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^4) - 8/5*I*(a^2 + 1)^2*x^2/(sqrt(b^2*x^
2 + 2*a*b*x + a^2 + 1)*b^3) - 5/2*(-I*a^3 - 3*a^2 + 3*I*a + 1)*a*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^3) -
5/8*(-3*I*a*b^2 - 3*b^2)*(a^2 + 1)*x^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^4) - 315/4*I*(a^2 + 1)*a^3*arcsin
h(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^5 + 14*(-3*I*a*b^2 - 3*b^2)*(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^5) - 231/4*I*(a^2 + 1)*a^4/(sqrt(b^2*x^2 + 2*a*b*x + a^2
+ 1)*b^5) - 29/6*(-3*I*a^2*b - 6*a*b + 3*I*b)*(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) + 3/2*(-I*a^3 - 3*a^2 + 3*I*a + 1)*(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) + 15/8*(-3*I*a*b^2 - 3*b^2)*(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^4) + 49/8*(-3*I*a*b^2 - 3*b^2)*(a^2 + 1)*a*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^5) - 4/3
*(-3*I*a^2*b - 6*a*b + 3*I*b)*(a^2 + 1)*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^4) + 315/8*(-3*I*a*b^2 - 3*b^
2)*a^4*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^7 - 35/2*(-3*I*a^2*b - 6*a*b + 3*I*b)*a^3
*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^6 + 105/8*I*(a^2 + 1)^2*a*arcsinh(2*(b^2*x + a*
b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^5 + 15/2*(-I*a^3 - 3*a^2 + 3*I*a + 1)*a^2*arcsinh(2*(b^2*x + a*b)/sqr
t(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^5 - 15/8*(-3*I*a*b^2 - 3*b^2)*(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^5) + 819/20*I*(a^2 + 1)^2*a^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^5) - 105/4*
(-3*I*a*b^2 - 3*b^2)*(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^7 + 15/2*(-3*
I*a^2*b - 6*a*b + 3*I*b)*(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 - 3/2*(-I
*a^3 - 3*a^2 + 3*I*a + 1)*(a^2 + 1)*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^5 - 105/4*(-
3*I*a*b^2 - 3*b^2)*(a^2 + 1)*a^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^7) + 35/3*(-3*I*a^2*b - 6*a*b + 3*I*b)*(
a^2 + 1)*a^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^6) - 16/5*I*(a^2 + 1)^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b
^5) - 5*(-I*a^3 - 3*a^2 + 3*I*a + 1)*(a^2 + 1)*a/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^5) + 15/8*(-3*I*a*b^2 -
3*b^2)*(a^2 + 1)^2*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^7 + 49/4*(-3*I*a*b^2 - 3*b^2)
*(a^2 + 1)^2*a/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^7) - 8/3*(-3*I*a^2*b - 6*a*b + 3*I*b)*(a^2 + 1)^2/(sqrt(b^
2*x^2 + 2*a*b*x + a^2 + 1)*b^6)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - i \left (\int \frac {i 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 a 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 {a^{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 b 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 {b^{3} x^{7}}{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^{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 \left (- \frac {3 i b^{2} 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}}\right )\, dx + \int \frac {3 a b^{2} 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 \frac {3 a^{2} b 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 \left (- \frac {6 i a b 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\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**4,x)

[Out]

-I*(Integral(I*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*a*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(a**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*b*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(b**3*x**7/(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**4/(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**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*a*b**2*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) + Int
egral(3*a**2*b*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(-6*I*a
*b*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))

________________________________________________________________________________________