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

Optimal. Leaf size=276 \[ \frac {\sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2} \left (-96 a^3-2 \left (-36 a^2-14 i a+13\right ) b x-86 i a^2+114 a+19 i\right )}{120 b^5}+\frac {\left (8 i a^4-16 a^3-24 i a^2+12 a+3 i\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{8 b^5}+\frac {\left (8 a^4+16 i a^3-24 a^2-12 i a+3\right ) \sinh ^{-1}(a+b x)}{8 b^5}-\frac {(8 a+i) x^2 \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{20 b^3}+\frac {x^3 \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{5 b^2} \]

[Out]

1/8*(3-12*I*a-24*a^2+16*I*a^3+8*a^4)*arcsinh(b*x+a)/b^5-1/20*(I+8*a)*x^2*(1+I*a+I*b*x)^(3/2)*(1-I*a-I*b*x)^(1/
2)/b^3+1/5*x^3*(1+I*a+I*b*x)^(3/2)*(1-I*a-I*b*x)^(1/2)/b^2+1/120*(1+I*a+I*b*x)^(3/2)*(19*I+114*a-86*I*a^2-96*a
^3-2*(13-14*I*a-36*a^2)*b*x)*(1-I*a-I*b*x)^(1/2)/b^5+1/8*(3*I+12*a-24*I*a^2-16*a^3+8*I*a^4)*(1-I*a-I*b*x)^(1/2
)*(1+I*a+I*b*x)^(1/2)/b^5

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Rubi [A]  time = 0.20, antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5095, 100, 153, 147, 50, 53, 619, 215} \[ \frac {\sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2} \left (-2 \left (-36 a^2-14 i a+13\right ) b x-96 a^3-86 i a^2+114 a+19 i\right )}{120 b^5}+\frac {\left (8 i a^4-16 a^3-24 i a^2+12 a+3 i\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{8 b^5}+\frac {\left (8 a^4+16 i a^3-24 a^2-12 i a+3\right ) \sinh ^{-1}(a+b x)}{8 b^5}+\frac {x^3 \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{5 b^2}-\frac {(8 a+i) x^2 \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{20 b^3} \]

Antiderivative was successfully verified.

[In]

Int[E^(I*ArcTan[a + b*x])*x^4,x]

[Out]

((3*I + 12*a - (24*I)*a^2 - 16*a^3 + (8*I)*a^4)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(8*b^5) - ((I + 8
*a)*x^2*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2))/(20*b^3) + (x^3*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)
^(3/2))/(5*b^2) + (Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2)*(19*I + 114*a - (86*I)*a^2 - 96*a^3 - 2*(13 -
(14*I)*a - 36*a^2)*b*x))/(120*b^5) + ((3 - (12*I)*a - 24*a^2 + (16*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 100

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

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^{i \tan ^{-1}(a+b x)} x^4 \, dx &=\int \frac {x^4 \sqrt {1+i a+i b x}}{\sqrt {1-i a-i b x}} \, dx\\ &=\frac {x^3 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}+\frac {\int \frac {x^2 \sqrt {1+i a+i b x} \left (-3 \left (1+a^2\right )-(i+8 a) b x\right )}{\sqrt {1-i a-i b x}} \, dx}{5 b^2}\\ &=-\frac {(i+8 a) x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}+\frac {x^3 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}+\frac {\int \frac {x \sqrt {1+i a+i b x} \left (-2 (i-a) (i+a) (i+8 a) b-\left (13-14 i a-36 a^2\right ) b^2 x\right )}{\sqrt {1-i a-i b x}} \, dx}{20 b^4}\\ &=-\frac {(i+8 a) x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}+\frac {x^3 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}+\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (19 i+114 a-86 i a^2-96 a^3-2 \left (13-14 i a-36 a^2\right ) b x\right )}{120 b^5}+\frac {\left (3-12 i a-24 a^2+16 i a^3+8 a^4\right ) \int \frac {\sqrt {1+i a+i b x}}{\sqrt {1-i a-i b x}} \, dx}{8 b^4}\\ &=\frac {\left (3 i+12 a-24 i a^2-16 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 {(i+8 a) x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}+\frac {x^3 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}+\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (19 i+114 a-86 i a^2-96 a^3-2 \left (13-14 i a-36 a^2\right ) b x\right )}{120 b^5}+\frac {\left (3-12 i a-24 a^2+16 i a^3+8 a^4\right ) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{8 b^4}\\ &=\frac {\left (3 i+12 a-24 i a^2-16 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 {(i+8 a) x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}+\frac {x^3 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}+\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (19 i+114 a-86 i a^2-96 a^3-2 \left (13-14 i a-36 a^2\right ) b x\right )}{120 b^5}+\frac {\left (3-12 i a-24 a^2+16 i a^3+8 a^4\right ) \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{8 b^4}\\ &=\frac {\left (3 i+12 a-24 i a^2-16 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 {(i+8 a) x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}+\frac {x^3 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}+\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (19 i+114 a-86 i a^2-96 a^3-2 \left (13-14 i a-36 a^2\right ) b x\right )}{120 b^5}+\frac {\left (3-12 i a-24 a^2+16 i a^3+8 a^4\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 {\left (3 i+12 a-24 i a^2-16 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 {(i+8 a) x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{20 b^3}+\frac {x^3 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{5 b^2}+\frac {\sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (19 i+114 a-86 i a^2-96 a^3-2 \left (13-14 i a-36 a^2\right ) b x\right )}{120 b^5}+\frac {\left (3-12 i a-24 a^2+16 i a^3+8 a^4\right ) \sinh ^{-1}(a+b x)}{8 b^5}\\ \end {align*}

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Mathematica [A]  time = 0.53, size = 217, normalized size = 0.79 \[ \frac {\sqrt [4]{-1} \left (8 a^4+16 i a^3-24 a^2-12 i a+3\right ) \sqrt {-i b} \sinh ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (a+b x+i)}}{\sqrt {-i b}}\right )}{4 b^{11/2}}+\frac {i \sqrt {a^2+2 a b x+b^2 x^2+1} \left (24 a^4+a^3 (-24 b x+250 i)+2 a^2 \left (12 b^2 x^2-65 i b x-166\right )+a \left (-24 b^3 x^3+70 i b^2 x^2+116 b x-275 i\right )+24 b^4 x^4-30 i b^3 x^3-32 b^2 x^2+45 i b x+64\right )}{120 b^5} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((I/120)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(64 + 24*a^4 + (45*I)*b*x - 32*b^2*x^2 - (30*I)*b^3*x^3 + 24*b^4*x^
4 + a^3*(250*I - 24*b*x) + 2*a^2*(-166 - (65*I)*b*x + 12*b^2*x^2) + a*(-275*I + 116*b*x + (70*I)*b^2*x^2 - 24*
b^3*x^3)))/b^5 + ((-1)^(1/4)*(3 - (12*I)*a - 24*a^2 + (16*I)*a^3 + 8*a^4)*Sqrt[(-I)*b]*ArcSinh[((1/2 + I/2)*Sq
rt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)*b]])/(4*b^(11/2))

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fricas [A]  time = 0.49, size = 175, normalized size = 0.63 \[ \frac {186 i \, a^{5} - 1345 \, a^{4} - 1730 i \, a^{3} + 1320 \, a^{2} - {\left (960 \, a^{4} + 1920 i \, a^{3} - 2880 \, a^{2} - 1440 i \, a + 360\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + {\left (192 i \, b^{4} x^{4} - 48 \, {\left (4 i \, a - 5\right )} b^{3} x^{3} + {\left (192 i \, a^{2} - 560 \, a - 256 i\right )} b^{2} x^{2} + 192 i \, a^{4} - 2000 \, a^{3} + {\left (-192 i \, a^{3} + 1040 \, a^{2} + 928 i \, a - 360\right )} b x - 2656 i \, a^{2} + 2200 \, a + 512 i\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 300 i \, a}{960 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2)*x^4,x, algorithm="fricas")

[Out]

1/960*(186*I*a^5 - 1345*a^4 - 1730*I*a^3 + 1320*a^2 - (960*a^4 + 1920*I*a^3 - 2880*a^2 - 1440*I*a + 360)*log(-
b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) + (192*I*b^4*x^4 - 48*(4*I*a - 5)*b^3*x^3 + (192*I*a^2 - 560*a -
256*I)*b^2*x^2 + 192*I*a^4 - 2000*a^3 + (-192*I*a^3 + 1040*a^2 + 928*I*a - 360)*b*x - 2656*I*a^2 + 2200*a + 51
2*I)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) + 300*I*a)/b^5

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giac [A]  time = 0.20, size = 214, normalized size = 0.78 \[ \frac {1}{120} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left ({\left (2 \, {\left (3 \, {\left (\frac {4 \, i x}{b} - \frac {4 \, a b^{7} i - 5 \, b^{7}}{b^{9}}\right )} x + \frac {12 \, a^{2} b^{6} i - 35 \, a b^{6} - 16 \, b^{6} i}{b^{9}}\right )} x - \frac {24 \, a^{3} b^{5} i - 130 \, a^{2} b^{5} - 116 \, a b^{5} i + 45 \, b^{5}}{b^{9}}\right )} x + \frac {24 \, a^{4} b^{4} i - 250 \, a^{3} b^{4} - 332 \, a^{2} b^{4} i + 275 \, a b^{4} + 64 \, b^{4} i}{b^{9}}\right )} - \frac {{\left (8 \, a^{4} + 16 \, a^{3} i - 24 \, a^{2} - 12 \, a i + 3\right )} \log \left (-a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{8 \, b^{4} {\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2)*x^4,x, algorithm="giac")

[Out]

1/120*sqrt((b*x + a)^2 + 1)*((2*(3*(4*i*x/b - (4*a*b^7*i - 5*b^7)/b^9)*x + (12*a^2*b^6*i - 35*a*b^6 - 16*b^6*i
)/b^9)*x - (24*a^3*b^5*i - 130*a^2*b^5 - 116*a*b^5*i + 45*b^5)/b^9)*x + (24*a^4*b^4*i - 250*a^3*b^4 - 332*a^2*
b^4*i + 275*a*b^4 + 64*b^4*i)/b^9) - 1/8*(8*a^4 + 16*a^3*i - 24*a^2 - 12*a*i + 3)*log(-a*b - (x*abs(b) - sqrt(
(b*x + a)^2 + 1))*abs(b))/(b^4*abs(b))

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maple [B]  time = 0.20, size = 656, normalized size = 2.38 \[ -\frac {7 a \,x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{12 b^{3}}+\frac {13 a^{2} x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{12 b^{4}}+\frac {a^{4} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b^{4} \sqrt {b^{2}}}-\frac {3 a^{2} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b^{4} \sqrt {b^{2}}}-\frac {83 i a^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{30 b^{5}}+\frac {29 i a x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{30 b^{4}}-\frac {3 i a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 b^{4} \sqrt {b^{2}}}-\frac {i a \,x^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{5 b^{2}}+\frac {i a^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{5 b^{5}}+\frac {i a^{2} x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{5 b^{3}}-\frac {4 i x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{15 b^{3}}+\frac {8 i \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{15 b^{5}}-\frac {i a^{3} x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{5 b^{4}}+\frac {2 i a^{3} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b^{4} \sqrt {b^{2}}}+\frac {x^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}-\frac {25 a^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{12 b^{5}}+\frac {55 a \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{24 b^{5}}+\frac {i x^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{5 b}-\frac {3 x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{8 b^{4}}+\frac {3 \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{4} \sqrt {b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7/12*a/b^3*x^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+13/12*a^2/b^4*x*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+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)-3*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)-83/30*I/b^5*a^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+29/30*I/b^4*a*x*(b^2*x^2+2*a*b*x+a
^2+1)^(1/2)-3/2*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)-1/5*I/b^2*a*x^3*
(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/5*I/b^5*a^4*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/5*I/b^3*a^2*x^2*(b^2*x^2+2*a*b*x+a
^2+1)^(1/2)-4/15*I/b^3*x^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+8/15*I/b^5*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-1/5*I/b^4*a^
3*x*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+2*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)+1/4*x^3/b^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-25/12*a^3/b^5*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+55/24*a/b^5*(b^2*x
^2+2*a*b*x+a^2+1)^(1/2)+1/5*I/b*x^4*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-3/8/b^4*x*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+3/8/
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)

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maxima [B]  time = 0.36, size = 749, normalized size = 2.71 \[ \frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x^{4}}{5 \, b} - \frac {9 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a x^{3}}{20 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (-i \, a - 1\right )} x^{3}}{4 \, b^{2}} + \frac {21 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2} x^{2}}{20 \, b^{3}} - \frac {7 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a {\left (i \, a + 1\right )} x^{2}}{12 \, b^{3}} - \frac {63 i \, a^{5} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{8 \, b^{5}} + \frac {35 \, a^{4} {\left (i \, a + 1\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{8 \, b^{5}} - \frac {21 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{3} x}{8 \, b^{4}} + \frac {35 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2} {\left (i \, a + 1\right )} x}{24 \, b^{4}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (4 i \, a^{2} + 4 i\right )} x^{2}}{15 \, b^{3}} + \frac {35 i \, {\left (a^{2} + 1\right )} a^{3} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{4 \, b^{5}} - \frac {15 \, {\left (a^{2} + 1\right )} a^{2} {\left (i \, a + 1\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{4 \, b^{5}} + \frac {63 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{4}}{8 \, b^{5}} - \frac {35 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{3} {\left (i \, a + 1\right )}}{8 \, b^{5}} + \frac {161 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )} a x}{120 \, b^{4}} - \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )} {\left (i \, a + 1\right )} x}{8 \, b^{4}} - \frac {15 i \, {\left (a^{2} + 1\right )}^{2} a \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{8 \, b^{5}} - \frac {3 \, {\left (a^{2} + 1\right )}^{2} {\left (-i \, a - 1\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{8 \, b^{5}} - \frac {49 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )} a^{2}}{8 \, b^{5}} + \frac {55 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )} a {\left (i \, a + 1\right )}}{24 \, b^{5}} + \frac {8 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a^{2} + 1\right )}^{2}}{15 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2)*x^4,x, algorithm="maxima")

[Out]

1/5*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*x^4/b - 9/20*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a*x^3/b^2 - 1/4*sqrt(
b^2*x^2 + 2*a*b*x + a^2 + 1)*(-I*a - 1)*x^3/b^2 + 21/20*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^2*x^2/b^3 - 7/12
*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a*(I*a + 1)*x^2/b^3 - 63/8*I*a^5*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 +
4*(a^2 + 1)*b^2))/b^5 + 35/8*a^4*(I*a + 1)*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^5 - 2
1/8*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^3*x/b^4 + 35/24*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^2*(I*a + 1)*x/b^
4 - 1/15*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(4*I*a^2 + 4*I)*x^2/b^3 + 35/4*I*(a^2 + 1)*a^3*arcsinh(2*(b^2*x + a
*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^5 - 15/4*(a^2 + 1)*a^2*(I*a + 1)*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2
*b^2 + 4*(a^2 + 1)*b^2))/b^5 + 63/8*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*a^4/b^5 - 35/8*sqrt(b^2*x^2 + 2*a*b*x
+ a^2 + 1)*a^3*(I*a + 1)/b^5 + 161/120*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)*a*x/b^4 - 3/8*sqrt(b^2*x^
2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)*(I*a + 1)*x/b^4 - 15/8*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 - 3/8*(a^2 + 1)^2*(-I*a - 1)*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b
^2))/b^5 - 49/8*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)*a^2/b^5 + 55/24*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1
)*(a^2 + 1)*a*(I*a + 1)/b^5 + 8/15*I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*(a^2 + 1)^2/b^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*(b*x+a))/(1+(b*x+a)**2)**(1/2)*x**4,x)

[Out]

I*(Integral(-I*x**4/sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1), x) + Integral(a*x**4/sqrt(a**2 + 2*a*b*x + b**2*x**2
+ 1), x) + Integral(b*x**5/sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1), x))

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