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

Optimal. Leaf size=249 \[ -\frac {i \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2} \left (-22 i a^2-2 (11-10 i a) b x+54 a+29 i\right )}{8 b^4}+\frac {3 \left (8 i a^3-36 a^2-44 i a+17\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{8 b^4}-\frac {3 \left (-8 a^3-36 i a^2+44 a+17 i\right ) \sinh ^{-1}(a+b x)}{8 b^4}-\frac {9 x^2 \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{4 b^2}-\frac {2 i x^3 (i a+i b x+1)^{3/2}}{b \sqrt {-i a-i b x+1}} \]

[Out]

-3/8*(17*I+44*a-36*I*a^2-8*a^3)*arcsinh(b*x+a)/b^4-2*I*x^3*(1+I*a+I*b*x)^(3/2)/b/(1-I*a-I*b*x)^(1/2)-9/4*x^2*(
1+I*a+I*b*x)^(3/2)*(1-I*a-I*b*x)^(1/2)/b^2-1/8*I*(1+I*a+I*b*x)^(3/2)*(29*I+54*a-22*I*a^2-2*(11-10*I*a)*b*x)*(1
-I*a-I*b*x)^(1/2)/b^4+3/8*(17-44*I*a-36*a^2+8*I*a^3)*(1-I*a-I*b*x)^(1/2)*(1+I*a+I*b*x)^(1/2)/b^4

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Rubi [A]  time = 0.24, antiderivative size = 249, 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, 97, 153, 147, 50, 53, 619, 215} \[ -\frac {i \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2} \left (-22 i a^2-2 (11-10 i a) b x+54 a+29 i\right )}{8 b^4}+\frac {3 \left (8 i a^3-36 a^2-44 i a+17\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{8 b^4}-\frac {3 \left (-8 a^3-36 i a^2+44 a+17 i\right ) \sinh ^{-1}(a+b x)}{8 b^4}-\frac {9 x^2 \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{4 b^2}-\frac {2 i x^3 (i a+i b x+1)^{3/2}}{b \sqrt {-i a-i b x+1}} \]

Antiderivative was successfully verified.

[In]

Int[E^((3*I)*ArcTan[a + b*x])*x^3,x]

[Out]

(3*(17 - (44*I)*a - 36*a^2 + (8*I)*a^3)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(8*b^4) - ((2*I)*x^3*(1 +
I*a + I*b*x)^(3/2))/(b*Sqrt[1 - I*a - I*b*x]) - (9*x^2*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2))/(4*b^2)
- ((I/8)*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2)*(29*I + 54*a - (22*I)*a^2 - 2*(11 - (10*I)*a)*b*x))/b^
4 - (3*(17*I + 44*a - (36*I)*a^2 - 8*a^3)*ArcSinh[a + b*x])/(8*b^4)

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^3 \, dx &=\int \frac {x^3 (1+i a+i b x)^{3/2}}{(1-i a-i b x)^{3/2}} \, dx\\ &=-\frac {2 i x^3 (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}+\frac {(2 i) \int \frac {x^2 \sqrt {1+i a+i b x} \left (3 (1+i a)+\frac {9 i b x}{2}\right )}{\sqrt {1-i a-i b x}} \, dx}{b}\\ &=-\frac {2 i x^3 (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}-\frac {9 x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}+\frac {i \int \frac {x \sqrt {1+i a+i b x} \left (-9 i \left (1+a^2\right ) b+\frac {3}{2} (11-10 i a) b^2 x\right )}{\sqrt {1-i a-i b x}} \, dx}{2 b^3}\\ &=-\frac {2 i x^3 (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}-\frac {9 x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}-\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (29 i+54 a-22 i a^2-2 (11-10 i a) b x\right )}{8 b^4}-\frac {\left (3 \left (17 i+44 a-36 i a^2-8 a^3\right )\right ) \int \frac {\sqrt {1+i a+i b x}}{\sqrt {1-i a-i b x}} \, dx}{8 b^3}\\ &=\frac {3 \left (17-44 i a-36 a^2+8 i a^3\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^4}-\frac {2 i x^3 (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}-\frac {9 x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}-\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (29 i+54 a-22 i a^2-2 (11-10 i a) b x\right )}{8 b^4}-\frac {\left (3 \left (17 i+44 a-36 i a^2-8 a^3\right )\right ) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{8 b^3}\\ &=\frac {3 \left (17-44 i a-36 a^2+8 i a^3\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^4}-\frac {2 i x^3 (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}-\frac {9 x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}-\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (29 i+54 a-22 i a^2-2 (11-10 i a) b x\right )}{8 b^4}-\frac {\left (3 \left (17 i+44 a-36 i a^2-8 a^3\right )\right ) \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{8 b^3}\\ &=\frac {3 \left (17-44 i a-36 a^2+8 i a^3\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^4}-\frac {2 i x^3 (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}-\frac {9 x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}-\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (29 i+54 a-22 i a^2-2 (11-10 i a) b x\right )}{8 b^4}-\frac {\left (3 \left (17 i+44 a-36 i a^2-8 a^3\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^5}\\ &=\frac {3 \left (17-44 i a-36 a^2+8 i a^3\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^4}-\frac {2 i x^3 (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}-\frac {9 x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}-\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2} \left (29 i+54 a-22 i a^2-2 (11-10 i a) b x\right )}{8 b^4}-\frac {3 \left (17 i+44 a-36 i a^2-8 a^3\right ) \sinh ^{-1}(a+b x)}{8 b^4}\\ \end {align*}

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Mathematica [A]  time = 0.28, size = 201, normalized size = 0.81 \[ \frac {3 \sqrt [4]{-1} \left (8 a^3+36 i a^2-44 a-17 i\right ) \sqrt {-i b} \sinh ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (a+b x+i)}}{\sqrt {-i b}}\right )}{4 b^{9/2}}+\frac {\sqrt {i a+i b x+1} \left (2 a^4+78 i a^3+a^2 (-233+22 i b x)-i a \left (10 b^2 x^2-54 i b x+237\right )-2 b^4 x^4+6 i b^3 x^3+11 b^2 x^2-29 i b x+80\right )}{8 b^4 \sqrt {-i (a+b x+i)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[1 + I*a + I*b*x]*(80 + (78*I)*a^3 + 2*a^4 - (29*I)*b*x + 11*b^2*x^2 + (6*I)*b^3*x^3 - 2*b^4*x^4 + a^2*(-
233 + (22*I)*b*x) - I*a*(237 - (54*I)*b*x + 10*b^2*x^2)))/(8*b^4*Sqrt[(-I)*(I + a + b*x)]) + (3*(-1)^(1/4)*(-1
7*I - 44*a + (36*I)*a^2 + 8*a^3)*Sqrt[(-I)*b]*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-I)*(I + a + b*x)])/Sqrt[(-I)
*b]])/(4*b^(9/2))

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fricas [A]  time = 0.51, size = 217, normalized size = 0.87 \[ \frac {15 i \, a^{5} - 495 \, a^{4} - 1664 i \, a^{3} + {\left (15 i \, a^{4} - 480 \, a^{3} - 1184 i \, a^{2} + 968 \, a + 256 i\right )} b x + 2152 \, a^{2} - {\left (192 \, a^{4} + 1056 i \, a^{3} + {\left (192 \, a^{3} + 864 i \, a^{2} - 1056 \, a - 408 i\right )} b x - 1920 \, a^{2} - 1464 i \, a + 408\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + {\left (-16 i \, b^{4} x^{4} - 48 \, b^{3} x^{3} + {\left (80 \, a + 88 i\right )} b^{2} x^{2} + 16 i \, a^{4} - 624 \, a^{3} - 8 \, {\left (22 \, a^{2} + 54 i \, a - 29\right )} b x - 1864 i \, a^{2} + 1896 \, a + 640 i\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 1224 i \, a - 256}{64 \, b^{5} x + {\left (64 \, a + 64 i\right )} b^{4}} \]

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^3,x, algorithm="fricas")

[Out]

(15*I*a^5 - 495*a^4 - 1664*I*a^3 + (15*I*a^4 - 480*a^3 - 1184*I*a^2 + 968*a + 256*I)*b*x + 2152*a^2 - (192*a^4
+ 1056*I*a^3 + (192*a^3 + 864*I*a^2 - 1056*a - 408*I)*b*x - 1920*a^2 - 1464*I*a + 408)*log(-b*x - a + sqrt(b^
2*x^2 + 2*a*b*x + a^2 + 1)) + (-16*I*b^4*x^4 - 48*b^3*x^3 + (80*a + 88*I)*b^2*x^2 + 16*I*a^4 - 624*a^3 - 8*(22
*a^2 + 54*I*a - 29)*b*x - 1864*I*a^2 + 1896*a + 640*I)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) + 1224*I*a - 256)/(64
*b^5*x + (64*a + 64*I)*b^4)

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giac [A]  time = 0.19, size = 293, normalized size = 1.18 \[ -\frac {1}{8} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left ({\left (2 \, {\left (\frac {i x}{b} - \frac {a b^{11} i - 4 \, b^{11}}{b^{13}}\right )} x + \frac {2 \, a^{2} b^{10} i - 20 \, a b^{10} - 19 \, b^{10} i}{b^{13}}\right )} x - \frac {2 \, a^{3} b^{9} i - 44 \, a^{2} b^{9} - 93 \, a b^{9} i + 48 \, b^{9}}{b^{13}}\right )} - \frac {{\left (8 \, a^{3} + 36 \, a^{2} i - 44 \, a - 17 \, i\right )} \log \left (3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b + a^{3} b + 2 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b i + 2 \, a^{2} b i + {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{3} {\left | b \right |} + 3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a^{2} {\left | b \right |} + 4 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a i {\left | b \right |} - a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{8 \, b^{3} {\left | b \right |}} \]

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^3,x, algorithm="giac")

[Out]

-1/8*sqrt((b*x + a)^2 + 1)*((2*(i*x/b - (a*b^11*i - 4*b^11)/b^13)*x + (2*a^2*b^10*i - 20*a*b^10 - 19*b^10*i)/b
^13)*x - (2*a^3*b^9*i - 44*a^2*b^9 - 93*a*b^9*i + 48*b^9)/b^13) - 1/8*(8*a^3 + 36*a^2*i - 44*a - 17*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) - sqr
t((b*x + a)^2 + 1))*a*i*abs(b) - a*b - (x*abs(b) - sqrt((b*x + a)^2 + 1))*abs(b))/(b^3*abs(b))

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maple [B]  time = 0.19, size = 711, normalized size = 2.86 \[ -\frac {3 x^{2} a^{2}}{2 b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {a \,x^{3}}{2 b \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {3 a^{3} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b^{3} \sqrt {b^{2}}}-\frac {33 a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 b^{3} \sqrt {b^{2}}}-\frac {265 i a^{2} x}{8 b^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {53 i a \,x^{2}}{8 b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {51 i \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{3} \sqrt {b^{2}}}-\frac {155 i a^{3}}{8 b^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {17 i x^{3}}{8 b \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {10}{b^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {27 i a^{2} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 b^{3} \sqrt {b^{2}}}-\frac {i b \,x^{5}}{4 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {i a^{4} x}{4 b^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {i a \,x^{4}}{4 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {i a^{5}}{4 b^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {51 i x}{8 b^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {157 i a}{8 b^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {5 x^{2}}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {x^{4}}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {53 a x}{2 b^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {25 a^{3} x}{2 b^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {a^{2}}{2 b^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {19 a^{4}}{2 b^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}} \]

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^3,x)

[Out]

-3/2*x^2/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a^2+1/2/b*a*x^3/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+3*a^3/b^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)-33/2*a/b^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)-265/8*I/b^3*a^2*x/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-53/8*I/b^2*a*x^2/(b^2*x^2+2*a*b*
x+a^2+1)^(1/2)-51/8*I/b^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)-155/8*I/b^4*a^
3/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+17/8*I/b*x^3/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+10/b^4/(b^2*x^2+2*a*b*x+a^2+1)^(1/2
)+27/2*I/b^3*a^2*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*I*b*x^5/(b^2*x^2+2*
a*b*x+a^2+1)^(1/2)+1/4*I/b^3*a^4*x/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-1/4*I*a*x^4/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+1/4
*I/b^4*a^5/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+51/8*I/b^3*x/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-157/8*I/b^4*a/(b^2*x^2+2*a
*b*x+a^2+1)^(1/2)+5*x^2/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-x^4/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)+53/2*a/b^3*x/(b^2*
x^2+2*a*b*x+a^2+1)^(1/2)-25/2*a^3/b^3/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*x+1/2*a^2/b^4/(b^2*x^2+2*a*b*x+a^2+1)^(1/2
)-19/2*a^4/b^4/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)

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maxima [B]  time = 0.37, size = 2321, normalized size = 9.32 \[ \text {result too large to display} \]

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^3,x, algorithm="maxima")

[Out]

-1/4*I*b*x^5/sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) + 315/4*I*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) + 3/4*I*a*x^4/sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) - 945/8*I*(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) - 21/8*I*a^2*x^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b) + 105/8*I*
(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^2) + 35*(-3*I*a*b^2 - 3*b^2)*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) - 15*(-3*I*a^2*b - 6*a*b + 3*I*b)*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) + 169/4*I*(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) + 6*(-I*a^3 - 3*a^2 + 3*I*a + 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) + 105/8*I*a^3*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - 1/8*(-5*I*a
^2 - 5*I)*x^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b) + 1/3*(-3*I*a*b^2 - 3*b^2)*x^4/(sqrt(b^2*x^2 + 2*a*b*x + a
^2 + 1)*b^2) - 14*I*(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^2) - 265/6*
(-3*I*a*b^2 - 3*b^2)*(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) + 31/2*
(-3*I*a^2*b - 6*a*b + 3*I*b)*(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)
- 15/8*I*(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) - 5*(-I*a^3 - 3*a^2 +
3*I*a + 1)*(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) - 49/8*I*(a^2 + 1)*a*
x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - 7/6*(-3*I*a*b^2 - 3*b^2)*a*x^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1
)*b^3) + 1/2*(-3*I*a^2*b - 6*a*b + 3*I*b)*x^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - 315/8*I*a^4*arcsinh(2*
(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^4 + 35/6*(-3*I*a*b^2 - 3*b^2)*(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) - 5/2*(-3*I*a^2*b - 6*a*b + 3*I*b)*(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) + 15/8*I*(a^2 + 1)^3*a/((a^2*b^2 - (a^2 + 1)*b^2)*sqr
t(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) + (-I*a^3 - 3*a^2 + 3*I*a + 1)*(a^2 + 1)*a^2/((a^2*b^2 - (a^2 + 1)*b^2)*sq
rt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) + 61/6*(-3*I*a*b^2 - 3*b^2)*(a^2 + 1)^2*a*x/((a^2*b^2 - (a^2 + 1)*b^2)*sq
rt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^3) - 3/2*(-3*I*a^2*b - 6*a*b + 3*I*b)*(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) + 35/6*(-3*I*a*b^2 - 3*b^2)*a^2*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 +
1)*b^4) - 5/2*(-3*I*a^2*b - 6*a*b + 3*I*b)*a*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^3) + (-I*a^3 - 3*a^2 + 3
*I*a + 1)*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) + 105/4*I*(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^4 - 29/6*(-3*I*a*b^2 - 3*b^2)*(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) + 105/4*I*(a^2 + 1)*a^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^4) + 3/2*(-3*I*a^
2*b - 6*a*b + 3*I*b)*(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) - 4/3*(-3
*I*a*b^2 - 3*b^2)*(a^2 + 1)*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^4) - 35/2*(-3*I*a*b^2 - 3*b^2)*a^3*arcsin
h(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^6 + 15/2*(-3*I*a^2*b - 6*a*b + 3*I*b)*a^2*arcsinh(2*(b
^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^5 - 15/8*I*(a^2 + 1)^2*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b
^2 + 4*(a^2 + 1)*b^2))/b^4 - 3*(-I*a^3 - 3*a^2 + 3*I*a + 1)*a*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2
+ 1)*b^2))/b^4 - 49/4*I*(a^2 + 1)^2*a/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^4) + 15/2*(-3*I*a*b^2 - 3*b^2)*(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*(-3*I*a^2*b - 6*a*b + 3*I*b)*(a
^2 + 1)*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^5 + 35/3*(-3*I*a*b^2 - 3*b^2)*(a^2 + 1)*
a^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^6) - 5*(-3*I*a^2*b - 6*a*b + 3*I*b)*(a^2 + 1)*a/(sqrt(b^2*x^2 + 2*a*b
*x + a^2 + 1)*b^5) + 2*(-I*a^3 - 3*a^2 + 3*I*a + 1)*(a^2 + 1)/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^4) - 8/3*(-
3*I*a*b^2 - 3*b^2)*(a^2 + 1)^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^6)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((x^3*(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^{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 \left (- \frac {3 a 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 {a^{3} 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 \left (- \frac {3 b 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 {b^{3} 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 \left (- \frac {3 i a^{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 \left (- \frac {3 i b^{2} 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 {3 a b^{2} 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 \frac {3 a^{2} b 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 {6 i a b 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\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**3,x)

[Out]

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

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