\(\int e^{\frac {3}{2} i \arctan (a+b x)} x^2 \, dx\) [235]

Optimal result

Integrand size = 18, antiderivative size = 391 \[ \int e^{\frac {3}{2} i \arctan (a+b x)} x^2 \, dx=-\frac {\left (17 i+36 a-24 i a^2\right ) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{3/4}}{24 b^3}-\frac {(3 i+8 a) \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{12 b^3}+\frac {x \sqrt [4]{1-i a-i b x} (1+i a+i b x)^{7/4}}{3 b^2}+\frac {\left (17 i+36 a-24 i a^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt {2} b^3}-\frac {\left (17 i+36 a-24 i a^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x}}\right )}{8 \sqrt {2} b^3}-\frac {\left (17 i+36 a-24 i a^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{1-i a-i b x}}{\sqrt [4]{1+i a+i b x} \left (1+\frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}}\right )}\right )}{8 \sqrt {2} b^3} \] Output:

-1/24*(17*I+36*a-24*I*a^2)*(1-I*a-I*b*x)^(1/4)*(1+I*a+I*b*x)^(3/4)/b^3-1/1 
2*(3*I+8*a)*(1-I*a-I*b*x)^(1/4)*(1+I*a+I*b*x)^(7/4)/b^3+1/3*x*(1-I*a-I*b*x 
)^(1/4)*(1+I*a+I*b*x)^(7/4)/b^2+1/16*(17*I+36*a-24*I*a^2)*arctan(1-2^(1/2) 
*(1-I*a-I*b*x)^(1/4)/(1+I*a+I*b*x)^(1/4))*2^(1/2)/b^3-1/16*(17*I+36*a-24*I 
*a^2)*arctan(1+2^(1/2)*(1-I*a-I*b*x)^(1/4)/(1+I*a+I*b*x)^(1/4))*2^(1/2)/b^ 
3-1/16*(17*I+36*a-24*I*a^2)*arctanh(2^(1/2)*(1-I*a-I*b*x)^(1/4)/(1+I*a+I*b 
*x)^(1/4)/(1+(1-I*a-I*b*x)^(1/2)/(1+I*a+I*b*x)^(1/2)))*2^(1/2)/b^3
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.12 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.31 \[ \int e^{\frac {3}{2} i \arctan (a+b x)} x^2 \, dx=\frac {\sqrt [4]{-i (i+a+b x)} \left (-i (1+i a+i b x)^{3/4} \left (3+8 a^2+7 i b x-4 b^2 x^2+a (-5 i+4 b x)\right )+2 i 2^{3/4} \left (-17+36 i a+24 a^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{4},\frac {5}{4},-\frac {1}{2} i (i+a+b x)\right )\right )}{12 b^3} \] Input:

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

Output:

(((-I)*(I + a + b*x))^(1/4)*((-I)*(1 + I*a + I*b*x)^(3/4)*(3 + 8*a^2 + (7* 
I)*b*x - 4*b^2*x^2 + a*(-5*I + 4*b*x)) + (2*I)*2^(3/4)*(-17 + (36*I)*a + 2 
4*a^2)*Hypergeometric2F1[-3/4, 1/4, 5/4, (-1/2*I)*(I + a + b*x)]))/(12*b^3 
)
 

Rubi [A] (warning: unable to verify)

Time = 0.89 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.06, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5618, 101, 27, 90, 60, 73, 770, 755, 1476, 1082, 217, 1479, 25, 27, 1103}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^2 e^{\frac {3}{2} i \arctan (a+b x)} \, dx\)

\(\Big \downarrow \) 5618

\(\displaystyle \int \frac {x^2 (i a+i b x+1)^{3/4}}{(-i a-i b x+1)^{3/4}}dx\)

\(\Big \downarrow \) 101

\(\displaystyle \frac {\int -\frac {(i a+i b x+1)^{3/4} \left (2 a^2+(8 a+3 i) b x+2\right )}{2 (-i a-i b x+1)^{3/4}}dx}{3 b^2}+\frac {x \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{3 b^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{3 b^2}-\frac {\int \frac {(i a+i b x+1)^{3/4} \left (2 \left (a^2+1\right )+(8 a+3 i) b x\right )}{(-i a-i b x+1)^{3/4}}dx}{6 b^2}\)

\(\Big \downarrow \) 90

\(\displaystyle \frac {x \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{3 b^2}-\frac {\frac {1}{4} \left (-24 a^2-36 i a+17\right ) \int \frac {(i a+i b x+1)^{3/4}}{(-i a-i b x+1)^{3/4}}dx+\frac {(8 a+3 i) \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{2 b}}{6 b^2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {x \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{3 b^2}-\frac {\frac {1}{4} \left (-24 a^2-36 i a+17\right ) \left (\frac {3}{2} \int \frac {1}{(-i a-i b x+1)^{3/4} \sqrt [4]{i a+i b x+1}}dx+\frac {i \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}{b}\right )+\frac {(8 a+3 i) \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{2 b}}{6 b^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {x \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{3 b^2}-\frac {\frac {1}{4} \left (-24 a^2-36 i a+17\right ) \left (\frac {6 i \int \frac {1}{\sqrt [4]{i a+i b x+1}}d\sqrt [4]{-i a-i b x+1}}{b}+\frac {i \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}{b}\right )+\frac {(8 a+3 i) \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{2 b}}{6 b^2}\)

\(\Big \downarrow \) 770

\(\displaystyle \frac {x \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{3 b^2}-\frac {\frac {1}{4} \left (-24 a^2-36 i a+17\right ) \left (\frac {6 i \int \frac {1}{-i a-i b x+2}d\frac {\sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}}{b}+\frac {i \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}{b}\right )+\frac {(8 a+3 i) \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{2 b}}{6 b^2}\)

\(\Big \downarrow \) 755

\(\displaystyle \frac {x \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{3 b^2}-\frac {\frac {1}{4} \left (-24 a^2-36 i a+17\right ) \left (\frac {6 i \left (\frac {1}{2} \int \frac {1-\sqrt {-i a-i b x+1}}{-i a-i b x+2}d\frac {\sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+\frac {1}{2} \int \frac {\sqrt {-i a-i b x+1}+1}{-i a-i b x+2}d\frac {\sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{b}+\frac {i \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}{b}\right )+\frac {(8 a+3 i) \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{2 b}}{6 b^2}\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {x \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{3 b^2}-\frac {\frac {1}{4} \left (-24 a^2-36 i a+17\right ) \left (\frac {6 i \left (\frac {1}{2} \int \frac {1-\sqrt {-i a-i b x+1}}{-i a-i b x+2}d\frac {\sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {-i a-i b x+1}-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1}d\frac {\sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+\frac {1}{2} \int \frac {1}{\sqrt {-i a-i b x+1}+\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1}d\frac {\sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )\right )}{b}+\frac {i \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}{b}\right )+\frac {(8 a+3 i) \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{2 b}}{6 b^2}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {x \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{3 b^2}-\frac {\frac {1}{4} \left (-24 a^2-36 i a+17\right ) \left (\frac {6 i \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\sqrt {-i a-i b x+1}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\sqrt {-i a-i b x+1}-1}d\left (\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{\sqrt {2}}\right )+\frac {1}{2} \int \frac {1-\sqrt {-i a-i b x+1}}{-i a-i b x+2}d\frac {\sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{b}+\frac {i \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}{b}\right )+\frac {(8 a+3 i) \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{2 b}}{6 b^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {x \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{3 b^2}-\frac {\frac {1}{4} \left (-24 a^2-36 i a+17\right ) \left (\frac {6 i \left (\frac {1}{2} \int \frac {1-\sqrt {-i a-i b x+1}}{-i a-i b x+2}d\frac {\sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{\sqrt {2}}\right )\right )}{b}+\frac {i \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}{b}\right )+\frac {(8 a+3 i) \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{2 b}}{6 b^2}\)

\(\Big \downarrow \) 1479

\(\displaystyle \frac {x \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{3 b^2}-\frac {\frac {1}{4} \left (-24 a^2-36 i a+17\right ) \left (\frac {6 i \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-\frac {2 \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}}{\sqrt {-i a-i b x+1}-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1}d\frac {\sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{\sqrt {-i a-i b x+1}+\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1}d\frac {\sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{\sqrt {2}}\right )\right )}{b}+\frac {i \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}{b}\right )+\frac {(8 a+3 i) \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{2 b}}{6 b^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{3 b^2}-\frac {\frac {1}{4} \left (-24 a^2-36 i a+17\right ) \left (\frac {6 i \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}}{\sqrt {-i a-i b x+1}-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1}d\frac {\sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{\sqrt {-i a-i b x+1}+\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1}d\frac {\sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{\sqrt {2}}\right )\right )}{b}+\frac {i \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}{b}\right )+\frac {(8 a+3 i) \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{2 b}}{6 b^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{3 b^2}-\frac {\frac {1}{4} \left (-24 a^2-36 i a+17\right ) \left (\frac {6 i \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-\frac {2 \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}}{\sqrt {-i a-i b x+1}-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1}d\frac {\sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1}{\sqrt {-i a-i b x+1}+\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1}d\frac {\sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )+\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{\sqrt {2}}\right )\right )}{b}+\frac {i \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}{b}\right )+\frac {(8 a+3 i) \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{2 b}}{6 b^2}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {x \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{3 b^2}-\frac {\frac {1}{4} \left (-24 a^2-36 i a+17\right ) \left (\frac {6 i \left (\frac {1}{2} \left (\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{\sqrt {2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\sqrt {-i a-i b x+1}+\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\sqrt {-i a-i b x+1}-\frac {\sqrt {2} \sqrt [4]{-i a-i b x+1}}{\sqrt [4]{i a+i b x+1}}+1\right )}{2 \sqrt {2}}\right )\right )}{b}+\frac {i \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{3/4}}{b}\right )+\frac {(8 a+3 i) \sqrt [4]{-i a-i b x+1} (i a+i b x+1)^{7/4}}{2 b}}{6 b^2}\)

Input:

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

Output:

(x*(1 - I*a - I*b*x)^(1/4)*(1 + I*a + I*b*x)^(7/4))/(3*b^2) - (((3*I + 8*a 
)*(1 - I*a - I*b*x)^(1/4)*(1 + I*a + I*b*x)^(7/4))/(2*b) + ((17 - (36*I)*a 
 - 24*a^2)*((I*(1 - I*a - I*b*x)^(1/4)*(1 + I*a + I*b*x)^(3/4))/b + ((6*I) 
*((-(ArcTan[1 - (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)] 
/Sqrt[2]) + ArcTan[1 + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x) 
^(1/4)]/Sqrt[2])/2 + (-1/2*Log[1 + Sqrt[1 - I*a - I*b*x] - (Sqrt[2]*(1 - I 
*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)]/Sqrt[2] + Log[1 + Sqrt[1 - I*a 
 - I*b*x] + (Sqrt[2]*(1 - I*a - I*b*x)^(1/4))/(1 + I*a + I*b*x)^(1/4)]/(2* 
Sqrt[2]))/2))/b))/4)/(6*b^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 60

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] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 90

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

rule 101

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( 
p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + 
 p + 3))), x] + Simp[1/(d*f*(n + p + 3))   Int[(c + d*x)^n*(e + f*x)^p*Simp 
[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f 
*(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, 
 c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 770

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n)   Subst[In 
t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, 
 b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 
/n]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 5618

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]
 

Maple [F]

\[\int {\left (\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}\right )}^{\frac {3}{2}} x^{2}d x\]

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 561 vs. \(2 (265) = 530\).

Time = 0.14 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.43 \[ \int e^{\frac {3}{2} i \arctan (a+b x)} x^2 \, dx=\frac {3 \, b^{3} \sqrt {\frac {576 i \, a^{4} - 1728 \, a^{3} - 2112 i \, a^{2} + 1224 \, a + 289 i}{b^{6}}} \log \left (\frac {b^{3} \sqrt {\frac {576 i \, a^{4} - 1728 \, a^{3} - 2112 i \, a^{2} + 1224 \, a + 289 i}{b^{6}}} + {\left (24 \, a^{2} + 36 i \, a - 17\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{24 \, a^{2} + 36 i \, a - 17}\right ) - 3 \, b^{3} \sqrt {\frac {576 i \, a^{4} - 1728 \, a^{3} - 2112 i \, a^{2} + 1224 \, a + 289 i}{b^{6}}} \log \left (-\frac {b^{3} \sqrt {\frac {576 i \, a^{4} - 1728 \, a^{3} - 2112 i \, a^{2} + 1224 \, a + 289 i}{b^{6}}} - {\left (24 \, a^{2} + 36 i \, a - 17\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{24 \, a^{2} + 36 i \, a - 17}\right ) - 3 \, b^{3} \sqrt {\frac {-576 i \, a^{4} + 1728 \, a^{3} + 2112 i \, a^{2} - 1224 \, a - 289 i}{b^{6}}} \log \left (\frac {b^{3} \sqrt {\frac {-576 i \, a^{4} + 1728 \, a^{3} + 2112 i \, a^{2} - 1224 \, a - 289 i}{b^{6}}} + {\left (24 \, a^{2} + 36 i \, a - 17\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{24 \, a^{2} + 36 i \, a - 17}\right ) + 3 \, b^{3} \sqrt {\frac {-576 i \, a^{4} + 1728 \, a^{3} + 2112 i \, a^{2} - 1224 \, a - 289 i}{b^{6}}} \log \left (-\frac {b^{3} \sqrt {\frac {-576 i \, a^{4} + 1728 \, a^{3} + 2112 i \, a^{2} - 1224 \, a - 289 i}{b^{6}}} - {\left (24 \, a^{2} + 36 i \, a - 17\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{24 \, a^{2} + 36 i \, a - 17}\right ) + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (8 i \, b^{2} x^{2} - 2 \, {\left (4 i \, a - 7\right )} b x + 8 i \, a^{2} - 46 \, a - 23 i\right )} \sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{48 \, b^{3}} \] Input:

integrate(((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(3/2)*x^2,x, algorithm="fric 
as")
 

Output:

1/48*(3*b^3*sqrt((576*I*a^4 - 1728*a^3 - 2112*I*a^2 + 1224*a + 289*I)/b^6) 
*log((b^3*sqrt((576*I*a^4 - 1728*a^3 - 2112*I*a^2 + 1224*a + 289*I)/b^6) + 
 (24*a^2 + 36*I*a - 17)*sqrt(I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/(b*x + a 
+ I)))/(24*a^2 + 36*I*a - 17)) - 3*b^3*sqrt((576*I*a^4 - 1728*a^3 - 2112*I 
*a^2 + 1224*a + 289*I)/b^6)*log(-(b^3*sqrt((576*I*a^4 - 1728*a^3 - 2112*I* 
a^2 + 1224*a + 289*I)/b^6) - (24*a^2 + 36*I*a - 17)*sqrt(I*sqrt(b^2*x^2 + 
2*a*b*x + a^2 + 1)/(b*x + a + I)))/(24*a^2 + 36*I*a - 17)) - 3*b^3*sqrt((- 
576*I*a^4 + 1728*a^3 + 2112*I*a^2 - 1224*a - 289*I)/b^6)*log((b^3*sqrt((-5 
76*I*a^4 + 1728*a^3 + 2112*I*a^2 - 1224*a - 289*I)/b^6) + (24*a^2 + 36*I*a 
 - 17)*sqrt(I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/(b*x + a + I)))/(24*a^2 + 
36*I*a - 17)) + 3*b^3*sqrt((-576*I*a^4 + 1728*a^3 + 2112*I*a^2 - 1224*a - 
289*I)/b^6)*log(-(b^3*sqrt((-576*I*a^4 + 1728*a^3 + 2112*I*a^2 - 1224*a - 
289*I)/b^6) - (24*a^2 + 36*I*a - 17)*sqrt(I*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 
 1)/(b*x + a + I)))/(24*a^2 + 36*I*a - 17)) + 2*sqrt(b^2*x^2 + 2*a*b*x + a 
^2 + 1)*(8*I*b^2*x^2 - 2*(4*I*a - 7)*b*x + 8*I*a^2 - 46*a - 23*I)*sqrt(I*s 
qrt(b^2*x^2 + 2*a*b*x + a^2 + 1)/(b*x + a + I)))/b^3
 

Sympy [F(-1)]

Timed out. \[ \int e^{\frac {3}{2} i \arctan (a+b x)} x^2 \, dx=\text {Timed out} \] Input:

integrate(((1+I*(b*x+a))/(1+(b*x+a)**2)**(1/2))**(3/2)*x**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int e^{\frac {3}{2} i \arctan (a+b x)} x^2 \, dx=\int { x^{2} \left (\frac {i \, b x + i \, a + 1}{\sqrt {{\left (b x + a\right )}^{2} + 1}}\right )^{\frac {3}{2}} \,d x } \] Input:

integrate(((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(3/2)*x^2,x, algorithm="maxi 
ma")
 

Output:

integrate(x^2*((I*b*x + I*a + 1)/sqrt((b*x + a)^2 + 1))^(3/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int e^{\frac {3}{2} i \arctan (a+b x)} x^2 \, dx=\text {Exception raised: TypeError} \] Input:

integrate(((1+I*(b*x+a))/(1+(b*x+a)^2)^(1/2))^(3/2)*x^2,x, algorithm="giac 
")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Warning, need to choose a branch fo 
r the root of a polynomial with parameters. This might be wrong.The choice 
 was done
 

Mupad [F(-1)]

Timed out. \[ \int e^{\frac {3}{2} i \arctan (a+b x)} x^2 \, dx=\int x^2\,{\left (\frac {1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}{\sqrt {{\left (a+b\,x\right )}^2+1}}\right )}^{3/2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int e^{\frac {3}{2} i \arctan (a+b x)} x^2 \, dx=\int {\left (\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}\right )}^{\frac {3}{2}} x^{2}d x \] Input:

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

Output:

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