3.2.82 \(\int e^{3 i \text {ArcTan}(a+b x)} x^2 \, dx\) [182]
Optimal. Leaf size=227 \[ \frac {\left (11 i+18 a-6 i a^2\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}+\frac {\left (11 i+18 a-6 i a^2\right ) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{6 b^3}-\frac {i (i+a)^2 (1+i a+i b x)^{5/2}}{b^3 \sqrt {1-i a-i b x}}+\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{5/2}}{3 b^3}+\frac {\left (11-18 i a-6 a^2\right ) \sinh ^{-1}(a+b x)}{2 b^3} \]
[Out]
1/2*(11-18*I*a-6*a^2)*arcsinh(b*x+a)/b^3-I*(I+a)^2*(1+I*a+I*b*x)^(5/2)/b^3/(1-I*a-I*b*x)^(1/2)+1/6*(11*I+18*a-
6*I*a^2)*(1+I*a+I*b*x)^(3/2)*(1-I*a-I*b*x)^(1/2)/b^3+1/3*I*(1+I*a+I*b*x)^(5/2)*(1-I*a-I*b*x)^(1/2)/b^3+1/2*(11
*I+18*a-6*I*a^2)*(1-I*a-I*b*x)^(1/2)*(1+I*a+I*b*x)^(1/2)/b^3
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Rubi [A]
time = 0.12, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5203, 91, 81,
52, 55, 633, 221} \begin {gather*} \frac {\left (-6 i a^2+18 a+11 i\right ) \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{6 b^3}+\frac {\left (-6 i a^2+18 a+11 i\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{2 b^3}+\frac {\left (-6 a^2-18 i a+11\right ) \sinh ^{-1}(a+b x)}{2 b^3}+\frac {i \sqrt {-i a-i b x+1} (i a+i b x+1)^{5/2}}{3 b^3}-\frac {i (a+i)^2 (i a+i b x+1)^{5/2}}{b^3 \sqrt {-i a-i b x+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[E^((3*I)*ArcTan[a + b*x])*x^2,x]
[Out]
((11*I + 18*a - (6*I)*a^2)*Sqrt[1 - I*a - I*b*x]*Sqrt[1 + I*a + I*b*x])/(2*b^3) + ((11*I + 18*a - (6*I)*a^2)*S
qrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(3/2))/(6*b^3) - (I*(I + a)^2*(1 + I*a + I*b*x)^(5/2))/(b^3*Sqrt[1 - I*
a - I*b*x]) + ((I/3)*Sqrt[1 - I*a - I*b*x]*(1 + I*a + I*b*x)^(5/2))/b^3 + ((11 - (18*I)*a - 6*a^2)*ArcSinh[a +
b*x])/(2*b^3)
Rule 52
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 55
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 81
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(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 91
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c - a*d
)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d*e - c*f)*(n + 1))), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
Rule 221
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 633
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 5203
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^2 \, dx &=\int \frac {x^2 (1+i a+i b x)^{3/2}}{(1-i a-i b x)^{3/2}} \, dx\\ &=-\frac {i (i+a)^2 (1+i a+i b x)^{5/2}}{b^3 \sqrt {1-i a-i b x}}-\frac {i \int \frac {(1+i a+i b x)^{3/2} \left ((3-2 i a) (i+a) b-b^2 x\right )}{\sqrt {1-i a-i b x}} \, dx}{b^3}\\ &=-\frac {i (i+a)^2 (1+i a+i b x)^{5/2}}{b^3 \sqrt {1-i a-i b x}}+\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{5/2}}{3 b^3}+\frac {\left (11-18 i a-6 a^2\right ) \int \frac {(1+i a+i b x)^{3/2}}{\sqrt {1-i a-i b x}} \, dx}{3 b^2}\\ &=\frac {\left (11 i+18 a-6 i a^2\right ) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{6 b^3}-\frac {i (i+a)^2 (1+i a+i b x)^{5/2}}{b^3 \sqrt {1-i a-i b x}}+\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{5/2}}{3 b^3}+\frac {\left (11-18 i a-6 a^2\right ) \int \frac {\sqrt {1+i a+i b x}}{\sqrt {1-i a-i b x}} \, dx}{2 b^2}\\ &=\frac {\left (11 i+18 a-6 i a^2\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}+\frac {\left (11 i+18 a-6 i a^2\right ) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{6 b^3}-\frac {i (i+a)^2 (1+i a+i b x)^{5/2}}{b^3 \sqrt {1-i a-i b x}}+\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{5/2}}{3 b^3}+\frac {\left (11-18 i a-6 a^2\right ) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{2 b^2}\\ &=\frac {\left (11 i+18 a-6 i a^2\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}+\frac {\left (11 i+18 a-6 i a^2\right ) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{6 b^3}-\frac {i (i+a)^2 (1+i a+i b x)^{5/2}}{b^3 \sqrt {1-i a-i b x}}+\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{5/2}}{3 b^3}+\frac {\left (11-18 i a-6 a^2\right ) \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{2 b^2}\\ &=\frac {\left (11 i+18 a-6 i a^2\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}+\frac {\left (11 i+18 a-6 i a^2\right ) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{6 b^3}-\frac {i (i+a)^2 (1+i a+i b x)^{5/2}}{b^3 \sqrt {1-i a-i b x}}+\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{5/2}}{3 b^3}+\frac {\left (11-18 i a-6 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{4 b^2}}} \, dx,x,2 a b+2 b^2 x\right )}{4 b^4}\\ &=\frac {\left (11 i+18 a-6 i a^2\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}+\frac {\left (11 i+18 a-6 i a^2\right ) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{6 b^3}-\frac {i (i+a)^2 (1+i a+i b x)^{5/2}}{b^3 \sqrt {1-i a-i b x}}+\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{5/2}}{3 b^3}+\frac {\left (11-18 i a-6 a^2\right ) \sinh ^{-1}(a+b x)}{2 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 160, normalized size = 0.70 \begin {gather*} \frac {\sqrt {1+i a+i b x} \left (52 i-53 i a^2-2 a^3+19 b x+7 i b^2 x^2-2 b^3 x^3+a (103-16 i b x)\right )}{6 b^3 \sqrt {-i (i+a+b x)}}+\frac {(-1)^{3/4} \left (-11+18 i a+6 a^2\right ) \sinh ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (i+a+b x)}}{\sqrt {-i b}}\right )}{\sqrt {-i b} b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[E^((3*I)*ArcTan[a + b*x])*x^2,x]
[Out]
(Sqrt[1 + I*a + I*b*x]*(52*I - (53*I)*a^2 - 2*a^3 + 19*b*x + (7*I)*b^2*x^2 - 2*b^3*x^3 + a*(103 - (16*I)*b*x))
)/(6*b^3*Sqrt[(-I)*(I + a + b*x)]) + ((-1)^(3/4)*(-11 + (18*I)*a + 6*a^2)*ArcSinh[((1/2 + I/2)*Sqrt[b]*Sqrt[(-
I)*(I + a + b*x)])/Sqrt[(-I)*b]])/(Sqrt[(-I)*b]*b^(5/2))
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1784 vs. \(2 (179 ) = 358\).
time = 0.16, size = 1785, normalized size = 7.86
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| method |
result |
size |
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| risch |
\(-\frac {i \left (2 b^{2} x^{2}-2 a b x -9 i b x +2 a^{2}+27 i a -28\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{6 b^{3}}+\frac {11 \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{2 b^{2} \sqrt {b^{2}}}-\frac {9 i \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right ) a}{b^{2} \sqrt {b^{2}}}-\frac {3 \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right ) a^{2}}{b^{2} \sqrt {b^{2}}}+\frac {4 \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}\, a^{2}}{b^{4} \left (x +\frac {i}{b}+\frac {a}{b}\right )}-\frac {4 \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}}{b^{4} \left (x +\frac {i}{b}+\frac {a}{b}\right )}+\frac {8 i \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}\, a}{b^{4} \left (x +\frac {i}{b}+\frac {a}{b}\right )}\) |
\(363\) |
| default |
\(\text {Expression too large to display}\) |
\(1785\) |
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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^2,x,method=_RETURNVERBOSE)
[Out]
-I*b^3*(1/3*x^4/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-7/3*a/b*(1/2*x^3/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-5/2*a/b*(
x^2/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-3*a/b*(-x/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-a/b*(-1/b^2/(b^2*x^2+2*a*b*x
+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2))+1/b^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))-2*(a^2+1)/b^2*(-1/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2
)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)))-3/2*(a^2+1)/b^2*(-x/b^2/(b^2
*x^2+2*a*b*x+a^2+1)^(1/2)-a/b*(-1/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2
*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2))+1/b^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)))-4/3*(a^2+1)/b^2*(x^2/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-3*a/b*(-x/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-a/b*(-
1/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1
/2))+1/b^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))-2*(a^2+1)/b^2*(-1/b^2/(b^2*x
^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2))))-3*(1+
I*a)*b^2*(1/2*x^3/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-5/2*a/b*(x^2/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-3*a/b*(-x/b
^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-a/b*(-1/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1
)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2))+1/b^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))-2*(a^2+1)/b^2*(-1/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)
/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)))-3/2*(a^2+1)/b^2*(-x/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-a/b*(-1/b^2/(b^2*x^2+2*
a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2))+1/b^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)))+3*I*(1+I*a)^2*b*(x^2/b^2/(b^2*x^2+2*a*b*x+a^
2+1)^(1/2)-3*a/b*(-x/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-a/b*(-1/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*
x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2))+1/b^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))-2*(a^2+1)/b^2*(-1/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*
b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)))+(1+I*a)^3*(-x/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-a/b*(-1
/b^2/(b^2*x^2+2*a*b*x+a^2+1)^(1/2)-2*a/b*(2*b^2*x+2*a*b)/(4*b^2*(a^2+1)-4*a^2*b^2)/(b^2*x^2+2*a*b*x+a^2+1)^(1/
2))+1/b^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))
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 1608 vs. \(2 (155) = 310\).
time = 0.29, size = 1608, normalized size = 7.08 \begin {gather*} \text {Too large to display} \end {gather*}
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^2,x, algorithm="maxima")
[Out]
-35*I*a^5*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 1/3*I*b*x^4/sqrt(b^2*x^2 + 2*a*b*x
+ a^2 + 1) + 265/6*I*(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)) + 7/6*I*a*
x^3/sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) - 35/6*I*(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) - 61/6*I*(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)) - 2*(-I*
a^3 - 3*a^2 + 3*I*a + 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)) + 45*(I*a*b^2 + b
^2)*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) - 18*(I*a^2*b + 2*a*b - I*b)*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) - 35/6*I*a^2*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^
2 + 1)*b) + 29/6*I*(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) + (-I*a^3 -
3*a^2 + 3*I*a + 1)*(a^2 + 1)*x/((a^2*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 93/2*(I*a*b^2
+ b^2)*(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*(I*a^2*b + 2*a*b
- I*b)*(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) - 4/3*(-I*a^2 - I)*x^2/(
sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b) - 3/2*(I*a*b^2 + b^2)*x^3/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) + 35/2*
I*a^3*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^3 + 15/2*(I*a*b^2 + b^2)*(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) - 3*(I*a^2*b + 2*a*b - I*b)*(a^2 + 1)*a^2/((a^2
*b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - (-I*a^3 - 3*a^2 + 3*I*a + 1)*(a^2 + 1)*a/((a^2*
b^2 - (a^2 + 1)*b^2)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b) + 9/2*(I*a*b^2 + b^2)*(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) + 15/2*(I*a*b^2 + b^2)*a*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 +
1)*b^3) - 3*(I*a^2*b + 2*a*b - I*b)*x^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^2) - 15/2*I*(a^2 + 1)*a*arcsinh(
2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^3 - 9/2*(I*a*b^2 + b^2)*(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) - 35/3*I*(a^2 + 1)*a^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^3)
- 45/2*(I*a*b^2 + b^2)*a^2*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^5 + 9*(I*a^2*b + 2*a
*b - I*b)*a*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^4 + (-I*a^3 - 3*a^2 + 3*I*a + 1)*arc
sinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^3 + 8/3*I*(a^2 + 1)^2/(sqrt(b^2*x^2 + 2*a*b*x + a^2
+ 1)*b^3) + 9/2*(I*a*b^2 + b^2)*(a^2 + 1)*arcsinh(2*(b^2*x + a*b)/sqrt(-4*a^2*b^2 + 4*(a^2 + 1)*b^2))/b^5 + 1
5*(I*a*b^2 + b^2)*(a^2 + 1)*a/(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^5) - 6*(I*a^2*b + 2*a*b - I*b)*(a^2 + 1)/(s
qrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*b^4)
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Fricas [A]
time = 2.11, size = 174, normalized size = 0.77 \begin {gather*} \frac {-7 i \, a^{4} + 166 \, a^{3} + {\left (-7 i \, a^{3} + 159 \, a^{2} + 249 i \, a - 96\right )} b x + 408 i \, a^{2} + 12 \, {\left (6 \, a^{3} + {\left (6 \, a^{2} + 18 i \, a - 11\right )} b x + 24 i \, a^{2} - 29 \, a - 11 i\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 4 \, {\left (2 i \, b^{3} x^{3} + 7 \, b^{2} x^{2} + 2 i \, a^{3} - {\left (16 \, a + 19 i\right )} b x - 53 \, a^{2} - 103 i \, a + 52\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 345 \, a - 96 i}{24 \, {\left (b^{4} x + {\left (a + i\right )} b^{3}\right )}} \end {gather*}
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^2,x, algorithm="fricas")
[Out]
1/24*(-7*I*a^4 + 166*a^3 + (-7*I*a^3 + 159*a^2 + 249*I*a - 96)*b*x + 408*I*a^2 + 12*(6*a^3 + (6*a^2 + 18*I*a -
11)*b*x + 24*I*a^2 - 29*a - 11*I)*log(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)) - 4*(2*I*b^3*x^3 + 7*b^2*
x^2 + 2*I*a^3 - (16*a + 19*I)*b*x - 53*a^2 - 103*I*a + 52)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) - 345*a - 96*I)/(
b^4*x + (a + I)*b^3)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - i \left (\int \frac {i x^{2}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 a x^{2}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {a^{3} x^{2}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 b 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 {b^{3} 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 {3 i a^{2} x^{2}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \left (- \frac {3 i b^{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 \frac {3 a b^{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}}\, dx + \int \frac {3 a^{2} b 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 {6 i a b 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\right ) \end {gather*}
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**2,x)
[Out]
-I*(Integral(I*x**2/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)
+ b**2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*a*x
**2/(a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**2*sqrt
(a**2 + 2*a*b*x + b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(a**3*x**2/(a**2*sqrt(a
**2 + 2*a*b*x + b**2*x**2 + 1) + 2*a*b*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1) + b**2*x**2*sqrt(a**2 + 2*a*b*x
+ b**2*x**2 + 1) + sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x) + Integral(-3*b*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(b**3*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*I*a**2*x**2/(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**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*b**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)), x) + Int
egral(3*a**2*b*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(-6*I*a
*b*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))
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Giac [A]
time = 0.47, size = 243, normalized size = 1.07 \begin {gather*} -\frac {1}{6} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left (x {\left (\frac {2 i \, x}{b} - \frac {2 i \, a b^{6} - 9 \, b^{6}}{b^{8}}\right )} - \frac {-2 i \, a^{2} b^{5} + 27 \, a b^{5} + 28 i \, b^{5}}{b^{8}}\right )} + \frac {{\left (6 \, a^{2} + 18 i \, a - 11\right )} \log \left (3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b + a^{3} b + {\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 |} + 2 i \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b + 2 i \, a^{2} b + 4 \, {\left (i \, x {\left | b \right |} - i \, \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a {\left | b \right |} - a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{6 \, b^{2} {\left | b \right |}} \end {gather*}
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^2,x, algorithm="giac")
[Out]
-1/6*sqrt((b*x + a)^2 + 1)*(x*(2*I*x/b - (2*I*a*b^6 - 9*b^6)/b^8) - (-2*I*a^2*b^5 + 27*a*b^5 + 28*I*b^5)/b^8)
+ 1/6*(6*a^2 + 18*I*a - 11)*log(3*(x*abs(b) - sqrt((b*x + a)^2 + 1))^2*a*b + a^3*b + (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) + 2*I*(x*abs(b) - sqrt((b*x + a)^2 + 1))^
2*b + 2*I*a^2*b + 4*(I*x*abs(b) - I*sqrt((b*x + a)^2 + 1))*a*abs(b) - a*b - (x*abs(b) - sqrt((b*x + a)^2 + 1))
*abs(b))/(b^2*abs(b))
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2\,{\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^2*(a*1i + b*x*1i + 1)^3)/((a + b*x)^2 + 1)^(3/2),x)
[Out]
int((x^2*(a*1i + b*x*1i + 1)^3)/((a + b*x)^2 + 1)^(3/2), x)
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