\(\int \frac {e^{5 i \arctan (a x)} x^2}{(c+a^2 c x^2)^{27/2}} \, dx\) [380]

Optimal result

Integrand size = 28, antiderivative size = 65 \[ \int \frac {e^{5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=-\frac {(i+5 a x) \sqrt {1+a^2 x^2}}{120 a^3 c^{13} (1-i a x)^{15} (1+i a x)^{10} \sqrt {c+a^2 c x^2}} \]

[Out]

-1/120*(I+5*a*x)*(a^2*x^2+1)^(1/2)/a^3/c^13/(1-I*a*x)^15/(1+I*a*x)^10/(a^2*c*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {5193, 5190, 82} \[ \int \frac {e^{5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=-\frac {(5 a x+i) \sqrt {a^2 x^2+1}}{120 a^3 c^{13} (1-i a x)^{15} (1+i a x)^{10} \sqrt {a^2 c x^2+c}} \]

[In]

Int[(E^((5*I)*ArcTan[a*x])*x^2)/(c + a^2*c*x^2)^(27/2),x]

[Out]

-1/120*((I + 5*a*x)*Sqrt[1 + a^2*x^2])/(a^3*c^13*(1 - I*a*x)^15*(1 + I*a*x)^10*Sqrt[c + a^2*c*x^2])

Rule 82

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

Rule 5190

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 - I
*a*x)^(p + I*(n/2))*(1 + I*a*x)^(p - I*(n/2)), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[d, a^2*c] && (Int
egerQ[p] || GtQ[c, 0])

Rule 5193

Int[E^(ArcTan[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Dist[c^IntPart[p]*((c + d
*x^2)^FracPart[p]/(1 + a^2*x^2)^FracPart[p]), Int[x^m*(1 + a^2*x^2)^p*E^(n*ArcTan[a*x]), x], x] /; FreeQ[{a, c
, d, m, n, p}, x] && EqQ[d, a^2*c] &&  !(IntegerQ[p] || GtQ[c, 0])

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+a^2 x^2} \int \frac {e^{5 i \arctan (a x)} x^2}{\left (1+a^2 x^2\right )^{27/2}} \, dx}{c^{13} \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {1+a^2 x^2} \int \frac {x^2}{(1-i a x)^{16} (1+i a x)^{11}} \, dx}{c^{13} \sqrt {c+a^2 c x^2}} \\ & = -\frac {(i+5 a x) \sqrt {1+a^2 x^2}}{120 a^3 c^{13} (1-i a x)^{15} (1+i a x)^{10} \sqrt {c+a^2 c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.97 \[ \int \frac {e^{5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\frac {(1-5 i a x) \sqrt {1+a^2 x^2}}{120 a^3 c^{13} (-i+a x)^{10} (i+a x)^{15} \sqrt {c+a^2 c x^2}} \]

[In]

Integrate[(E^((5*I)*ArcTan[a*x])*x^2)/(c + a^2*c*x^2)^(27/2),x]

[Out]

((1 - (5*I)*a*x)*Sqrt[1 + a^2*x^2])/(120*a^3*c^13*(-I + a*x)^10*(I + a*x)^15*Sqrt[c + a^2*c*x^2])

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.88

method	result	size
default	\(-\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (5 i a x -1\right )}{120 \sqrt {a^{2} x^{2}+1}\, c^{14} a^{3} \left (a x +i\right )^{15} \left (-a x +i\right )^{10}}\)	\(57\)
gosper	\(\frac {\left (-a x +i\right ) \left (a x +i\right ) \left (5 a x +i\right ) \left (i a x +1\right )^{5}}{120 a^{3} \left (a^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a^{2} c \,x^{2}+c \right )^{\frac {27}{2}}}\)	\(58\)
risch	\(\frac {\sqrt {a^{2} x^{2}+1}\, \left (-\frac {i x}{24 a^{2}}+\frac {1}{120 a^{3}}\right )}{c^{13} \sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (a x +i\right )^{15} \left (a x -i\right )^{10}}\)	\(58\)

[In]

int((1+I*a*x)^5/(a^2*x^2+1)^(5/2)*x^2/(a^2*c*x^2+c)^(27/2),x,method=_RETURNVERBOSE)

[Out]

-1/120/(a^2*x^2+1)^(1/2)*(c*(a^2*x^2+1))^(1/2)*(5*I*a*x-1)/c^14/a^3/(I+a*x)^15/(I-a*x)^10

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. 496 vs. \(2 (53) = 106\).

Time = 0.36 (sec) , antiderivative size = 496, normalized size of antiderivative = 7.63 \[ \int \frac {e^{5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\frac {{\left (i \, a^{22} x^{25} - 5 \, a^{21} x^{24} - 40 \, a^{19} x^{22} - 50 i \, a^{18} x^{21} - 126 \, a^{17} x^{20} - 280 i \, a^{16} x^{19} - 160 \, a^{15} x^{18} - 765 i \, a^{14} x^{17} + 105 \, a^{13} x^{16} - 1248 i \, a^{12} x^{15} + 720 \, a^{11} x^{14} - 1260 i \, a^{10} x^{13} + 1260 \, a^{9} x^{12} - 720 i \, a^{8} x^{11} + 1248 \, a^{7} x^{10} - 105 i \, a^{6} x^{9} + 765 \, a^{5} x^{8} + 160 i \, a^{4} x^{7} + 280 \, a^{3} x^{6} + 126 i \, a^{2} x^{5} + 50 \, a x^{4} + 40 i \, x^{3}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1}}{120 \, {\left (a^{27} c^{14} x^{27} + 5 i \, a^{26} c^{14} x^{26} + a^{25} c^{14} x^{25} + 45 i \, a^{24} c^{14} x^{24} - 50 \, a^{23} c^{14} x^{23} + 166 i \, a^{22} c^{14} x^{22} - 330 \, a^{21} c^{14} x^{21} + 286 i \, a^{20} c^{14} x^{20} - 1045 \, a^{19} c^{14} x^{19} + 55 i \, a^{18} c^{14} x^{18} - 2013 \, a^{17} c^{14} x^{17} - 825 i \, a^{16} c^{14} x^{16} - 2508 \, a^{15} c^{14} x^{15} - 1980 i \, a^{14} c^{14} x^{14} - 1980 \, a^{13} c^{14} x^{13} - 2508 i \, a^{12} c^{14} x^{12} - 825 \, a^{11} c^{14} x^{11} - 2013 i \, a^{10} c^{14} x^{10} + 55 \, a^{9} c^{14} x^{9} - 1045 i \, a^{8} c^{14} x^{8} + 286 \, a^{7} c^{14} x^{7} - 330 i \, a^{6} c^{14} x^{6} + 166 \, a^{5} c^{14} x^{5} - 50 i \, a^{4} c^{14} x^{4} + 45 \, a^{3} c^{14} x^{3} + i \, a^{2} c^{14} x^{2} + 5 \, a c^{14} x + i \, c^{14}\right )}} \]

[In]

integrate((1+I*a*x)^5/(a^2*x^2+1)^(5/2)*x^2/(a^2*c*x^2+c)^(27/2),x, algorithm="fricas")

[Out]

1/120*(I*a^22*x^25 - 5*a^21*x^24 - 40*a^19*x^22 - 50*I*a^18*x^21 - 126*a^17*x^20 - 280*I*a^16*x^19 - 160*a^15*
x^18 - 765*I*a^14*x^17 + 105*a^13*x^16 - 1248*I*a^12*x^15 + 720*a^11*x^14 - 1260*I*a^10*x^13 + 1260*a^9*x^12 -
720*I*a^8*x^11 + 1248*a^7*x^10 - 105*I*a^6*x^9 + 765*a^5*x^8 + 160*I*a^4*x^7 + 280*a^3*x^6 + 126*I*a^2*x^5 +
50*a*x^4 + 40*I*x^3)*sqrt(a^2*c*x^2 + c)*sqrt(a^2*x^2 + 1)/(a^27*c^14*x^27 + 5*I*a^26*c^14*x^26 + a^25*c^14*x^
25 + 45*I*a^24*c^14*x^24 - 50*a^23*c^14*x^23 + 166*I*a^22*c^14*x^22 - 330*a^21*c^14*x^21 + 286*I*a^20*c^14*x^2
0 - 1045*a^19*c^14*x^19 + 55*I*a^18*c^14*x^18 - 2013*a^17*c^14*x^17 - 825*I*a^16*c^14*x^16 - 2508*a^15*c^14*x^
15 - 1980*I*a^14*c^14*x^14 - 1980*a^13*c^14*x^13 - 2508*I*a^12*c^14*x^12 - 825*a^11*c^14*x^11 - 2013*I*a^10*c^
14*x^10 + 55*a^9*c^14*x^9 - 1045*I*a^8*c^14*x^8 + 286*a^7*c^14*x^7 - 330*I*a^6*c^14*x^6 + 166*a^5*c^14*x^5 - 5
0*I*a^4*c^14*x^4 + 45*a^3*c^14*x^3 + I*a^2*c^14*x^2 + 5*a*c^14*x + I*c^14)

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\text {Timed out} \]

[In]

integrate((1+I*a*x)**5/(a**2*x**2+1)**(5/2)*x**2/(a**2*c*x**2+c)**(27/2),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {e^{5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((1+I*a*x)^5/(a^2*x^2+1)^(5/2)*x^2/(a^2*c*x^2+c)^(27/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [F]

\[ \int \frac {e^{5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{5} x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {27}{2}} {\left (a^{2} x^{2} + 1\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((1+I*a*x)^5/(a^2*x^2+1)^(5/2)*x^2/(a^2*c*x^2+c)^(27/2),x, algorithm="giac")

[Out]

integrate((I*a*x + 1)^5*x^2/((a^2*c*x^2 + c)^(27/2)*(a^2*x^2 + 1)^(5/2)), x)

Mupad [B] (verification not implemented)

Time = 2.47 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.71 \[ \int \frac {e^{5 i \arctan (a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx=-\frac {c\,{\left (a\,x-\mathrm {i}\right )}^5\,\left (5\,a\,x+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{120\,a^3\,{\left (c\,\left (a^2\,x^2+1\right )\right )}^{29/2}\,\sqrt {a^2\,x^2+1}} \]

[In]

int((x^2*(a*x*1i + 1)^5)/((c + a^2*c*x^2)^(27/2)*(a^2*x^2 + 1)^(5/2)),x)

[Out]

-(c*(a*x - 1i)^5*(5*a*x + 1i)*1i)/(120*a^3*(c*(a^2*x^2 + 1))^(29/2)*(a^2*x^2 + 1)^(1/2))