3.380 \(\int \frac{e^{5 i \tan ^{-1}(a x)} x^2}{(c+a^2 c x^2)^{27/2}} \, dx\)

Optimal. Leaf size=65 \[ -\frac{(5 a x+i) \sqrt{a^2 x^2+1}}{120 a^3 c^{13} (1-i a x)^{15} (1+i a x)^{10} \sqrt{a^2 c x^2+c}} \]

[Out]

-((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])

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Rubi [A]  time = 0.201334, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {5085, 5082, 81} \[ -\frac{(5 a x+i) \sqrt{a^2 x^2+1}}{120 a^3 c^{13} (1-i a x)^{15} (1+i a x)^{10} \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((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])

Rule 5085

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])

Rule 5082

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 81

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]

Rubi steps

\begin{align*} \int \frac{e^{5 i \tan ^{-1}(a x)} x^2}{\left (c+a^2 c x^2\right )^{27/2}} \, dx &=\frac{\sqrt{1+a^2 x^2} \int \frac{e^{5 i \tan ^{-1}(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]  time = 0.481836, size = 63, normalized size = 0.97 \[ \frac{(1-5 i a x) \sqrt{a^2 x^2+1}}{120 a^3 c^{13} (a x-i)^{10} (a x+i)^{15} \sqrt{a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[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])

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Maple [A]  time = 0.098, size = 58, normalized size = 0.9 \begin{align*}{\frac{ \left ( -ax+i \right ) \left ( ax+i \right ) \left ( i+5\,ax \right ) \left ( 1+iax \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}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

[Out]

1/120*(a*x+I)*(-a*x+I)*(I+5*a*x)*(1+I*a*x)^5/a^3/(a^2*x^2+1)^(5/2)/(a^2*c*x^2+c)^(27/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 4.18018, size = 1369, normalized size = 21.06 \begin{align*} \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 \, a^{27} c^{14} x^{27} + 600 i \, a^{26} c^{14} x^{26} + 120 \, a^{25} c^{14} x^{25} + 5400 i \, a^{24} c^{14} x^{24} - 6000 \, a^{23} c^{14} x^{23} + 19920 i \, a^{22} c^{14} x^{22} - 39600 \, a^{21} c^{14} x^{21} + 34320 i \, a^{20} c^{14} x^{20} - 125400 \, a^{19} c^{14} x^{19} + 6600 i \, a^{18} c^{14} x^{18} - 241560 \, a^{17} c^{14} x^{17} - 99000 i \, a^{16} c^{14} x^{16} - 300960 \, a^{15} c^{14} x^{15} - 237600 i \, a^{14} c^{14} x^{14} - 237600 \, a^{13} c^{14} x^{13} - 300960 i \, a^{12} c^{14} x^{12} - 99000 \, a^{11} c^{14} x^{11} - 241560 i \, a^{10} c^{14} x^{10} + 6600 \, a^{9} c^{14} x^{9} - 125400 i \, a^{8} c^{14} x^{8} + 34320 \, a^{7} c^{14} x^{7} - 39600 i \, a^{6} c^{14} x^{6} + 19920 \, a^{5} c^{14} x^{5} - 6000 i \, a^{4} c^{14} x^{4} + 5400 \, a^{3} c^{14} x^{3} + 120 i \, a^{2} c^{14} x^{2} + 600 \, a c^{14} x + 120 i \, c^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

(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)/(120*a^27*c^14*x^27 + 600*I*a^26*c^14*x^26 + 120*a^25*c^1
4*x^25 + 5400*I*a^24*c^14*x^24 - 6000*a^23*c^14*x^23 + 19920*I*a^22*c^14*x^22 - 39600*a^21*c^14*x^21 + 34320*I
*a^20*c^14*x^20 - 125400*a^19*c^14*x^19 + 6600*I*a^18*c^14*x^18 - 241560*a^17*c^14*x^17 - 99000*I*a^16*c^14*x^
16 - 300960*a^15*c^14*x^15 - 237600*I*a^14*c^14*x^14 - 237600*a^13*c^14*x^13 - 300960*I*a^12*c^14*x^12 - 99000
*a^11*c^14*x^11 - 241560*I*a^10*c^14*x^10 + 6600*a^9*c^14*x^9 - 125400*I*a^8*c^14*x^8 + 34320*a^7*c^14*x^7 - 3
9600*I*a^6*c^14*x^6 + 19920*a^5*c^14*x^5 - 6000*I*a^4*c^14*x^4 + 5400*a^3*c^14*x^3 + 120*I*a^2*c^14*x^2 + 600*
a*c^14*x + 120*I*c^14)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)