3.3 \(\int x^5 \tan ^{-1}(\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}) \, dx\)

Optimal. Leaf size=144 \[ \frac {5 d^3 \sqrt {-e} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{96 e^{7/2}}+\frac {5 d^2 x \sqrt {d+e x^2}}{96 (-e)^{5/2}}+\frac {1}{6} x^6 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )+\frac {x^5 \sqrt {d+e x^2}}{36 \sqrt {-e}}+\frac {5 d x^3 \sqrt {d+e x^2}}{144 (-e)^{3/2}} \]

[Out]

1/6*x^6*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))+5/96*d^3*arctanh(x*e^(1/2)/(e*x^2+d)^(1/2))*(-e)^(1/2)/e^(7/2)+5/
96*d^2*x*(e*x^2+d)^(1/2)/(-e)^(5/2)+5/144*d*x^3*(e*x^2+d)^(1/2)/(-e)^(3/2)+1/36*x^5*(e*x^2+d)^(1/2)/(-e)^(1/2)

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Rubi [A]  time = 0.08, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {5151, 321, 217, 206} \[ \frac {5 d^3 \sqrt {-e} \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{96 e^{7/2}}+\frac {5 d^2 x \sqrt {d+e x^2}}{96 (-e)^{5/2}}+\frac {x^5 \sqrt {d+e x^2}}{36 \sqrt {-e}}+\frac {5 d x^3 \sqrt {d+e x^2}}{144 (-e)^{3/2}}+\frac {1}{6} x^6 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^5*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]],x]

[Out]

(5*d^2*x*Sqrt[d + e*x^2])/(96*(-e)^(5/2)) + (5*d*x^3*Sqrt[d + e*x^2])/(144*(-e)^(3/2)) + (x^5*Sqrt[d + e*x^2])
/(36*Sqrt[-e]) + (x^6*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/6 + (5*d^3*Sqrt[-e]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e
*x^2]])/(96*e^(7/2))

Rule 206

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 5151

Int[ArcTan[((c_.)*(x_))/Sqrt[(a_.) + (b_.)*(x_)^2]]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*ArcTa
n[(c*x)/Sqrt[a + b*x^2]])/(d*(m + 1)), x] - Dist[c/(d*(m + 1)), Int[(d*x)^(m + 1)/Sqrt[a + b*x^2], x], x] /; F
reeQ[{a, b, c, d, m}, x] && EqQ[b + c^2, 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int x^5 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx &=\frac {1}{6} x^6 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {1}{6} \sqrt {-e} \int \frac {x^6}{\sqrt {d+e x^2}} \, dx\\ &=\frac {x^5 \sqrt {d+e x^2}}{36 \sqrt {-e}}+\frac {1}{6} x^6 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {(5 d) \int \frac {x^4}{\sqrt {d+e x^2}} \, dx}{36 \sqrt {-e}}\\ &=\frac {5 d x^3 \sqrt {d+e x^2}}{144 (-e)^{3/2}}+\frac {x^5 \sqrt {d+e x^2}}{36 \sqrt {-e}}+\frac {1}{6} x^6 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {\left (5 d^2\right ) \int \frac {x^2}{\sqrt {d+e x^2}} \, dx}{48 (-e)^{3/2}}\\ &=\frac {5 d^2 x \sqrt {d+e x^2}}{96 (-e)^{5/2}}+\frac {5 d x^3 \sqrt {d+e x^2}}{144 (-e)^{3/2}}+\frac {x^5 \sqrt {d+e x^2}}{36 \sqrt {-e}}+\frac {1}{6} x^6 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {\left (5 d^3\right ) \int \frac {1}{\sqrt {d+e x^2}} \, dx}{96 (-e)^{5/2}}\\ &=\frac {5 d^2 x \sqrt {d+e x^2}}{96 (-e)^{5/2}}+\frac {5 d x^3 \sqrt {d+e x^2}}{144 (-e)^{3/2}}+\frac {x^5 \sqrt {d+e x^2}}{36 \sqrt {-e}}+\frac {1}{6} x^6 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {\left (5 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-e x^2} \, dx,x,\frac {x}{\sqrt {d+e x^2}}\right )}{96 (-e)^{5/2}}\\ &=\frac {5 d^2 x \sqrt {d+e x^2}}{96 (-e)^{5/2}}+\frac {5 d x^3 \sqrt {d+e x^2}}{144 (-e)^{3/2}}+\frac {x^5 \sqrt {d+e x^2}}{36 \sqrt {-e}}+\frac {1}{6} x^6 \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {5 d^3 \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{96 (-e)^{5/2} \sqrt {e}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 86, normalized size = 0.60 \[ \frac {3 \left (5 d^3+16 e^3 x^6\right ) \tan ^{-1}\left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )+\sqrt {-e} x \sqrt {d+e x^2} \left (-15 d^2+10 d e x^2-8 e^2 x^4\right )}{288 e^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]],x]

[Out]

(Sqrt[-e]*x*Sqrt[d + e*x^2]*(-15*d^2 + 10*d*e*x^2 - 8*e^2*x^4) + 3*(5*d^3 + 16*e^3*x^6)*ArcTan[(Sqrt[-e]*x)/Sq
rt[d + e*x^2]])/(288*e^3)

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fricas [A]  time = 0.46, size = 76, normalized size = 0.53 \[ -\frac {{\left (8 \, e^{2} x^{5} - 10 \, d e x^{3} + 15 \, d^{2} x\right )} \sqrt {e x^{2} + d} \sqrt {-e} - 3 \, {\left (16 \, e^{3} x^{6} + 5 \, d^{3}\right )} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right )}{288 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2)),x, algorithm="fricas")

[Out]

-1/288*((8*e^2*x^5 - 10*d*e*x^3 + 15*d^2*x)*sqrt(e*x^2 + d)*sqrt(-e) - 3*(16*e^3*x^6 + 5*d^3)*arctan(sqrt(-e)*
x/sqrt(e*x^2 + d)))/e^3

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giac [A]  time = 0.26, size = 88, normalized size = 0.61 \[ \frac {1}{6} \, x^{6} \arctan \left (\frac {x \sqrt {-e}}{\sqrt {x^{2} e + d}}\right ) - \frac {5}{96} \, d^{3} \arcsin \left (\frac {x e}{\sqrt {-d e}}\right ) e^{\left (-3\right )} - \frac {1}{288} \, {\left (2 \, {\left (4 \, x^{2} e^{\left (-1\right )} - 5 \, d e^{\left (-2\right )}\right )} x^{2} + 15 \, d^{2} e^{\left (-3\right )}\right )} \sqrt {-x^{2} e^{2} - d e} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2)),x, algorithm="giac")

[Out]

1/6*x^6*arctan(x*sqrt(-e)/sqrt(x^2*e + d)) - 5/96*d^3*arcsin(x*e/sqrt(-d*e))*e^(-3) - 1/288*(2*(4*x^2*e^(-1) -
5*d*e^(-2))*x^2 + 15*d^2*e^(-3))*sqrt(-x^2*e^2 - d*e)*x

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maple [A]  time = 0.05, size = 211, normalized size = 1.47 \[ \frac {x^{6} \arctan \left (\frac {x \sqrt {-e}}{\sqrt {e \,x^{2}+d}}\right )}{6}+\frac {\sqrt {-e}\, x^{7} \sqrt {e \,x^{2}+d}}{48 d}-\frac {7 \sqrt {-e}\, x^{5} \sqrt {e \,x^{2}+d}}{288 e}+\frac {35 \sqrt {-e}\, d \,x^{3} \sqrt {e \,x^{2}+d}}{1152 e^{2}}-\frac {5 \sqrt {-e}\, d^{2} x \sqrt {e \,x^{2}+d}}{128 e^{3}}+\frac {5 \sqrt {-e}\, d^{3} \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{96 e^{\frac {7}{2}}}-\frac {\sqrt {-e}\, x^{5} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{48 d e}+\frac {5 \sqrt {-e}\, x^{3} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{288 e^{2}}-\frac {5 \sqrt {-e}\, d x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{384 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2)),x)

[Out]

1/6*x^6*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2))+1/48*(-e)^(1/2)/d*x^7*(e*x^2+d)^(1/2)-7/288*(-e)^(1/2)/e*x^5*(e*x
^2+d)^(1/2)+35/1152*(-e)^(1/2)/e^2*d*x^3*(e*x^2+d)^(1/2)-5/128*(-e)^(1/2)/e^3*d^2*x*(e*x^2+d)^(1/2)+5/96*(-e)^
(1/2)/e^(7/2)*d^3*ln(x*e^(1/2)+(e*x^2+d)^(1/2))-1/48*(-e)^(1/2)/d*x^5*(e*x^2+d)^(3/2)/e+5/288*(-e)^(1/2)/e^2*x
^3*(e*x^2+d)^(3/2)-5/384*(-e)^(1/2)*d/e^3*x*(e*x^2+d)^(3/2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arctan(x*(-e)^(1/2)/(e*x^2+d)^(1/2)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: sign: argument cannot be imaginary
; found sqrt(-_SAGE_VAR_e)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^5\,\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*atan(((-e)^(1/2)*x)/(d + e*x^2)^(1/2)),x)

[Out]

int(x^5*atan(((-e)^(1/2)*x)/(d + e*x^2)^(1/2)), x)

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sympy [A]  time = 7.25, size = 129, normalized size = 0.90 \[ \begin {cases} \frac {5 i d^{3} \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{96 e^{3}} - \frac {5 i d^{2} x \sqrt {d + e x^{2}}}{96 e^{\frac {5}{2}}} + \frac {5 i d x^{3} \sqrt {d + e x^{2}}}{144 e^{\frac {3}{2}}} + \frac {i x^{6} \operatorname {atanh}{\left (\frac {\sqrt {e} x}{\sqrt {d + e x^{2}}} \right )}}{6} - \frac {i x^{5} \sqrt {d + e x^{2}}}{36 \sqrt {e}} & \text {for}\: e \neq 0 \\0 & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*atan(x*(-e)**(1/2)/(e*x**2+d)**(1/2)),x)

[Out]

Piecewise((5*I*d**3*atanh(sqrt(e)*x/sqrt(d + e*x**2))/(96*e**3) - 5*I*d**2*x*sqrt(d + e*x**2)/(96*e**(5/2)) +
5*I*d*x**3*sqrt(d + e*x**2)/(144*e**(3/2)) + I*x**6*atanh(sqrt(e)*x/sqrt(d + e*x**2))/6 - I*x**5*sqrt(d + e*x*
*2)/(36*sqrt(e)), Ne(e, 0)), (0, True))

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