\(\int x^{7/2} \arctan (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}) \, dx\) [25]

Optimal result

Integrand size = 27, antiderivative size = 326 \[ \int x^{7/2} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {28 d x^{3/2} \sqrt {d+e x^2}}{405 (-e)^{3/2}}+\frac {4 x^{7/2} \sqrt {d+e x^2}}{81 \sqrt {-e}}-\frac {28 d^2 \sqrt {-e} \sqrt {x} \sqrt {d+e x^2}}{135 e^{5/2} \left (\sqrt {d}+\sqrt {e} x\right )}+\frac {2}{9} x^{9/2} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )+\frac {28 d^{9/4} \sqrt {-e} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{135 e^{11/4} \sqrt {d+e x^2}}-\frac {14 d^{9/4} \sqrt {-e} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right ),\frac {1}{2}\right )}{135 e^{11/4} \sqrt {d+e x^2}} \] Output:

28/405*d*x^(3/2)*(e*x^2+d)^(1/2)/(-e)^(3/2)+4/81*x^(7/2)*(e*x^2+d)^(1/2)/( 
-e)^(1/2)-28/135*d^2*(-e)^(1/2)*x^(1/2)*(e*x^2+d)^(1/2)/e^(5/2)/(d^(1/2)+e 
^(1/2)*x)+2/9*x^(9/2)*arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2))+28/135*d^(9/4)* 
(-e)^(1/2)*(d^(1/2)+e^(1/2)*x)*((e*x^2+d)/(d^(1/2)+e^(1/2)*x)^2)^(1/2)*Ell 
ipticE(sin(2*arctan(e^(1/4)*x^(1/2)/d^(1/4))),1/2*2^(1/2))/e^(11/4)/(e*x^2 
+d)^(1/2)-14/135*d^(9/4)*(-e)^(1/2)*(d^(1/2)+e^(1/2)*x)*((e*x^2+d)/(d^(1/2 
)+e^(1/2)*x)^2)^(1/2)*InverseJacobiAM(2*arctan(e^(1/4)*x^(1/2)/d^(1/4)),1/ 
2*2^(1/2))/e^(11/4)/(e*x^2+d)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.11 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.43 \[ \int x^{7/2} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {2 x^{3/2} \left (2 \sqrt {-e} \left (7 d^2+2 d e x^2-5 e^2 x^4\right )+45 e^2 x^3 \sqrt {d+e x^2} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-14 d^2 \sqrt {-e} \sqrt {1+\frac {e x^2}{d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {e x^2}{d}\right )\right )}{405 e^2 \sqrt {d+e x^2}} \] Input:

Integrate[x^(7/2)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]],x]
 

Output:

(2*x^(3/2)*(2*Sqrt[-e]*(7*d^2 + 2*d*e*x^2 - 5*e^2*x^4) + 45*e^2*x^3*Sqrt[d 
 + e*x^2]*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]] - 14*d^2*Sqrt[-e]*Sqrt[1 + 
(e*x^2)/d]*Hypergeometric2F1[1/2, 3/4, 7/4, -((e*x^2)/d)]))/(405*e^2*Sqrt[ 
d + e*x^2])
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.98, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {5674, 262, 262, 266, 834, 27, 761, 1510}

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.

\(\displaystyle \int x^{7/2} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx\)

\(\Big \downarrow \) 5674

\(\displaystyle \frac {2}{9} x^{9/2} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {2}{9} \sqrt {-e} \int \frac {x^{9/2}}{\sqrt {e x^2+d}}dx\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {2}{9} x^{9/2} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {2}{9} \sqrt {-e} \left (\frac {2 x^{7/2} \sqrt {d+e x^2}}{9 e}-\frac {7 d \int \frac {x^{5/2}}{\sqrt {e x^2+d}}dx}{9 e}\right )\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {2}{9} x^{9/2} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {2}{9} \sqrt {-e} \left (\frac {2 x^{7/2} \sqrt {d+e x^2}}{9 e}-\frac {7 d \left (\frac {2 x^{3/2} \sqrt {d+e x^2}}{5 e}-\frac {3 d \int \frac {\sqrt {x}}{\sqrt {e x^2+d}}dx}{5 e}\right )}{9 e}\right )\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {2}{9} x^{9/2} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {2}{9} \sqrt {-e} \left (\frac {2 x^{7/2} \sqrt {d+e x^2}}{9 e}-\frac {7 d \left (\frac {2 x^{3/2} \sqrt {d+e x^2}}{5 e}-\frac {6 d \int \frac {x}{\sqrt {e x^2+d}}d\sqrt {x}}{5 e}\right )}{9 e}\right )\)

\(\Big \downarrow \) 834

\(\displaystyle \frac {2}{9} x^{9/2} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {2}{9} \sqrt {-e} \left (\frac {2 x^{7/2} \sqrt {d+e x^2}}{9 e}-\frac {7 d \left (\frac {2 x^{3/2} \sqrt {d+e x^2}}{5 e}-\frac {6 d \left (\frac {\sqrt {d} \int \frac {1}{\sqrt {e x^2+d}}d\sqrt {x}}{\sqrt {e}}-\frac {\sqrt {d} \int \frac {\sqrt {d}-\sqrt {e} x}{\sqrt {d} \sqrt {e x^2+d}}d\sqrt {x}}{\sqrt {e}}\right )}{5 e}\right )}{9 e}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{9} x^{9/2} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {2}{9} \sqrt {-e} \left (\frac {2 x^{7/2} \sqrt {d+e x^2}}{9 e}-\frac {7 d \left (\frac {2 x^{3/2} \sqrt {d+e x^2}}{5 e}-\frac {6 d \left (\frac {\sqrt {d} \int \frac {1}{\sqrt {e x^2+d}}d\sqrt {x}}{\sqrt {e}}-\frac {\int \frac {\sqrt {d}-\sqrt {e} x}{\sqrt {e x^2+d}}d\sqrt {x}}{\sqrt {e}}\right )}{5 e}\right )}{9 e}\right )\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {2}{9} x^{9/2} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {2}{9} \sqrt {-e} \left (\frac {2 x^{7/2} \sqrt {d+e x^2}}{9 e}-\frac {7 d \left (\frac {2 x^{3/2} \sqrt {d+e x^2}}{5 e}-\frac {6 d \left (\frac {\sqrt [4]{d} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right ),\frac {1}{2}\right )}{2 e^{3/4} \sqrt {d+e x^2}}-\frac {\int \frac {\sqrt {d}-\sqrt {e} x}{\sqrt {e x^2+d}}d\sqrt {x}}{\sqrt {e}}\right )}{5 e}\right )}{9 e}\right )\)

\(\Big \downarrow \) 1510

\(\displaystyle \frac {2}{9} x^{9/2} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right )-\frac {2}{9} \sqrt {-e} \left (\frac {2 x^{7/2} \sqrt {d+e x^2}}{9 e}-\frac {7 d \left (\frac {2 x^{3/2} \sqrt {d+e x^2}}{5 e}-\frac {6 d \left (\frac {\sqrt [4]{d} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right ),\frac {1}{2}\right )}{2 e^{3/4} \sqrt {d+e x^2}}-\frac {\frac {\sqrt [4]{d} \left (\sqrt {d}+\sqrt {e} x\right ) \sqrt {\frac {d+e x^2}{\left (\sqrt {d}+\sqrt {e} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt {x}}{\sqrt [4]{d}}\right )|\frac {1}{2}\right )}{\sqrt [4]{e} \sqrt {d+e x^2}}-\frac {\sqrt {x} \sqrt {d+e x^2}}{\sqrt {d}+\sqrt {e} x}}{\sqrt {e}}\right )}{5 e}\right )}{9 e}\right )\)

Input:

Int[x^(7/2)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]],x]
 

Output:

(2*x^(9/2)*ArcTan[(Sqrt[-e]*x)/Sqrt[d + e*x^2]])/9 - (2*Sqrt[-e]*((2*x^(7/ 
2)*Sqrt[d + e*x^2])/(9*e) - (7*d*((2*x^(3/2)*Sqrt[d + e*x^2])/(5*e) - (6*d 
*(-((-((Sqrt[x]*Sqrt[d + e*x^2])/(Sqrt[d] + Sqrt[e]*x)) + (d^(1/4)*(Sqrt[d 
] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*EllipticE[2*ArcTa 
n[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(e^(1/4)*Sqrt[d + e*x^2]))/Sqrt[e]) + 
(d^(1/4)*(Sqrt[d] + Sqrt[e]*x)*Sqrt[(d + e*x^2)/(Sqrt[d] + Sqrt[e]*x)^2]*E 
llipticF[2*ArcTan[(e^(1/4)*Sqrt[x])/d^(1/4)], 1/2])/(2*e^(3/4)*Sqrt[d + e* 
x^2])))/(5*e)))/(9*e)))/9
 

Defintions of rubi rules used

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 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 834

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S 
imp[1/q   Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q   Int[(1 - q*x^2)/Sqrt[a 
 + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 1510

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

rule 5674

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

Maple [F]

\[\int x^{\frac {7}{2}} \arctan \left (\frac {\sqrt {-e}\, x}{\sqrt {e \,x^{2}+d}}\right )d x\]

Input:

int(x^(7/2)*arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2)),x)
 

Output:

int(x^(7/2)*arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2)),x)
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.29 \[ \int x^{7/2} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\frac {2 \, {\left (45 \, e^{3} x^{\frac {9}{2}} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) + 42 \, d^{2} \sqrt {-e} \sqrt {e} {\rm weierstrassZeta}\left (-\frac {4 \, d}{e}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, d}{e}, 0, x\right )\right ) - 2 \, {\left (5 \, e^{2} x^{3} - 7 \, d e x\right )} \sqrt {e x^{2} + d} \sqrt {-e} \sqrt {x}\right )}}{405 \, e^{3}} \] Input:

integrate(x^(7/2)*arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2)),x, algorithm="frica 
s")
 

Output:

2/405*(45*e^3*x^(9/2)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) + 42*d^2*sqrt(-e) 
*sqrt(e)*weierstrassZeta(-4*d/e, 0, weierstrassPInverse(-4*d/e, 0, x)) - 2 
*(5*e^2*x^3 - 7*d*e*x)*sqrt(e*x^2 + d)*sqrt(-e)*sqrt(x))/e^3
 

Sympy [F(-1)]

Timed out. \[ \int x^{7/2} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\text {Timed out} \] Input:

integrate(x**(7/2)*atan((-e)**(1/2)*x/(e*x**2+d)**(1/2)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^{7/2} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int { x^{\frac {7}{2}} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) \,d x } \] Input:

integrate(x^(7/2)*arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2)),x, algorithm="maxim 
a")
 

Output:

2/9*x^(9/2)*arctan2(sqrt(-e)*x, sqrt(e*x^2 + d)) - 2*d*sqrt(-e)*integrate( 
-1/9*x*e^(1/2*log(e*x^2 + d) + 7/2*log(x))/(e^2*x^4 + d*e*x^2 - (e*x^2 + d 
)^2), x)
 

Giac [F]

\[ \int x^{7/2} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int { x^{\frac {7}{2}} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) \,d x } \] Input:

integrate(x^(7/2)*arctan((-e)^(1/2)*x/(e*x^2+d)^(1/2)),x, algorithm="giac" 
)
 

Output:

integrate(x^(7/2)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^{7/2} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int x^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {-e}\,x}{\sqrt {e\,x^2+d}}\right ) \,d x \] Input:

int(x^(7/2)*atan(((-e)^(1/2)*x)/(d + e*x^2)^(1/2)),x)
 

Output:

int(x^(7/2)*atan(((-e)^(1/2)*x)/(d + e*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int x^{7/2} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {d+e x^2}}\right ) \, dx=\int \sqrt {x}\, \mathit {atan} \left (\frac {\sqrt {e}\, i x}{\sqrt {e \,x^{2}+d}}\right ) x^{3}d x \] Input:

int(x^(7/2)*atan((-e)^(1/2)*x/(e*x^2+d)^(1/2)),x)
 

Output:

int(sqrt(x)*atan((sqrt(e)*i*x)/sqrt(d + e*x**2))*x**3,x)