\(\int \frac {(d+e x)^5}{\sqrt {d^2-e^2 x^2}} \, dx\) [826]

Optimal result

Integrand size = 24, antiderivative size = 182 \[ \int \frac {(d+e x)^5}{\sqrt {d^2-e^2 x^2}} \, dx=-\frac {63 d^4 \sqrt {d^2-e^2 x^2}}{8 e}-\frac {21 d^3 (d+e x) \sqrt {d^2-e^2 x^2}}{8 e}-\frac {21 d^2 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{20 e}-\frac {9 d (d+e x)^3 \sqrt {d^2-e^2 x^2}}{20 e}-\frac {(d+e x)^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {63 d^5 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e} \]

[Out]

63/8*d^5*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e-63/8*d^4*(-e^2*x^2+d^2)^(1/2)/e-21/8*d^3*(e*x+d)*(-e^2*x^2+d^2)^(1
/2)/e-21/20*d^2*(e*x+d)^2*(-e^2*x^2+d^2)^(1/2)/e-9/20*d*(e*x+d)^3*(-e^2*x^2+d^2)^(1/2)/e-1/5*(e*x+d)^4*(-e^2*x
^2+d^2)^(1/2)/e

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {685, 655, 223, 209} \[ \int \frac {(d+e x)^5}{\sqrt {d^2-e^2 x^2}} \, dx=\frac {63 d^5 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e}-\frac {21 d^2 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{20 e}-\frac {9 d (d+e x)^3 \sqrt {d^2-e^2 x^2}}{20 e}-\frac {(d+e x)^4 \sqrt {d^2-e^2 x^2}}{5 e}-\frac {63 d^4 \sqrt {d^2-e^2 x^2}}{8 e}-\frac {21 d^3 (d+e x) \sqrt {d^2-e^2 x^2}}{8 e} \]

[In]

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

[Out]

(-63*d^4*Sqrt[d^2 - e^2*x^2])/(8*e) - (21*d^3*(d + e*x)*Sqrt[d^2 - e^2*x^2])/(8*e) - (21*d^2*(d + e*x)^2*Sqrt[
d^2 - e^2*x^2])/(20*e) - (9*d*(d + e*x)^3*Sqrt[d^2 - e^2*x^2])/(20*e) - ((d + e*x)^4*Sqrt[d^2 - e^2*x^2])/(5*e
) + (63*d^5*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(8*e)

Rule 209

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

Rule 223

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 655

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[e*((a + c*x^2)^(p + 1)/(2*c*(p + 1))),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 685

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*((a + c*x^2)^(p
+ 1)/(c*(m + 2*p + 1))), x] + Dist[2*c*d*((m + p)/(c*(m + 2*p + 1))), Int[(d + e*x)^(m - 1)*(a + c*x^2)^p, x]
, x] /; FreeQ[{a, c, d, e, p}, x] && EqQ[c*d^2 + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p
]

Rubi steps \begin{align*} \text {integral}& = -\frac {(d+e x)^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {1}{5} (9 d) \int \frac {(d+e x)^4}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = -\frac {9 d (d+e x)^3 \sqrt {d^2-e^2 x^2}}{20 e}-\frac {(d+e x)^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {1}{20} \left (63 d^2\right ) \int \frac {(d+e x)^3}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = -\frac {21 d^2 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{20 e}-\frac {9 d (d+e x)^3 \sqrt {d^2-e^2 x^2}}{20 e}-\frac {(d+e x)^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {1}{4} \left (21 d^3\right ) \int \frac {(d+e x)^2}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = -\frac {21 d^3 (d+e x) \sqrt {d^2-e^2 x^2}}{8 e}-\frac {21 d^2 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{20 e}-\frac {9 d (d+e x)^3 \sqrt {d^2-e^2 x^2}}{20 e}-\frac {(d+e x)^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {1}{8} \left (63 d^4\right ) \int \frac {d+e x}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = -\frac {63 d^4 \sqrt {d^2-e^2 x^2}}{8 e}-\frac {21 d^3 (d+e x) \sqrt {d^2-e^2 x^2}}{8 e}-\frac {21 d^2 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{20 e}-\frac {9 d (d+e x)^3 \sqrt {d^2-e^2 x^2}}{20 e}-\frac {(d+e x)^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {1}{8} \left (63 d^5\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = -\frac {63 d^4 \sqrt {d^2-e^2 x^2}}{8 e}-\frac {21 d^3 (d+e x) \sqrt {d^2-e^2 x^2}}{8 e}-\frac {21 d^2 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{20 e}-\frac {9 d (d+e x)^3 \sqrt {d^2-e^2 x^2}}{20 e}-\frac {(d+e x)^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {1}{8} \left (63 d^5\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right ) \\ & = -\frac {63 d^4 \sqrt {d^2-e^2 x^2}}{8 e}-\frac {21 d^3 (d+e x) \sqrt {d^2-e^2 x^2}}{8 e}-\frac {21 d^2 (d+e x)^2 \sqrt {d^2-e^2 x^2}}{20 e}-\frac {9 d (d+e x)^3 \sqrt {d^2-e^2 x^2}}{20 e}-\frac {(d+e x)^4 \sqrt {d^2-e^2 x^2}}{5 e}+\frac {63 d^5 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.57 \[ \int \frac {(d+e x)^5}{\sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {d^2-e^2 x^2} \left (488 d^4+275 d^3 e x+144 d^2 e^2 x^2+50 d e^3 x^3+8 e^4 x^4\right )+630 d^5 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{40 e} \]

[In]

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

[Out]

-1/40*(Sqrt[d^2 - e^2*x^2]*(488*d^4 + 275*d^3*e*x + 144*d^2*e^2*x^2 + 50*d*e^3*x^3 + 8*e^4*x^4) + 630*d^5*ArcT
an[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/e

Maple [A] (verified)

Time = 2.40 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.52

method	result	size
risch	\(-\frac {\left (8 e^{4} x^{4}+50 d \,e^{3} x^{3}+144 d^{2} e^{2} x^{2}+275 d^{3} e x +488 d^{4}\right ) \sqrt {-x^{2} e^{2}+d^{2}}}{40 e}+\frac {63 d^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{8 \sqrt {e^{2}}}\)	\(94\)
default	\(\frac {d^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}+e^{5} \left (-\frac {x^{4} \sqrt {-x^{2} e^{2}+d^{2}}}{5 e^{2}}+\frac {4 d^{2} \left (-\frac {x^{2} \sqrt {-x^{2} e^{2}+d^{2}}}{3 e^{2}}-\frac {2 d^{2} \sqrt {-x^{2} e^{2}+d^{2}}}{3 e^{4}}\right )}{5 e^{2}}\right )+5 d \,e^{4} \left (-\frac {x^{3} \sqrt {-x^{2} e^{2}+d^{2}}}{4 e^{2}}+\frac {3 d^{2} \left (-\frac {x \sqrt {-x^{2} e^{2}+d^{2}}}{2 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{2 e^{2} \sqrt {e^{2}}}\right )}{4 e^{2}}\right )-\frac {5 d^{4} \sqrt {-x^{2} e^{2}+d^{2}}}{e}+10 d^{2} e^{3} \left (-\frac {x^{2} \sqrt {-x^{2} e^{2}+d^{2}}}{3 e^{2}}-\frac {2 d^{2} \sqrt {-x^{2} e^{2}+d^{2}}}{3 e^{4}}\right )+10 d^{3} e^{2} \left (-\frac {x \sqrt {-x^{2} e^{2}+d^{2}}}{2 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{2 e^{2} \sqrt {e^{2}}}\right )\)	\(345\)

[In]

int((e*x+d)^5/(-e^2*x^2+d^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/40*(8*e^4*x^4+50*d*e^3*x^3+144*d^2*e^2*x^2+275*d^3*e*x+488*d^4)/e*(-e^2*x^2+d^2)^(1/2)+63/8*d^5/(e^2)^(1/2)
*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.52 \[ \int \frac {(d+e x)^5}{\sqrt {d^2-e^2 x^2}} \, dx=-\frac {630 \, d^{5} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (8 \, e^{4} x^{4} + 50 \, d e^{3} x^{3} + 144 \, d^{2} e^{2} x^{2} + 275 \, d^{3} e x + 488 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{40 \, e} \]

[In]

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

[Out]

-1/40*(630*d^5*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (8*e^4*x^4 + 50*d*e^3*x^3 + 144*d^2*e^2*x^2 + 275*d
^3*e*x + 488*d^4)*sqrt(-e^2*x^2 + d^2))/e

Sympy [A] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.84 \[ \int \frac {(d+e x)^5}{\sqrt {d^2-e^2 x^2}} \, dx=\begin {cases} \frac {63 d^{5} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{8} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {61 d^{4}}{5 e} - \frac {55 d^{3} x}{8} - \frac {18 d^{2} e x^{2}}{5} - \frac {5 d e^{2} x^{3}}{4} - \frac {e^{3} x^{4}}{5}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {\begin {cases} d^{5} x & \text {for}\: e = 0 \\\frac {\left (d + e x\right )^{6}}{6 e} & \text {otherwise} \end {cases}}{\sqrt {d^{2}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((63*d**5*Piecewise((log(-2*e**2*x + 2*sqrt(-e**2)*sqrt(d**2 - e**2*x**2))/sqrt(-e**2), Ne(d**2, 0)),
(x*log(x)/sqrt(-e**2*x**2), True))/8 + sqrt(d**2 - e**2*x**2)*(-61*d**4/(5*e) - 55*d**3*x/8 - 18*d**2*e*x**2/
5 - 5*d*e**2*x**3/4 - e**3*x**4/5), Ne(e**2, 0)), (Piecewise((d**5*x, Eq(e, 0)), ((d + e*x)**6/(6*e), True))/s
qrt(d**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.74 \[ \int \frac {(d+e x)^5}{\sqrt {d^2-e^2 x^2}} \, dx=-\frac {1}{5} \, \sqrt {-e^{2} x^{2} + d^{2}} e^{3} x^{4} - \frac {5}{4} \, \sqrt {-e^{2} x^{2} + d^{2}} d e^{2} x^{3} - \frac {18}{5} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e x^{2} + \frac {63 \, d^{5} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{8 \, \sqrt {e^{2}}} - \frac {55}{8} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3} x - \frac {61 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4}}{5 \, e} \]

[In]

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

[Out]

-1/5*sqrt(-e^2*x^2 + d^2)*e^3*x^4 - 5/4*sqrt(-e^2*x^2 + d^2)*d*e^2*x^3 - 18/5*sqrt(-e^2*x^2 + d^2)*d^2*e*x^2 +
63/8*d^5*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) - 55/8*sqrt(-e^2*x^2 + d^2)*d^3*x - 61/5*sqrt(-e^2*x^2 + d^2)*
d^4/e

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.43 \[ \int \frac {(d+e x)^5}{\sqrt {d^2-e^2 x^2}} \, dx=\frac {63 \, d^{5} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{8 \, {\left | e \right |}} - \frac {1}{40} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (\frac {488 \, d^{4}}{e} + {\left (275 \, d^{3} + 2 \, {\left (72 \, d^{2} e + {\left (4 \, e^{3} x + 25 \, d e^{2}\right )} x\right )} x\right )} x\right )} \]

[In]

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

[Out]

63/8*d^5*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) - 1/40*sqrt(-e^2*x^2 + d^2)*(488*d^4/e + (275*d^3 + 2*(72*d^2*e +
(4*e^3*x + 25*d*e^2)*x)*x)*x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^5}{\sqrt {d^2-e^2 x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^5}{\sqrt {d^2-e^2\,x^2}} \,d x \]

[In]

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

[Out]

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