\(\int \frac {1}{(d+e x)^5 \sqrt {d^2-e^2 x^2}} \, dx\) [835]

Optimal result

Integrand size = 24, antiderivative size = 166 \[ \int \frac {1}{(d+e x)^5 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {d^2-e^2 x^2}}{9 d e (d+e x)^5}-\frac {4 \sqrt {d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac {4 \sqrt {d^2-e^2 x^2}}{105 d^3 e (d+e x)^3}-\frac {8 \sqrt {d^2-e^2 x^2}}{315 d^4 e (d+e x)^2}-\frac {8 \sqrt {d^2-e^2 x^2}}{315 d^5 e (d+e x)} \]

[Out]

-1/9*(-e^2*x^2+d^2)^(1/2)/d/e/(e*x+d)^5-4/63*(-e^2*x^2+d^2)^(1/2)/d^2/e/(e*x+d)^4-4/105*(-e^2*x^2+d^2)^(1/2)/d
^3/e/(e*x+d)^3-8/315*(-e^2*x^2+d^2)^(1/2)/d^4/e/(e*x+d)^2-8/315*(-e^2*x^2+d^2)^(1/2)/d^5/e/(e*x+d)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {673, 665} \[ \int \frac {1}{(d+e x)^5 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {4 \sqrt {d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac {\sqrt {d^2-e^2 x^2}}{9 d e (d+e x)^5}-\frac {8 \sqrt {d^2-e^2 x^2}}{315 d^5 e (d+e x)}-\frac {8 \sqrt {d^2-e^2 x^2}}{315 d^4 e (d+e x)^2}-\frac {4 \sqrt {d^2-e^2 x^2}}{105 d^3 e (d+e x)^3} \]

[In]

Int[1/((d + e*x)^5*Sqrt[d^2 - e^2*x^2]),x]

[Out]

-1/9*Sqrt[d^2 - e^2*x^2]/(d*e*(d + e*x)^5) - (4*Sqrt[d^2 - e^2*x^2])/(63*d^2*e*(d + e*x)^4) - (4*Sqrt[d^2 - e^
2*x^2])/(105*d^3*e*(d + e*x)^3) - (8*Sqrt[d^2 - e^2*x^2])/(315*d^4*e*(d + e*x)^2) - (8*Sqrt[d^2 - e^2*x^2])/(3
15*d^5*e*(d + e*x))

Rule 665

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

Rule 673

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {d^2-e^2 x^2}}{9 d e (d+e x)^5}+\frac {4 \int \frac {1}{(d+e x)^4 \sqrt {d^2-e^2 x^2}} \, dx}{9 d} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{9 d e (d+e x)^5}-\frac {4 \sqrt {d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}+\frac {4 \int \frac {1}{(d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx}{21 d^2} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{9 d e (d+e x)^5}-\frac {4 \sqrt {d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac {4 \sqrt {d^2-e^2 x^2}}{105 d^3 e (d+e x)^3}+\frac {8 \int \frac {1}{(d+e x)^2 \sqrt {d^2-e^2 x^2}} \, dx}{105 d^3} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{9 d e (d+e x)^5}-\frac {4 \sqrt {d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac {4 \sqrt {d^2-e^2 x^2}}{105 d^3 e (d+e x)^3}-\frac {8 \sqrt {d^2-e^2 x^2}}{315 d^4 e (d+e x)^2}+\frac {8 \int \frac {1}{(d+e x) \sqrt {d^2-e^2 x^2}} \, dx}{315 d^4} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{9 d e (d+e x)^5}-\frac {4 \sqrt {d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac {4 \sqrt {d^2-e^2 x^2}}{105 d^3 e (d+e x)^3}-\frac {8 \sqrt {d^2-e^2 x^2}}{315 d^4 e (d+e x)^2}-\frac {8 \sqrt {d^2-e^2 x^2}}{315 d^5 e (d+e x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.45 \[ \int \frac {1}{(d+e x)^5 \sqrt {d^2-e^2 x^2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-83 d^4-100 d^3 e x-84 d^2 e^2 x^2-40 d e^3 x^3-8 e^4 x^4\right )}{315 d^5 e (d+e x)^5} \]

[In]

Integrate[1/((d + e*x)^5*Sqrt[d^2 - e^2*x^2]),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-83*d^4 - 100*d^3*e*x - 84*d^2*e^2*x^2 - 40*d*e^3*x^3 - 8*e^4*x^4))/(315*d^5*e*(d + e*x)
^5)

Maple [A] (verified)

Time = 2.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.43

method	result	size
trager	\(-\frac {\left (8 e^{4} x^{4}+40 d \,e^{3} x^{3}+84 d^{2} e^{2} x^{2}+100 d^{3} e x +83 d^{4}\right ) \sqrt {-x^{2} e^{2}+d^{2}}}{315 d^{5} \left (e x +d \right )^{5} e}\)	\(71\)
gosper	\(-\frac {\left (-e x +d \right ) \left (8 e^{4} x^{4}+40 d \,e^{3} x^{3}+84 d^{2} e^{2} x^{2}+100 d^{3} e x +83 d^{4}\right )}{315 \left (e x +d \right )^{4} d^{5} e \sqrt {-x^{2} e^{2}+d^{2}}}\)	\(77\)
default	\(\frac {-\frac {\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{9 d e \left (x +\frac {d}{e}\right )^{5}}+\frac {4 e \left (-\frac {\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{7 d e \left (x +\frac {d}{e}\right )^{4}}+\frac {3 e \left (-\frac {\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5 d e \left (x +\frac {d}{e}\right )^{3}}+\frac {2 e \left (-\frac {\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{3 d e \left (x +\frac {d}{e}\right )^{2}}-\frac {\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{3 d^{2} \left (x +\frac {d}{e}\right )}\right )}{5 d}\right )}{7 d}\right )}{9 d}}{e^{5}}\)	\(249\)

[In]

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

[Out]

-1/315*(8*e^4*x^4+40*d*e^3*x^3+84*d^2*e^2*x^2+100*d^3*e*x+83*d^4)/d^5/(e*x+d)^5/e*(-e^2*x^2+d^2)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(d+e x)^5 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {83 \, e^{5} x^{5} + 415 \, d e^{4} x^{4} + 830 \, d^{2} e^{3} x^{3} + 830 \, d^{3} e^{2} x^{2} + 415 \, d^{4} e x + 83 \, d^{5} + {\left (8 \, e^{4} x^{4} + 40 \, d e^{3} x^{3} + 84 \, d^{2} e^{2} x^{2} + 100 \, d^{3} e x + 83 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{315 \, {\left (d^{5} e^{6} x^{5} + 5 \, d^{6} e^{5} x^{4} + 10 \, d^{7} e^{4} x^{3} + 10 \, d^{8} e^{3} x^{2} + 5 \, d^{9} e^{2} x + d^{10} e\right )}} \]

[In]

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

[Out]

-1/315*(83*e^5*x^5 + 415*d*e^4*x^4 + 830*d^2*e^3*x^3 + 830*d^3*e^2*x^2 + 415*d^4*e*x + 83*d^5 + (8*e^4*x^4 + 4
0*d*e^3*x^3 + 84*d^2*e^2*x^2 + 100*d^3*e*x + 83*d^4)*sqrt(-e^2*x^2 + d^2))/(d^5*e^6*x^5 + 5*d^6*e^5*x^4 + 10*d
^7*e^4*x^3 + 10*d^8*e^3*x^2 + 5*d^9*e^2*x + d^10*e)

Sympy [F]

\[ \int \frac {1}{(d+e x)^5 \sqrt {d^2-e^2 x^2}} \, dx=\int \frac {1}{\sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{5}}\, dx \]

[In]

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

[Out]

Integral(1/(sqrt(-(-d + e*x)*(d + e*x))*(d + e*x)**5), x)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.62 \[ \int \frac {1}{(d+e x)^5 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {-e^{2} x^{2} + d^{2}}}{9 \, {\left (d e^{6} x^{5} + 5 \, d^{2} e^{5} x^{4} + 10 \, d^{3} e^{4} x^{3} + 10 \, d^{4} e^{3} x^{2} + 5 \, d^{5} e^{2} x + d^{6} e\right )}} - \frac {4 \, \sqrt {-e^{2} x^{2} + d^{2}}}{63 \, {\left (d^{2} e^{5} x^{4} + 4 \, d^{3} e^{4} x^{3} + 6 \, d^{4} e^{3} x^{2} + 4 \, d^{5} e^{2} x + d^{6} e\right )}} - \frac {4 \, \sqrt {-e^{2} x^{2} + d^{2}}}{105 \, {\left (d^{3} e^{4} x^{3} + 3 \, d^{4} e^{3} x^{2} + 3 \, d^{5} e^{2} x + d^{6} e\right )}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}}}{315 \, {\left (d^{4} e^{3} x^{2} + 2 \, d^{5} e^{2} x + d^{6} e\right )}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}}}{315 \, {\left (d^{5} e^{2} x + d^{6} e\right )}} \]

[In]

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

[Out]

-1/9*sqrt(-e^2*x^2 + d^2)/(d*e^6*x^5 + 5*d^2*e^5*x^4 + 10*d^3*e^4*x^3 + 10*d^4*e^3*x^2 + 5*d^5*e^2*x + d^6*e)
- 4/63*sqrt(-e^2*x^2 + d^2)/(d^2*e^5*x^4 + 4*d^3*e^4*x^3 + 6*d^4*e^3*x^2 + 4*d^5*e^2*x + d^6*e) - 4/105*sqrt(-
e^2*x^2 + d^2)/(d^3*e^4*x^3 + 3*d^4*e^3*x^2 + 3*d^5*e^2*x + d^6*e) - 8/315*sqrt(-e^2*x^2 + d^2)/(d^4*e^3*x^2 +
2*d^5*e^2*x + d^6*e) - 8/315*sqrt(-e^2*x^2 + d^2)/(d^5*e^2*x + d^6*e)

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(d+e x)^5 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {-\frac {128 i \, \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d^{5}} + \frac {35 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {9}{2}} + 180 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {7}{2}} + 378 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} + 420 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} + 315 \, \sqrt {\frac {2 \, d}{e x + d} - 1}}{d^{5} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}}{5040 \, {\left | e \right |}} \]

[In]

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

[Out]

-1/5040*(-128*I*sgn(1/(e*x + d))*sgn(e)/d^5 + (35*(2*d/(e*x + d) - 1)^(9/2) + 180*(2*d/(e*x + d) - 1)^(7/2) +
378*(2*d/(e*x + d) - 1)^(5/2) + 420*(2*d/(e*x + d) - 1)^(3/2) + 315*sqrt(2*d/(e*x + d) - 1))/(d^5*sgn(1/(e*x +
d))*sgn(e)))/abs(e)

Mupad [B] (verification not implemented)

Time = 9.77 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(d+e x)^5 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {d^2-e^2\,x^2}}{9\,d\,e\,{\left (d+e\,x\right )}^5}-\frac {4\,\sqrt {d^2-e^2\,x^2}}{63\,d^2\,e\,{\left (d+e\,x\right )}^4}-\frac {4\,\sqrt {d^2-e^2\,x^2}}{105\,d^3\,e\,{\left (d+e\,x\right )}^3}-\frac {8\,\sqrt {d^2-e^2\,x^2}}{315\,d^4\,e\,{\left (d+e\,x\right )}^2}-\frac {8\,\sqrt {d^2-e^2\,x^2}}{315\,d^5\,e\,\left (d+e\,x\right )} \]

[In]

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

[Out]

- (d^2 - e^2*x^2)^(1/2)/(9*d*e*(d + e*x)^5) - (4*(d^2 - e^2*x^2)^(1/2))/(63*d^2*e*(d + e*x)^4) - (4*(d^2 - e^2
*x^2)^(1/2))/(105*d^3*e*(d + e*x)^3) - (8*(d^2 - e^2*x^2)^(1/2))/(315*d^4*e*(d + e*x)^2) - (8*(d^2 - e^2*x^2)^
(1/2))/(315*d^5*e*(d + e*x))