\(\int \frac {1}{(d+e x)^2 (d^2-e^2 x^2)^{5/2}} \, dx\) [843]

Optimal result

Integrand size = 24, antiderivative size = 115 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 x}{21 d^6 \sqrt {d^2-e^2 x^2}} \]

[Out]

4/21*x/d^4/(-e^2*x^2+d^2)^(3/2)-1/7/d/e/(e*x+d)^2/(-e^2*x^2+d^2)^(3/2)-1/7/d^2/e/(e*x+d)/(-e^2*x^2+d^2)^(3/2)+
8/21*x/d^6/(-e^2*x^2+d^2)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {673, 198, 197} \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 x}{21 d^6 \sqrt {d^2-e^2 x^2}}+\frac {4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}} \]

[In]

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

[Out]

(4*x)/(21*d^4*(d^2 - e^2*x^2)^(3/2)) - 1/(7*d*e*(d + e*x)^2*(d^2 - e^2*x^2)^(3/2)) - 1/(7*d^2*e*(d + e*x)*(d^2
- e^2*x^2)^(3/2)) + (8*x)/(21*d^6*Sqrt[d^2 - e^2*x^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

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 {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {5 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx}{7 d} \\ & = -\frac {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {4 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{7 d^2} \\ & = \frac {4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{21 d^4} \\ & = \frac {4 x}{21 d^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {1}{7 d^2 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 x}{21 d^6 \sqrt {d^2-e^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-6 d^5+9 d^4 e x+24 d^3 e^2 x^2+4 d^2 e^3 x^3-16 d e^4 x^4-8 e^5 x^5\right )}{21 d^6 e (d-e x)^2 (d+e x)^4} \]

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-6*d^5 + 9*d^4*e*x + 24*d^3*e^2*x^2 + 4*d^2*e^3*x^3 - 16*d*e^4*x^4 - 8*e^5*x^5))/(21*d^6
*e*(d - e*x)^2*(d + e*x)^4)

Maple [A] (verified)

Time = 2.39 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.77

method	result	size
gosper	\(-\frac {\left (-e x +d \right ) \left (8 e^{5} x^{5}+16 x^{4} d \,e^{4}-4 d^{2} e^{3} x^{3}-24 d^{3} e^{2} x^{2}-9 d^{4} e x +6 d^{5}\right )}{21 \left (e x +d \right ) d^{6} e \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}\)	\(88\)
trager	\(-\frac {\left (8 e^{5} x^{5}+16 x^{4} d \,e^{4}-4 d^{2} e^{3} x^{3}-24 d^{3} e^{2} x^{2}-9 d^{4} e x +6 d^{5}\right ) \sqrt {-x^{2} e^{2}+d^{2}}}{21 d^{6} \left (e x +d \right )^{4} \left (-e x +d \right )^{2} e}\)	\(90\)
default	\(\frac {-\frac {1}{7 d e \left (x +\frac {d}{e}\right )^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {5 e \left (-\frac {1}{5 d e \left (x +\frac {d}{e}\right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {4 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{6 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e^{2} d^{4} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}\right )}{7 d}}{e^{2}}\)	\(216\)

[In]

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

[Out]

-1/21*(-e*x+d)*(8*e^5*x^5+16*d*e^4*x^4-4*d^2*e^3*x^3-24*d^3*e^2*x^2-9*d^4*e*x+6*d^5)/(e*x+d)/d^6/e/(-e^2*x^2+d
^2)^(5/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (99) = 198\).

Time = 0.29 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.77 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {6 \, e^{6} x^{6} + 12 \, d e^{5} x^{5} - 6 \, d^{2} e^{4} x^{4} - 24 \, d^{3} e^{3} x^{3} - 6 \, d^{4} e^{2} x^{2} + 12 \, d^{5} e x + 6 \, d^{6} + {\left (8 \, e^{5} x^{5} + 16 \, d e^{4} x^{4} - 4 \, d^{2} e^{3} x^{3} - 24 \, d^{3} e^{2} x^{2} - 9 \, d^{4} e x + 6 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{21 \, {\left (d^{6} e^{7} x^{6} + 2 \, d^{7} e^{6} x^{5} - d^{8} e^{5} x^{4} - 4 \, d^{9} e^{4} x^{3} - d^{10} e^{3} x^{2} + 2 \, d^{11} e^{2} x + d^{12} e\right )}} \]

[In]

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

[Out]

-1/21*(6*e^6*x^6 + 12*d*e^5*x^5 - 6*d^2*e^4*x^4 - 24*d^3*e^3*x^3 - 6*d^4*e^2*x^2 + 12*d^5*e*x + 6*d^6 + (8*e^5
*x^5 + 16*d*e^4*x^4 - 4*d^2*e^3*x^3 - 24*d^3*e^2*x^2 - 9*d^4*e*x + 6*d^5)*sqrt(-e^2*x^2 + d^2))/(d^6*e^7*x^6 +
2*d^7*e^6*x^5 - d^8*e^5*x^4 - 4*d^9*e^4*x^3 - d^10*e^3*x^2 + 2*d^11*e^2*x + d^12*e)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.36 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {1}{7 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{3} x^{2} + 2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} e\right )}} - \frac {1}{7 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} e\right )}} + \frac {4 \, x}{21 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4}} + \frac {8 \, x}{21 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{6}} \]

[In]

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

[Out]

-1/7/((-e^2*x^2 + d^2)^(3/2)*d*e^3*x^2 + 2*(-e^2*x^2 + d^2)^(3/2)*d^2*e^2*x + (-e^2*x^2 + d^2)^(3/2)*d^3*e) -
1/7/((-e^2*x^2 + d^2)^(3/2)*d^2*e^2*x + (-e^2*x^2 + d^2)^(3/2)*d^3*e) + 4/21*x/((-e^2*x^2 + d^2)^(3/2)*d^4) +
8/21*x/(sqrt(-e^2*x^2 + d^2)*d^6)

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.32 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.10 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {e^{5} {\left (\frac {14 \, {\left (\frac {15 \, d}{e x + d} - 7\right )}}{d^{6} e^{5} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )} - \frac {3 \, d^{36} e^{30} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {7}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{6} \mathrm {sgn}\left (e\right )^{6} + 21 \, d^{36} e^{30} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{6} \mathrm {sgn}\left (e\right )^{6} + 70 \, d^{36} e^{30} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{6} \mathrm {sgn}\left (e\right )^{6} + 210 \, d^{36} e^{30} \sqrt {\frac {2 \, d}{e x + d} - 1} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{6} \mathrm {sgn}\left (e\right )^{6}}{d^{42} e^{35} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{7} \mathrm {sgn}\left (e\right )^{7}}\right )} + \frac {256 i \, \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d^{6}}}{672 \, {\left | e \right |}} \]

[In]

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

[Out]

1/672*(e^5*(14*(15*d/(e*x + d) - 7)/(d^6*e^5*(2*d/(e*x + d) - 1)^(3/2)*sgn(1/(e*x + d))*sgn(e)) - (3*d^36*e^30
*(2*d/(e*x + d) - 1)^(7/2)*sgn(1/(e*x + d))^6*sgn(e)^6 + 21*d^36*e^30*(2*d/(e*x + d) - 1)^(5/2)*sgn(1/(e*x + d
))^6*sgn(e)^6 + 70*d^36*e^30*(2*d/(e*x + d) - 1)^(3/2)*sgn(1/(e*x + d))^6*sgn(e)^6 + 210*d^36*e^30*sqrt(2*d/(e
*x + d) - 1)*sgn(1/(e*x + d))^6*sgn(e)^6)/(d^42*e^35*sgn(1/(e*x + d))^7*sgn(e)^7)) + 256*I*sgn(1/(e*x + d))*sg
n(e)/d^6)/abs(e)

Mupad [B] (verification not implemented)

Time = 9.89 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.21 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {11\,x}{42\,d^4}-\frac {5}{28\,d^3\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}}{28\,d^3\,e\,{\left (d+e\,x\right )}^4}-\frac {\sqrt {d^2-e^2\,x^2}}{14\,d^4\,e\,{\left (d+e\,x\right )}^3}+\frac {8\,x\,\sqrt {d^2-e^2\,x^2}}{21\,d^6\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \]

[In]

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

[Out]

((d^2 - e^2*x^2)^(1/2)*((11*x)/(42*d^4) - 5/(28*d^3*e)))/((d + e*x)^2*(d - e*x)^2) - (d^2 - e^2*x^2)^(1/2)/(28
*d^3*e*(d + e*x)^4) - (d^2 - e^2*x^2)^(1/2)/(14*d^4*e*(d + e*x)^3) + (8*x*(d^2 - e^2*x^2)^(1/2))/(21*d^6*(d +
e*x)*(d - e*x))