3.855 \(\int \frac {1}{(d+e x) (d^2-e^2 x^2)^{7/2}} \, dx\)

Optimal. Leaf size=106 \[ -\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {16 x}{35 d^7 \sqrt {d^2-e^2 x^2}}+\frac {8 x}{35 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {6 x}{35 d^3 \left (d^2-e^2 x^2\right )^{5/2}} \]

[Out]

6/35*x/d^3/(-e^2*x^2+d^2)^(5/2)-1/7/d/e/(e*x+d)/(-e^2*x^2+d^2)^(5/2)+8/35*x/d^5/(-e^2*x^2+d^2)^(3/2)+16/35*x/d
^7/(-e^2*x^2+d^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {659, 192, 191} \[ \frac {16 x}{35 d^7 \sqrt {d^2-e^2 x^2}}+\frac {8 x}{35 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {6 x}{35 d^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(6*x)/(35*d^3*(d^2 - e^2*x^2)^(5/2)) - 1/(7*d*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) + (8*x)/(35*d^5*(d^2 - e^2*x^
2)^(3/2)) + (16*x)/(35*d^7*Sqrt[d^2 - e^2*x^2])

Rule 191

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 192

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 659

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*} \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {6 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{7 d}\\ &=\frac {6 x}{35 d^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {24 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{35 d^3}\\ &=\frac {6 x}{35 d^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {8 x}{35 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{35 d^5}\\ &=\frac {6 x}{35 d^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {8 x}{35 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {16 x}{35 d^7 \sqrt {d^2-e^2 x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 104, normalized size = 0.98 \[ \frac {\sqrt {d^2-e^2 x^2} \left (-5 d^6+30 d^5 e x+30 d^4 e^2 x^2-40 d^3 e^3 x^3-40 d^2 e^4 x^4+16 d e^5 x^5+16 e^6 x^6\right )}{35 d^7 e (d-e x)^3 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)*(d^2 - e^2*x^2)^(7/2)),x]

[Out]

(Sqrt[d^2 - e^2*x^2]*(-5*d^6 + 30*d^5*e*x + 30*d^4*e^2*x^2 - 40*d^3*e^3*x^3 - 40*d^2*e^4*x^4 + 16*d*e^5*x^5 +
16*e^6*x^6))/(35*d^7*e*(d - e*x)^3*(d + e*x)^4)

________________________________________________________________________________________

fricas [B]  time = 1.17, size = 236, normalized size = 2.23 \[ -\frac {5 \, e^{7} x^{7} + 5 \, d e^{6} x^{6} - 15 \, d^{2} e^{5} x^{5} - 15 \, d^{3} e^{4} x^{4} + 15 \, d^{4} e^{3} x^{3} + 15 \, d^{5} e^{2} x^{2} - 5 \, d^{6} e x - 5 \, d^{7} + {\left (16 \, e^{6} x^{6} + 16 \, d e^{5} x^{5} - 40 \, d^{2} e^{4} x^{4} - 40 \, d^{3} e^{3} x^{3} + 30 \, d^{4} e^{2} x^{2} + 30 \, d^{5} e x - 5 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{35 \, {\left (d^{7} e^{8} x^{7} + d^{8} e^{7} x^{6} - 3 \, d^{9} e^{6} x^{5} - 3 \, d^{10} e^{5} x^{4} + 3 \, d^{11} e^{4} x^{3} + 3 \, d^{12} e^{3} x^{2} - d^{13} e^{2} x - d^{14} e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/35*(5*e^7*x^7 + 5*d*e^6*x^6 - 15*d^2*e^5*x^5 - 15*d^3*e^4*x^4 + 15*d^4*e^3*x^3 + 15*d^5*e^2*x^2 - 5*d^6*e*x
- 5*d^7 + (16*e^6*x^6 + 16*d*e^5*x^5 - 40*d^2*e^4*x^4 - 40*d^3*e^3*x^3 + 30*d^4*e^2*x^2 + 30*d^5*e*x - 5*d^6)
*sqrt(-e^2*x^2 + d^2))/(d^7*e^8*x^7 + d^8*e^7*x^6 - 3*d^9*e^6*x^5 - 3*d^10*e^5*x^4 + 3*d^11*e^4*x^3 + 3*d^12*e
^3*x^2 - d^13*e^2*x - d^14*e)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to transpose Error: Bad Argument Value

________________________________________________________________________________________

maple [A]  time = 0.05, size = 92, normalized size = 0.87 \[ -\frac {\left (-e x +d \right ) \left (-16 e^{6} x^{6}-16 e^{5} x^{5} d +40 e^{4} x^{4} d^{2}+40 e^{3} x^{3} d^{3}-30 e^{2} x^{2} d^{4}-30 x \,d^{5} e +5 d^{6}\right )}{35 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{7} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/35*(-e*x+d)*(-16*e^6*x^6-16*d*e^5*x^5+40*d^2*e^4*x^4+40*d^3*e^3*x^3-30*d^4*e^2*x^2-30*d^5*e*x+5*d^6)/d^7/e/
(-e^2*x^2+d^2)^(7/2)

________________________________________________________________________________________

maxima [A]  time = 1.43, size = 105, normalized size = 0.99 \[ -\frac {1}{7 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e\right )}} + \frac {6 \, x}{35 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}} + \frac {8 \, x}{35 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5}} + \frac {16 \, x}{35 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7/((-e^2*x^2 + d^2)^(5/2)*d*e^2*x + (-e^2*x^2 + d^2)^(5/2)*d^2*e) + 6/35*x/((-e^2*x^2 + d^2)^(5/2)*d^3) + 8
/35*x/((-e^2*x^2 + d^2)^(3/2)*d^5) + 16/35*x/(sqrt(-e^2*x^2 + d^2)*d^7)

________________________________________________________________________________________

mupad [B]  time = 0.63, size = 155, normalized size = 1.46 \[ \frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {17\,x}{70\,d^3}-\frac {1}{7\,d^2\,e}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}+\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {8\,x}{35\,d^5}+\frac {1}{56\,d^4\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}}{56\,d^4\,e\,{\left (d+e\,x\right )}^4}+\frac {16\,x\,\sqrt {d^2-e^2\,x^2}}{35\,d^7\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d^2 - e^2*x^2)^(1/2)*((17*x)/(70*d^3) - 1/(7*d^2*e)))/((d + e*x)^3*(d - e*x)^3) + ((d^2 - e^2*x^2)^(1/2)*((8
*x)/(35*d^5) + 1/(56*d^4*e)))/((d + e*x)^2*(d - e*x)^2) - (d^2 - e^2*x^2)^(1/2)/(56*d^4*e*(d + e*x)^4) + (16*x
*(d^2 - e^2*x^2)^(1/2))/(35*d^7*(d + e*x)*(d - e*x))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (d + e x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________