3.2.35 \(\int \frac {1}{x^3 (d+e x) (d^2-e^2 x^2)^{3/2}} \, dx\) [135]

3.2.35.1 Optimal result

Integrand size = 27, antiderivative size = 152 \[ \int \frac {1}{x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {5 d-4 e x}{3 d^4 x^2 \sqrt {d^2-e^2 x^2}}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d^5 x^2}+\frac {8 e \sqrt {d^2-e^2 x^2}}{3 d^6 x}-\frac {5 e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^6} \]

output

-5/2*e^2*arctanh((-e^2*x^2+d^2)^(1/2)/d)/d^6+1/3*(-4*e*x+5*d)/d^4/x^2/(-e^ 
2*x^2+d^2)^(1/2)+1/3/d^2/x^2/(e*x+d)/(-e^2*x^2+d^2)^(1/2)-5/2*(-e^2*x^2+d^ 
2)^(1/2)/d^5/x^2+8/3*e*(-e^2*x^2+d^2)^(1/2)/d^6/x
 

3.2.35.2 Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {\frac {d \sqrt {d^2-e^2 x^2} \left (3 d^4-3 d^3 e x-23 d^2 e^2 x^2+d e^3 x^3+16 e^4 x^4\right )}{x^2 (-d+e x) (d+e x)^2}-15 \sqrt {d^2} e^2 \log (x)+15 \sqrt {d^2} e^2 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{6 d^7} \]

input

Integrate[1/(x^3*(d + e*x)*(d^2 - e^2*x^2)^(3/2)),x]
 

output

((d*Sqrt[d^2 - e^2*x^2]*(3*d^4 - 3*d^3*e*x - 23*d^2*e^2*x^2 + d*e^3*x^3 + 
16*e^4*x^4))/(x^2*(-d + e*x)*(d + e*x)^2) - 15*Sqrt[d^2]*e^2*Log[x] + 15*S 
qrt[d^2]*e^2*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/(6*d^7)
 

3.2.35.3 Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {569, 25, 532, 25, 2338, 27, 534, 243, 73, 221}

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.

\(\displaystyle \int \frac {1}{x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 569

\(\displaystyle \frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\int -\frac {5 d-4 e x}{x^3 \left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {5 d-4 e x}{x^3 \left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 532

\(\displaystyle \frac {\frac {e^2 (5 d-4 e x)}{d^4 \sqrt {d^2-e^2 x^2}}-\frac {\int -\frac {\frac {5 e^2 x^2}{d}-4 e x+5 d}{x^3 \sqrt {d^2-e^2 x^2}}dx}{d^2}}{3 d^2}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {\frac {5 e^2 x^2}{d}-4 e x+5 d}{x^3 \sqrt {d^2-e^2 x^2}}dx}{d^2}+\frac {e^2 (5 d-4 e x)}{d^4 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 2338

\(\displaystyle \frac {\frac {-\frac {\int \frac {d e (8 d-15 e x)}{x^2 \sqrt {d^2-e^2 x^2}}dx}{2 d^2}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^2}}{d^2}+\frac {e^2 (5 d-4 e x)}{d^4 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {-\frac {e \int \frac {8 d-15 e x}{x^2 \sqrt {d^2-e^2 x^2}}dx}{2 d}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^2}}{d^2}+\frac {e^2 (5 d-4 e x)}{d^4 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 534

\(\displaystyle \frac {\frac {-\frac {e \left (-15 e \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx-\frac {8 \sqrt {d^2-e^2 x^2}}{d x}\right )}{2 d}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^2}}{d^2}+\frac {e^2 (5 d-4 e x)}{d^4 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\frac {-\frac {e \left (-\frac {15}{2} e \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2-\frac {8 \sqrt {d^2-e^2 x^2}}{d x}\right )}{2 d}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^2}}{d^2}+\frac {e^2 (5 d-4 e x)}{d^4 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {-\frac {e \left (\frac {15 \int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}}{e}-\frac {8 \sqrt {d^2-e^2 x^2}}{d x}\right )}{2 d}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^2}}{d^2}+\frac {e^2 (5 d-4 e x)}{d^4 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {-\frac {e \left (\frac {15 e \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d}-\frac {8 \sqrt {d^2-e^2 x^2}}{d x}\right )}{2 d}-\frac {5 \sqrt {d^2-e^2 x^2}}{2 d x^2}}{d^2}+\frac {e^2 (5 d-4 e x)}{d^4 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {1}{3 d^2 x^2 (d+e x) \sqrt {d^2-e^2 x^2}}\)

input

Int[1/(x^3*(d + e*x)*(d^2 - e^2*x^2)^(3/2)),x]
 

output

1/(3*d^2*x^2*(d + e*x)*Sqrt[d^2 - e^2*x^2]) + ((e^2*(5*d - 4*e*x))/(d^4*Sq 
rt[d^2 - e^2*x^2]) + ((-5*Sqrt[d^2 - e^2*x^2])/(2*d*x^2) - (e*((-8*Sqrt[d^ 
2 - e^2*x^2])/(d*x) + (15*e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d))/(2*d))/d^2 
)/(3*d^2)
 

3.2.35.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

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

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 532

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe 
ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol 
ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) 
*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[x^m 
*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), 
x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, 
 -1] && IntegerQ[2*p]
 

rule 534

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d   Int[ 
x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 
0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
 

rule 569

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

rule 2338

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ 
Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S 
imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( 
m + 1))   Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( 
m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt 
Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
 

3.2.35.4 Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.36

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-2 e x +d \right )}{2 d^{6} x^{2}}-\frac {5 e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{5} \sqrt {d^{2}}}+\frac {23 e \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{12 d^{6} \left (x +\frac {d}{e}\right )}-\frac {e \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{4 d^{6} \left (x -\frac {d}{e}\right )}+\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{6 d^{5} \left (x +\frac {d}{e}\right )^{2}}\)	\(206\)
default	\(\frac {-\frac {1}{2 d^{2} x^{2} \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {3 e^{2} \left (\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}\right )}{2 d^{2}}}{d}+\frac {e^{2} \left (\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}\right )}{d^{3}}-\frac {e \left (-\frac {1}{d^{2} x \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {2 e^{2} x}{d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{d^{2}}-\frac {e^{2} \left (-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{3 e \,d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{d^{3}}\)	\(325\)

input

int(1/x^3/(e*x+d)/(-e^2*x^2+d^2)^(3/2),x,method=_RETURNVERBOSE)
 

output

-1/2*(-e^2*x^2+d^2)^(1/2)*(-2*e*x+d)/d^6/x^2-5/2/d^5*e^2/(d^2)^(1/2)*ln((2 
*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)+23/12/d^6*e/(x+d/e)*(-(x+d/e)^ 
2*e^2+2*d*e*(x+d/e))^(1/2)-1/4/d^6*e/(x-d/e)*(-(x-d/e)^2*e^2-2*d*e*(x-d/e) 
)^(1/2)+1/6/d^5/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)
 

3.2.35.5 Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {14 \, e^{5} x^{5} + 14 \, d e^{4} x^{4} - 14 \, d^{2} e^{3} x^{3} - 14 \, d^{3} e^{2} x^{2} + 15 \, {\left (e^{5} x^{5} + d e^{4} x^{4} - d^{2} e^{3} x^{3} - d^{3} e^{2} x^{2}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (16 \, e^{4} x^{4} + d e^{3} x^{3} - 23 \, d^{2} e^{2} x^{2} - 3 \, d^{3} e x + 3 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6 \, {\left (d^{6} e^{3} x^{5} + d^{7} e^{2} x^{4} - d^{8} e x^{3} - d^{9} x^{2}\right )}} \]

input

integrate(1/x^3/(e*x+d)/(-e^2*x^2+d^2)^(3/2),x, algorithm="fricas")
 

output

1/6*(14*e^5*x^5 + 14*d*e^4*x^4 - 14*d^2*e^3*x^3 - 14*d^3*e^2*x^2 + 15*(e^5 
*x^5 + d*e^4*x^4 - d^2*e^3*x^3 - d^3*e^2*x^2)*log(-(d - sqrt(-e^2*x^2 + d^ 
2))/x) + (16*e^4*x^4 + d*e^3*x^3 - 23*d^2*e^2*x^2 - 3*d^3*e*x + 3*d^4)*sqr 
t(-e^2*x^2 + d^2))/(d^6*e^3*x^5 + d^7*e^2*x^4 - d^8*e*x^3 - d^9*x^2)
 

3.2.35.6 Sympy [F]

\[ \int \frac {1}{x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{3} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]

input

integrate(1/x**3/(e*x+d)/(-e**2*x**2+d**2)**(3/2),x)
 

output

Integral(1/(x**3*(-(-d + e*x)*(d + e*x))**(3/2)*(d + e*x)), x)
 

3.2.35.7 Maxima [F]

\[ \int \frac {1}{x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )} x^{3}} \,d x } \]

input

integrate(1/x^3/(e*x+d)/(-e^2*x^2+d^2)^(3/2),x, algorithm="maxima")
 

output

integrate(1/((-e^2*x^2 + d^2)^(3/2)*(e*x + d)*x^3), x)
 

3.2.35.8 Giac [F]

\[ \int \frac {1}{x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )} x^{3}} \,d x } \]

input

integrate(1/x^3/(e*x+d)/(-e^2*x^2+d^2)^(3/2),x, algorithm="giac")
 

output

integrate(1/((-e^2*x^2 + d^2)^(3/2)*(e*x + d)*x^3), x)
 

3.2.35.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^3 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{x^3\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \]

input

int(1/(x^3*(d^2 - e^2*x^2)^(3/2)*(d + e*x)),x)
 

output

int(1/(x^3*(d^2 - e^2*x^2)^(3/2)*(d + e*x)), x)