\(\int \frac {(d^2-e^2 x^2)^{5/2}}{x^5 (d+e x)^4} \, dx\) [208]

Optimal result

Integrand size = 27, antiderivative size = 170 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^5 (d+e x)^4} \, dx=\frac {8 e^4 (d-e x)}{d^3 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}+\frac {4 e \sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {31 e^2 \sqrt {d^2-e^2 x^2}}{8 d^2 x^2}+\frac {32 e^3 \sqrt {d^2-e^2 x^2}}{3 d^3 x}-\frac {95 e^4 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^3} \]

[Out]

-95/8*e^4*arctanh((-e^2*x^2+d^2)^(1/2)/d)/d^3+8*e^4*(-e*x+d)/d^3/(-e^2*x^2+d^2)^(1/2)-1/4*(-e^2*x^2+d^2)^(1/2)
/x^4+4/3*e*(-e^2*x^2+d^2)^(1/2)/d/x^3-31/8*e^2*(-e^2*x^2+d^2)^(1/2)/d^2/x^2+32/3*e^3*(-e^2*x^2+d^2)^(1/2)/d^3/
x

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {866, 1819, 1821, 821, 272, 65, 214} \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^5 (d+e x)^4} \, dx=-\frac {95 e^4 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^3}-\frac {31 e^2 \sqrt {d^2-e^2 x^2}}{8 d^2 x^2}-\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}+\frac {4 e \sqrt {d^2-e^2 x^2}}{3 d x^3}+\frac {8 e^4 (d-e x)}{d^3 \sqrt {d^2-e^2 x^2}}+\frac {32 e^3 \sqrt {d^2-e^2 x^2}}{3 d^3 x} \]

[In]

Int[(d^2 - e^2*x^2)^(5/2)/(x^5*(d + e*x)^4),x]

[Out]

(8*e^4*(d - e*x))/(d^3*Sqrt[d^2 - e^2*x^2]) - Sqrt[d^2 - e^2*x^2]/(4*x^4) + (4*e*Sqrt[d^2 - e^2*x^2])/(3*d*x^3
) - (31*e^2*Sqrt[d^2 - e^2*x^2])/(8*d^2*x^2) + (32*e^3*Sqrt[d^2 - e^2*x^2])/(3*d^3*x) - (95*e^4*ArcTanh[Sqrt[d
^2 - e^2*x^2]/d])/(8*d^3)

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[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] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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 272

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

Rule 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0
] && EqQ[Simplify[m + 2*p + 3], 0]

Rule 866

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

Rule 1819

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1821

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]}, Simp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Dist[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] && LtQ[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])

Rubi steps \begin{align*} \text {integral}& = \int \frac {(d-e x)^4}{x^5 \left (d^2-e^2 x^2\right )^{3/2}} \, dx \\ & = \frac {8 e^4 (d-e x)}{d^3 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-d^4+4 d^3 e x-7 d^2 e^2 x^2+8 d e^3 x^3-8 e^4 x^4}{x^5 \sqrt {d^2-e^2 x^2}} \, dx}{d^2} \\ & = \frac {8 e^4 (d-e x)}{d^3 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}+\frac {\int \frac {-16 d^5 e+31 d^4 e^2 x-32 d^3 e^3 x^2+32 d^2 e^4 x^3}{x^4 \sqrt {d^2-e^2 x^2}} \, dx}{4 d^4} \\ & = \frac {8 e^4 (d-e x)}{d^3 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}+\frac {4 e \sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {\int \frac {-93 d^6 e^2+128 d^5 e^3 x-96 d^4 e^4 x^2}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{12 d^6} \\ & = \frac {8 e^4 (d-e x)}{d^3 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}+\frac {4 e \sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {31 e^2 \sqrt {d^2-e^2 x^2}}{8 d^2 x^2}+\frac {\int \frac {-256 d^7 e^3+285 d^6 e^4 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{24 d^8} \\ & = \frac {8 e^4 (d-e x)}{d^3 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}+\frac {4 e \sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {31 e^2 \sqrt {d^2-e^2 x^2}}{8 d^2 x^2}+\frac {32 e^3 \sqrt {d^2-e^2 x^2}}{3 d^3 x}+\frac {\left (95 e^4\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{8 d^2} \\ & = \frac {8 e^4 (d-e x)}{d^3 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}+\frac {4 e \sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {31 e^2 \sqrt {d^2-e^2 x^2}}{8 d^2 x^2}+\frac {32 e^3 \sqrt {d^2-e^2 x^2}}{3 d^3 x}+\frac {\left (95 e^4\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d^2-e^2 x}} \, dx,x,x^2\right )}{16 d^2} \\ & = \frac {8 e^4 (d-e x)}{d^3 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}+\frac {4 e \sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {31 e^2 \sqrt {d^2-e^2 x^2}}{8 d^2 x^2}+\frac {32 e^3 \sqrt {d^2-e^2 x^2}}{3 d^3 x}-\frac {\left (95 e^2\right ) \text {Subst}\left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^2}{e^2}} \, dx,x,\sqrt {d^2-e^2 x^2}\right )}{8 d^2} \\ & = \frac {8 e^4 (d-e x)}{d^3 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}+\frac {4 e \sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {31 e^2 \sqrt {d^2-e^2 x^2}}{8 d^2 x^2}+\frac {32 e^3 \sqrt {d^2-e^2 x^2}}{3 d^3 x}-\frac {95 e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.68 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^5 (d+e x)^4} \, dx=\frac {\frac {\sqrt {d^2-e^2 x^2} \left (-6 d^4+26 d^3 e x-61 d^2 e^2 x^2+163 d e^3 x^3+448 e^4 x^4\right )}{x^4 (d+e x)}+570 e^4 \text {arctanh}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{24 d^3} \]

[In]

Integrate[(d^2 - e^2*x^2)^(5/2)/(x^5*(d + e*x)^4),x]

[Out]

((Sqrt[d^2 - e^2*x^2]*(-6*d^4 + 26*d^3*e*x - 61*d^2*e^2*x^2 + 163*d*e^3*x^3 + 448*e^4*x^4))/(x^4*(d + e*x)) +
570*e^4*ArcTanh[(Sqrt[-e^2]*x - Sqrt[d^2 - e^2*x^2])/d])/(24*d^3)

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.85

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-256 e^{3} x^{3}+93 d \,e^{2} x^{2}-32 d^{2} e x +6 d^{3}\right )}{24 d^{3} x^{4}}-\frac {95 e^{4} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 d^{2} \sqrt {d^{2}}}+\frac {8 e^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{d^{3} \left (x +\frac {d}{e}\right )}\)	\(144\)
default	\(\text {Expression too large to display}\)	\(1799\)

[In]

int((-e^2*x^2+d^2)^(5/2)/x^5/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

-1/24*(-e^2*x^2+d^2)^(1/2)*(-256*e^3*x^3+93*d*e^2*x^2-32*d^2*e*x+6*d^3)/d^3/x^4-95/8/d^2*e^4/(d^2)^(1/2)*ln((2
*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)+8/d^3*e^3/(x+d/e)*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.80 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^5 (d+e x)^4} \, dx=\frac {192 \, e^{5} x^{5} + 192 \, d e^{4} x^{4} + 285 \, {\left (e^{5} x^{5} + d e^{4} x^{4}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (448 \, e^{4} x^{4} + 163 \, d e^{3} x^{3} - 61 \, d^{2} e^{2} x^{2} + 26 \, d^{3} e x - 6 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{24 \, {\left (d^{3} e x^{5} + d^{4} x^{4}\right )}} \]

[In]

integrate((-e^2*x^2+d^2)^(5/2)/x^5/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/24*(192*e^5*x^5 + 192*d*e^4*x^4 + 285*(e^5*x^5 + d*e^4*x^4)*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (448*e^4*x^
4 + 163*d*e^3*x^3 - 61*d^2*e^2*x^2 + 26*d^3*e*x - 6*d^4)*sqrt(-e^2*x^2 + d^2))/(d^3*e*x^5 + d^4*x^4)

Sympy [F]

\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^5 (d+e x)^4} \, dx=\int \frac {\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}{x^{5} \left (d + e x\right )^{4}}\, dx \]

[In]

integrate((-e**2*x**2+d**2)**(5/2)/x**5/(e*x+d)**4,x)

[Out]

Integral((-(-d + e*x)*(d + e*x))**(5/2)/(x**5*(d + e*x)**4), x)

Maxima [F]

\[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^5 (d+e x)^4} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{4} x^{5}} \,d x } \]

[In]

integrate((-e^2*x^2+d^2)^(5/2)/x^5/(e*x+d)^4,x, algorithm="maxima")

[Out]

integrate((-e^2*x^2 + d^2)^(5/2)/((e*x + d)^4*x^5), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 392 vs. \(2 (149) = 298\).

Time = 0.33 (sec) , antiderivative size = 392, normalized size of antiderivative = 2.31 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^5 (d+e x)^4} \, dx=\frac {{\left (3 \, e^{5} - \frac {29 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{3}}{x} + \frac {160 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e}{x^{2}} - \frac {864 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e x^{3}} - \frac {4128 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e^{3} x^{4}}\right )} e^{8} x^{4}}{192 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{3} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )} {\left | e \right |}} - \frac {95 \, e^{5} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{8 \, d^{3} {\left | e \right |}} + \frac {\frac {1056 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{9} e^{5} {\left | e \right |}}{x} - \frac {192 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{9} e^{3} {\left | e \right |}}{x^{2}} + \frac {32 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{9} e {\left | e \right |}}{x^{3}} - \frac {3 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{9} {\left | e \right |}}{e x^{4}}}{192 \, d^{12} e^{4}} \]

[In]

integrate((-e^2*x^2+d^2)^(5/2)/x^5/(e*x+d)^4,x, algorithm="giac")

[Out]

1/192*(3*e^5 - 29*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e^3/x + 160*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*e/x^2
- 864*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3/(e*x^3) - 4128*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4/(e^3*x^4))*e^
8*x^4/((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d^3*((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) + 1)*abs(e)) - 9
5/8*e^5*log(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/(d^3*abs(e)) + 1/192*(1056*(d*e + sq
rt(-e^2*x^2 + d^2)*abs(e))*d^9*e^5*abs(e)/x - 192*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^9*e^3*abs(e)/x^2 + 3
2*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^9*e*abs(e)/x^3 - 3*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d^9*abs(e)/
(e*x^4))/(d^12*e^4)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^5 (d+e x)^4} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^5\,{\left (d+e\,x\right )}^4} \,d x \]

[In]

int((d^2 - e^2*x^2)^(5/2)/(x^5*(d + e*x)^4),x)

[Out]

int((d^2 - e^2*x^2)^(5/2)/(x^5*(d + e*x)^4), x)