\(\int \frac {(d+e x)^2}{x^5 \sqrt {d^2-e^2 x^2}} \, dx\) [42]

Optimal result

Integrand size = 27, antiderivative size = 140 \[ \int \frac {(d+e x)^2}{x^5 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}-\frac {2 e \sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {7 e^2 \sqrt {d^2-e^2 x^2}}{8 d^2 x^2}-\frac {4 e^3 \sqrt {d^2-e^2 x^2}}{3 d^3 x}-\frac {7 e^4 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1821, 849, 821, 272, 65, 214} \[ \int \frac {(d+e x)^2}{x^5 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {7 e^4 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^3}-\frac {7 e^2 \sqrt {d^2-e^2 x^2}}{8 d^2 x^2}-\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}-\frac {2 e \sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {4 e^3 \sqrt {d^2-e^2 x^2}}{3 d^3 x} \]

[In]

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

[Out]

-1/4*Sqrt[d^2 - e^2*x^2]/x^4 - (2*e*Sqrt[d^2 - e^2*x^2])/(3*d*x^3) - (7*e^2*Sqrt[d^2 - e^2*x^2])/(8*d^2*x^2) -
(4*e^3*Sqrt[d^2 - e^2*x^2])/(3*d^3*x) - (7*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 849

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)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

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}& = -\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}-\frac {\int \frac {-8 d^3 e-7 d^2 e^2 x}{x^4 \sqrt {d^2-e^2 x^2}} \, dx}{4 d^2} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}-\frac {2 e \sqrt {d^2-e^2 x^2}}{3 d x^3}+\frac {\int \frac {21 d^4 e^2+16 d^3 e^3 x}{x^3 \sqrt {d^2-e^2 x^2}} \, dx}{12 d^4} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}-\frac {2 e \sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {7 e^2 \sqrt {d^2-e^2 x^2}}{8 d^2 x^2}-\frac {\int \frac {-32 d^5 e^3-21 d^4 e^4 x}{x^2 \sqrt {d^2-e^2 x^2}} \, dx}{24 d^6} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}-\frac {2 e \sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {7 e^2 \sqrt {d^2-e^2 x^2}}{8 d^2 x^2}-\frac {4 e^3 \sqrt {d^2-e^2 x^2}}{3 d^3 x}+\frac {\left (7 e^4\right ) \int \frac {1}{x \sqrt {d^2-e^2 x^2}} \, dx}{8 d^2} \\ & = -\frac {\sqrt {d^2-e^2 x^2}}{4 x^4}-\frac {2 e \sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {7 e^2 \sqrt {d^2-e^2 x^2}}{8 d^2 x^2}-\frac {4 e^3 \sqrt {d^2-e^2 x^2}}{3 d^3 x}+\frac {\left (7 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 {\sqrt {d^2-e^2 x^2}}{4 x^4}-\frac {2 e \sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {7 e^2 \sqrt {d^2-e^2 x^2}}{8 d^2 x^2}-\frac {4 e^3 \sqrt {d^2-e^2 x^2}}{3 d^3 x}-\frac {\left (7 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 {\sqrt {d^2-e^2 x^2}}{4 x^4}-\frac {2 e \sqrt {d^2-e^2 x^2}}{3 d x^3}-\frac {7 e^2 \sqrt {d^2-e^2 x^2}}{8 d^2 x^2}-\frac {4 e^3 \sqrt {d^2-e^2 x^2}}{3 d^3 x}-\frac {7 e^4 \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{8 d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.74 \[ \int \frac {(d+e x)^2}{x^5 \sqrt {d^2-e^2 x^2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-6 d^3-16 d^2 e x-21 d e^2 x^2-32 e^3 x^3\right )}{24 d^3 x^4}+\frac {7 e^4 \text {arctanh}\left (\frac {\sqrt {-e^2} x}{d}-\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{4 d^3} \]

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(-6*d^3 - 16*d^2*e*x - 21*d*e^2*x^2 - 32*e^3*x^3))/(24*d^3*x^4) + (7*e^4*ArcTanh[(Sqrt[-e
^2]*x)/d - Sqrt[d^2 - e^2*x^2]/d])/(4*d^3)

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.71

method	result	size
risch	\(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (32 e^{3} x^{3}+21 d \,e^{2} x^{2}+16 d^{2} e x +6 d^{3}\right )}{24 d^{3} x^{4}}-\frac {7 e^{4} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 d^{2} \sqrt {d^{2}}}\)	\(99\)
default	\(d^{2} \left (-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{4 d^{2} x^{4}}+\frac {3 e^{2} \left (-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{2 d^{2} x^{2}}-\frac {e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{2} \sqrt {d^{2}}}\right )}{4 d^{2}}\right )+e^{2} \left (-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{2 d^{2} x^{2}}-\frac {e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{2} \sqrt {d^{2}}}\right )+2 d e \left (-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{2} x^{3}}-\frac {2 e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}{3 d^{4} x}\right )\)	\(229\)

[In]

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

[Out]

-1/24*(-e^2*x^2+d^2)^(1/2)*(32*e^3*x^3+21*d*e^2*x^2+16*d^2*e*x+6*d^3)/d^3/x^4-7/8/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)

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.62 \[ \int \frac {(d+e x)^2}{x^5 \sqrt {d^2-e^2 x^2}} \, dx=\frac {21 \, e^{4} x^{4} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (32 \, e^{3} x^{3} + 21 \, d e^{2} x^{2} + 16 \, d^{2} e x + 6 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{24 \, d^{3} x^{4}} \]

[In]

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

[Out]

1/24*(21*e^4*x^4*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (32*e^3*x^3 + 21*d*e^2*x^2 + 16*d^2*e*x + 6*d^3)*sqrt(-e
^2*x^2 + d^2))/(d^3*x^4)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.33 (sec) , antiderivative size = 449, normalized size of antiderivative = 3.21 \[ \int \frac {(d+e x)^2}{x^5 \sqrt {d^2-e^2 x^2}} \, dx=d^{2} \left (\begin {cases} - \frac {1}{4 e x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e}{8 d^{2} x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {3 e^{3}}{8 d^{4} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {3 e^{4} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{8 d^{5}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i}{4 e x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e}{8 d^{2} x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {3 i e^{3}}{8 d^{4} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {3 i e^{4} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{8 d^{5}} & \text {otherwise} \end {cases}\right ) + 2 d e \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2} x^{2}} - \frac {2 e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{4}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2} x^{2}} - \frac {2 i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{4}} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{2 d^{2} x} - \frac {e^{2} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{2 d^{3}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i}{2 e x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e}{2 d^{2} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{2} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{2 d^{3}} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

d**2*Piecewise((-1/(4*e*x**5*sqrt(d**2/(e**2*x**2) - 1)) - e/(8*d**2*x**3*sqrt(d**2/(e**2*x**2) - 1)) + 3*e**3
/(8*d**4*x*sqrt(d**2/(e**2*x**2) - 1)) - 3*e**4*acosh(d/(e*x))/(8*d**5), Abs(d**2/(e**2*x**2)) > 1), (I/(4*e*x
**5*sqrt(-d**2/(e**2*x**2) + 1)) + I*e/(8*d**2*x**3*sqrt(-d**2/(e**2*x**2) + 1)) - 3*I*e**3/(8*d**4*x*sqrt(-d*
*2/(e**2*x**2) + 1)) + 3*I*e**4*asin(d/(e*x))/(8*d**5), True)) + 2*d*e*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1
)/(3*d**2*x**2) - 2*e**3*sqrt(d**2/(e**2*x**2) - 1)/(3*d**4), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e*
*2*x**2) + 1)/(3*d**2*x**2) - 2*I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(3*d**4), True)) + e**2*Piecewise((-e*sqrt(
d**2/(e**2*x**2) - 1)/(2*d**2*x) - e**2*acosh(d/(e*x))/(2*d**3), Abs(d**2/(e**2*x**2)) > 1), (I/(2*e*x**3*sqrt
(-d**2/(e**2*x**2) + 1)) - I*e/(2*d**2*x*sqrt(-d**2/(e**2*x**2) + 1)) + I*e**2*asin(d/(e*x))/(2*d**3), True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.95 \[ \int \frac {(d+e x)^2}{x^5 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {7 \, e^{4} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{8 \, d^{3}} - \frac {4 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{3}}{3 \, d^{3} x} - \frac {7 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{2}}{8 \, d^{2} x^{2}} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} e}{3 \, d x^{3}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{4 \, x^{4}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (120) = 240\).

Time = 0.31 (sec) , antiderivative size = 329, normalized size of antiderivative = 2.35 \[ \int \frac {(d+e x)^2}{x^5 \sqrt {d^2-e^2 x^2}} \, dx=\frac {{\left (3 \, e^{5} + \frac {16 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{3}}{x} + \frac {48 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e}{x^{2}} + \frac {144 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e x^{3}}\right )} e^{8} x^{4}}{192 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{3} {\left | e \right |}} - \frac {7 \, e^{5} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{8 \, d^{3} {\left | e \right |}} - \frac {\frac {144 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{9} e^{5} {\left | e \right |}}{x} + \frac {48 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{9} e^{3} {\left | e \right |}}{x^{2}} + \frac {16 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{9} e {\left | e \right |}}{x^{3}} + \frac {3 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{9} {\left | e \right |}}{e x^{4}}}{192 \, d^{12} e^{4}} \]

[In]

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

[Out]

1/192*(3*e^5 + 16*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e^3/x + 48*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*e/x^2 +
144*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3/(e*x^3))*e^8*x^4/((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d^3*abs(e))
- 7/8*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*(144*(d*e +
sqrt(-e^2*x^2 + d^2)*abs(e))*d^9*e^5*abs(e)/x + 48*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^9*e^3*abs(e)/x^2 +
16*(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. \[ \int \frac {(d+e x)^2}{x^5 \sqrt {d^2-e^2 x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{x^5\,\sqrt {d^2-e^2\,x^2}} \,d x \]

[In]

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

[Out]

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