\(\int \frac {(d^2-e^2 x^2)^{7/2}}{(d+e x)^2} \, dx\) [804]

Optimal result

Integrand size = 24, antiderivative size = 132 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^2} \, dx=\frac {7}{16} d^4 x \sqrt {d^2-e^2 x^2}+\frac {7}{24} d^2 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7 d \left (d^2-e^2 x^2\right )^{5/2}}{30 e}+\frac {(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac {7 d^6 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e} \]

[Out]

7/24*d^2*x*(-e^2*x^2+d^2)^(3/2)+7/30*d*(-e^2*x^2+d^2)^(5/2)/e+1/6*(-e*x+d)*(-e^2*x^2+d^2)^(5/2)/e+7/16*d^6*arc
tan(e*x/(-e^2*x^2+d^2)^(1/2))/e+7/16*d^4*x*(-e^2*x^2+d^2)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {669, 685, 655, 201, 223, 209} \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^2} \, dx=\frac {7 d^6 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e}+\frac {7}{24} d^2 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7 d \left (d^2-e^2 x^2\right )^{5/2}}{30 e}+\frac {(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac {7}{16} d^4 x \sqrt {d^2-e^2 x^2} \]

[In]

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

[Out]

(7*d^4*x*Sqrt[d^2 - e^2*x^2])/16 + (7*d^2*x*(d^2 - e^2*x^2)^(3/2))/24 + (7*d*(d^2 - e^2*x^2)^(5/2))/(30*e) + (
(d - e*x)*(d^2 - e^2*x^2)^(5/2))/(6*e) + (7*d^6*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/(16*e)

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 655

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[e*((a + c*x^2)^(p + 1)/(2*c*(p + 1))),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 669

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^(2*m)/a^m, Int[(a + c*x^2)^(m + p
)/(d - e*x)^m, x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && IntegerQ[m]
&& RationalQ[p] && (LtQ[0, -m, p] || LtQ[p, -m, 0]) && NeQ[m, 2] && NeQ[m, -1]

Rule 685

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

Rubi steps \begin{align*} \text {integral}& = \int (d-e x)^2 \left (d^2-e^2 x^2\right )^{3/2} \, dx \\ & = \frac {(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac {1}{6} (7 d) \int (d-e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx \\ & = \frac {7 d \left (d^2-e^2 x^2\right )^{5/2}}{30 e}+\frac {(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac {1}{6} \left (7 d^2\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx \\ & = \frac {7}{24} d^2 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7 d \left (d^2-e^2 x^2\right )^{5/2}}{30 e}+\frac {(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac {1}{8} \left (7 d^4\right ) \int \sqrt {d^2-e^2 x^2} \, dx \\ & = \frac {7}{16} d^4 x \sqrt {d^2-e^2 x^2}+\frac {7}{24} d^2 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7 d \left (d^2-e^2 x^2\right )^{5/2}}{30 e}+\frac {(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac {1}{16} \left (7 d^6\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx \\ & = \frac {7}{16} d^4 x \sqrt {d^2-e^2 x^2}+\frac {7}{24} d^2 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7 d \left (d^2-e^2 x^2\right )^{5/2}}{30 e}+\frac {(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac {1}{16} \left (7 d^6\right ) \text {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right ) \\ & = \frac {7}{16} d^4 x \sqrt {d^2-e^2 x^2}+\frac {7}{24} d^2 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {7 d \left (d^2-e^2 x^2\right )^{5/2}}{30 e}+\frac {(d-e x) \left (d^2-e^2 x^2\right )^{5/2}}{6 e}+\frac {7 d^6 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.86 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^2} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (96 d^5+135 d^4 e x-192 d^3 e^2 x^2+10 d^2 e^3 x^3+96 d e^4 x^4-40 e^5 x^5\right )-210 d^6 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{240 e} \]

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(96*d^5 + 135*d^4*e*x - 192*d^3*e^2*x^2 + 10*d^2*e^3*x^3 + 96*d*e^4*x^4 - 40*e^5*x^5) - 2
10*d^6*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/(240*e)

Maple [A] (verified)

Time = 2.21 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.80

method	result	size
risch	\(\frac {\left (-40 e^{5} x^{5}+96 x^{4} d \,e^{4}+10 d^{2} e^{3} x^{3}-192 d^{3} e^{2} x^{2}+135 d^{4} e x +96 d^{5}\right ) \sqrt {-x^{2} e^{2}+d^{2}}}{240 e}+\frac {7 d^{6} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{16 \sqrt {e^{2}}}\)	\(105\)
default	\(\frac {\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{2}}+\frac {7 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{7}+d e \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{12 e^{2}}+\frac {5 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )\right )}{5 d}}{e^{2}}\)	\(300\)

[In]

int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^2,x,method=_RETURNVERBOSE)

[Out]

1/240*(-40*e^5*x^5+96*d*e^4*x^4+10*d^2*e^3*x^3-192*d^3*e^2*x^2+135*d^4*e*x+96*d^5)/e*(-e^2*x^2+d^2)^(1/2)+7/16
*d^6/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.80 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^2} \, dx=-\frac {210 \, d^{6} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (40 \, e^{5} x^{5} - 96 \, d e^{4} x^{4} - 10 \, d^{2} e^{3} x^{3} + 192 \, d^{3} e^{2} x^{2} - 135 \, d^{4} e x - 96 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{240 \, e} \]

[In]

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

[Out]

-1/240*(210*d^6*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (40*e^5*x^5 - 96*d*e^4*x^4 - 10*d^2*e^3*x^3 + 192*
d^3*e^2*x^2 - 135*d^4*e*x - 96*d^5)*sqrt(-e^2*x^2 + d^2))/e

Sympy [A] (verification not implemented)

Time = 2.13 (sec) , antiderivative size = 328, normalized size of antiderivative = 2.48 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^2} \, dx=d^{4} \left (\begin {cases} \frac {d^{2} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {d^{2} - e^{2} x^{2}}}{2} & \text {for}\: e^{2} \neq 0 \\x \sqrt {d^{2}} & \text {otherwise} \end {cases}\right ) - 2 d^{3} e \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{2}}{3 e^{2}} + \frac {x^{2}}{3}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{2} \sqrt {d^{2}}}{2} & \text {otherwise} \end {cases}\right ) + 2 d e^{3} \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {2 d^{4}}{15 e^{4}} - \frac {d^{2} x^{2}}{15 e^{2}} + \frac {x^{4}}{5}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{4} \sqrt {d^{2}}}{4} & \text {otherwise} \end {cases}\right ) - e^{4} \left (\begin {cases} \frac {d^{6} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{16 e^{4}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{4} x}{16 e^{4}} - \frac {d^{2} x^{3}}{24 e^{2}} + \frac {x^{5}}{6}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{5} \sqrt {d^{2}}}{5} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

d**4*Piecewise((d**2*Piecewise((log(-2*e**2*x + 2*sqrt(-e**2)*sqrt(d**2 - e**2*x**2))/sqrt(-e**2), Ne(d**2, 0)
), (x*log(x)/sqrt(-e**2*x**2), True))/2 + x*sqrt(d**2 - e**2*x**2)/2, Ne(e**2, 0)), (x*sqrt(d**2), True)) - 2*
d**3*e*Piecewise((sqrt(d**2 - e**2*x**2)*(-d**2/(3*e**2) + x**2/3), Ne(e**2, 0)), (x**2*sqrt(d**2)/2, True)) +
2*d*e**3*Piecewise((sqrt(d**2 - e**2*x**2)*(-2*d**4/(15*e**4) - d**2*x**2/(15*e**2) + x**4/5), Ne(e**2, 0)),
(x**4*sqrt(d**2)/4, True)) - e**4*Piecewise((d**6*Piecewise((log(-2*e**2*x + 2*sqrt(-e**2)*sqrt(d**2 - e**2*x*
*2))/sqrt(-e**2), Ne(d**2, 0)), (x*log(x)/sqrt(-e**2*x**2), True))/(16*e**4) + sqrt(d**2 - e**2*x**2)*(-d**4*x
/(16*e**4) - d**2*x**3/(24*e**2) + x**5/6), Ne(e**2, 0)), (x**5*sqrt(d**2)/5, True))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.05 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^2} \, dx=-\frac {7 i \, d^{6} \arcsin \left (\frac {e x}{d} + 2\right )}{16 \, e} + \frac {7}{16} \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{4} x + \frac {7 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{5}}{8 \, e} + \frac {7}{24} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} x + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{6 \, {\left (e^{2} x + d e\right )}} + \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{30 \, e} \]

[In]

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

[Out]

-7/16*I*d^6*arcsin(e*x/d + 2)/e + 7/16*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^4*x + 7/8*sqrt(e^2*x^2 + 4*d*e*x + 3*
d^2)*d^5/e + 7/24*(-e^2*x^2 + d^2)^(3/2)*d^2*x + 1/6*(-e^2*x^2 + d^2)^(7/2)/(e^2*x + d*e) + 7/30*(-e^2*x^2 + d
^2)^(5/2)*d/e

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (113) = 226\).

Time = 0.32 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.88 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^2} \, dx=-\frac {{\left (6720 \, d^{7} e^{7} \arctan \left (\sqrt {\frac {2 \, d}{e x + d} - 1}\right ) \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + \frac {{\left (105 \, d^{7} e^{7} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {11}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + 595 \, d^{7} e^{7} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {9}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 1686 \, d^{7} e^{7} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {7}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 1386 \, d^{7} e^{7} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 595 \, d^{7} e^{7} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 105 \, d^{7} e^{7} \sqrt {\frac {2 \, d}{e x + d} - 1} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )\right )} {\left (e x + d\right )}^{6}}{d^{6}}\right )} {\left | e \right |}}{7680 \, d e^{9}} \]

[In]

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

[Out]

-1/7680*(6720*d^7*e^7*arctan(sqrt(2*d/(e*x + d) - 1))*sgn(1/(e*x + d))*sgn(e) + (105*d^7*e^7*(2*d/(e*x + d) -
1)^(11/2)*sgn(1/(e*x + d))*sgn(e) + 595*d^7*e^7*(2*d/(e*x + d) - 1)^(9/2)*sgn(1/(e*x + d))*sgn(e) - 1686*d^7*e
^7*(2*d/(e*x + d) - 1)^(7/2)*sgn(1/(e*x + d))*sgn(e) - 1386*d^7*e^7*(2*d/(e*x + d) - 1)^(5/2)*sgn(1/(e*x + d))
*sgn(e) - 595*d^7*e^7*(2*d/(e*x + d) - 1)^(3/2)*sgn(1/(e*x + d))*sgn(e) - 105*d^7*e^7*sqrt(2*d/(e*x + d) - 1)*
sgn(1/(e*x + d))*sgn(e))*(e*x + d)^6/d^6)*abs(e)/(d*e^9)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^2} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{7/2}}{{\left (d+e\,x\right )}^2} \,d x \]

[In]

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

[Out]

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