\(\int \frac {(d^2-e^2 x^2)^{7/2}}{d+e x} \, dx\) [803]

Optimal result

Integrand size = 24, antiderivative size = 124 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{d+e x} \, dx=\frac {5}{16} d^5 x \sqrt {d^2-e^2 x^2}+\frac {5}{24} d^3 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac {5 d^7 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e} \]

[Out]

5/24*d^3*x*(-e^2*x^2+d^2)^(3/2)+1/6*d*x*(-e^2*x^2+d^2)^(5/2)+1/7*(-e^2*x^2+d^2)^(7/2)/e+5/16*d^7*arctan(e*x/(-
e^2*x^2+d^2)^(1/2))/e+5/16*d^5*x*(-e^2*x^2+d^2)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {679, 201, 223, 209} \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{d+e x} \, dx=\frac {5 d^7 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e}+\frac {1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac {5}{16} d^5 x \sqrt {d^2-e^2 x^2}+\frac {5}{24} d^3 x \left (d^2-e^2 x^2\right )^{3/2} \]

[In]

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

[Out]

(5*d^5*x*Sqrt[d^2 - e^2*x^2])/16 + (5*d^3*x*(d^2 - e^2*x^2)^(3/2))/24 + (d*x*(d^2 - e^2*x^2)^(5/2))/6 + (d^2 -
e^2*x^2)^(7/2)/(7*e) + (5*d^7*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 679

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

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

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.07 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{d+e x} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (48 d^6+231 d^5 e x-144 d^4 e^2 x^2-182 d^3 e^3 x^3+144 d^2 e^4 x^4+56 d e^5 x^5-48 e^6 x^6\right )}{336 e}-\frac {5 d^7 \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{16 \sqrt {-e^2}} \]

[In]

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

[Out]

(Sqrt[d^2 - e^2*x^2]*(48*d^6 + 231*d^5*e*x - 144*d^4*e^2*x^2 - 182*d^3*e^3*x^3 + 144*d^2*e^4*x^4 + 56*d*e^5*x^
5 - 48*e^6*x^6))/(336*e) - (5*d^7*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/(16*Sqrt[-e^2])

Maple [A] (verified)

Time = 2.23 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94

method	result	size
risch	\(\frac {\left (-48 e^{6} x^{6}+56 d \,e^{5} x^{5}+144 d^{2} e^{4} x^{4}-182 x^{3} d^{3} e^{3}-144 d^{4} e^{2} x^{2}+231 d^{5} e x +48 d^{6}\right ) \sqrt {-x^{2} e^{2}+d^{2}}}{336 e}+\frac {5 d^{7} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{16 \sqrt {e^{2}}}\)	\(116\)
default	\(\frac {\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{7}+d e \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{12 e^{2}}+\frac {5 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{e}\)	\(248\)

[In]

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

[Out]

1/336*(-48*e^6*x^6+56*d*e^5*x^5+144*d^2*e^4*x^4-182*d^3*e^3*x^3-144*d^4*e^2*x^2+231*d^5*e*x+48*d^6)/e*(-e^2*x^
2+d^2)^(1/2)+5/16*d^7/(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 = 0.38 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{d+e x} \, dx=-\frac {210 \, d^{7} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (48 \, e^{6} x^{6} - 56 \, d e^{5} x^{5} - 144 \, d^{2} e^{4} x^{4} + 182 \, d^{3} e^{3} x^{3} + 144 \, d^{4} e^{2} x^{2} - 231 \, d^{5} e x - 48 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{336 \, e} \]

[In]

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

[Out]

-1/336*(210*d^7*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (48*e^6*x^6 - 56*d*e^5*x^5 - 144*d^2*e^4*x^4 + 182
*d^3*e^3*x^3 + 144*d^4*e^2*x^2 - 231*d^5*e*x - 48*d^6)*sqrt(-e^2*x^2 + d^2))/e

Sympy [A] (verification not implemented)

Time = 1.61 (sec) , antiderivative size = 520, normalized size of antiderivative = 4.19 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{d+e x} \, dx=d^{5} \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 ) - d^{4} 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^{3} e^{2} \left (\begin {cases} \frac {d^{4} \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 )}{8 e^{2}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{2} x}{8 e^{2}} + \frac {x^{3}}{4}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{3} \sqrt {d^{2}}}{3} & \text {otherwise} \end {cases}\right ) + 2 d^{2} 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 ) + d 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 ) - e^{5} \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {8 d^{6}}{105 e^{6}} - \frac {4 d^{4} x^{2}}{105 e^{4}} - \frac {d^{2} x^{4}}{35 e^{2}} + \frac {x^{6}}{7}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{6} \sqrt {d^{2}}}{6} & \text {otherwise} \end {cases}\right ) \]

[In]

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

[Out]

d**5*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)) - d*
*4*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**3*e**2*Piecewise((d**4*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))/(8*e**2) + sqrt(d**2 - e**2*x**2)*(-d**2*x/(8*e**2) + x**3/4), Ne(
e**2, 0)), (x**3*sqrt(d**2)/3, True)) + 2*d**2*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)) + d*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*sq
rt(d**2)/5, True)) - e**5*Piecewise((sqrt(d**2 - e**2*x**2)*(-8*d**6/(105*e**6) - 4*d**4*x**2/(105*e**4) - d**
2*x**4/(35*e**2) + x**6/7), Ne(e**2, 0)), (x**6*sqrt(d**2)/6, True))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.04 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{d+e x} \, dx=-\frac {5 i \, d^{7} \arcsin \left (\frac {e x}{d} + 2\right )}{16 \, e} + \frac {5}{16} \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{5} x + \frac {5 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{6}}{8 \, e} + \frac {5}{24} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} x + \frac {1}{6} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d x + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{7 \, e} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.83 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{d+e x} \, dx=\frac {5 \, d^{7} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{16 \, {\left | e \right |}} + \frac {1}{336} \, {\left (\frac {48 \, d^{6}}{e} + {\left (231 \, d^{5} - 2 \, {\left (72 \, d^{4} e + {\left (91 \, d^{3} e^{2} - 4 \, {\left (18 \, d^{2} e^{3} - {\left (6 \, e^{5} x - 7 \, d e^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}} \]

[In]

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

[Out]

5/16*d^7*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) + 1/336*(48*d^6/e + (231*d^5 - 2*(72*d^4*e + (91*d^3*e^2 - 4*(18*d
^2*e^3 - (6*e^5*x - 7*d*e^4)*x)*x)*x)*x)*x)*sqrt(-e^2*x^2 + d^2)

Mupad [F(-1)]

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

[In]

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

[Out]

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