Integrand size = 24, antiderivative size = 124 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{d+e x} \, dx=\frac {5}{16} d^5 x \sqrt {d^2-e^2 x^2}+\frac {5}{24} d^3 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {1}{6} d x \left (d^2-e^2 x^2\right )^{5/2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}+\frac {5 d^7 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{16 e} \] Output:
5/16*d^5*x*(-e^2*x^2+d^2)^(1/2)+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
Time = 0.28 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.07 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{d+e x} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (48 d^6+231 d^5 e x-144 d^4 e^2 x^2-182 d^3 e^3 x^3+144 d^2 e^4 x^4+56 d e^5 x^5-48 e^6 x^6\right )}{336 e}-\frac {5 d^7 \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{16 \sqrt {-e^2}} \] Input:
Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x),x]
Output:
(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])
Time = 0.38 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {466, 211, 211, 211, 224, 216}
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.
\(\displaystyle \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{d+e x} \, dx\) |
\(\Big \downarrow \) 466 |
\(\displaystyle d \int \left (d^2-e^2 x^2\right )^{5/2}dx+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle d \left (\frac {5}{6} d^2 \int \left (d^2-e^2 x^2\right )^{3/2}dx+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle d \left (\frac {5}{6} d^2 \left (\frac {3}{4} d^2 \int \sqrt {d^2-e^2 x^2}dx+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle d \left (\frac {5}{6} d^2 \left (\frac {3}{4} d^2 \left (\frac {1}{2} d^2 \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle d \left (\frac {5}{6} d^2 \left (\frac {3}{4} d^2 \left (\frac {1}{2} d^2 \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle d \left (\frac {5}{6} d^2 \left (\frac {3}{4} d^2 \left (\frac {d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{7 e}\) |
Input:
Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x),x]
Output:
(d^2 - e^2*x^2)^(7/2)/(7*e) + d*((x*(d^2 - e^2*x^2)^(5/2))/6 + (5*d^2*((x* (d^2 - e^2*x^2)^(3/2))/4 + (3*d^2*((x*Sqrt[d^2 - e^2*x^2])/2 + (d^2*ArcTan [(e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e)))/4))/6)
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (c + d*x)^(n + 1)*((a + b*x^2)^p/(d*(n + 2*p + 1))), x] - Simp[2*b*c*(p/(d^ 2*(n + 2*p + 1))) Int[(c + d*x)^(n + 1)*(a + b*x^2)^(p - 1), x], x] /; Fr eeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[p, 0] && (LeQ[-2, n, 0 ] || EqQ[n + p + 1, 0]) && NeQ[n + 2*p + 1, 0] && IntegerQ[2*p]
Time = 0.24 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94
method | result | size |
risch | \(\frac {\left (-48 e^{6} x^{6}+56 d \,e^{5} x^{5}+144 d^{2} e^{4} x^{4}-182 d^{3} e^{3} x^{3}-144 d^{4} e^{2} x^{2}+231 d^{5} e x +48 d^{6}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{336 e}+\frac {5 d^{7} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{16 \sqrt {e^{2}}}\) | \(116\) |
default | \(\frac {\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{7}+d e \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{12 e^{2}}+\frac {5 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{e}\) | \(248\) |
Input:
int((-e^2*x^2+d^2)^(7/2)/(e*x+d),x,method=_RETURNVERBOSE)
Output:
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)*arct an((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))
Time = 0.11 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{d+e x} \, dx=-\frac {210 \, d^{7} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (48 \, e^{6} x^{6} - 56 \, d e^{5} x^{5} - 144 \, d^{2} e^{4} x^{4} + 182 \, d^{3} e^{3} x^{3} + 144 \, d^{4} e^{2} x^{2} - 231 \, d^{5} e x - 48 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{336 \, e} \] Input:
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d),x, algorithm="fricas")
Output:
-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
Time = 1.72 (sec) , antiderivative size = 520, normalized size of antiderivative = 4.19 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{d+e x} \, dx =\text {Too large to display} \] Input:
integrate((-e**2*x**2+d**2)**(7/2)/(e*x+d),x)
Output:
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((lo g(-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*Pi ecewise((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)) - 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))
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.04 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{d+e x} \, dx=-\frac {5 i \, d^{7} \arcsin \left (\frac {e x}{d} + 2\right )}{16 \, e} + \frac {5}{16} \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{5} x + \frac {5 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{6}}{8 \, e} + \frac {5}{24} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} x + \frac {1}{6} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d x + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{7 \, e} \] Input:
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d),x, algorithm="maxima")
Output:
-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
Time = 0.14 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.83 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{d+e x} \, dx=\frac {5 \, d^{7} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{16 \, {\left | e \right |}} + \frac {1}{336} \, {\left (\frac {48 \, d^{6}}{e} + {\left (231 \, d^{5} - 2 \, {\left (72 \, d^{4} e + {\left (91 \, d^{3} e^{2} - 4 \, {\left (18 \, d^{2} e^{3} - {\left (6 \, e^{5} x - 7 \, d e^{4}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}} \] Input:
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d),x, algorithm="giac")
Output:
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)
Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{d+e x} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{7/2}}{d+e\,x} \,d x \] Input:
int((d^2 - e^2*x^2)^(7/2)/(d + e*x),x)
Output:
int((d^2 - e^2*x^2)^(7/2)/(d + e*x), x)
Time = 0.20 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.42 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{d+e x} \, dx=\frac {105 \mathit {asin} \left (\frac {e x}{d}\right ) d^{7}+48 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{6}+231 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{5} e x -144 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{4} e^{2} x^{2}-182 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3} e^{3} x^{3}+144 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e^{4} x^{4}+56 \sqrt {-e^{2} x^{2}+d^{2}}\, d \,e^{5} x^{5}-48 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{6} x^{6}-48 d^{7}}{336 e} \] Input:
int((-e^2*x^2+d^2)^(7/2)/(e*x+d),x)
Output:
(105*asin((e*x)/d)*d**7 + 48*sqrt(d**2 - e**2*x**2)*d**6 + 231*sqrt(d**2 - e**2*x**2)*d**5*e*x - 144*sqrt(d**2 - e**2*x**2)*d**4*e**2*x**2 - 182*sqr t(d**2 - e**2*x**2)*d**3*e**3*x**3 + 144*sqrt(d**2 - e**2*x**2)*d**2*e**4* x**4 + 56*sqrt(d**2 - e**2*x**2)*d*e**5*x**5 - 48*sqrt(d**2 - e**2*x**2)*e **6*x**6 - 48*d**7)/(336*e)