Integrand size = 24, antiderivative size = 145 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^6} \, dx=\frac {35 d \sqrt {d^2-e^2 x^2}}{2 e}+\frac {35 \left (d^2-e^2 x^2\right )^{3/2}}{6 e (d+e x)}+\frac {14 \left (d^2-e^2 x^2\right )^{5/2}}{3 e (d+e x)^3}-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{3 e (d+e x)^5}+\frac {35 d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e} \]
35/6*(-e^2*x^2+d^2)^(3/2)/e/(e*x+d)+14/3*(-e^2*x^2+d^2)^(5/2)/e/(e*x+d)^3- 2/3*(-e^2*x^2+d^2)^(7/2)/e/(e*x+d)^5+35/2*d^2*arctan(e*x/(-e^2*x^2+d^2)^(1 /2))/e+35/2*d*(-e^2*x^2+d^2)^(1/2)/e
Time = 0.55 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.70 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^6} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (164 d^3+229 d^2 e x+30 d e^2 x^2-3 e^3 x^3\right )}{6 e (d+e x)^2}-\frac {35 d^2 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{e} \]
(Sqrt[d^2 - e^2*x^2]*(164*d^3 + 229*d^2*e*x + 30*d*e^2*x^2 - 3*e^3*x^3))/( 6*e*(d + e*x)^2) - (35*d^2*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])] )/e
Time = 0.31 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.97, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {465, 463, 25, 2346, 25, 27, 455, 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.
| \(\displaystyle \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^6} \, dx\) |
| \(\Big \downarrow \) 465 |
| \(\displaystyle -\frac {7}{3} \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4}dx-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{3 e (d+e x)^5}\) |
| \(\Big \downarrow \) 463 |
| \(\displaystyle -\frac {7}{3} \left (\int -\frac {7 d^2-4 e x d+e^2 x^2}{\sqrt {d^2-e^2 x^2}}dx-\frac {8 d^2 \sqrt {d^2-e^2 x^2}}{e (d+e x)}\right )-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{3 e (d+e x)^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {7}{3} \left (-\int \frac {7 d^2-4 e x d+e^2 x^2}{\sqrt {d^2-e^2 x^2}}dx-\frac {8 d^2 \sqrt {d^2-e^2 x^2}}{e (d+e x)}\right )-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{3 e (d+e x)^5}\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle -\frac {7}{3} \left (\frac {\int -\frac {d e^2 (15 d-8 e x)}{\sqrt {d^2-e^2 x^2}}dx}{2 e^2}-\frac {8 d^2 \sqrt {d^2-e^2 x^2}}{e (d+e x)}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{3 e (d+e x)^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {7}{3} \left (-\frac {\int \frac {d e^2 (15 d-8 e x)}{\sqrt {d^2-e^2 x^2}}dx}{2 e^2}-\frac {8 d^2 \sqrt {d^2-e^2 x^2}}{e (d+e x)}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{3 e (d+e x)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {7}{3} \left (-\frac {1}{2} d \int \frac {15 d-8 e x}{\sqrt {d^2-e^2 x^2}}dx-\frac {8 d^2 \sqrt {d^2-e^2 x^2}}{e (d+e x)}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{3 e (d+e x)^5}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle -\frac {7}{3} \left (-\frac {1}{2} d \left (15 d \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+\frac {8 \sqrt {d^2-e^2 x^2}}{e}\right )-\frac {8 d^2 \sqrt {d^2-e^2 x^2}}{e (d+e x)}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{3 e (d+e x)^5}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle -\frac {7}{3} \left (-\frac {1}{2} d \left (15 d \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}+\frac {8 \sqrt {d^2-e^2 x^2}}{e}\right )-\frac {8 d^2 \sqrt {d^2-e^2 x^2}}{e (d+e x)}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{3 e (d+e x)^5}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {7}{3} \left (-\frac {1}{2} d \left (\frac {15 d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}+\frac {8 \sqrt {d^2-e^2 x^2}}{e}\right )-\frac {8 d^2 \sqrt {d^2-e^2 x^2}}{e (d+e x)}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )-\frac {2 \left (d^2-e^2 x^2\right )^{7/2}}{3 e (d+e x)^5}\) |
(-2*(d^2 - e^2*x^2)^(7/2))/(3*e*(d + e*x)^5) - (7*((x*Sqrt[d^2 - e^2*x^2]) /2 - (8*d^2*Sqrt[d^2 - e^2*x^2])/(e*(d + e*x)) - (d*((8*Sqrt[d^2 - e^2*x^2 ])/e + (15*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e))/2))/3
3.9.8.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-(-c)^(-n - 2))*d^(2*n + 3)*(Sqrt[a + b*x^2]/(2^(n + 1)*b^(n + 2)*(c + d*x ))), x] - Simp[d^(2*n + 2)/b^(n + 1) Int[(1/Sqrt[a + b*x^2])*ExpandToSum[ (2^(-n - 1)*(-c)^(-n - 1) - (-c + d*x)^(-n - 1))/(c + d*x), x], x], x] /; F reeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[n, 0] && EqQ[n + p, -3/2]
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 + p + 1))), x] - Simp[b*(p/(d^2*(n + p + 1))) Int[(c + d*x)^(n + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[p, 0] && (LtQ[n, -2] || EqQ[n + 2*p + 1, 0]) && NeQ[n + p + 1, 0] && IntegerQ[2*p]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1)) Int[(a + b*x^2)^p*ExpandToS um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && !LeQ[p, -1]
Time = 2.48 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.04
| method | result | size |
| risch | \(\frac {\left (-e x +12 d \right ) \sqrt {-x^{2} e^{2}+d^{2}}}{2 e}+\frac {35 d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}-\frac {16 d^{3} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{3 e^{3} \left (x +\frac {d}{e}\right )^{2}}+\frac {80 d^{2} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{3 e^{2} \left (x +\frac {d}{e}\right )}\) | \(151\) |
| default | \(\frac {-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{6}}-\frac {e \left (-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{d e \left (x +\frac {d}{e}\right )^{5}}-\frac {4 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{d e \left (x +\frac {d}{e}\right )^{4}}+\frac {5 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{3}}+\frac {2 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{2}}+\frac {7 e \left (\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{7}+d e \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{12 e^{2}}+\frac {5 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right ) \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )\right )}{5 d}\right )}{d}\right )}{d}\right )}{d}\right )}{d}}{e^{6}}\) | \(507\) |
1/2*(-e*x+12*d)/e*(-e^2*x^2+d^2)^(1/2)+35/2*d^2/(e^2)^(1/2)*arctan((e^2)^( 1/2)*x/(-e^2*x^2+d^2)^(1/2))-16/3*d^3/e^3/(x+d/e)^2*(-e^2*(x+d/e)^2+2*d*e* (x+d/e))^(1/2)+80/3*d^2/e^2/(x+d/e)*(-e^2*(x+d/e)^2+2*d*e*(x+d/e))^(1/2)
Time = 0.30 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.99 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^6} \, dx=\frac {164 \, d^{2} e^{2} x^{2} + 328 \, d^{3} e x + 164 \, d^{4} - 210 \, {\left (d^{2} e^{2} x^{2} + 2 \, d^{3} e x + d^{4}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (3 \, e^{3} x^{3} - 30 \, d e^{2} x^{2} - 229 \, d^{2} e x - 164 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \]
1/6*(164*d^2*e^2*x^2 + 328*d^3*e*x + 164*d^4 - 210*(d^2*e^2*x^2 + 2*d^3*e* x + d^4)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (3*e^3*x^3 - 30*d*e^2 *x^2 - 229*d^2*e*x - 164*d^3)*sqrt(-e^2*x^2 + d^2))/(e^3*x^2 + 2*d*e^2*x + d^2*e)
\[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^6} \, dx=\int \frac {\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}{\left (d + e x\right )^{6}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (125) = 250\).
Time = 0.30 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.87 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^6} \, dx=\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{2 \, {\left (e^{6} x^{5} + 5 \, d e^{5} x^{4} + 10 \, d^{2} e^{4} x^{3} + 10 \, d^{3} e^{3} x^{2} + 5 \, d^{4} e^{2} x + d^{5} e\right )}} + \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{2 \, {\left (e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e\right )}} - \frac {35 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{6 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} - \frac {35 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}}{3 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} + \frac {35 \, d^{2} \arcsin \left (\frac {e x}{d}\right )}{2 \, e} + \frac {245 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{6 \, {\left (e^{2} x + d e\right )}} \]
1/2*(-e^2*x^2 + d^2)^(7/2)/(e^6*x^5 + 5*d*e^5*x^4 + 10*d^2*e^4*x^3 + 10*d^ 3*e^3*x^2 + 5*d^4*e^2*x + d^5*e) + 7/2*(-e^2*x^2 + d^2)^(5/2)*d/(e^5*x^4 + 4*d*e^4*x^3 + 6*d^2*e^3*x^2 + 4*d^3*e^2*x + d^4*e) - 35/6*(-e^2*x^2 + d^2 )^(3/2)*d^2/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e) - 35/3*sqrt(-e^2 *x^2 + d^2)*d^3/(e^3*x^2 + 2*d*e^2*x + d^2*e) + 35/2*d^2*arcsin(e*x/d)/e + 245/6*sqrt(-e^2*x^2 + d^2)*d^2/(e^2*x + d*e)
Time = 0.45 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.07 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^6} \, dx=\frac {35 \, d^{2} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{2 \, {\left | e \right |}} - \frac {1}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (x - \frac {12 \, d}{e}\right )} - \frac {32 \, {\left (4 \, d^{2} + \frac {9 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{2}}{e^{2} x} + \frac {3 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{2}}{e^{4} x^{2}}\right )}}{3 \, {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )}^{3} {\left | e \right |}} \]
35/2*d^2*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) - 1/2*sqrt(-e^2*x^2 + d^2)*(x - 12*d/e) - 32/3*(4*d^2 + 9*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^2/(e^2*x ) + 3*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^2/(e^4*x^2))/(((d*e + sqrt(- e^2*x^2 + d^2)*abs(e))/(e^2*x) + 1)^3*abs(e))
Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^6} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{7/2}}{{\left (d+e\,x\right )}^6} \,d x \]