Integrand size = 27, antiderivative size = 162 \[ \int \frac {x^7}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {x^6 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d-7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d-35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d-35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}+\frac {7 d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^8} \]
1/5*x^6*(-e*x+d)/e^2/(-e^2*x^2+d^2)^(5/2)-1/15*x^4*(-7*e*x+6*d)/e^4/(-e^2* x^2+d^2)^(3/2)+7/2*d^2*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^8+1/15*x^2*(-35* e*x+24*d)/e^6/(-e^2*x^2+d^2)^(1/2)+1/10*(-35*e*x+32*d)*(-e^2*x^2+d^2)^(1/2 )/e^8
Time = 0.55 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.86 \[ \int \frac {x^7}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {\frac {\sqrt {d^2-e^2 x^2} \left (96 d^6-9 d^5 e x-249 d^4 e^2 x^2-4 d^3 e^3 x^3+176 d^2 e^4 x^4+15 d e^5 x^5-15 e^6 x^6\right )}{(d-e x)^2 (d+e x)^3}-210 d^2 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{30 e^8} \]
((Sqrt[d^2 - e^2*x^2]*(96*d^6 - 9*d^5*e*x - 249*d^4*e^2*x^2 - 4*d^3*e^3*x^ 3 + 176*d^2*e^4*x^4 + 15*d*e^5*x^5 - 15*e^6*x^6))/((d - e*x)^2*(d + e*x)^3 ) - 210*d^2*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/(30*e^8)
Time = 0.52 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.28, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {568, 530, 25, 2345, 27, 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.
| \(\displaystyle \int \frac {x^7}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 568 |
| \(\displaystyle \frac {x^6}{5 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {x^5 (6 d-7 e x)}{\left (d^2-e^2 x^2\right )^{5/2}}dx}{5 e^2}\) |
| \(\Big \downarrow \) 530 |
| \(\displaystyle \frac {x^6}{5 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {d^4 (6 d-7 e x)}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int -\frac {\frac {7 d^6}{e^5}-\frac {18 x d^5}{e^4}+\frac {21 x^2 d^4}{e^3}-\frac {18 x^3 d^3}{e^2}+\frac {21 x^4 d^2}{e}}{\left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2}}{5 e^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x^6}{5 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {\int \frac {\frac {7 d^6}{e^5}-\frac {18 x d^5}{e^4}+\frac {21 x^2 d^4}{e^3}-\frac {18 x^3 d^3}{e^2}+\frac {21 x^4 d^2}{e}}{\left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2}+\frac {d^4 (6 d-7 e x)}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}}{5 e^2}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {x^6}{5 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {-\frac {\int \frac {3 \left (\frac {14 d^6}{e^5}-\frac {6 x d^5}{e^4}+\frac {7 x^2 d^4}{e^3}\right )}{\sqrt {d^2-e^2 x^2}}dx}{d^2}-\frac {d^4 (36 d-49 e x)}{e^6 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {d^4 (6 d-7 e x)}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}}{5 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^6}{5 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {-\frac {3 \int \frac {\frac {14 d^6}{e^5}-\frac {6 x d^5}{e^4}+\frac {7 x^2 d^4}{e^3}}{\sqrt {d^2-e^2 x^2}}dx}{d^2}-\frac {d^4 (36 d-49 e x)}{e^6 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {d^4 (6 d-7 e x)}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}}{5 e^2}\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle \frac {x^6}{5 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {-\frac {3 \left (-\frac {\int -\frac {d^5 (35 d-12 e x)}{e^3 \sqrt {d^2-e^2 x^2}}dx}{2 e^2}-\frac {7 d^4 x \sqrt {d^2-e^2 x^2}}{2 e^5}\right )}{d^2}-\frac {d^4 (36 d-49 e x)}{e^6 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {d^4 (6 d-7 e x)}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}}{5 e^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x^6}{5 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {-\frac {3 \left (\frac {\int \frac {d^5 (35 d-12 e x)}{e^3 \sqrt {d^2-e^2 x^2}}dx}{2 e^2}-\frac {7 d^4 x \sqrt {d^2-e^2 x^2}}{2 e^5}\right )}{d^2}-\frac {d^4 (36 d-49 e x)}{e^6 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {d^4 (6 d-7 e x)}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}}{5 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^6}{5 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {-\frac {3 \left (\frac {d^5 \int \frac {35 d-12 e x}{\sqrt {d^2-e^2 x^2}}dx}{2 e^5}-\frac {7 d^4 x \sqrt {d^2-e^2 x^2}}{2 e^5}\right )}{d^2}-\frac {d^4 (36 d-49 e x)}{e^6 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {d^4 (6 d-7 e x)}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}}{5 e^2}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {x^6}{5 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {-\frac {3 \left (\frac {d^5 \left (35 d \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+\frac {12 \sqrt {d^2-e^2 x^2}}{e}\right )}{2 e^5}-\frac {7 d^4 x \sqrt {d^2-e^2 x^2}}{2 e^5}\right )}{d^2}-\frac {d^4 (36 d-49 e x)}{e^6 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {d^4 (6 d-7 e x)}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}}{5 e^2}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {x^6}{5 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {-\frac {3 \left (\frac {d^5 \left (35 d \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}+\frac {12 \sqrt {d^2-e^2 x^2}}{e}\right )}{2 e^5}-\frac {7 d^4 x \sqrt {d^2-e^2 x^2}}{2 e^5}\right )}{d^2}-\frac {d^4 (36 d-49 e x)}{e^6 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {d^4 (6 d-7 e x)}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}}{5 e^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {x^6}{5 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {-\frac {3 \left (\frac {d^5 \left (\frac {35 d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}+\frac {12 \sqrt {d^2-e^2 x^2}}{e}\right )}{2 e^5}-\frac {7 d^4 x \sqrt {d^2-e^2 x^2}}{2 e^5}\right )}{d^2}-\frac {d^4 (36 d-49 e x)}{e^6 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {d^4 (6 d-7 e x)}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}}{5 e^2}\) |
x^6/(5*e^2*(d + e*x)*(d^2 - e^2*x^2)^(3/2)) - ((d^4*(6*d - 7*e*x))/(3*e^6* (d^2 - e^2*x^2)^(3/2)) + (-((d^4*(36*d - 49*e*x))/(e^6*Sqrt[d^2 - e^2*x^2] )) - (3*((-7*d^4*x*Sqrt[d^2 - e^2*x^2])/(2*e^5) + (d^5*((12*Sqrt[d^2 - e^2 *x^2])/e + (35*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e))/(2*e^5)))/d^2)/(3* d^2))/(5*e^2)
3.2.36.3.1 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[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symb ol] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Co eff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Po lynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x )*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Qx + e*(2*p + 3), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 0] && LtQ[p, -1] && EqQ[n, 1] && IntegerQ[2*p]
Int[((x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)), x_Symbol] : > Simp[x^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*p*(c + d*x))), x] + Simp[1/(2*d^ 2*p) Int[x^(m - 2)*(a + b*x^2)^p*(c*(m - 1) - d*m*x), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && IGtQ[m, 1] && LtQ[p, -1]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(294\) vs. \(2(142)=284\).
Time = 0.44 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.82
| method | result | size |
| risch | \(\frac {\left (-e x +2 d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{2 e^{8}}+\frac {7 d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{7} \sqrt {e^{2}}}-\frac {7 d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{15 e^{10} \left (x +\frac {d}{e}\right )^{2}}+\frac {773 d^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{240 e^{9} \left (x +\frac {d}{e}\right )}+\frac {d^{3} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{24 e^{10} \left (x -\frac {d}{e}\right )^{2}}+\frac {31 d^{2} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{48 e^{9} \left (x -\frac {d}{e}\right )}+\frac {d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{20 e^{11} \left (x +\frac {d}{e}\right )^{3}}\) | \(295\) |
| default | \(\frac {-\frac {x^{5}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {5 d^{2} \left (\frac {x^{3}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}\right )}{2 e^{2}}}{e}+\frac {d^{6} \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{7}}+\frac {d^{2} \left (\frac {x^{3}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}\right )}{e^{3}}+\frac {d^{4} \left (\frac {x}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {d^{2} \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{2}}\right )}{e^{5}}-\frac {d \left (-\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {4 d^{2} \left (\frac {x^{2}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {2 d^{2}}{3 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}\right )}{e^{2}}\right )}{e^{2}}-\frac {d^{3} \left (\frac {x^{2}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {2 d^{2}}{3 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}\right )}{e^{4}}-\frac {d^{5}}{3 e^{8} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {d^{7} \left (-\frac {1}{5 d e \left (x +\frac {d}{e}\right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {4 e \left (-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{6 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{3 e^{2} d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}\right )}{e^{8}}\) | \(651\) |
1/2*(-e*x+2*d)/e^8*(-e^2*x^2+d^2)^(1/2)+7/2*d^2/e^7/(e^2)^(1/2)*arctan((e^ 2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-7/15*d^3/e^10/(x+d/e)^2*(-(x+d/e)^2*e^2+2 *d*e*(x+d/e))^(1/2)+773/240*d^2/e^9/(x+d/e)*(-(x+d/e)^2*e^2+2*d*e*(x+d/e)) ^(1/2)+1/24*d^3/e^10/(x-d/e)^2*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)+31/48* d^2/e^9/(x-d/e)*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)+1/20*d^4/e^11/(x+d/e) ^3*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)
Time = 0.32 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.69 \[ \int \frac {x^7}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {96 \, d^{2} e^{5} x^{5} + 96 \, d^{3} e^{4} x^{4} - 192 \, d^{4} e^{3} x^{3} - 192 \, d^{5} e^{2} x^{2} + 96 \, d^{6} e x + 96 \, d^{7} - 210 \, {\left (d^{2} e^{5} x^{5} + d^{3} e^{4} x^{4} - 2 \, d^{4} e^{3} x^{3} - 2 \, d^{5} e^{2} x^{2} + d^{6} e x + d^{7}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (15 \, e^{6} x^{6} - 15 \, d e^{5} x^{5} - 176 \, d^{2} e^{4} x^{4} + 4 \, d^{3} e^{3} x^{3} + 249 \, d^{4} e^{2} x^{2} + 9 \, d^{5} e x - 96 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{30 \, {\left (e^{13} x^{5} + d e^{12} x^{4} - 2 \, d^{2} e^{11} x^{3} - 2 \, d^{3} e^{10} x^{2} + d^{4} e^{9} x + d^{5} e^{8}\right )}} \]
1/30*(96*d^2*e^5*x^5 + 96*d^3*e^4*x^4 - 192*d^4*e^3*x^3 - 192*d^5*e^2*x^2 + 96*d^6*e*x + 96*d^7 - 210*(d^2*e^5*x^5 + d^3*e^4*x^4 - 2*d^4*e^3*x^3 - 2 *d^5*e^2*x^2 + d^6*e*x + d^7)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (15*e^6*x^6 - 15*d*e^5*x^5 - 176*d^2*e^4*x^4 + 4*d^3*e^3*x^3 + 249*d^4*e^2 *x^2 + 9*d^5*e*x - 96*d^6)*sqrt(-e^2*x^2 + d^2))/(e^13*x^5 + d*e^12*x^4 - 2*d^2*e^11*x^3 - 2*d^3*e^10*x^2 + d^4*e^9*x + d^5*e^8)
\[ \int \frac {x^7}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int \frac {x^{7}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (143) = 286\).
Time = 0.30 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.78 \[ \int \frac {x^7}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {d^{6}}{5 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{9} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{8}\right )}} - \frac {x^{5}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}} + \frac {d x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}} + \frac {25 \, d^{2} x^{3}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5}} - \frac {65 \, d^{3} x^{2}}{6 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{6}} - \frac {164 \, d^{4} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{7}} - \frac {7 \, d x^{2}}{6 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{6}} + \frac {53 \, d^{5}}{6 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{8}} + \frac {229 \, d^{2} x}{30 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{7}} + \frac {7 \, d^{2} \arcsin \left (\frac {e x}{d}\right )}{2 \, e^{8}} - \frac {14 \, d^{3}}{3 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{8}} - \frac {7 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{6 \, e^{8}} \]
1/5*d^6/((-e^2*x^2 + d^2)^(3/2)*e^9*x + (-e^2*x^2 + d^2)^(3/2)*d*e^8) - 1/ 2*x^5/((-e^2*x^2 + d^2)^(3/2)*e^3) + d*x^4/((-e^2*x^2 + d^2)^(3/2)*e^4) + 25/2*d^2*x^3/((-e^2*x^2 + d^2)^(3/2)*e^5) - 65/6*d^3*x^2/((-e^2*x^2 + d^2) ^(3/2)*e^6) - 164/15*d^4*x/((-e^2*x^2 + d^2)^(3/2)*e^7) - 7/6*d*x^2/(sqrt( -e^2*x^2 + d^2)*e^6) + 53/6*d^5/((-e^2*x^2 + d^2)^(3/2)*e^8) + 229/30*d^2* x/(sqrt(-e^2*x^2 + d^2)*e^7) + 7/2*d^2*arcsin(e*x/d)/e^8 - 14/3*d^3/(sqrt( -e^2*x^2 + d^2)*e^8) - 7/6*sqrt(-e^2*x^2 + d^2)*d/e^8
\[ \int \frac {x^7}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int { \frac {x^{7}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}} \,d x } \]
Timed out. \[ \int \frac {x^7}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int \frac {x^7}{{\left (d^2-e^2\,x^2\right )}^{5/2}\,\left (d+e\,x\right )} \,d x \]