Integrand size = 31, antiderivative size = 277 \[ \int \frac {(d+e x)^3 (f+g x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {(e f+d g)^5 (d+e x)^3}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(2 e f-23 d g) (e f+d g)^4 (d+e x)^2}{15 d^2 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {(e f+d g)^3 \left (2 e^2 f^2-21 d e f g+127 d^2 g^2\right ) (d+e x)}{15 d^3 e^6 \sqrt {d^2-e^2 x^2}}+\frac {g^4 (10 e f+7 d g) \sqrt {d^2-e^2 x^2}}{2 e^6}-\frac {g^5 (d-e x) \sqrt {d^2-e^2 x^2}}{2 e^6}-\frac {g^3 \left (20 e^2 f^2+30 d e f g+13 d^2 g^2\right ) \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^6} \] Output:
1/5*(d*g+e*f)^5*(e*x+d)^3/d/e^6/(-e^2*x^2+d^2)^(5/2)+1/15*(-23*d*g+2*e*f)* (d*g+e*f)^4*(e*x+d)^2/d^2/e^6/(-e^2*x^2+d^2)^(3/2)+1/15*(d*g+e*f)^3*(127*d ^2*g^2-21*d*e*f*g+2*e^2*f^2)*(e*x+d)/d^3/e^6/(-e^2*x^2+d^2)^(1/2)+1/2*g^4* (7*d*g+10*e*f)*(-e^2*x^2+d^2)^(1/2)/e^6-1/2*g^5*(-e*x+d)*(-e^2*x^2+d^2)^(1 /2)/e^6-1/2*g^3*(13*d^2*g^2+30*d*e*f*g+20*e^2*f^2)*arctan(e*x/(-e^2*x^2+d^ 2)^(1/2))/e^6
Time = 4.68 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.09 \[ \int \frac {(d+e x)^3 (f+g x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\frac {\sqrt {d^2-e^2 x^2} \left (304 d^7 g^5+4 e^7 f^5 x^2+3 d^6 e g^4 (240 f-239 g x)-6 d e^6 f^4 x (2 f+5 g x)+2 d^2 e^5 f^3 \left (7 f^2+45 f g x+70 g^2 x^2\right )+d^5 e^2 g^3 \left (440 f^2-1710 f g x+479 g^2 x^2\right )+5 d^4 e^3 g^2 \left (8 f^3-204 f^2 g x+234 f g^2 x^2-9 g^3 x^3\right )-5 d^3 e^4 g \left (6 f^4+24 f^3 g x-128 f^2 g^2 x^2+30 f g^3 x^3+3 g^4 x^4\right )\right )}{d^3 (d-e x)^3}+30 g^3 \left (20 e^2 f^2+30 d e f g+13 d^2 g^2\right ) \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{30 e^6} \] Input:
Integrate[((d + e*x)^3*(f + g*x)^5)/(d^2 - e^2*x^2)^(7/2),x]
Output:
((Sqrt[d^2 - e^2*x^2]*(304*d^7*g^5 + 4*e^7*f^5*x^2 + 3*d^6*e*g^4*(240*f - 239*g*x) - 6*d*e^6*f^4*x*(2*f + 5*g*x) + 2*d^2*e^5*f^3*(7*f^2 + 45*f*g*x + 70*g^2*x^2) + d^5*e^2*g^3*(440*f^2 - 1710*f*g*x + 479*g^2*x^2) + 5*d^4*e^ 3*g^2*(8*f^3 - 204*f^2*g*x + 234*f*g^2*x^2 - 9*g^3*x^3) - 5*d^3*e^4*g*(6*f ^4 + 24*f^3*g*x - 128*f^2*g^2*x^2 + 30*f*g^3*x^3 + 3*g^4*x^4)))/(d^3*(d - e*x)^3) + 30*g^3*(20*e^2*f^2 + 30*d*e*f*g + 13*d^2*g^2)*ArcTan[(e*x)/(Sqrt [d^2] - Sqrt[d^2 - e^2*x^2])])/(30*e^6)
Time = 1.43 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.08, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {691, 25, 2166, 25, 2166, 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.
| \(\displaystyle \int \frac {(d+e x)^3 (f+g x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 691 |
| \(\displaystyle \frac {(d+e x)^3 (d g+e f)^5}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int -\frac {(d+e x)^2 \left (-\frac {5 d x^4 g^5}{e}-\frac {5 d (5 e f+d g) x^3 g^4}{e^2}-\frac {5 d \left (10 e^2 f^2+5 d e g f+d^2 g^2\right ) x^2 g^3}{e^3}-\frac {5 d \left (10 e^3 f^3+10 d e^2 g f^2+5 d^2 e g^2 f+d^3 g^3\right ) x g^2}{e^4}+\frac {2 e^5 f^5-15 d e^4 g f^4-30 d^2 e^3 g^2 f^3-30 d^3 e^2 g^3 f^2-15 d^4 e g^4 f-3 d^5 g^5}{e^5}\right )}{\left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {(d+e x)^2 \left (-\frac {5 d x^4 g^5}{e}-\frac {5 d (5 e f+d g) x^3 g^4}{e^2}-\frac {5 d \left (10 e^2 f^2+5 d e g f+d^2 g^2\right ) x^2 g^3}{e^3}-\frac {5 d \left (10 e^3 f^3+10 d e^2 g f^2+5 d^2 e g^2 f+d^3 g^3\right ) x g^2}{e^4}+\frac {2 e^5 f^5-15 d e^4 g f^4-30 d^2 e^3 g^2 f^3-30 d^3 e^2 g^3 f^2-15 d^4 e g^4 f-3 d^5 g^5}{e^5}\right )}{\left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d}+\frac {(d+e x)^3 (d g+e f)^5}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 2166 |
| \(\displaystyle \frac {\frac {(d+e x)^2 (2 e f-23 d g) (d g+e f)^4}{3 d e^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int -\frac {(d+e x) \left (\frac {15 d^2 x^3 g^5}{e^2}+\frac {15 d^2 (5 e f+2 d g) x^2 g^4}{e^3}+\frac {15 d^2 \left (10 e^2 f^2+10 d e g f+3 d^2 g^2\right ) x g^3}{e^4}+\frac {2 e^5 f^5-15 d e^4 g f^4+70 d^2 e^3 g^2 f^3+170 d^3 e^2 g^3 f^2+135 d^4 e g^4 f+37 d^5 g^5}{e^5}\right )}{\left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d}}{5 d}+\frac {(d+e x)^3 (d g+e f)^5}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {(d+e x) \left (\frac {15 d^2 x^3 g^5}{e^2}+\frac {15 d^2 (5 e f+2 d g) x^2 g^4}{e^3}+\frac {15 d^2 \left (10 e^2 f^2+10 d e g f+3 d^2 g^2\right ) x g^3}{e^4}+\frac {2 e^5 f^5-15 d e^4 g f^4+70 d^2 e^3 g^2 f^3+170 d^3 e^2 g^3 f^2+135 d^4 e g^4 f+37 d^5 g^5}{e^5}\right )}{\left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d}+\frac {(d+e x)^2 (2 e f-23 d g) (d g+e f)^4}{3 d e^6 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d}+\frac {(d+e x)^3 (d g+e f)^5}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 2166 |
| \(\displaystyle \frac {\frac {\frac {(d+e x) (d g+e f)^3 \left (127 d^2 g^2-21 d e f g+2 e^2 f^2\right )}{d e^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {15 \left (\frac {d^3 x^2 g^5}{e^3}+\frac {d^3 (5 e f+3 d g) x g^4}{e^4}+\frac {d^3 \left (10 e^2 f^2+15 d e g f+6 d^2 g^2\right ) g^3}{e^5}\right )}{\sqrt {d^2-e^2 x^2}}dx}{d}}{3 d}+\frac {(d+e x)^2 (2 e f-23 d g) (d g+e f)^4}{3 d e^6 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d}+\frac {(d+e x)^3 (d g+e f)^5}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {(d+e x) (d g+e f)^3 \left (127 d^2 g^2-21 d e f g+2 e^2 f^2\right )}{d e^6 \sqrt {d^2-e^2 x^2}}-\frac {15 \int \frac {\frac {d^3 x^2 g^5}{e^3}+\frac {d^3 (5 e f+3 d g) x g^4}{e^4}+\frac {d^3 \left (10 e^2 f^2+15 d e g f+6 d^2 g^2\right ) g^3}{e^5}}{\sqrt {d^2-e^2 x^2}}dx}{d}}{3 d}+\frac {(d+e x)^2 (2 e f-23 d g) (d g+e f)^4}{3 d e^6 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d}+\frac {(d+e x)^3 (d g+e f)^5}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle \frac {\frac {\frac {(d+e x) (d g+e f)^3 \left (127 d^2 g^2-21 d e f g+2 e^2 f^2\right )}{d e^6 \sqrt {d^2-e^2 x^2}}-\frac {15 \left (-\frac {\int -\frac {d^3 g^3 \left (20 e^2 f^2+30 d e g f+13 d^2 g^2+2 e g (5 e f+3 d g) x\right )}{e^3 \sqrt {d^2-e^2 x^2}}dx}{2 e^2}-\frac {d^3 g^5 x \sqrt {d^2-e^2 x^2}}{2 e^5}\right )}{d}}{3 d}+\frac {(d+e x)^2 (2 e f-23 d g) (d g+e f)^4}{3 d e^6 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d}+\frac {(d+e x)^3 (d g+e f)^5}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {(d+e x) (d g+e f)^3 \left (127 d^2 g^2-21 d e f g+2 e^2 f^2\right )}{d e^6 \sqrt {d^2-e^2 x^2}}-\frac {15 \left (\frac {\int \frac {d^3 g^3 \left (20 e^2 f^2+30 d e g f+13 d^2 g^2+2 e g (5 e f+3 d g) x\right )}{e^3 \sqrt {d^2-e^2 x^2}}dx}{2 e^2}-\frac {d^3 g^5 x \sqrt {d^2-e^2 x^2}}{2 e^5}\right )}{d}}{3 d}+\frac {(d+e x)^2 (2 e f-23 d g) (d g+e f)^4}{3 d e^6 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d}+\frac {(d+e x)^3 (d g+e f)^5}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {(d+e x) (d g+e f)^3 \left (127 d^2 g^2-21 d e f g+2 e^2 f^2\right )}{d e^6 \sqrt {d^2-e^2 x^2}}-\frac {15 \left (\frac {d^3 g^3 \int \frac {20 e^2 f^2+30 d e g f+13 d^2 g^2+2 e g (5 e f+3 d g) x}{\sqrt {d^2-e^2 x^2}}dx}{2 e^5}-\frac {d^3 g^5 x \sqrt {d^2-e^2 x^2}}{2 e^5}\right )}{d}}{3 d}+\frac {(d+e x)^2 (2 e f-23 d g) (d g+e f)^4}{3 d e^6 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d}+\frac {(d+e x)^3 (d g+e f)^5}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {\frac {\frac {(d+e x) (d g+e f)^3 \left (127 d^2 g^2-21 d e f g+2 e^2 f^2\right )}{d e^6 \sqrt {d^2-e^2 x^2}}-\frac {15 \left (\frac {d^3 g^3 \left (\left (13 d^2 g^2+30 d e f g+20 e^2 f^2\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx-\frac {2 g \sqrt {d^2-e^2 x^2} (3 d g+5 e f)}{e}\right )}{2 e^5}-\frac {d^3 g^5 x \sqrt {d^2-e^2 x^2}}{2 e^5}\right )}{d}}{3 d}+\frac {(d+e x)^2 (2 e f-23 d g) (d g+e f)^4}{3 d e^6 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d}+\frac {(d+e x)^3 (d g+e f)^5}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {\frac {(d+e x) (d g+e f)^3 \left (127 d^2 g^2-21 d e f g+2 e^2 f^2\right )}{d e^6 \sqrt {d^2-e^2 x^2}}-\frac {15 \left (\frac {d^3 g^3 \left (\left (13 d^2 g^2+30 d e f g+20 e^2 f^2\right ) \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}-\frac {2 g \sqrt {d^2-e^2 x^2} (3 d g+5 e f)}{e}\right )}{2 e^5}-\frac {d^3 g^5 x \sqrt {d^2-e^2 x^2}}{2 e^5}\right )}{d}}{3 d}+\frac {(d+e x)^2 (2 e f-23 d g) (d g+e f)^4}{3 d e^6 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d}+\frac {(d+e x)^3 (d g+e f)^5}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {\frac {(d+e x) (d g+e f)^3 \left (127 d^2 g^2-21 d e f g+2 e^2 f^2\right )}{d e^6 \sqrt {d^2-e^2 x^2}}-\frac {15 \left (\frac {d^3 g^3 \left (\frac {\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right ) \left (13 d^2 g^2+30 d e f g+20 e^2 f^2\right )}{e}-\frac {2 g \sqrt {d^2-e^2 x^2} (3 d g+5 e f)}{e}\right )}{2 e^5}-\frac {d^3 g^5 x \sqrt {d^2-e^2 x^2}}{2 e^5}\right )}{d}}{3 d}+\frac {(d+e x)^2 (2 e f-23 d g) (d g+e f)^4}{3 d e^6 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d}+\frac {(d+e x)^3 (d g+e f)^5}{5 d e^6 \left (d^2-e^2 x^2\right )^{5/2}}\) |
Input:
Int[((d + e*x)^3*(f + g*x)^5)/(d^2 - e^2*x^2)^(7/2),x]
Output:
((e*f + d*g)^5*(d + e*x)^3)/(5*d*e^6*(d^2 - e^2*x^2)^(5/2)) + (((2*e*f - 2 3*d*g)*(e*f + d*g)^4*(d + e*x)^2)/(3*d*e^6*(d^2 - e^2*x^2)^(3/2)) + (((e*f + d*g)^3*(2*e^2*f^2 - 21*d*e*f*g + 127*d^2*g^2)*(d + e*x))/(d*e^6*Sqrt[d^ 2 - e^2*x^2]) - (15*(-1/2*(d^3*g^5*x*Sqrt[d^2 - e^2*x^2])/e^5 + (d^3*g^3*( (-2*g*(5*e*f + 3*d*g)*Sqrt[d^2 - e^2*x^2])/e + ((20*e^2*f^2 + 30*d*e*f*g + 13*d^2*g^2)*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e))/(2*e^5)))/d)/(3*d))/(5 *d)
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[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_) ^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(f + g*x)^n, a*e + c*d* x, x], R = PolynomialRemainder[(f + g*x)^n, a*e + c*d*x, x]}, Simp[(-d)*R*( d + e*x)^m*((a + c*x^2)^(p + 1)/(2*a*e*(p + 1))), x] + Simp[d/(2*a*(p + 1)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*e*(p + 1)*Q + R*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, c, d, e, f, g}, x] && IGtQ[n, 1] && IGtQ[m, 0] && LtQ[p, -1] && EqQ[c*d^2 + a*e^2, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[Pq, a*e + b*d*x, x], R = PolynomialRemainde r[Pq, a*e + b*d*x, x]}, Simp[(-d)*R*(d + e*x)^m*((a + b*x^2)^(p + 1)/(2*a*e *(p + 1))), x] + Simp[d/(2*a*(p + 1)) Int[(d + e*x)^(m - 1)*(a + b*x^2)^( p + 1)*ExpandToSum[2*a*e*(p + 1)*Qx + R*(m + 2*p + 2), x], x], x]] /; FreeQ [{a, b, d, e}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 0] && GtQ[m, 0]
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. \(603\) vs. \(2(253)=506\).
Time = 1.62 (sec) , antiderivative size = 604, normalized size of antiderivative = 2.18
| method | result | size |
| risch | \(\frac {g^{4} \left (e g x +6 d g +10 e f \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{2 e^{6}}-\frac {\frac {\left (2 g^{5} d^{5}+10 e f \,g^{4} d^{4}+20 e^{2} f^{2} g^{3} d^{3}+20 e^{3} f^{3} g^{2} d^{2}+10 f^{4} g \,e^{4} d +2 f^{5} e^{5}\right ) \left (\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{5 d e \left (x -\frac {d}{e}\right )^{3}}-\frac {2 e \left (\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{3 d e \left (x -\frac {d}{e}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{3 d^{2} \left (x -\frac {d}{e}\right )}\right )}{5 d}\right )}{e^{3}}+\frac {13 d^{2} g^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}+\frac {20 e^{2} f^{2} g^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}+\frac {10 g \left (g^{4} d^{4}+4 d^{3} e f \,g^{3}+6 d^{2} e^{2} f^{2} g^{2}+4 d \,e^{3} f^{3} g +e^{4} f^{4}\right ) \left (\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{3 d e \left (x -\frac {d}{e}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{3 d^{2} \left (x -\frac {d}{e}\right )}\right )}{e^{2}}+\frac {20 g^{2} \left (d^{3} g^{3}+3 d^{2} e f \,g^{2}+3 d \,e^{2} f^{2} g +e^{3} f^{3}\right ) \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{e^{2} d \left (x -\frac {d}{e}\right )}+\frac {30 d e f \,g^{4} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}}{2 e^{5}}\) | \(604\) |
| default | \(\text {Expression too large to display}\) | \(1004\) |
Input:
int((e*x+d)^3*(g*x+f)^5/(-e^2*x^2+d^2)^(7/2),x,method=_RETURNVERBOSE)
Output:
1/2*g^4*(e*g*x+6*d*g+10*e*f)/e^6*(-e^2*x^2+d^2)^(1/2)-1/2/e^5*((2*d^5*g^5+ 10*d^4*e*f*g^4+20*d^3*e^2*f^2*g^3+20*d^2*e^3*f^3*g^2+10*d*e^4*f^4*g+2*e^5* f^5)/e^3*(1/5/d/e/(x-d/e)^3*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)-2/5*e/d*( 1/3/d/e/(x-d/e)^2*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)-1/3/d^2/(x-d/e)*(-( x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)))+13*d^2*g^5/(e^2)^(1/2)*arctan((e^2)^(1 /2)*x/(-e^2*x^2+d^2)^(1/2))+20*e^2*f^2*g^3/(e^2)^(1/2)*arctan((e^2)^(1/2)* x/(-e^2*x^2+d^2)^(1/2))+10*g*(d^4*g^4+4*d^3*e*f*g^3+6*d^2*e^2*f^2*g^2+4*d* e^3*f^3*g+e^4*f^4)/e^2*(1/3/d/e/(x-d/e)^2*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^( 1/2)-1/3/d^2/(x-d/e)*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2))+20*g^2*(d^3*g^3 +3*d^2*e*f*g^2+3*d*e^2*f^2*g+e^3*f^3)/e^2/d/(x-d/e)*(-(x-d/e)^2*e^2-2*d*e* (x-d/e))^(1/2)+30*d*e*f*g^4/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2 )^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 807 vs. \(2 (254) = 508\).
Time = 0.13 (sec) , antiderivative size = 807, normalized size of antiderivative = 2.91 \[ \int \frac {(d+e x)^3 (f+g x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^3*(g*x+f)^5/(-e^2*x^2+d^2)^(7/2),x, algorithm="fricas")
Output:
-1/30*(14*d^3*e^5*f^5 - 30*d^4*e^4*f^4*g + 40*d^5*e^3*f^3*g^2 + 440*d^6*e^ 2*f^2*g^3 + 720*d^7*e*f*g^4 + 304*d^8*g^5 - 2*(7*e^8*f^5 - 15*d*e^7*f^4*g + 20*d^2*e^6*f^3*g^2 + 220*d^3*e^5*f^2*g^3 + 360*d^4*e^4*f*g^4 + 152*d^5*e ^3*g^5)*x^3 + 6*(7*d*e^7*f^5 - 15*d^2*e^6*f^4*g + 20*d^3*e^5*f^3*g^2 + 220 *d^4*e^4*f^2*g^3 + 360*d^5*e^3*f*g^4 + 152*d^6*e^2*g^5)*x^2 - 6*(7*d^2*e^6 *f^5 - 15*d^3*e^5*f^4*g + 20*d^4*e^4*f^3*g^2 + 220*d^5*e^3*f^2*g^3 + 360*d ^6*e^2*f*g^4 + 152*d^7*e*g^5)*x + 30*(20*d^6*e^2*f^2*g^3 + 30*d^7*e*f*g^4 + 13*d^8*g^5 - (20*d^3*e^5*f^2*g^3 + 30*d^4*e^4*f*g^4 + 13*d^5*e^3*g^5)*x^ 3 + 3*(20*d^4*e^4*f^2*g^3 + 30*d^5*e^3*f*g^4 + 13*d^6*e^2*g^5)*x^2 - 3*(20 *d^5*e^3*f^2*g^3 + 30*d^6*e^2*f*g^4 + 13*d^7*e*g^5)*x)*arctan(-(d - sqrt(- e^2*x^2 + d^2))/(e*x)) - (15*d^3*e^4*g^5*x^4 - 14*d^2*e^5*f^5 + 30*d^3*e^4 *f^4*g - 40*d^4*e^3*f^3*g^2 - 440*d^5*e^2*f^2*g^3 - 720*d^6*e*f*g^4 - 304* d^7*g^5 + 15*(10*d^3*e^4*f*g^4 + 3*d^4*e^3*g^5)*x^3 - (4*e^7*f^5 - 30*d*e^ 6*f^4*g + 140*d^2*e^5*f^3*g^2 + 640*d^3*e^4*f^2*g^3 + 1170*d^4*e^3*f*g^4 + 479*d^5*e^2*g^5)*x^2 + 3*(4*d*e^6*f^5 - 30*d^2*e^5*f^4*g + 40*d^3*e^4*f^3 *g^2 + 340*d^4*e^3*f^2*g^3 + 570*d^5*e^2*f*g^4 + 239*d^6*e*g^5)*x)*sqrt(-e ^2*x^2 + d^2))/(d^3*e^9*x^3 - 3*d^4*e^8*x^2 + 3*d^5*e^7*x - d^6*e^6)
\[ \int \frac {(d+e x)^3 (f+g x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {\left (d + e x\right )^{3} \left (f + g x\right )^{5}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \] Input:
integrate((e*x+d)**3*(g*x+f)**5/(-e**2*x**2+d**2)**(7/2),x)
Output:
Integral((d + e*x)**3*(f + g*x)**5/(-(-d + e*x)*(d + e*x))**(7/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 1603 vs. \(2 (254) = 508\).
Time = 0.14 (sec) , antiderivative size = 1603, normalized size of antiderivative = 5.79 \[ \int \frac {(d+e x)^3 (f+g x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^3*(g*x+f)^5/(-e^2*x^2+d^2)^(7/2),x, algorithm="maxima")
Output:
-1/2*e*g^5*x^7/(-e^2*x^2 + d^2)^(5/2) + 7/30*d^2*e*g^5*x*(15*x^4/((-e^2*x^ 2 + d^2)^(5/2)*e^2) - 20*d^2*x^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 8*d^4/((-e ^2*x^2 + d^2)^(5/2)*e^6)) - 7/6*d^2*g^5*x*(3*x^2/((-e^2*x^2 + d^2)^(3/2)*e ^2) - 2*d^2/((-e^2*x^2 + d^2)^(3/2)*e^4))/e + 1/5*d*f^5*x/(-e^2*x^2 + d^2) ^(5/2) + 3/5*d^2*f^5/((-e^2*x^2 + d^2)^(5/2)*e) + d^3*f^4*g/((-e^2*x^2 + d ^2)^(5/2)*e^2) + 4/15*f^5*x/((-e^2*x^2 + d^2)^(3/2)*d) + 14/15*d^4*g^5*x/( (-e^2*x^2 + d^2)^(3/2)*e^5) + 1/15*(10*e^3*f^2*g^3 + 15*d*e^2*f*g^4 + 3*d^ 2*e*g^5)*x*(15*x^4/((-e^2*x^2 + d^2)^(5/2)*e^2) - 20*d^2*x^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 8*d^4/((-e^2*x^2 + d^2)^(5/2)*e^6)) + 8/15*f^5*x/(sqrt(- e^2*x^2 + d^2)*d^3) - 49/30*d^2*g^5*x/(sqrt(-e^2*x^2 + d^2)*e^5) - (5*e^3* f*g^4 + 3*d*e^2*g^5)*x^6/((-e^2*x^2 + d^2)^(5/2)*e^2) - 7/2*d^2*g^5*arcsin (e^2*x/(d*sqrt(e^2)))/(sqrt(e^2)*e^5) - 1/3*(10*e^3*f^2*g^3 + 15*d*e^2*f*g ^4 + 3*d^2*e*g^5)*x*(3*x^2/((-e^2*x^2 + d^2)^(3/2)*e^2) - 2*d^2/((-e^2*x^2 + d^2)^(3/2)*e^4))/e^2 + 6*(5*e^3*f*g^4 + 3*d*e^2*g^5)*d^2*x^4/((-e^2*x^2 + d^2)^(5/2)*e^4) + (10*e^3*f^3*g^2 + 30*d*e^2*f^2*g^3 + 15*d^2*e*f*g^4 + d^3*g^5)*x^4/((-e^2*x^2 + d^2)^(5/2)*e^2) + 5/2*(e^3*f^4*g + 6*d*e^2*f^3* g^2 + 6*d^2*e*f^2*g^3 + d^3*f*g^4)*x^3/((-e^2*x^2 + d^2)^(5/2)*e^2) - 8*(5 *e^3*f*g^4 + 3*d*e^2*g^5)*d^4*x^2/((-e^2*x^2 + d^2)^(5/2)*e^6) - 4/3*(10*e ^3*f^3*g^2 + 30*d*e^2*f^2*g^3 + 15*d^2*e*f*g^4 + d^3*g^5)*d^2*x^2/((-e^2*x ^2 + d^2)^(5/2)*e^4) + 1/3*(e^3*f^5 + 15*d*e^2*f^4*g + 30*d^2*e*f^3*g^2...
Leaf count of result is larger than twice the leaf count of optimal. 969 vs. \(2 (254) = 508\).
Time = 0.18 (sec) , antiderivative size = 969, normalized size of antiderivative = 3.50 \[ \int \frac {(d+e x)^3 (f+g x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^3*(g*x+f)^5/(-e^2*x^2+d^2)^(7/2),x, algorithm="giac")
Output:
1/2*sqrt(-e^2*x^2 + d^2)*(g^5*x/e^5 + 2*(5*e^11*f*g^4 + 3*d*e^10*g^5)/e^16 ) - 1/2*(20*e^2*f^2*g^3 + 30*d*e*f*g^4 + 13*d^2*g^5)*arcsin(e*x/d)*sgn(d)* sgn(e)/(e^5*abs(e)) + 2/15*(7*e^5*f^5 - 15*d*e^4*f^4*g + 20*d^2*e^3*f^3*g^ 2 + 220*d^3*e^2*f^2*g^3 + 285*d^4*e*f*g^4 + 107*d^5*g^5 - 20*(d*e + sqrt(- e^2*x^2 + d^2)*abs(e))*e^3*f^5/x + 75*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))* d*e^2*f^4*g/x - 100*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^2*e*f^3*g^2/x - 950*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^3*f^2*g^3/x - 1200*(d*e + sqrt(- e^2*x^2 + d^2)*abs(e))*d^4*f*g^4/(e*x) - 445*(d*e + sqrt(-e^2*x^2 + d^2)*a bs(e))*d^5*g^5/(e^2*x) + 40*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*e*f^5/x^ 2 - 75*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d*f^4*g/x^2 + 200*(d*e + sqrt (-e^2*x^2 + d^2)*abs(e))^2*d^2*f^3*g^2/(e*x^2) + 1450*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^3*f^2*g^3/(e^2*x^2) + 1800*(d*e + sqrt(-e^2*x^2 + d^2) *abs(e))^2*d^4*f*g^4/(e^3*x^2) + 665*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2 *d^5*g^5/(e^4*x^2) - 30*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*f^5/(e*x^3) + 75*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d*f^4*g/(e^2*x^3) - 750*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^3*f^2*g^3/(e^4*x^3) - 1050*(d*e + sqrt(-e ^2*x^2 + d^2)*abs(e))^3*d^4*f*g^4/(e^5*x^3) - 405*(d*e + sqrt(-e^2*x^2 + d ^2)*abs(e))^3*d^5*g^5/(e^6*x^3) + 15*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4 *f^5/(e^3*x^4) + 150*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d^3*f^2*g^3/(e^ 6*x^4) + 225*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d^4*f*g^4/(e^7*x^4) ...
Timed out. \[ \int \frac {(d+e x)^3 (f+g x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^5\,{\left (d+e\,x\right )}^3}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \] Input:
int(((f + g*x)^5*(d + e*x)^3)/(d^2 - e^2*x^2)^(7/2),x)
Output:
int(((f + g*x)^5*(d + e*x)^3)/(d^2 - e^2*x^2)^(7/2), x)
Time = 0.29 (sec) , antiderivative size = 1448, normalized size of antiderivative = 5.23 \[ \int \frac {(d+e x)^3 (f+g x)^5}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx =\text {Too large to display} \] Input:
int((e*x+d)^3*(g*x+f)^5/(-e^2*x^2+d^2)^(7/2),x)
Output:
( - 195*sqrt(d**2 - e**2*x**2)*asin((e*x)/d)*d**7*g**5 - 450*sqrt(d**2 - e **2*x**2)*asin((e*x)/d)*d**6*e*f*g**4 + 390*sqrt(d**2 - e**2*x**2)*asin((e *x)/d)*d**6*e*g**5*x - 300*sqrt(d**2 - e**2*x**2)*asin((e*x)/d)*d**5*e**2* f**2*g**3 + 900*sqrt(d**2 - e**2*x**2)*asin((e*x)/d)*d**5*e**2*f*g**4*x - 195*sqrt(d**2 - e**2*x**2)*asin((e*x)/d)*d**5*e**2*g**5*x**2 + 600*sqrt(d* *2 - e**2*x**2)*asin((e*x)/d)*d**4*e**3*f**2*g**3*x - 450*sqrt(d**2 - e**2 *x**2)*asin((e*x)/d)*d**4*e**3*f*g**4*x**2 - 300*sqrt(d**2 - e**2*x**2)*as in((e*x)/d)*d**3*e**4*f**2*g**3*x**2 + 195*asin((e*x)/d)*d**8*g**5 + 450*a sin((e*x)/d)*d**7*e*f*g**4 - 585*asin((e*x)/d)*d**7*e*g**5*x + 300*asin((e *x)/d)*d**6*e**2*f**2*g**3 - 1350*asin((e*x)/d)*d**6*e**2*f*g**4*x + 585*a sin((e*x)/d)*d**6*e**2*g**5*x**2 - 900*asin((e*x)/d)*d**5*e**3*f**2*g**3*x + 1350*asin((e*x)/d)*d**5*e**3*f*g**4*x**2 - 195*asin((e*x)/d)*d**5*e**3* g**5*x**3 + 900*asin((e*x)/d)*d**4*e**4*f**2*g**3*x**2 - 450*asin((e*x)/d) *d**4*e**4*f*g**4*x**3 - 300*asin((e*x)/d)*d**3*e**5*f**2*g**3*x**3 - 78*s qrt(d**2 - e**2*x**2)*d**7*g**5 - 180*sqrt(d**2 - e**2*x**2)*d**6*e*f*g**4 + 265*sqrt(d**2 - e**2*x**2)*d**6*e*g**5*x - 120*sqrt(d**2 - e**2*x**2)*d **5*e**2*f**2*g**3 + 630*sqrt(d**2 - e**2*x**2)*d**5*e**2*f*g**4*x - 253*s qrt(d**2 - e**2*x**2)*d**5*e**2*g**5*x**2 + 380*sqrt(d**2 - e**2*x**2)*d** 4*e**3*f**2*g**3*x - 630*sqrt(d**2 - e**2*x**2)*d**4*e**3*f*g**4*x**2 + 45 *sqrt(d**2 - e**2*x**2)*d**4*e**3*g**5*x**3 + 40*sqrt(d**2 - e**2*x**2)...