Integrand size = 30, antiderivative size = 263 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 (b d-a e)^3 (a+b x) \sqrt {d+e x}}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (b d-a e) (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{7/2}}{7 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (b d-a e)^{7/2} (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
2/3*(-a*e+b*d)^2*(b*x+a)*(e*x+d)^(3/2)/b^3/((b*x+a)^2)^(1/2)+2/5*(-a*e+b*d )*(b*x+a)*(e*x+d)^(5/2)/b^2/((b*x+a)^2)^(1/2)+2/7*(b*x+a)*(e*x+d)^(7/2)/b/ ((b*x+a)^2)^(1/2)-2*(-a*e+b*d)^(7/2)*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/2) /(-a*e+b*d)^(1/2))/b^(9/2)/((b*x+a)^2)^(1/2)+2*(-a*e+b*d)^3*(b*x+a)*(e*x+d )^(1/2)/b^4/((b*x+a)^2)^(1/2)
Time = 0.18 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.64 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 (a+b x) \left (\sqrt {b} \sqrt {d+e x} \left (-105 a^3 e^3+35 a^2 b e^2 (10 d+e x)-7 a b^2 e \left (58 d^2+16 d e x+3 e^2 x^2\right )+b^3 \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )\right )+105 (-b d+a e)^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{105 b^{9/2} \sqrt {(a+b x)^2}} \]
(2*(a + b*x)*(Sqrt[b]*Sqrt[d + e*x]*(-105*a^3*e^3 + 35*a^2*b*e^2*(10*d + e *x) - 7*a*b^2*e*(58*d^2 + 16*d*e*x + 3*e^2*x^2) + b^3*(176*d^3 + 122*d^2*e *x + 66*d*e^2*x^2 + 15*e^3*x^3)) + 105*(-(b*d) + a*e)^(7/2)*ArcTan[(Sqrt[b ]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]]))/(105*b^(9/2)*Sqrt[(a + b*x)^2])
Time = 0.28 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.67, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1102, 27, 60, 60, 60, 60, 73, 221}
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.
| \(\displaystyle \int \frac {(d+e x)^{7/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx\) |
| \(\Big \downarrow \) 1102 |
| \(\displaystyle \frac {b (a+b x) \int \frac {(d+e x)^{7/2}}{b (a+b x)}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {(d+e x)^{7/2}}{a+b x}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(b d-a e) \int \frac {(d+e x)^{5/2}}{a+b x}dx}{b}+\frac {2 (d+e x)^{7/2}}{7 b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \int \frac {(d+e x)^{3/2}}{a+b x}dx}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{b}+\frac {2 (d+e x)^{7/2}}{7 b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \int \frac {\sqrt {d+e x}}{a+b x}dx}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{b}+\frac {2 (d+e x)^{7/2}}{7 b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{b}+\frac {2 (d+e x)^{7/2}}{7 b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {2 (b d-a e) \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b e}+\frac {2 \sqrt {d+e x}}{b}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{b}+\frac {2 (d+e x)^{7/2}}{7 b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(a+b x) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {(b d-a e) \left (\frac {2 \sqrt {d+e x}}{b}-\frac {2 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{3/2}}\right )}{b}+\frac {2 (d+e x)^{3/2}}{3 b}\right )}{b}+\frac {2 (d+e x)^{5/2}}{5 b}\right )}{b}+\frac {2 (d+e x)^{7/2}}{7 b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*((2*(d + e*x)^(7/2))/(7*b) + ((b*d - a*e)*((2*(d + e*x)^(5/2))/ (5*b) + ((b*d - a*e)*((2*(d + e*x)^(3/2))/(3*b) + ((b*d - a*e)*((2*Sqrt[d + e*x])/b - (2*Sqrt[b*d - a*e]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(3/2)))/b))/b))/b))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
3.18.3.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*F racPart[p])) Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0]
Time = 2.39 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.90
| method | result | size |
| risch | \(-\frac {2 \left (-15 e^{3} x^{3} b^{3}+21 x^{2} a \,b^{2} e^{3}-66 x^{2} b^{3} d \,e^{2}-35 a^{2} b \,e^{3} x +112 x a \,b^{2} d \,e^{2}-122 b^{3} d^{2} e x +105 a^{3} e^{3}-350 a^{2} b d \,e^{2}+406 a \,b^{2} d^{2} e -176 b^{3} d^{3}\right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{105 b^{4} \left (b x +a \right )}+\frac {2 \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b^{4} \sqrt {\left (a e -b d \right ) b}\, \left (b x +a \right )}\) | \(237\) |
| default | \(\frac {2 \left (b x +a \right ) \left (15 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {7}{2}} b^{3}-21 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {5}{2}} a \,b^{2} e +21 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {5}{2}} b^{3} d +35 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} a^{2} b \,e^{2}-70 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} a \,b^{2} d e +35 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} b^{3} d^{2}+105 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{4} e^{4}-420 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{3} b d \,e^{3}+630 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} b^{2} d^{2} e^{2}-420 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a \,b^{3} d^{3} e +105 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) b^{4} d^{4}-105 \sqrt {e x +d}\, a^{3} e^{3} \sqrt {\left (a e -b d \right ) b}+315 \sqrt {e x +d}\, a^{2} d \,e^{2} b \sqrt {\left (a e -b d \right ) b}-315 \sqrt {e x +d}\, a \,d^{2} e \,b^{2} \sqrt {\left (a e -b d \right ) b}+105 \sqrt {e x +d}\, d^{3} b^{3} \sqrt {\left (a e -b d \right ) b}\right )}{105 \sqrt {\left (b x +a \right )^{2}}\, b^{4} \sqrt {\left (a e -b d \right ) b}}\) | \(462\) |
-2/105*(-15*b^3*e^3*x^3+21*a*b^2*e^3*x^2-66*b^3*d*e^2*x^2-35*a^2*b*e^3*x+1 12*a*b^2*d*e^2*x-122*b^3*d^2*e*x+105*a^3*e^3-350*a^2*b*d*e^2+406*a*b^2*d^2 *e-176*b^3*d^3)*(e*x+d)^(1/2)/b^4*((b*x+a)^2)^(1/2)/(b*x+a)+2*(a^4*e^4-4*a ^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)/b^4/((a*e-b*d)*b)^(1/2 )*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*((b*x+a)^2)^(1/2)/(b*x+a)
Time = 0.38 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.61 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\left [-\frac {105 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e + 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) - 2 \, {\left (15 \, b^{3} e^{3} x^{3} + 176 \, b^{3} d^{3} - 406 \, a b^{2} d^{2} e + 350 \, a^{2} b d e^{2} - 105 \, a^{3} e^{3} + 3 \, {\left (22 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + {\left (122 \, b^{3} d^{2} e - 112 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{105 \, b^{4}}, -\frac {2 \, {\left (105 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (15 \, b^{3} e^{3} x^{3} + 176 \, b^{3} d^{3} - 406 \, a b^{2} d^{2} e + 350 \, a^{2} b d e^{2} - 105 \, a^{3} e^{3} + 3 \, {\left (22 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + {\left (122 \, b^{3} d^{2} e - 112 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}\right )}}{105 \, b^{4}}\right ] \]
[-1/105*(105*(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*sqrt((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e + 2*sqrt(e*x + d)*b*sqrt((b*d - a*e)/b ))/(b*x + a)) - 2*(15*b^3*e^3*x^3 + 176*b^3*d^3 - 406*a*b^2*d^2*e + 350*a^ 2*b*d*e^2 - 105*a^3*e^3 + 3*(22*b^3*d*e^2 - 7*a*b^2*e^3)*x^2 + (122*b^3*d^ 2*e - 112*a*b^2*d*e^2 + 35*a^2*b*e^3)*x)*sqrt(e*x + d))/b^4, -2/105*(105*( b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*sqrt(-(b*d - a*e)/b)*ar ctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (15*b^3*e^3*x^3 + 176*b^3*d^3 - 406*a*b^2*d^2*e + 350*a^2*b*d*e^2 - 105*a^3*e^3 + 3*(22*b^ 3*d*e^2 - 7*a*b^2*e^3)*x^2 + (122*b^3*d^2*e - 112*a*b^2*d*e^2 + 35*a^2*b*e ^3)*x)*sqrt(e*x + d))/b^4]
Timed out. \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\text {Timed out} \]
\[ \int \frac {(d+e x)^{7/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{\sqrt {{\left (b x + a\right )}^{2}}} \,d x } \]
Time = 0.30 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.30 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 \, {\left (b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{4}} + \frac {2 \, {\left (15 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{6} \mathrm {sgn}\left (b x + a\right ) + 21 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{6} d \mathrm {sgn}\left (b x + a\right ) + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{6} d^{2} \mathrm {sgn}\left (b x + a\right ) + 105 \, \sqrt {e x + d} b^{6} d^{3} \mathrm {sgn}\left (b x + a\right ) - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{5} e \mathrm {sgn}\left (b x + a\right ) - 70 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{5} d e \mathrm {sgn}\left (b x + a\right ) - 315 \, \sqrt {e x + d} a b^{5} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 315 \, \sqrt {e x + d} a^{2} b^{4} d e^{2} \mathrm {sgn}\left (b x + a\right ) - 105 \, \sqrt {e x + d} a^{3} b^{3} e^{3} \mathrm {sgn}\left (b x + a\right )\right )}}{105 \, b^{7}} \]
2*(b^4*d^4*sgn(b*x + a) - 4*a*b^3*d^3*e*sgn(b*x + a) + 6*a^2*b^2*d^2*e^2*s gn(b*x + a) - 4*a^3*b*d*e^3*sgn(b*x + a) + a^4*e^4*sgn(b*x + a))*arctan(sq rt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^4) + 2/105*(15 *(e*x + d)^(7/2)*b^6*sgn(b*x + a) + 21*(e*x + d)^(5/2)*b^6*d*sgn(b*x + a) + 35*(e*x + d)^(3/2)*b^6*d^2*sgn(b*x + a) + 105*sqrt(e*x + d)*b^6*d^3*sgn( b*x + a) - 21*(e*x + d)^(5/2)*a*b^5*e*sgn(b*x + a) - 70*(e*x + d)^(3/2)*a* b^5*d*e*sgn(b*x + a) - 315*sqrt(e*x + d)*a*b^5*d^2*e*sgn(b*x + a) + 35*(e* x + d)^(3/2)*a^2*b^4*e^2*sgn(b*x + a) + 315*sqrt(e*x + d)*a^2*b^4*d*e^2*sg n(b*x + a) - 105*sqrt(e*x + d)*a^3*b^3*e^3*sgn(b*x + a))/b^7
Timed out. \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{7/2}}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \]