Integrand size = 30, antiderivative size = 223 \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {5 e}{4 (b d-a e)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{2 (b d-a e) (a+b x) \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {15 e^2 (a+b x)}{4 (b d-a e)^3 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {15 \sqrt {b} e^2 (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 (b d-a e)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
-15/4*e^2*(b*x+a)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))*b^(1/2)/ (-a*e+b*d)^(7/2)/((b*x+a)^2)^(1/2)+5/4*e/(-a*e+b*d)^2/(e*x+d)^(1/2)/((b*x+ a)^2)^(1/2)-1/2/(-a*e+b*d)/(b*x+a)/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2)+15/4*e^ 2*(b*x+a)/(-a*e+b*d)^3/(e*x+d)^(1/2)/((b*x+a)^2)^(1/2)
Time = 0.41 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {\sqrt {-b d+a e} \left (8 a^2 e^2+a b e (9 d+25 e x)+b^2 \left (-2 d^2+5 d e x+15 e^2 x^2\right )\right )+15 \sqrt {b} e^2 (a+b x)^2 \sqrt {d+e x} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{4 (-b d+a e)^{7/2} (a+b x) \sqrt {(a+b x)^2} \sqrt {d+e x}} \]
-1/4*(Sqrt[-(b*d) + a*e]*(8*a^2*e^2 + a*b*e*(9*d + 25*e*x) + b^2*(-2*d^2 + 5*d*e*x + 15*e^2*x^2)) + 15*Sqrt[b]*e^2*(a + b*x)^2*Sqrt[d + e*x]*ArcTan[ (Sqrt[b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/((-(b*d) + a*e)^(7/2)*(a + b* x)*Sqrt[(a + b*x)^2]*Sqrt[d + e*x])
Time = 0.30 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.83, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {1102, 27, 52, 52, 61, 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.
| \(\displaystyle \int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2} (d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1102 |
| \(\displaystyle \frac {b^3 (a+b x) \int \frac {1}{b^3 (a+b x)^3 (d+e x)^{3/2}}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {1}{(a+b x)^3 (d+e x)^{3/2}}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {5 e \int \frac {1}{(a+b x)^2 (d+e x)^{3/2}}dx}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {5 e \left (-\frac {3 e \int \frac {1}{(a+b x) (d+e x)^{3/2}}dx}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {5 e \left (-\frac {3 e \left (\frac {b \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b d-a e}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {5 e \left (-\frac {3 e \left (\frac {2 b \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{e (b d-a e)}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {5 e \left (-\frac {3 e \left (\frac {2}{\sqrt {d+e x} (b d-a e)}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
((a + b*x)*(-1/2*1/((b*d - a*e)*(a + b*x)^2*Sqrt[d + e*x]) - (5*e*(-(1/((b *d - a*e)*(a + b*x)*Sqrt[d + e*x])) - (3*e*(2/((b*d - a*e)*Sqrt[d + e*x]) - (2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*e) ^(3/2)))/(2*(b*d - a*e))))/(4*(b*d - a*e))))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
3.18.17.3.1 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 + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[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 = 2.29 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.28
| method | result | size |
| default | \(-\frac {\left (15 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) b^{3} e^{2} x^{2} \sqrt {e x +d}+30 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a \,b^{2} e^{2} x \sqrt {e x +d}+15 \sqrt {\left (a e -b d \right ) b}\, b^{2} e^{2} x^{2}+15 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) \sqrt {e x +d}\, a^{2} b \,e^{2}+25 \sqrt {\left (a e -b d \right ) b}\, a b \,e^{2} x +5 \sqrt {\left (a e -b d \right ) b}\, b^{2} d e x +8 \sqrt {\left (a e -b d \right ) b}\, a^{2} e^{2}+9 \sqrt {\left (a e -b d \right ) b}\, a b d e -2 \sqrt {\left (a e -b d \right ) b}\, b^{2} d^{2}\right ) \left (b x +a \right )}{4 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, \left (a e -b d \right )^{3} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(285\) |
-1/4*(15*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*b^3*e^2*x^2*(e*x+d)^( 1/2)+30*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))*a*b^2*e^2*x*(e*x+d)^(1 /2)+15*((a*e-b*d)*b)^(1/2)*b^2*e^2*x^2+15*arctan(b*(e*x+d)^(1/2)/((a*e-b*d )*b)^(1/2))*(e*x+d)^(1/2)*a^2*b*e^2+25*((a*e-b*d)*b)^(1/2)*a*b*e^2*x+5*((a *e-b*d)*b)^(1/2)*b^2*d*e*x+8*((a*e-b*d)*b)^(1/2)*a^2*e^2+9*((a*e-b*d)*b)^( 1/2)*a*b*d*e-2*((a*e-b*d)*b)^(1/2)*b^2*d^2)*(b*x+a)/(e*x+d)^(1/2)/((a*e-b* d)*b)^(1/2)/(a*e-b*d)^3/((b*x+a)^2)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (155) = 310\).
Time = 0.41 (sec) , antiderivative size = 782, normalized size of antiderivative = 3.51 \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\left [-\frac {15 \, {\left (b^{2} e^{3} x^{3} + a^{2} d e^{2} + {\left (b^{2} d e^{2} + 2 \, a b e^{3}\right )} x^{2} + {\left (2 \, a b d e^{2} + a^{2} e^{3}\right )} x\right )} \sqrt {\frac {b}{b d - a e}} \log \left (\frac {b e x + 2 \, b d - a e + 2 \, {\left (b d - a e\right )} \sqrt {e x + d} \sqrt {\frac {b}{b d - a e}}}{b x + a}\right ) - 2 \, {\left (15 \, b^{2} e^{2} x^{2} - 2 \, b^{2} d^{2} + 9 \, a b d e + 8 \, a^{2} e^{2} + 5 \, {\left (b^{2} d e + 5 \, a b e^{2}\right )} x\right )} \sqrt {e x + d}}{8 \, {\left (a^{2} b^{3} d^{4} - 3 \, a^{3} b^{2} d^{3} e + 3 \, a^{4} b d^{2} e^{2} - a^{5} d e^{3} + {\left (b^{5} d^{3} e - 3 \, a b^{4} d^{2} e^{2} + 3 \, a^{2} b^{3} d e^{3} - a^{3} b^{2} e^{4}\right )} x^{3} + {\left (b^{5} d^{4} - a b^{4} d^{3} e - 3 \, a^{2} b^{3} d^{2} e^{2} + 5 \, a^{3} b^{2} d e^{3} - 2 \, a^{4} b e^{4}\right )} x^{2} + {\left (2 \, a b^{4} d^{4} - 5 \, a^{2} b^{3} d^{3} e + 3 \, a^{3} b^{2} d^{2} e^{2} + a^{4} b d e^{3} - a^{5} e^{4}\right )} x\right )}}, -\frac {15 \, {\left (b^{2} e^{3} x^{3} + a^{2} d e^{2} + {\left (b^{2} d e^{2} + 2 \, a b e^{3}\right )} x^{2} + {\left (2 \, a b d e^{2} + a^{2} e^{3}\right )} x\right )} \sqrt {-\frac {b}{b d - a e}} \arctan \left (-\frac {{\left (b d - a e\right )} \sqrt {e x + d} \sqrt {-\frac {b}{b d - a e}}}{b e x + b d}\right ) - {\left (15 \, b^{2} e^{2} x^{2} - 2 \, b^{2} d^{2} + 9 \, a b d e + 8 \, a^{2} e^{2} + 5 \, {\left (b^{2} d e + 5 \, a b e^{2}\right )} x\right )} \sqrt {e x + d}}{4 \, {\left (a^{2} b^{3} d^{4} - 3 \, a^{3} b^{2} d^{3} e + 3 \, a^{4} b d^{2} e^{2} - a^{5} d e^{3} + {\left (b^{5} d^{3} e - 3 \, a b^{4} d^{2} e^{2} + 3 \, a^{2} b^{3} d e^{3} - a^{3} b^{2} e^{4}\right )} x^{3} + {\left (b^{5} d^{4} - a b^{4} d^{3} e - 3 \, a^{2} b^{3} d^{2} e^{2} + 5 \, a^{3} b^{2} d e^{3} - 2 \, a^{4} b e^{4}\right )} x^{2} + {\left (2 \, a b^{4} d^{4} - 5 \, a^{2} b^{3} d^{3} e + 3 \, a^{3} b^{2} d^{2} e^{2} + a^{4} b d e^{3} - a^{5} e^{4}\right )} x\right )}}\right ] \]
[-1/8*(15*(b^2*e^3*x^3 + a^2*d*e^2 + (b^2*d*e^2 + 2*a*b*e^3)*x^2 + (2*a*b* d*e^2 + a^2*e^3)*x)*sqrt(b/(b*d - a*e))*log((b*e*x + 2*b*d - a*e + 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a)) - 2*(15*b^2*e^2*x^2 - 2*b^2*d^2 + 9*a*b*d*e + 8*a^2*e^2 + 5*(b^2*d*e + 5*a*b*e^2)*x)*sqrt(e*x + d))/(a^2*b^3*d^4 - 3*a^3*b^2*d^3*e + 3*a^4*b*d^2*e^2 - a^5*d*e^3 + (b^5*d ^3*e - 3*a*b^4*d^2*e^2 + 3*a^2*b^3*d*e^3 - a^3*b^2*e^4)*x^3 + (b^5*d^4 - a *b^4*d^3*e - 3*a^2*b^3*d^2*e^2 + 5*a^3*b^2*d*e^3 - 2*a^4*b*e^4)*x^2 + (2*a *b^4*d^4 - 5*a^2*b^3*d^3*e + 3*a^3*b^2*d^2*e^2 + a^4*b*d*e^3 - a^5*e^4)*x) , -1/4*(15*(b^2*e^3*x^3 + a^2*d*e^2 + (b^2*d*e^2 + 2*a*b*e^3)*x^2 + (2*a*b *d*e^2 + a^2*e^3)*x)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(e*x + d )*sqrt(-b/(b*d - a*e))/(b*e*x + b*d)) - (15*b^2*e^2*x^2 - 2*b^2*d^2 + 9*a* b*d*e + 8*a^2*e^2 + 5*(b^2*d*e + 5*a*b*e^2)*x)*sqrt(e*x + d))/(a^2*b^3*d^4 - 3*a^3*b^2*d^3*e + 3*a^4*b*d^2*e^2 - a^5*d*e^3 + (b^5*d^3*e - 3*a*b^4*d^ 2*e^2 + 3*a^2*b^3*d*e^3 - a^3*b^2*e^4)*x^3 + (b^5*d^4 - a*b^4*d^3*e - 3*a^ 2*b^3*d^2*e^2 + 5*a^3*b^2*d*e^3 - 2*a^4*b*e^4)*x^2 + (2*a*b^4*d^4 - 5*a^2* b^3*d^3*e + 3*a^3*b^2*d^2*e^2 + a^4*b*d*e^3 - a^5*e^4)*x)]
\[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (d + e x\right )^{\frac {3}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.37 \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {15 \, b e^{2} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, {\left (b^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b d e^{2} \mathrm {sgn}\left (b x + a\right ) - a^{3} e^{3} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, e^{2}}{{\left (b^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b d e^{2} \mathrm {sgn}\left (b x + a\right ) - a^{3} e^{3} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {e x + d}} + \frac {7 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{2} e^{2} - 9 \, \sqrt {e x + d} b^{2} d e^{2} + 9 \, \sqrt {e x + d} a b e^{3}}{4 \, {\left (b^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b d e^{2} \mathrm {sgn}\left (b x + a\right ) - a^{3} e^{3} \mathrm {sgn}\left (b x + a\right )\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{2}} \]
15/4*b*e^2*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^3*d^3*sgn(b*x + a) - 3*a*b^2*d^2*e*sgn(b*x + a) + 3*a^2*b*d*e^2*sgn(b*x + a) - a^3*e^3*s gn(b*x + a))*sqrt(-b^2*d + a*b*e)) + 2*e^2/((b^3*d^3*sgn(b*x + a) - 3*a*b^ 2*d^2*e*sgn(b*x + a) + 3*a^2*b*d*e^2*sgn(b*x + a) - a^3*e^3*sgn(b*x + a))* sqrt(e*x + d)) + 1/4*(7*(e*x + d)^(3/2)*b^2*e^2 - 9*sqrt(e*x + d)*b^2*d*e^ 2 + 9*sqrt(e*x + d)*a*b*e^3)/((b^3*d^3*sgn(b*x + a) - 3*a*b^2*d^2*e*sgn(b* x + a) + 3*a^2*b*d*e^2*sgn(b*x + a) - a^3*e^3*sgn(b*x + a))*((e*x + d)*b - b*d + a*e)^2)
Timed out. \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^{3/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]