Integrand size = 33, antiderivative size = 140 \[ \int \frac {a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {15 e^2}{4 (b d-a e)^3 \sqrt {d+e x}}-\frac {1}{2 (b d-a e) (a+b x)^2 \sqrt {d+e x}}+\frac {5 e}{4 (b d-a e)^2 (a+b x) \sqrt {d+e x}}-\frac {15 \sqrt {b} e^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 (b d-a e)^{7/2}} \]
-15/4*e^2*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))*b^(1/2)/(-a*e+b* d)^(7/2)+15/4*e^2/(-a*e+b*d)^3/(e*x+d)^(1/2)-1/2/(-a*e+b*d)/(b*x+a)^2/(e*x +d)^(1/2)+5/4*e/(-a*e+b*d)^2/(b*x+a)/(e*x+d)^(1/2)
Time = 0.15 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.90 \[ \int \frac {a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {1}{4} \left (\frac {8 a^2 e^2+a b e (9 d+25 e x)+b^2 \left (-2 d^2+5 d e x+15 e^2 x^2\right )}{(b d-a e)^3 (a+b x)^2 \sqrt {d+e x}}-\frac {15 \sqrt {b} e^2 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{7/2}}\right ) \]
((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))/ ((b*d - a*e)^3*(a + b*x)^2*Sqrt[d + e*x]) - (15*Sqrt[b]*e^2*ArcTan[(Sqrt[b ]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(-(b*d) + a*e)^(7/2))/4
Time = 0.26 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {1184, 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.
\(\displaystyle \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^2 (d+e x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle b^4 \int \frac {1}{b^4 (a+b x)^3 (d+e x)^{3/2}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {1}{(a+b x)^3 (d+e x)^{3/2}}dx\) |
\(\Big \downarrow \) 52 |
\(\displaystyle -\frac {5 e \int \frac {1}{(a+b x)^2 (d+e x)^{3/2}}dx}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle -\frac {5 e \left (-\frac {3 e \int \frac {1}{(a+b x) (d+e x)^{3/2}}dx}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {5 e \left (-\frac {3 e \left (\frac {b \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b d-a e}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {5 e \left (-\frac {3 e \left (\frac {2 b \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{e (b d-a e)}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {5 e \left (-\frac {3 e \left (\frac {2}{\sqrt {d+e x} (b d-a e)}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 \sqrt {d+e x} (b d-a e)}\) |
-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))
3.21.80.3.1 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_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.35 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(2 e^{2} \left (-\frac {1}{\left (a e -b d \right )^{3} \sqrt {e x +d}}-\frac {b \left (\frac {\frac {7 b \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {9 a e}{8}-\frac {9 b d}{8}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {15 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{3}}\right )\) | \(122\) |
default | \(2 e^{2} \left (-\frac {1}{\left (a e -b d \right )^{3} \sqrt {e x +d}}-\frac {b \left (\frac {\frac {7 b \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {9 a e}{8}-\frac {9 b d}{8}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {15 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{3}}\right )\) | \(122\) |
pseudoelliptic | \(-\frac {15 \left (b \,e^{2} \sqrt {e x +d}\, \left (b x +a \right )^{2} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )+\frac {8 \left (\left (\frac {15}{8} e^{2} x^{2}+\frac {5}{8} d e x -\frac {1}{4} d^{2}\right ) b^{2}+\frac {9 e \left (\frac {25 e x}{9}+d \right ) a b}{8}+e^{2} a^{2}\right ) \sqrt {\left (a e -b d \right ) b}}{15}\right )}{4 \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}\, \left (b x +a \right )^{2} \left (a e -b d \right )^{3}}\) | \(137\) |
2*e^2*(-1/(a*e-b*d)^3/(e*x+d)^(1/2)-1/(a*e-b*d)^3*b*((7/8*b*(e*x+d)^(3/2)+ (9/8*a*e-9/8*b*d)*(e*x+d)^(1/2))/(b*(e*x+d)+a*e-b*d)^2+15/8/((a*e-b*d)*b)^ (1/2)*arctan(b*(e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (116) = 232\).
Time = 0.40 (sec) , antiderivative size = 782, normalized size of antiderivative = 5.59 \[ \int \frac {a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^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)]
Timed out. \[ \int \frac {a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (116) = 232\).
Time = 0.27 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.67 \[ \int \frac {a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {15 \, b e^{2} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, e^{2}}{{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt {e x + d}} + \frac {7 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{2} e^{2} - 9 \, \sqrt {e x + d} b^{2} d e^{2} + 9 \, \sqrt {e x + d} a b e^{3}}{4 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{2}} \]
15/4*b*e^2*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^3*d^3 - 3*a*b^ 2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*sqrt(-b^2*d + a*b*e)) + 2*e^2/((b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*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 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*((e*x + d)*b - b*d + a *e)^2)
Time = 11.02 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.46 \[ \int \frac {a+b x}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {\frac {2\,e^2}{a\,e-b\,d}+\frac {15\,b^2\,e^2\,{\left (d+e\,x\right )}^2}{4\,{\left (a\,e-b\,d\right )}^3}+\frac {25\,b\,e^2\,\left (d+e\,x\right )}{4\,{\left (a\,e-b\,d\right )}^2}}{b^2\,{\left (d+e\,x\right )}^{5/2}-\left (2\,b^2\,d-2\,a\,b\,e\right )\,{\left (d+e\,x\right )}^{3/2}+\sqrt {d+e\,x}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}-\frac {15\,\sqrt {b}\,e^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{{\left (a\,e-b\,d\right )}^{7/2}}\right )}{4\,{\left (a\,e-b\,d\right )}^{7/2}} \]
- ((2*e^2)/(a*e - b*d) + (15*b^2*e^2*(d + e*x)^2)/(4*(a*e - b*d)^3) + (25* b*e^2*(d + e*x))/(4*(a*e - b*d)^2))/(b^2*(d + e*x)^(5/2) - (2*b^2*d - 2*a* b*e)*(d + e*x)^(3/2) + (d + e*x)^(1/2)*(a^2*e^2 + b^2*d^2 - 2*a*b*d*e)) - (15*b^(1/2)*e^2*atan((b^(1/2)*(d + e*x)^(1/2)*(a^3*e^3 - b^3*d^3 + 3*a*b^2 *d^2*e - 3*a^2*b*d*e^2))/(a*e - b*d)^(7/2)))/(4*(a*e - b*d)^(7/2))