Integrand size = 33, antiderivative size = 180 \[ \int \frac {a+b x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {\sqrt {d+e x}}{4 (b d-a e) (a+b x)^4}+\frac {7 e \sqrt {d+e x}}{24 (b d-a e)^2 (a+b x)^3}-\frac {35 e^2 \sqrt {d+e x}}{96 (b d-a e)^3 (a+b x)^2}+\frac {35 e^3 \sqrt {d+e x}}{64 (b d-a e)^4 (a+b x)}-\frac {35 e^4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 \sqrt {b} (b d-a e)^{9/2}} \]
-35/64*e^4*arctanh(b^(1/2)*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/(-a*e+b*d)^(9/2 )/b^(1/2)-1/4*(e*x+d)^(1/2)/(-a*e+b*d)/(b*x+a)^4+7/24*e*(e*x+d)^(1/2)/(-a* e+b*d)^2/(b*x+a)^3-35/96*e^2*(e*x+d)^(1/2)/(-a*e+b*d)^3/(b*x+a)^2+35/64*e^ 3*(e*x+d)^(1/2)/(-a*e+b*d)^4/(b*x+a)
Time = 0.19 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.92 \[ \int \frac {a+b x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {1}{192} \left (\frac {\sqrt {d+e x} \left (279 a^3 e^3+a^2 b e^2 (-326 d+511 e x)+a b^2 e \left (200 d^2-252 d e x+385 e^2 x^2\right )+b^3 \left (-48 d^3+56 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )\right )}{(b d-a e)^4 (a+b x)^4}+\frac {105 e^4 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{\sqrt {b} (-b d+a e)^{9/2}}\right ) \]
((Sqrt[d + e*x]*(279*a^3*e^3 + a^2*b*e^2*(-326*d + 511*e*x) + a*b^2*e*(200 *d^2 - 252*d*e*x + 385*e^2*x^2) + b^3*(-48*d^3 + 56*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3)))/((b*d - a*e)^4*(a + b*x)^4) + (105*e^4*ArcTan[(Sqrt[b]*S qrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(Sqrt[b]*(-(b*d) + a*e)^(9/2)))/192
Time = 0.31 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.19, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {1184, 27, 52, 52, 52, 52, 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.
| \(\displaystyle \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^3 \sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle b^6 \int \frac {1}{b^6 (a+b x)^5 \sqrt {d+e x}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{(a+b x)^5 \sqrt {d+e x}}dx\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {7 e \int \frac {1}{(a+b x)^4 \sqrt {d+e x}}dx}{8 (b d-a e)}-\frac {\sqrt {d+e x}}{4 (a+b x)^4 (b d-a e)}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {7 e \left (-\frac {5 e \int \frac {1}{(a+b x)^3 \sqrt {d+e x}}dx}{6 (b d-a e)}-\frac {\sqrt {d+e x}}{3 (a+b x)^3 (b d-a e)}\right )}{8 (b d-a e)}-\frac {\sqrt {d+e x}}{4 (a+b x)^4 (b d-a e)}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \int \frac {1}{(a+b x)^2 \sqrt {d+e x}}dx}{4 (b d-a e)}-\frac {\sqrt {d+e x}}{2 (a+b x)^2 (b d-a e)}\right )}{6 (b d-a e)}-\frac {\sqrt {d+e x}}{3 (a+b x)^3 (b d-a e)}\right )}{8 (b d-a e)}-\frac {\sqrt {d+e x}}{4 (a+b x)^4 (b d-a e)}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \left (-\frac {e \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{2 (b d-a e)}-\frac {\sqrt {d+e x}}{(a+b x) (b d-a e)}\right )}{4 (b d-a e)}-\frac {\sqrt {d+e x}}{2 (a+b x)^2 (b d-a e)}\right )}{6 (b d-a e)}-\frac {\sqrt {d+e x}}{3 (a+b x)^3 (b d-a e)}\right )}{8 (b d-a e)}-\frac {\sqrt {d+e x}}{4 (a+b x)^4 (b d-a e)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \left (-\frac {\int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{b d-a e}-\frac {\sqrt {d+e x}}{(a+b x) (b d-a e)}\right )}{4 (b d-a e)}-\frac {\sqrt {d+e x}}{2 (a+b x)^2 (b d-a e)}\right )}{6 (b d-a e)}-\frac {\sqrt {d+e x}}{3 (a+b x)^3 (b d-a e)}\right )}{8 (b d-a e)}-\frac {\sqrt {d+e x}}{4 (a+b x)^4 (b d-a e)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {7 e \left (-\frac {5 e \left (-\frac {3 e \left (\frac {e \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{\sqrt {b} (b d-a e)^{3/2}}-\frac {\sqrt {d+e x}}{(a+b x) (b d-a e)}\right )}{4 (b d-a e)}-\frac {\sqrt {d+e x}}{2 (a+b x)^2 (b d-a e)}\right )}{6 (b d-a e)}-\frac {\sqrt {d+e x}}{3 (a+b x)^3 (b d-a e)}\right )}{8 (b d-a e)}-\frac {\sqrt {d+e x}}{4 (a+b x)^4 (b d-a e)}\) |
-1/4*Sqrt[d + e*x]/((b*d - a*e)*(a + b*x)^4) - (7*e*(-1/3*Sqrt[d + e*x]/(( b*d - a*e)*(a + b*x)^3) - (5*e*(-1/2*Sqrt[d + e*x]/((b*d - a*e)*(a + b*x)^ 2) - (3*e*(-(Sqrt[d + e*x]/((b*d - a*e)*(a + b*x))) + (e*ArcTanh[(Sqrt[b]* Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(Sqrt[b]*(b*d - a*e)^(3/2))))/(4*(b*d - a *e))))/(6*(b*d - a*e))))/(8*(b*d - a*e))
3.21.89.3.1 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] :> 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 = 0.38 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.94
| method | result | size |
| pseudoelliptic | \(\frac {\frac {35 e^{4} \left (b x +a \right )^{4} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{64}+\frac {93 \sqrt {\left (a e -b d \right ) b}\, \left (\frac {\left (35 e^{3} x^{3}-\frac {70}{3} d \,e^{2} x^{2}+\frac {56}{3} d^{2} e x -16 d^{3}\right ) b^{3}}{93}+\frac {200 e \left (\frac {77}{40} e^{2} x^{2}-\frac {63}{50} d e x +d^{2}\right ) a \,b^{2}}{279}-\frac {326 e^{2} \left (-\frac {511 e x}{326}+d \right ) a^{2} b}{279}+a^{3} e^{3}\right ) \sqrt {e x +d}}{64}}{\left (a e -b d \right )^{4} \left (b x +a \right )^{4} \sqrt {\left (a e -b d \right ) b}}\) | \(169\) |
| derivativedivides | \(2 e^{4} \left (\frac {\sqrt {e x +d}}{8 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {\frac {7 \sqrt {e x +d}}{48 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {7 \left (\frac {5 \sqrt {e x +d}}{24 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {5 \left (\frac {3 \sqrt {e x +d}}{8 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )}+\frac {3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \left (a e -b d \right ) \sqrt {\left (a e -b d \right ) b}}\right )}{6 \left (a e -b d \right )}\right )}{8 \left (a e -b d \right )}}{a e -b d}\right )\) | \(236\) |
| default | \(2 e^{4} \left (\frac {\sqrt {e x +d}}{8 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {\frac {7 \sqrt {e x +d}}{48 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {7 \left (\frac {5 \sqrt {e x +d}}{24 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {5 \left (\frac {3 \sqrt {e x +d}}{8 \left (a e -b d \right ) \left (b \left (e x +d \right )+a e -b d \right )}+\frac {3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \left (a e -b d \right ) \sqrt {\left (a e -b d \right ) b}}\right )}{6 \left (a e -b d \right )}\right )}{8 \left (a e -b d \right )}}{a e -b d}\right )\) | \(236\) |
93/64/((a*e-b*d)*b)^(1/2)*(35/93*e^4*(b*x+a)^4*arctan(b*(e*x+d)^(1/2)/((a* e-b*d)*b)^(1/2))+((a*e-b*d)*b)^(1/2)*(1/93*(35*e^3*x^3-70/3*d*e^2*x^2+56/3 *d^2*e*x-16*d^3)*b^3+200/279*e*(77/40*e^2*x^2-63/50*d*e*x+d^2)*a*b^2-326/2 79*e^2*(-511/326*e*x+d)*a^2*b+a^3*e^3)*(e*x+d)^(1/2))/(a*e-b*d)^4/(b*x+a)^ 4
Leaf count of result is larger than twice the leaf count of optimal. 656 vs. \(2 (152) = 304\).
Time = 0.70 (sec) , antiderivative size = 1325, normalized size of antiderivative = 7.36 \[ \int \frac {a+b x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \]
[1/384*(105*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6*a^2*b^2*e^4*x^2 + 4*a^3*b*e ^4*x + a^4*e^4)*sqrt(b^2*d - a*b*e)*log((b*e*x + 2*b*d - a*e - 2*sqrt(b^2* d - a*b*e)*sqrt(e*x + d))/(b*x + a)) - 2*(48*b^5*d^4 - 248*a*b^4*d^3*e + 5 26*a^2*b^3*d^2*e^2 - 605*a^3*b^2*d*e^3 + 279*a^4*b*e^4 - 105*(b^5*d*e^3 - a*b^4*e^4)*x^3 + 35*(2*b^5*d^2*e^2 - 13*a*b^4*d*e^3 + 11*a^2*b^3*e^4)*x^2 - 7*(8*b^5*d^3*e - 44*a*b^4*d^2*e^2 + 109*a^2*b^3*d*e^3 - 73*a^3*b^2*e^4)* x)*sqrt(e*x + d))/(a^4*b^6*d^5 - 5*a^5*b^5*d^4*e + 10*a^6*b^4*d^3*e^2 - 10 *a^7*b^3*d^2*e^3 + 5*a^8*b^2*d*e^4 - a^9*b*e^5 + (b^10*d^5 - 5*a*b^9*d^4*e + 10*a^2*b^8*d^3*e^2 - 10*a^3*b^7*d^2*e^3 + 5*a^4*b^6*d*e^4 - a^5*b^5*e^5 )*x^4 + 4*(a*b^9*d^5 - 5*a^2*b^8*d^4*e + 10*a^3*b^7*d^3*e^2 - 10*a^4*b^6*d ^2*e^3 + 5*a^5*b^5*d*e^4 - a^6*b^4*e^5)*x^3 + 6*(a^2*b^8*d^5 - 5*a^3*b^7*d ^4*e + 10*a^4*b^6*d^3*e^2 - 10*a^5*b^5*d^2*e^3 + 5*a^6*b^4*d*e^4 - a^7*b^3 *e^5)*x^2 + 4*(a^3*b^7*d^5 - 5*a^4*b^6*d^4*e + 10*a^5*b^5*d^3*e^2 - 10*a^6 *b^4*d^2*e^3 + 5*a^7*b^3*d*e^4 - a^8*b^2*e^5)*x), 1/192*(105*(b^4*e^4*x^4 + 4*a*b^3*e^4*x^3 + 6*a^2*b^2*e^4*x^2 + 4*a^3*b*e^4*x + a^4*e^4)*sqrt(-b^2 *d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) - (48 *b^5*d^4 - 248*a*b^4*d^3*e + 526*a^2*b^3*d^2*e^2 - 605*a^3*b^2*d*e^3 + 279 *a^4*b*e^4 - 105*(b^5*d*e^3 - a*b^4*e^4)*x^3 + 35*(2*b^5*d^2*e^2 - 13*a*b^ 4*d*e^3 + 11*a^2*b^3*e^4)*x^2 - 7*(8*b^5*d^3*e - 44*a*b^4*d^2*e^2 + 109*a^ 2*b^3*d*e^3 - 73*a^3*b^2*e^4)*x)*sqrt(e*x + d))/(a^4*b^6*d^5 - 5*a^5*b^...
Timed out. \[ \int \frac {a+b x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {a+b x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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. 331 vs. \(2 (152) = 304\).
Time = 0.27 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.84 \[ \int \frac {a+b x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {35 \, e^{4} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt {-b^{2} d + a b e}} + \frac {105 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{3} e^{4} - 385 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{3} d e^{4} + 511 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{3} d^{2} e^{4} - 279 \, \sqrt {e x + d} b^{3} d^{3} e^{4} + 385 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{2} e^{5} - 1022 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{2} d e^{5} + 837 \, \sqrt {e x + d} a b^{2} d^{2} e^{5} + 511 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b e^{6} - 837 \, \sqrt {e x + d} a^{2} b d e^{6} + 279 \, \sqrt {e x + d} a^{3} e^{7}}{192 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{4}} \]
35/64*e^4*arctan(sqrt(e*x + d)*b/sqrt(-b^2*d + a*b*e))/((b^4*d^4 - 4*a*b^3 *d^3*e + 6*a^2*b^2*d^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*sqrt(-b^2*d + a*b*e) ) + 1/192*(105*(e*x + d)^(7/2)*b^3*e^4 - 385*(e*x + d)^(5/2)*b^3*d*e^4 + 5 11*(e*x + d)^(3/2)*b^3*d^2*e^4 - 279*sqrt(e*x + d)*b^3*d^3*e^4 + 385*(e*x + d)^(5/2)*a*b^2*e^5 - 1022*(e*x + d)^(3/2)*a*b^2*d*e^5 + 837*sqrt(e*x + d )*a*b^2*d^2*e^5 + 511*(e*x + d)^(3/2)*a^2*b*e^6 - 837*sqrt(e*x + d)*a^2*b* d*e^6 + 279*sqrt(e*x + d)*a^3*e^7)/((b^4*d^4 - 4*a*b^3*d^3*e + 6*a^2*b^2*d ^2*e^2 - 4*a^3*b*d*e^3 + a^4*e^4)*((e*x + d)*b - b*d + a*e)^4)
Time = 11.05 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.71 \[ \int \frac {a+b x}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {\frac {93\,e^4\,\sqrt {d+e\,x}}{64\,\left (a\,e-b\,d\right )}+\frac {385\,b^2\,e^4\,{\left (d+e\,x\right )}^{5/2}}{192\,{\left (a\,e-b\,d\right )}^3}+\frac {35\,b^3\,e^4\,{\left (d+e\,x\right )}^{7/2}}{64\,{\left (a\,e-b\,d\right )}^4}+\frac {511\,b\,e^4\,{\left (d+e\,x\right )}^{3/2}}{192\,{\left (a\,e-b\,d\right )}^2}}{b^4\,{\left (d+e\,x\right )}^4-\left (4\,b^4\,d-4\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^3-\left (d+e\,x\right )\,\left (-4\,a^3\,b\,e^3+12\,a^2\,b^2\,d\,e^2-12\,a\,b^3\,d^2\,e+4\,b^4\,d^3\right )+a^4\,e^4+b^4\,d^4+{\left (d+e\,x\right )}^2\,\left (6\,a^2\,b^2\,e^2-12\,a\,b^3\,d\,e+6\,b^4\,d^2\right )+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e-4\,a^3\,b\,d\,e^3}+\frac {35\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{64\,\sqrt {b}\,{\left (a\,e-b\,d\right )}^{9/2}} \]
((93*e^4*(d + e*x)^(1/2))/(64*(a*e - b*d)) + (385*b^2*e^4*(d + e*x)^(5/2)) /(192*(a*e - b*d)^3) + (35*b^3*e^4*(d + e*x)^(7/2))/(64*(a*e - b*d)^4) + ( 511*b*e^4*(d + e*x)^(3/2))/(192*(a*e - b*d)^2))/(b^4*(d + e*x)^4 - (4*b^4* d - 4*a*b^3*e)*(d + e*x)^3 - (d + e*x)*(4*b^4*d^3 - 4*a^3*b*e^3 + 12*a^2*b ^2*d*e^2 - 12*a*b^3*d^2*e) + a^4*e^4 + b^4*d^4 + (d + e*x)^2*(6*b^4*d^2 + 6*a^2*b^2*e^2 - 12*a*b^3*d*e) + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3*e - 4*a^3* b*d*e^3) + (35*e^4*atan((b^(1/2)*(d + e*x)^(1/2))/(a*e - b*d)^(1/2)))/(64* b^(1/2)*(a*e - b*d)^(9/2))