Integrand size = 33, antiderivative size = 167 \[ \int \frac {a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {35 e^2}{12 (b d-a e)^3 (d+e x)^{3/2}}-\frac {1}{2 (b d-a e) (a+b x)^2 (d+e x)^{3/2}}+\frac {7 e}{4 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}+\frac {35 b e^2}{4 (b d-a e)^4 \sqrt {d+e x}}-\frac {35 b^{3/2} e^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 (b d-a e)^{9/2}} \]
35/12*e^2/(-a*e+b*d)^3/(e*x+d)^(3/2)-1/2/(-a*e+b*d)/(b*x+a)^2/(e*x+d)^(3/2 )+7/4*e/(-a*e+b*d)^2/(b*x+a)/(e*x+d)^(3/2)-35/4*b^(3/2)*e^2*arctanh(b^(1/2 )*(e*x+d)^(1/2)/(-a*e+b*d)^(1/2))/(-a*e+b*d)^(9/2)+35/4*b*e^2/(-a*e+b*d)^4 /(e*x+d)^(1/2)
Time = 0.18 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.01 \[ \int \frac {a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {-8 a^3 e^3+8 a^2 b e^2 (10 d+7 e x)+a b^2 e \left (39 d^2+238 d e x+175 e^2 x^2\right )+b^3 \left (-6 d^3+21 d^2 e x+140 d e^2 x^2+105 e^3 x^3\right )}{12 (b d-a e)^4 (a+b x)^2 (d+e x)^{3/2}}+\frac {35 b^{3/2} e^2 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{4 (-b d+a e)^{9/2}} \]
(-8*a^3*e^3 + 8*a^2*b*e^2*(10*d + 7*e*x) + a*b^2*e*(39*d^2 + 238*d*e*x + 1 75*e^2*x^2) + b^3*(-6*d^3 + 21*d^2*e*x + 140*d*e^2*x^2 + 105*e^3*x^3))/(12 *(b*d - a*e)^4*(a + b*x)^2*(d + e*x)^(3/2)) + (35*b^(3/2)*e^2*ArcTan[(Sqrt [b]*Sqrt[d + e*x])/Sqrt[-(b*d) + a*e]])/(4*(-(b*d) + a*e)^(9/2))
Time = 0.30 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.17, 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, 61, 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.
\(\displaystyle \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^2 (d+e x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle b^4 \int \frac {1}{b^4 (a+b x)^3 (d+e x)^{5/2}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {1}{(a+b x)^3 (d+e x)^{5/2}}dx\) |
\(\Big \downarrow \) 52 |
\(\displaystyle -\frac {7 e \int \frac {1}{(a+b x)^2 (d+e x)^{5/2}}dx}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle -\frac {7 e \left (-\frac {5 e \int \frac {1}{(a+b x) (d+e x)^{5/2}}dx}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {7 e \left (-\frac {5 e \left (\frac {b \int \frac {1}{(a+b x) (d+e x)^{3/2}}dx}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {7 e \left (-\frac {5 e \left (\frac {b \left (\frac {b \int \frac {1}{(a+b x) \sqrt {d+e x}}dx}{b d-a e}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {7 e \left (-\frac {5 e \left (\frac {b \left (\frac {2 b \int \frac {1}{a+\frac {b (d+e x)}{e}-\frac {b d}{e}}d\sqrt {d+e x}}{e (b d-a e)}+\frac {2}{\sqrt {d+e x} (b d-a e)}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {7 e \left (-\frac {5 e \left (\frac {b \left (\frac {2}{\sqrt {d+e x} (b d-a e)}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{3/2}}\right )}{b d-a e}+\frac {2}{3 (d+e x)^{3/2} (b d-a e)}\right )}{2 (b d-a e)}-\frac {1}{(a+b x) (d+e x)^{3/2} (b d-a e)}\right )}{4 (b d-a e)}-\frac {1}{2 (a+b x)^2 (d+e x)^{3/2} (b d-a e)}\) |
-1/2*1/((b*d - a*e)*(a + b*x)^2*(d + e*x)^(3/2)) - (7*e*(-(1/((b*d - a*e)* (a + b*x)*(d + e*x)^(3/2))) - (5*e*(2/(3*(b*d - a*e)*(d + e*x)^(3/2)) + (b *(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)))/(b*d - a*e)))/(2*(b*d - a*e))))/( 4*(b*d - a*e))
3.21.81.3.1 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 = 0.38 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(2 e^{2} \left (-\frac {1}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}+\frac {3 b}{\left (a e -b d \right )^{4} \sqrt {e x +d}}+\frac {b^{2} \left (\frac {\frac {11 b \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {13 a e}{8}-\frac {13 b d}{8}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {35 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{4}}\right )\) | \(143\) |
default | \(2 e^{2} \left (-\frac {1}{3 \left (a e -b d \right )^{3} \left (e x +d \right )^{\frac {3}{2}}}+\frac {3 b}{\left (a e -b d \right )^{4} \sqrt {e x +d}}+\frac {b^{2} \left (\frac {\frac {11 b \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {13 a e}{8}-\frac {13 b d}{8}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {35 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{4}}\right )\) | \(143\) |
pseudoelliptic | \(-\frac {2 \left (-\frac {105 b^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}} \left (b x +a \right )^{2} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8}+\sqrt {\left (a e -b d \right ) b}\, \left (\left (-\frac {105}{8} e^{3} x^{3}-\frac {35}{2} d \,e^{2} x^{2}-\frac {21}{8} d^{2} e x +\frac {3}{4} d^{3}\right ) b^{3}-\frac {39 e a \left (\frac {175}{39} e^{2} x^{2}+\frac {238}{39} d e x +d^{2}\right ) b^{2}}{8}-10 e^{2} \left (\frac {7 e x}{10}+d \right ) a^{2} b +a^{3} e^{3}\right )\right )}{3 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} \left (b x +a \right )^{2} \left (a e -b d \right )^{4}}\) | \(178\) |
2*e^2*(-1/3/(a*e-b*d)^3/(e*x+d)^(3/2)+3/(a*e-b*d)^4*b/(e*x+d)^(1/2)+1/(a*e -b*d)^4*b^2*((11/8*b*(e*x+d)^(3/2)+(13/8*a*e-13/8*b*d)*(e*x+d)^(1/2))/(b*( e*x+d)+a*e-b*d)^2+35/8/((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. 608 vs. \(2 (139) = 278\).
Time = 0.37 (sec) , antiderivative size = 1226, normalized size of antiderivative = 7.34 \[ \int \frac {a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Too large to display} \]
[1/24*(105*(b^3*e^4*x^4 + a^2*b*d^2*e^2 + 2*(b^3*d*e^3 + a*b^2*e^4)*x^3 + (b^3*d^2*e^2 + 4*a*b^2*d*e^3 + a^2*b*e^4)*x^2 + 2*(a*b^2*d^2*e^2 + a^2*b*d *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*(105*b^3*e^3*x^3 - 6*b^3*d^3 + 39*a*b^2*d^2*e + 80*a^2*b*d*e^2 - 8*a^3*e^3 + 35*(4*b^3*d*e^2 + 5*a*b^2 *e^3)*x^2 + 7*(3*b^3*d^2*e + 34*a*b^2*d*e^2 + 8*a^2*b*e^3)*x)*sqrt(e*x + d ))/(a^2*b^4*d^6 - 4*a^3*b^3*d^5*e + 6*a^4*b^2*d^4*e^2 - 4*a^5*b*d^3*e^3 + a^6*d^2*e^4 + (b^6*d^4*e^2 - 4*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 - 4*a^3*b ^3*d*e^5 + a^4*b^2*e^6)*x^4 + 2*(b^6*d^5*e - 3*a*b^5*d^4*e^2 + 2*a^2*b^4*d ^3*e^3 + 2*a^3*b^3*d^2*e^4 - 3*a^4*b^2*d*e^5 + a^5*b*e^6)*x^3 + (b^6*d^6 - 9*a^2*b^4*d^4*e^2 + 16*a^3*b^3*d^3*e^3 - 9*a^4*b^2*d^2*e^4 + a^6*e^6)*x^2 + 2*(a*b^5*d^6 - 3*a^2*b^4*d^5*e + 2*a^3*b^3*d^4*e^2 + 2*a^4*b^2*d^3*e^3 - 3*a^5*b*d^2*e^4 + a^6*d*e^5)*x), -1/12*(105*(b^3*e^4*x^4 + a^2*b*d^2*e^2 + 2*(b^3*d*e^3 + a*b^2*e^4)*x^3 + (b^3*d^2*e^2 + 4*a*b^2*d*e^3 + a^2*b*e^ 4)*x^2 + 2*(a*b^2*d^2*e^2 + a^2*b*d*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)) - (105*b^3*e^ 3*x^3 - 6*b^3*d^3 + 39*a*b^2*d^2*e + 80*a^2*b*d*e^2 - 8*a^3*e^3 + 35*(4*b^ 3*d*e^2 + 5*a*b^2*e^3)*x^2 + 7*(3*b^3*d^2*e + 34*a*b^2*d*e^2 + 8*a^2*b*e^3 )*x)*sqrt(e*x + d))/(a^2*b^4*d^6 - 4*a^3*b^3*d^5*e + 6*a^4*b^2*d^4*e^2 - 4 *a^5*b*d^3*e^3 + a^6*d^2*e^4 + (b^6*d^4*e^2 - 4*a*b^5*d^3*e^3 + 6*a^2*b...
Timed out. \[ \int \frac {a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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. 298 vs. \(2 (139) = 278\).
Time = 0.27 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.78 \[ \int \frac {a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {35 \, b^{2} e^{2} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} \sqrt {-b^{2} d + a b e}} + \frac {2 \, {\left (9 \, {\left (e x + d\right )} b e^{2} + b d e^{2} - a e^{3}\right )}}{3 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}}} + \frac {11 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{3} e^{2} - 13 \, \sqrt {e x + d} b^{3} d e^{2} + 13 \, \sqrt {e x + d} a b^{2} e^{3}}{4 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{2}} \]
35/4*b^2*e^2*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)) + 2/3*(9*(e*x + d)*b*e^2 + b*d*e^2 - a*e^3)/((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)^(3/2)) + 1/4*(11 *(e*x + d)^(3/2)*b^3*e^2 - 13*sqrt(e*x + d)*b^3*d*e^2 + 13*sqrt(e*x + d)*a *b^2*e^3)/((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)^2)
Time = 11.12 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.46 \[ \int \frac {a+b x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {\frac {175\,b^2\,e^2\,{\left (d+e\,x\right )}^2}{12\,{\left (a\,e-b\,d\right )}^3}-\frac {2\,e^2}{3\,\left (a\,e-b\,d\right )}+\frac {35\,b^3\,e^2\,{\left (d+e\,x\right )}^3}{4\,{\left (a\,e-b\,d\right )}^4}+\frac {14\,b\,e^2\,\left (d+e\,x\right )}{3\,{\left (a\,e-b\,d\right )}^2}}{b^2\,{\left (d+e\,x\right )}^{7/2}-\left (2\,b^2\,d-2\,a\,b\,e\right )\,{\left (d+e\,x\right )}^{5/2}+{\left (d+e\,x\right )}^{3/2}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}+\frac {35\,b^{3/2}\,e^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{{\left (a\,e-b\,d\right )}^{9/2}}\right )}{4\,{\left (a\,e-b\,d\right )}^{9/2}} \]
((175*b^2*e^2*(d + e*x)^2)/(12*(a*e - b*d)^3) - (2*e^2)/(3*(a*e - b*d)) + (35*b^3*e^2*(d + e*x)^3)/(4*(a*e - b*d)^4) + (14*b*e^2*(d + e*x))/(3*(a*e - b*d)^2))/(b^2*(d + e*x)^(7/2) - (2*b^2*d - 2*a*b*e)*(d + e*x)^(5/2) + (d + e*x)^(3/2)*(a^2*e^2 + b^2*d^2 - 2*a*b*d*e)) + (35*b^(3/2)*e^2*atan((b^( 1/2)*(d + e*x)^(1/2)*(a^4*e^4 + b^4*d^4 + 6*a^2*b^2*d^2*e^2 - 4*a*b^3*d^3* e - 4*a^3*b*d*e^3))/(a*e - b*d)^(9/2)))/(4*(a*e - b*d)^(9/2))