Integrand size = 33, antiderivative size = 210 \[ \int \frac {a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {e^2}{(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {1}{3 (b d-a e) (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e}{2 (b d-a e)^2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e^3 (a+b x) \log (a+b x)}{(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^3 (a+b x) \log (d+e x)}{(b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}} \] Output:
-e^2/(-a*e+b*d)^3/((b*x+a)^2)^(1/2)-1/3/(-a*e+b*d)/(b*x+a)^2/((b*x+a)^2)^( 1/2)+1/2*e/(-a*e+b*d)^2/(b*x+a)/((b*x+a)^2)^(1/2)-e^3*(b*x+a)*ln(b*x+a)/(- a*e+b*d)^4/((b*x+a)^2)^(1/2)+e^3*(b*x+a)*ln(e*x+d)/(-a*e+b*d)^4/((b*x+a)^2 )^(1/2)
Time = 1.11 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.55 \[ \int \frac {a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {-\left ((b d-a e) \left (11 a^2 e^2+a b e (-7 d+15 e x)+b^2 \left (2 d^2-3 d e x+6 e^2 x^2\right )\right )\right )-6 e^3 (a+b x)^3 \log (a+b x)+6 e^3 (a+b x)^3 \log (d+e x)}{6 (b d-a e)^4 \left ((a+b x)^2\right )^{3/2}} \] Input:
Integrate[(a + b*x)/((d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
Output:
(-((b*d - a*e)*(11*a^2*e^2 + a*b*e*(-7*d + 15*e*x) + b^2*(2*d^2 - 3*d*e*x + 6*e^2*x^2))) - 6*e^3*(a + b*x)^3*Log[a + b*x] + 6*e^3*(a + b*x)^3*Log[d + e*x])/(6*(b*d - a*e)^4*((a + b*x)^2)^(3/2))
Time = 0.54 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.63, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 27, 54, 2009}
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.
| \(\displaystyle \int \frac {a+b x}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2} (d+e x)} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {b^5 (a+b x) \int \frac {1}{b^5 (a+b x)^4 (d+e x)}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+b x) \int \frac {1}{(a+b x)^4 (d+e x)}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {(a+b x) \int \left (\frac {e^4}{(b d-a e)^4 (d+e x)}-\frac {b e^3}{(b d-a e)^4 (a+b x)}+\frac {b e^2}{(b d-a e)^3 (a+b x)^2}-\frac {b e}{(b d-a e)^2 (a+b x)^3}+\frac {b}{(b d-a e) (a+b x)^4}\right )dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(a+b x) \left (-\frac {e^3 \log (a+b x)}{(b d-a e)^4}+\frac {e^3 \log (d+e x)}{(b d-a e)^4}-\frac {e^2}{(a+b x) (b d-a e)^3}+\frac {e}{2 (a+b x)^2 (b d-a e)^2}-\frac {1}{3 (a+b x)^3 (b d-a e)}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
Input:
Int[(a + b*x)/((d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]
Output:
((a + b*x)*(-1/3*1/((b*d - a*e)*(a + b*x)^3) + e/(2*(b*d - a*e)^2*(a + b*x )^2) - e^2/((b*d - a*e)^3*(a + b*x)) - (e^3*Log[a + b*x])/(b*d - a*e)^4 + (e^3*Log[d + e*x])/(b*d - a*e)^4))/Sqrt[a^2 + 2*a*b*x + b^2*x^2]
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] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^ IntPart[p]*(b/2 + c*x)^(2*FracPart[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, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Time = 1.40 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.20
| method | result | size |
| default | \(-\frac {\left (6 \ln \left (b x +a \right ) x^{3} b^{3} e^{3}-6 \ln \left (e x +d \right ) b^{3} e^{3} x^{3}+18 \ln \left (b x +a \right ) x^{2} a \,b^{2} e^{3}-18 \ln \left (e x +d \right ) a \,b^{2} e^{3} x^{2}+18 \ln \left (b x +a \right ) x \,a^{2} b \,e^{3}-18 \ln \left (e x +d \right ) a^{2} b \,e^{3} x -6 x^{2} a \,b^{2} e^{3}+6 x^{2} b^{3} d \,e^{2}+6 \ln \left (b x +a \right ) a^{3} e^{3}-6 \ln \left (e x +d \right ) a^{3} e^{3}-15 a^{2} b \,e^{3} x +18 x a \,b^{2} d \,e^{2}-3 b^{3} d^{2} e x -11 e^{3} a^{3}+18 a^{2} b d \,e^{2}-9 a \,b^{2} d^{2} e +2 b^{3} d^{3}\right ) \left (b x +a \right )^{2}}{6 \left (a e -b d \right )^{4} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(251\) |
| risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\frac {b^{2} e^{2} x^{2}}{e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}}+\frac {\left (5 a e -b d \right ) b e x}{2 e^{3} a^{3}-6 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -2 b^{3} d^{3}}+\frac {11 e^{2} a^{2}-7 a b d e +2 b^{2} d^{2}}{6 e^{3} a^{3}-18 a^{2} b d \,e^{2}+18 a \,b^{2} d^{2} e -6 b^{3} d^{3}}\right )}{\left (b x +a \right )^{4}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, e^{3} \ln \left (-e x -d \right )}{\left (b x +a \right ) \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}-\frac {\sqrt {\left (b x +a \right )^{2}}\, e^{3} \ln \left (b x +a \right )}{\left (b x +a \right ) \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}\) | \(341\) |
Input:
int((b*x+a)/(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/6*(6*ln(b*x+a)*x^3*b^3*e^3-6*ln(e*x+d)*b^3*e^3*x^3+18*ln(b*x+a)*x^2*a*b ^2*e^3-18*ln(e*x+d)*a*b^2*e^3*x^2+18*ln(b*x+a)*x*a^2*b*e^3-18*ln(e*x+d)*a^ 2*b*e^3*x-6*x^2*a*b^2*e^3+6*x^2*b^3*d*e^2+6*ln(b*x+a)*a^3*e^3-6*ln(e*x+d)* a^3*e^3-15*a^2*b*e^3*x+18*x*a*b^2*d*e^2-3*b^3*d^2*e*x-11*e^3*a^3+18*a^2*b* d*e^2-9*a*b^2*d^2*e+2*b^3*d^3)*(b*x+a)^2/(a*e-b*d)^4/((b*x+a)^2)^(5/2)
Leaf count of result is larger than twice the leaf count of optimal. 425 vs. \(2 (151) = 302\).
Time = 0.08 (sec) , antiderivative size = 425, normalized size of antiderivative = 2.02 \[ \int \frac {a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {2 \, b^{3} d^{3} - 9 \, a b^{2} d^{2} e + 18 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3} + 6 \, {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} - 3 \, {\left (b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + 5 \, a^{2} b e^{3}\right )} x + 6 \, {\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \log \left (b x + a\right ) - 6 \, {\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \log \left (e x + d\right )}{6 \, {\left (a^{3} b^{4} d^{4} - 4 \, a^{4} b^{3} d^{3} e + 6 \, a^{5} b^{2} d^{2} e^{2} - 4 \, a^{6} b d e^{3} + a^{7} e^{4} + {\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} x^{3} + 3 \, {\left (a b^{6} d^{4} - 4 \, a^{2} b^{5} d^{3} e + 6 \, a^{3} b^{4} d^{2} e^{2} - 4 \, a^{4} b^{3} d e^{3} + a^{5} b^{2} e^{4}\right )} x^{2} + 3 \, {\left (a^{2} b^{5} d^{4} - 4 \, a^{3} b^{4} d^{3} e + 6 \, a^{4} b^{3} d^{2} e^{2} - 4 \, a^{5} b^{2} d e^{3} + a^{6} b e^{4}\right )} x\right )}} \] Input:
integrate((b*x+a)/(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas ")
Output:
-1/6*(2*b^3*d^3 - 9*a*b^2*d^2*e + 18*a^2*b*d*e^2 - 11*a^3*e^3 + 6*(b^3*d*e ^2 - a*b^2*e^3)*x^2 - 3*(b^3*d^2*e - 6*a*b^2*d*e^2 + 5*a^2*b*e^3)*x + 6*(b ^3*e^3*x^3 + 3*a*b^2*e^3*x^2 + 3*a^2*b*e^3*x + a^3*e^3)*log(b*x + a) - 6*( b^3*e^3*x^3 + 3*a*b^2*e^3*x^2 + 3*a^2*b*e^3*x + a^3*e^3)*log(e*x + d))/(a^ 3*b^4*d^4 - 4*a^4*b^3*d^3*e + 6*a^5*b^2*d^2*e^2 - 4*a^6*b*d*e^3 + a^7*e^4 + (b^7*d^4 - 4*a*b^6*d^3*e + 6*a^2*b^5*d^2*e^2 - 4*a^3*b^4*d*e^3 + a^4*b^3 *e^4)*x^3 + 3*(a*b^6*d^4 - 4*a^2*b^5*d^3*e + 6*a^3*b^4*d^2*e^2 - 4*a^4*b^3 *d*e^3 + a^5*b^2*e^4)*x^2 + 3*(a^2*b^5*d^4 - 4*a^3*b^4*d^3*e + 6*a^4*b^3*d ^2*e^2 - 4*a^5*b^2*d*e^3 + a^6*b*e^4)*x)
\[ \int \frac {a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {a + b x}{\left (d + e x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((b*x+a)/(e*x+d)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
Output:
Integral((a + b*x)/((d + e*x)*((a + b*x)**2)**(5/2)), x)
Exception generated. \[ \int \frac {a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a)/(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima ")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(((2*a*b)/e>0)', see `assume?` fo r more det
Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (151) = 302\).
Time = 0.20 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.48 \[ \int \frac {a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {b e^{3} \log \left ({\left | b x + a \right |}\right )}{b^{5} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{4} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{3} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b^{2} d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} b e^{4} \mathrm {sgn}\left (b x + a\right )} + \frac {e^{4} \log \left ({\left | e x + d \right |}\right )}{b^{4} d^{4} e \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {2 \, b^{3} d^{3} - 9 \, a b^{2} d^{2} e + 18 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3} + 6 \, {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} - 3 \, {\left (b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + 5 \, a^{2} b e^{3}\right )} x}{6 \, {\left (b d - a e\right )}^{4} {\left (b x + a\right )}^{3} \mathrm {sgn}\left (b x + a\right )} \] Input:
integrate((b*x+a)/(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")
Output:
-b*e^3*log(abs(b*x + a))/(b^5*d^4*sgn(b*x + a) - 4*a*b^4*d^3*e*sgn(b*x + a ) + 6*a^2*b^3*d^2*e^2*sgn(b*x + a) - 4*a^3*b^2*d*e^3*sgn(b*x + a) + a^4*b* e^4*sgn(b*x + a)) + e^4*log(abs(e*x + d))/(b^4*d^4*e*sgn(b*x + a) - 4*a*b^ 3*d^3*e^2*sgn(b*x + a) + 6*a^2*b^2*d^2*e^3*sgn(b*x + a) - 4*a^3*b*d*e^4*sg n(b*x + a) + a^4*e^5*sgn(b*x + a)) - 1/6*(2*b^3*d^3 - 9*a*b^2*d^2*e + 18*a ^2*b*d*e^2 - 11*a^3*e^3 + 6*(b^3*d*e^2 - a*b^2*e^3)*x^2 - 3*(b^3*d^2*e - 6 *a*b^2*d*e^2 + 5*a^2*b*e^3)*x)/((b*d - a*e)^4*(b*x + a)^3*sgn(b*x + a))
Timed out. \[ \int \frac {a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {a+b\,x}{\left (d+e\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \] Input:
int((a + b*x)/((d + e*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)),x)
Output:
int((a + b*x)/((d + e*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2)), x)
Time = 0.22 (sec) , antiderivative size = 499, normalized size of antiderivative = 2.38 \[ \int \frac {a+b x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {-6 \,\mathrm {log}\left (b x +a \right ) a^{4} e^{3}-18 \,\mathrm {log}\left (b x +a \right ) a^{3} b \,e^{3} x -18 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{2} e^{3} x^{2}-6 \,\mathrm {log}\left (b x +a \right ) a \,b^{3} e^{3} x^{3}+6 \,\mathrm {log}\left (e x +d \right ) a^{4} e^{3}+18 \,\mathrm {log}\left (e x +d \right ) a^{3} b \,e^{3} x +18 \,\mathrm {log}\left (e x +d \right ) a^{2} b^{2} e^{3} x^{2}+6 \,\mathrm {log}\left (e x +d \right ) a \,b^{3} e^{3} x^{3}+9 a^{4} e^{3}-16 a^{3} b d \,e^{2}+9 a^{3} b \,e^{3} x +9 a^{2} b^{2} d^{2} e -12 a^{2} b^{2} d \,e^{2} x -2 a \,b^{3} d^{3}+3 a \,b^{3} d^{2} e x -2 a \,b^{3} e^{3} x^{3}+2 b^{4} d \,e^{2} x^{3}}{6 a \left (a^{4} b^{3} e^{4} x^{3}-4 a^{3} b^{4} d \,e^{3} x^{3}+6 a^{2} b^{5} d^{2} e^{2} x^{3}-4 a \,b^{6} d^{3} e \,x^{3}+b^{7} d^{4} x^{3}+3 a^{5} b^{2} e^{4} x^{2}-12 a^{4} b^{3} d \,e^{3} x^{2}+18 a^{3} b^{4} d^{2} e^{2} x^{2}-12 a^{2} b^{5} d^{3} e \,x^{2}+3 a \,b^{6} d^{4} x^{2}+3 a^{6} b \,e^{4} x -12 a^{5} b^{2} d \,e^{3} x +18 a^{4} b^{3} d^{2} e^{2} x -12 a^{3} b^{4} d^{3} e x +3 a^{2} b^{5} d^{4} x +a^{7} e^{4}-4 a^{6} b d \,e^{3}+6 a^{5} b^{2} d^{2} e^{2}-4 a^{4} b^{3} d^{3} e +a^{3} b^{4} d^{4}\right )} \] Input:
int((b*x+a)/(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
Output:
( - 6*log(a + b*x)*a**4*e**3 - 18*log(a + b*x)*a**3*b*e**3*x - 18*log(a + b*x)*a**2*b**2*e**3*x**2 - 6*log(a + b*x)*a*b**3*e**3*x**3 + 6*log(d + e*x )*a**4*e**3 + 18*log(d + e*x)*a**3*b*e**3*x + 18*log(d + e*x)*a**2*b**2*e* *3*x**2 + 6*log(d + e*x)*a*b**3*e**3*x**3 + 9*a**4*e**3 - 16*a**3*b*d*e**2 + 9*a**3*b*e**3*x + 9*a**2*b**2*d**2*e - 12*a**2*b**2*d*e**2*x - 2*a*b**3 *d**3 + 3*a*b**3*d**2*e*x - 2*a*b**3*e**3*x**3 + 2*b**4*d*e**2*x**3)/(6*a* (a**7*e**4 - 4*a**6*b*d*e**3 + 3*a**6*b*e**4*x + 6*a**5*b**2*d**2*e**2 - 1 2*a**5*b**2*d*e**3*x + 3*a**5*b**2*e**4*x**2 - 4*a**4*b**3*d**3*e + 18*a** 4*b**3*d**2*e**2*x - 12*a**4*b**3*d*e**3*x**2 + a**4*b**3*e**4*x**3 + a**3 *b**4*d**4 - 12*a**3*b**4*d**3*e*x + 18*a**3*b**4*d**2*e**2*x**2 - 4*a**3* b**4*d*e**3*x**3 + 3*a**2*b**5*d**4*x - 12*a**2*b**5*d**3*e*x**2 + 6*a**2* b**5*d**2*e**2*x**3 + 3*a*b**6*d**4*x**2 - 4*a*b**6*d**3*e*x**3 + b**7*d** 4*x**3))