Integrand size = 33, antiderivative size = 196 \[ \int \frac {A+B x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {B d-A e}{(b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {A b-a B}{2 b (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {e (B d-A e) (a+b x) \log (a+b x)}{(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e (B d-A e) (a+b x) \log (d+e x)}{(b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}} \] Output:
-(-A*e+B*d)/(-a*e+b*d)^2/((b*x+a)^2)^(1/2)-1/2*(A*b-B*a)/b/(-a*e+b*d)/(b*x +a)/((b*x+a)^2)^(1/2)-e*(-A*e+B*d)*(b*x+a)*ln(b*x+a)/(-a*e+b*d)^3/((b*x+a) ^2)^(1/2)+e*(-A*e+B*d)*(b*x+a)*ln(e*x+d)/(-a*e+b*d)^3/((b*x+a)^2)^(1/2)
Time = 1.12 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.67 \[ \int \frac {A+B x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {-\left ((b d-a e) \left (B \left (a b d+a^2 e+2 b^2 d x\right )+A b (-3 a e+b (d-2 e x))\right )\right )+2 b e (-B d+A e) (a+b x)^2 \log (a+b x)+2 b e (B d-A e) (a+b x)^2 \log (d+e x)}{2 b (b d-a e)^3 (a+b x) \sqrt {(a+b x)^2}} \] Input:
Integrate[(A + B*x)/((d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]
Output:
(-((b*d - a*e)*(B*(a*b*d + a^2*e + 2*b^2*d*x) + A*b*(-3*a*e + b*(d - 2*e*x )))) + 2*b*e*(-(B*d) + A*e)*(a + b*x)^2*Log[a + b*x] + 2*b*e*(B*d - A*e)*( a + b*x)^2*Log[d + e*x])/(2*b*(b*d - a*e)^3*(a + b*x)*Sqrt[(a + b*x)^2])
Time = 0.57 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.71, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1187, 27, 86, 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.
\(\displaystyle \int \frac {A+B x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2} (d+e x)} \, dx\) |
\(\Big \downarrow \) 1187 |
\(\displaystyle \frac {b^3 (a+b x) \int \frac {A+B x}{b^3 (a+b x)^3 (d+e x)}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(a+b x) \int \frac {A+B x}{(a+b x)^3 (d+e x)}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \frac {(a+b x) \int \left (-\frac {(A e-B d) e^2}{(b d-a e)^3 (d+e x)}+\frac {b (A e-B d) e}{(b d-a e)^3 (a+b x)}+\frac {b (B d-A e)}{(b d-a e)^2 (a+b x)^2}+\frac {A b-a B}{(b d-a e) (a+b x)^3}\right )dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(a+b x) \left (-\frac {A b-a B}{2 b (a+b x)^2 (b d-a e)}-\frac {B d-A e}{(a+b x) (b d-a e)^2}-\frac {e \log (a+b x) (B d-A e)}{(b d-a e)^3}+\frac {e (B d-A e) \log (d+e x)}{(b d-a e)^3}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
Input:
Int[(A + B*x)/((d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]
Output:
((a + b*x)*(-1/2*(A*b - a*B)/(b*(b*d - a*e)*(a + b*x)^2) - (B*d - A*e)/((b *d - a*e)^2*(a + b*x)) - (e*(B*d - A*e)*Log[a + b*x])/(b*d - a*e)^3 + (e*( B*d - A*e)*Log[d + e*x])/(b*d - a*e)^3))/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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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 = 1.71 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.29
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (\frac {b \left (A e -B d \right ) x}{e^{2} a^{2}-2 a b d e +b^{2} d^{2}}+\frac {3 A a b e -A \,b^{2} d -B e \,a^{2}-B a b d}{2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b}\right )}{\left (b x +a \right )^{3}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, e \left (A e -B d \right ) \ln \left (-e x -d \right )}{\left (b x +a \right ) \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}-\frac {\sqrt {\left (b x +a \right )^{2}}\, e \left (A e -B d \right ) \ln \left (b x +a \right )}{\left (b x +a \right ) \left (e^{3} a^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}\) | \(253\) |
default | \(-\frac {\left (2 A \ln \left (b x +a \right ) b^{3} e^{2} x^{2}-2 A \ln \left (e x +d \right ) b^{3} e^{2} x^{2}-2 B \ln \left (b x +a \right ) b^{3} d e \,x^{2}+2 B \ln \left (e x +d \right ) b^{3} d e \,x^{2}+4 A \ln \left (b x +a \right ) a \,b^{2} e^{2} x -4 A \ln \left (e x +d \right ) a \,b^{2} e^{2} x -4 B \ln \left (b x +a \right ) a \,b^{2} d e x +4 B \ln \left (e x +d \right ) a \,b^{2} d e x +2 A \ln \left (b x +a \right ) a^{2} b \,e^{2}-2 A \ln \left (e x +d \right ) a^{2} b \,e^{2}-2 A a \,b^{2} e^{2} x +2 A \,b^{3} d e x -2 B \ln \left (b x +a \right ) a^{2} b d e +2 B \ln \left (e x +d \right ) a^{2} b d e +2 B a \,b^{2} d e x -2 B \,b^{3} d^{2} x -3 A b \,e^{2} a^{2}+4 A a \,b^{2} d e -A \,b^{3} d^{2}+B \,e^{2} a^{3}-B a \,b^{2} d^{2}\right ) \left (b x +a \right )}{2 b \left (a e -b d \right )^{3} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(315\) |
Input:
int((B*x+A)/(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
((b*x+a)^2)^(1/2)/(b*x+a)^3*(b*(A*e-B*d)/(a^2*e^2-2*a*b*d*e+b^2*d^2)*x+1/2 *(3*A*a*b*e-A*b^2*d-B*a^2*e-B*a*b*d)/(a^2*e^2-2*a*b*d*e+b^2*d^2)/b)+((b*x+ a)^2)^(1/2)/(b*x+a)*e*(A*e-B*d)/(a^3*e^3-3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d ^3)*ln(-e*x-d)-((b*x+a)^2)^(1/2)/(b*x+a)*e*(A*e-B*d)/(a^3*e^3-3*a^2*b*d*e^ 2+3*a*b^2*d^2*e-b^3*d^3)*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (150) = 300\).
Time = 0.09 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.84 \[ \int \frac {A+B x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {4 \, A a b^{2} d e - {\left (B a b^{2} + A b^{3}\right )} d^{2} + {\left (B a^{3} - 3 \, A a^{2} b\right )} e^{2} - 2 \, {\left (B b^{3} d^{2} + A a b^{2} e^{2} - {\left (B a b^{2} + A b^{3}\right )} d e\right )} x - 2 \, {\left (B a^{2} b d e - A a^{2} b e^{2} + {\left (B b^{3} d e - A b^{3} e^{2}\right )} x^{2} + 2 \, {\left (B a b^{2} d e - A a b^{2} e^{2}\right )} x\right )} \log \left (b x + a\right ) + 2 \, {\left (B a^{2} b d e - A a^{2} b e^{2} + {\left (B b^{3} d e - A b^{3} e^{2}\right )} x^{2} + 2 \, {\left (B a b^{2} d e - A a b^{2} e^{2}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (a^{2} b^{4} d^{3} - 3 \, a^{3} b^{3} d^{2} e + 3 \, a^{4} b^{2} d e^{2} - a^{5} b e^{3} + {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} x^{2} + 2 \, {\left (a b^{5} d^{3} - 3 \, a^{2} b^{4} d^{2} e + 3 \, a^{3} b^{3} d e^{2} - a^{4} b^{2} e^{3}\right )} x\right )}} \] Input:
integrate((B*x+A)/(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas ")
Output:
1/2*(4*A*a*b^2*d*e - (B*a*b^2 + A*b^3)*d^2 + (B*a^3 - 3*A*a^2*b)*e^2 - 2*( B*b^3*d^2 + A*a*b^2*e^2 - (B*a*b^2 + A*b^3)*d*e)*x - 2*(B*a^2*b*d*e - A*a^ 2*b*e^2 + (B*b^3*d*e - A*b^3*e^2)*x^2 + 2*(B*a*b^2*d*e - A*a*b^2*e^2)*x)*l og(b*x + a) + 2*(B*a^2*b*d*e - A*a^2*b*e^2 + (B*b^3*d*e - A*b^3*e^2)*x^2 + 2*(B*a*b^2*d*e - A*a*b^2*e^2)*x)*log(e*x + d))/(a^2*b^4*d^3 - 3*a^3*b^3*d ^2*e + 3*a^4*b^2*d*e^2 - a^5*b*e^3 + (b^6*d^3 - 3*a*b^5*d^2*e + 3*a^2*b^4* d*e^2 - a^3*b^3*e^3)*x^2 + 2*(a*b^5*d^3 - 3*a^2*b^4*d^2*e + 3*a^3*b^3*d*e^ 2 - a^4*b^2*e^3)*x)
\[ \int \frac {A+B x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {A + B x}{\left (d + e x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((B*x+A)/(e*x+d)/(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
Output:
Integral((A + B*x)/((d + e*x)*((a + b*x)**2)**(3/2)), x)
Exception generated. \[ \int \frac {A+B x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((B*x+A)/(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^(3/2),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
Time = 0.17 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.47 \[ \int \frac {A+B x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {{\left (B b d e - A b e^{2}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{3} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b^{2} d e^{2} \mathrm {sgn}\left (b x + a\right ) - a^{3} b e^{3} \mathrm {sgn}\left (b x + a\right )} + \frac {{\left (B d e^{2} - A e^{3}\right )} \log \left ({\left | e x + d \right |}\right )}{b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b d e^{3} \mathrm {sgn}\left (b x + a\right ) - a^{3} e^{4} \mathrm {sgn}\left (b x + a\right )} - \frac {B a b^{2} d^{2} + A b^{3} d^{2} - 4 \, A a b^{2} d e - B a^{3} e^{2} + 3 \, A a^{2} b e^{2} + 2 \, {\left (B b^{3} d^{2} - B a b^{2} d e - A b^{3} d e + A a b^{2} e^{2}\right )} x}{2 \, {\left (b d - a e\right )}^{3} {\left (b x + a\right )}^{2} b \mathrm {sgn}\left (b x + a\right )} \] Input:
integrate((B*x+A)/(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")
Output:
-(B*b*d*e - A*b*e^2)*log(abs(b*x + a))/(b^4*d^3*sgn(b*x + a) - 3*a*b^3*d^2 *e*sgn(b*x + a) + 3*a^2*b^2*d*e^2*sgn(b*x + a) - a^3*b*e^3*sgn(b*x + a)) + (B*d*e^2 - A*e^3)*log(abs(e*x + d))/(b^3*d^3*e*sgn(b*x + a) - 3*a*b^2*d^2 *e^2*sgn(b*x + a) + 3*a^2*b*d*e^3*sgn(b*x + a) - a^3*e^4*sgn(b*x + a)) - 1 /2*(B*a*b^2*d^2 + A*b^3*d^2 - 4*A*a*b^2*d*e - B*a^3*e^2 + 3*A*a^2*b*e^2 + 2*(B*b^3*d^2 - B*a*b^2*d*e - A*b^3*d*e + A*a*b^2*e^2)*x)/((b*d - a*e)^3*(b *x + a)^2*b*sgn(b*x + a))
Timed out. \[ \int \frac {A+B x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x}{\left (d+e\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \] Input:
int((A + B*x)/((d + e*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2)),x)
Output:
int((A + B*x)/((d + e*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2)), x)
Time = 0.21 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.59 \[ \int \frac {A+B x}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {-\mathrm {log}\left (b x +a \right ) a^{2} e -\mathrm {log}\left (b x +a \right ) a b e x +\mathrm {log}\left (e x +d \right ) a^{2} e +\mathrm {log}\left (e x +d \right ) a b e x -a b e x +b^{2} d x}{a \left (a^{2} b \,e^{2} x -2 a \,b^{2} d e x +b^{3} d^{2} x +a^{3} e^{2}-2 a^{2} b d e +a \,b^{2} d^{2}\right )} \] Input:
int((B*x+A)/(e*x+d)/(b^2*x^2+2*a*b*x+a^2)^(3/2),x)
Output:
( - log(a + b*x)*a**2*e - log(a + b*x)*a*b*e*x + log(d + e*x)*a**2*e + log (d + e*x)*a*b*e*x - a*b*e*x + b**2*d*x)/(a*(a**3*e**2 - 2*a**2*b*d*e + a** 2*b*e**2*x + a*b**2*d**2 - 2*a*b**2*d*e*x + b**3*d**2*x))