Integrand size = 28, antiderivative size = 124 \[ \int \frac {(d+e x)^2}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {e (2 b d-a e) x (a+b x)}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {e^2 x^2 (a+b x)}{2 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {(b d-a e)^2 (a+b x) \log (a+b x)}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \] Output:
e*(-a*e+2*b*d)*x*(b*x+a)/b^2/((b*x+a)^2)^(1/2)+1/2*e^2*x^2*(b*x+a)/b/((b*x +a)^2)^(1/2)+(-a*e+b*d)^2*(b*x+a)*ln(b*x+a)/b^3/((b*x+a)^2)^(1/2)
Time = 1.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.48 \[ \int \frac {(d+e x)^2}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {(a+b x) \left (b e x (4 b d-2 a e+b e x)+2 (b d-a e)^2 \log (a+b x)\right )}{2 b^3 \sqrt {(a+b x)^2}} \] Input:
Integrate[(d + e*x)^2/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
Output:
((a + b*x)*(b*e*x*(4*b*d - 2*a*e + b*e*x) + 2*(b*d - a*e)^2*Log[a + b*x])) /(2*b^3*Sqrt[(a + b*x)^2])
Time = 0.37 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.60, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1102, 27, 49, 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.
\(\displaystyle \int \frac {(d+e x)^2}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx\) |
\(\Big \downarrow \) 1102 |
\(\displaystyle \frac {b (a+b x) \int \frac {(d+e x)^2}{b (a+b x)}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(a+b x) \int \frac {(d+e x)^2}{a+b x}dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {(a+b x) \int \left (\frac {(b d-a e)^2}{b^2 (a+b x)}+\frac {e (b d-a e)}{b^2}+\frac {e (d+e x)}{b}\right )dx}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(a+b x) \left (\frac {(b d-a e)^2 \log (a+b x)}{b^3}+\frac {e x (b d-a e)}{b^2}+\frac {(d+e x)^2}{2 b}\right )}{\sqrt {a^2+2 a b x+b^2 x^2}}\) |
Input:
Int[(d + e*x)^2/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
Output:
((a + b*x)*((e*(b*d - a*e)*x)/b^2 + (d + e*x)^2/(2*b) + ((b*d - a*e)^2*Log [a + b*x])/b^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_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*F racPart[p])) Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0]
Time = 0.86 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.70
method | result | size |
default | \(\frac {\left (b x +a \right ) \left (b^{2} e^{2} x^{2}+2 \ln \left (b x +a \right ) a^{2} e^{2}-4 \ln \left (b x +a \right ) a b d e +2 \ln \left (b x +a \right ) b^{2} d^{2}-2 x a b \,e^{2}+4 b^{2} d e x \right )}{2 \sqrt {\left (b x +a \right )^{2}}\, b^{3}}\) | \(87\) |
risch | \(-\frac {\sqrt {\left (b x +a \right )^{2}}\, e \left (-\frac {1}{2} b e \,x^{2}+a e x -2 b d x \right )}{\left (b x +a \right ) b^{2}}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \ln \left (b x +a \right )}{\left (b x +a \right ) b^{3}}\) | \(88\) |
Input:
int((e*x+d)^2/((b*x+a)^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2*(b*x+a)*(b^2*e^2*x^2+2*ln(b*x+a)*a^2*e^2-4*ln(b*x+a)*a*b*d*e+2*ln(b*x+ a)*b^2*d^2-2*x*a*b*e^2+4*b^2*d*e*x)/((b*x+a)^2)^(1/2)/b^3
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.51 \[ \int \frac {(d+e x)^2}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {b^{2} e^{2} x^{2} + 2 \, {\left (2 \, b^{2} d e - a b e^{2}\right )} x + 2 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left (b x + a\right )}{2 \, b^{3}} \] Input:
integrate((e*x+d)^2/((b*x+a)^2)^(1/2),x, algorithm="fricas")
Output:
1/2*(b^2*e^2*x^2 + 2*(2*b^2*d*e - a*b*e^2)*x + 2*(b^2*d^2 - 2*a*b*d*e + a^ 2*e^2)*log(b*x + a))/b^3
Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (88) = 176\).
Time = 0.89 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.14 \[ \int \frac {(d+e x)^2}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\begin {cases} \left (\frac {e^{2} x}{2 b^{2}} + \frac {- \frac {3 a e^{2}}{2 b} + 2 d e}{b^{2}}\right ) \sqrt {a^{2} + 2 a b x + b^{2} x^{2}} + \frac {\left (\frac {a}{b} + x\right ) \left (- \frac {a^{2} e^{2}}{2 b^{2}} - \frac {a \left (- \frac {3 a e^{2}}{2 b} + 2 d e\right )}{b} + d^{2}\right ) \log {\left (\frac {a}{b} + x \right )}}{\sqrt {b^{2} \left (\frac {a}{b} + x\right )^{2}}} & \text {for}\: b^{2} \neq 0 \\\frac {2 d^{2} \sqrt {a^{2} + 2 a b x} + \frac {2 d e \left (- a^{2} \sqrt {a^{2} + 2 a b x} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3}\right )}{a b} + \frac {e^{2} \left (a^{4} \sqrt {a^{2} + 2 a b x} - \frac {2 a^{2} \left (a^{2} + 2 a b x\right )^{\frac {3}{2}}}{3} + \frac {\left (a^{2} + 2 a b x\right )^{\frac {5}{2}}}{5}\right )}{2 a^{2} b^{2}}}{2 a b} & \text {for}\: a b \neq 0 \\\frac {d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}}{\sqrt {a^{2}}} & \text {otherwise} \end {cases} \] Input:
integrate((e*x+d)**2/((b*x+a)**2)**(1/2),x)
Output:
Piecewise(((e**2*x/(2*b**2) + (-3*a*e**2/(2*b) + 2*d*e)/b**2)*sqrt(a**2 + 2*a*b*x + b**2*x**2) + (a/b + x)*(-a**2*e**2/(2*b**2) - a*(-3*a*e**2/(2*b) + 2*d*e)/b + d**2)*log(a/b + x)/sqrt(b**2*(a/b + x)**2), Ne(b**2, 0)), (( 2*d**2*sqrt(a**2 + 2*a*b*x) + 2*d*e*(-a**2*sqrt(a**2 + 2*a*b*x) + (a**2 + 2*a*b*x)**(3/2)/3)/(a*b) + e**2*(a**4*sqrt(a**2 + 2*a*b*x) - 2*a**2*(a**2 + 2*a*b*x)**(3/2)/3 + (a**2 + 2*a*b*x)**(5/2)/5)/(2*a**2*b**2))/(2*a*b), N e(a*b, 0)), ((d**2*x + d*e*x**2 + e**2*x**3/3)/sqrt(a**2), True))
Time = 0.03 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.77 \[ \int \frac {(d+e x)^2}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {e^{2} x^{2}}{2 \, b} - \frac {a e^{2} x}{b^{2}} + \frac {d^{2} \log \left (x + \frac {a}{b}\right )}{b} - \frac {2 \, a d e \log \left (x + \frac {a}{b}\right )}{b^{2}} + \frac {a^{2} e^{2} \log \left (x + \frac {a}{b}\right )}{b^{3}} + \frac {2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} d e}{b^{2}} \] Input:
integrate((e*x+d)^2/((b*x+a)^2)^(1/2),x, algorithm="maxima")
Output:
1/2*e^2*x^2/b - a*e^2*x/b^2 + d^2*log(x + a/b)/b - 2*a*d*e*log(x + a/b)/b^ 2 + a^2*e^2*log(x + a/b)/b^3 + 2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*d*e/b^2
Time = 0.11 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.77 \[ \int \frac {(d+e x)^2}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {b e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 4 \, b d e x \mathrm {sgn}\left (b x + a\right ) - 2 \, a e^{2} x \mathrm {sgn}\left (b x + a\right )}{2 \, b^{2}} + \frac {{\left (b^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b d e \mathrm {sgn}\left (b x + a\right ) + a^{2} e^{2} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3}} \] Input:
integrate((e*x+d)^2/((b*x+a)^2)^(1/2),x, algorithm="giac")
Output:
1/2*(b*e^2*x^2*sgn(b*x + a) + 4*b*d*e*x*sgn(b*x + a) - 2*a*e^2*x*sgn(b*x + a))/b^2 + (b^2*d^2*sgn(b*x + a) - 2*a*b*d*e*sgn(b*x + a) + a^2*e^2*sgn(b* x + a))*log(abs(b*x + a))/b^3
Timed out. \[ \int \frac {(d+e x)^2}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \] Input:
int((d + e*x)^2/((a + b*x)^2)^(1/2),x)
Output:
int((d + e*x)^2/((a + b*x)^2)^(1/2), x)
Time = 0.23 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.58 \[ \int \frac {(d+e x)^2}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 \,\mathrm {log}\left (b x +a \right ) a^{2} e^{2}-4 \,\mathrm {log}\left (b x +a \right ) a b d e +2 \,\mathrm {log}\left (b x +a \right ) b^{2} d^{2}-2 a b \,e^{2} x +4 b^{2} d e x +b^{2} e^{2} x^{2}}{2 b^{3}} \] Input:
int((e*x+d)^2/((b*x+a)^2)^(1/2),x)
Output:
(2*log(a + b*x)*a**2*e**2 - 4*log(a + b*x)*a*b*d*e + 2*log(a + b*x)*b**2*d **2 - 2*a*b*e**2*x + 4*b**2*d*e*x + b**2*e**2*x**2)/(2*b**3)