Integrand size = 19, antiderivative size = 54 \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^2} \, dx=-\frac {\left (a+b \sqrt {c+d x}\right )^2}{x}-\frac {2 a b d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{\sqrt {c}}+b^2 d \log (x) \] Output:
-(a+b*(d*x+c)^(1/2))^2/x-2*a*b*d*arctanh((d*x+c)^(1/2)/c^(1/2))/c^(1/2)+b^ 2*d*ln(x)
Time = 0.14 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^2} \, dx=-\frac {a^2+b^2 c+2 a b \sqrt {c+d x}}{x}-\frac {2 a b d \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{\sqrt {c}}+b^2 d \log (-d x) \] Input:
Integrate[(a + b*Sqrt[c + d*x])^2/x^2,x]
Output:
-((a^2 + b^2*c + 2*a*b*Sqrt[c + d*x])/x) - (2*a*b*d*ArcTanh[Sqrt[c + d*x]/ Sqrt[c]])/Sqrt[c] + b^2*d*Log[-(d*x)]
Time = 0.38 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.22, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {896, 1732, 531, 27, 452, 219, 240}
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.
\(\displaystyle \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^2} \, dx\) |
\(\Big \downarrow \) 896 |
\(\displaystyle d \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{d^2 x^2}d(c+d x)\) |
\(\Big \downarrow \) 1732 |
\(\displaystyle 2 d \int \frac {\sqrt {c+d x} \left (a+b \sqrt {c+d x}\right )^2}{d^2 x^2}d\sqrt {c+d x}\) |
\(\Big \downarrow \) 531 |
\(\displaystyle 2 d \left (\frac {\int \frac {2 b c \left (a+b \sqrt {c+d x}\right )}{d x}d\sqrt {c+d x}}{2 c}-\frac {\left (a+b \sqrt {c+d x}\right )^2}{2 d x}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 d \left (-b \int -\frac {a+b \sqrt {c+d x}}{d x}d\sqrt {c+d x}-\frac {\left (a+b \sqrt {c+d x}\right )^2}{2 d x}\right )\) |
\(\Big \downarrow \) 452 |
\(\displaystyle 2 d \left (-b \left (a \int -\frac {1}{d x}d\sqrt {c+d x}+b \int -\frac {\sqrt {c+d x}}{d x}d\sqrt {c+d x}\right )-\frac {\left (a+b \sqrt {c+d x}\right )^2}{2 d x}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 2 d \left (-b \left (b \int -\frac {\sqrt {c+d x}}{d x}d\sqrt {c+d x}+\frac {a \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\left (a+b \sqrt {c+d x}\right )^2}{2 d x}\right )\) |
\(\Big \downarrow \) 240 |
\(\displaystyle 2 d \left (-b \left (\frac {a \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {1}{2} b \log (-d x)\right )-\frac {\left (a+b \sqrt {c+d x}\right )^2}{2 d x}\right )\) |
Input:
Int[(a + b*Sqrt[c + d*x])^2/x^2,x]
Output:
2*d*(-1/2*(a + b*Sqrt[c + d*x])^2/(d*x) - b*((a*ArcTanh[Sqrt[c + d*x]/Sqrt [c]])/Sqrt[c] - (b*Log[-(d*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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[c Int[1/ (a + b*x^2), x], x] + Simp[d Int[x/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c^2 + a*d^2, 0]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symb ol] :> With[{Qx = PolynomialQuotient[x^m, a + b*x^2, x], e = Coeff[Polynomi alRemainder[x^m, a + b*x^2, x], x, 0], f = Coeff[PolynomialRemainder[x^m, a + b*x^2, x], x, 1]}, Simp[(c + d*x)^n*(a*f - b*e*x)*((a + b*x^2)^(p + 1)/( 2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(c + d*x)^(n - 1)*(a + b *x^2)^(p + 1)*ExpandToSum[2*a*b*(p + 1)*(c + d*x)*Qx - a*d*f*n + b*c*e*(2*p + 3) + b*d*e*(n + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGt Q[n, 0] && IGtQ[m, 0] && LtQ[p, -1] && GtQ[n, 1] && IntegerQ[2*p]
Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1) Subst[Int[Si mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symb ol] :> With[{g = Denominator[n]}, Simp[g Subst[Int[x^(g - 1)*(d + e*x^(g* n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p, q} , x] && EqQ[n2, 2*n] && FractionQ[n]
Time = 0.36 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.17
method | result | size |
default | \(b^{2} \left (d \ln \left (x \right )-\frac {c}{x}\right )+4 a b d \left (-\frac {\sqrt {d x +c}}{2 d x}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}\right )-\frac {a^{2}}{x}\) | \(63\) |
derivativedivides | \(2 d \left (-\frac {a b \sqrt {d x +c}+\frac {b^{2} c}{2}+\frac {a^{2}}{2}}{d x}+b \left (\frac {b \ln \left (-d x \right )}{2}-\frac {a \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )\right )\) | \(64\) |
Input:
int((a+b*(d*x+c)^(1/2))^2/x^2,x,method=_RETURNVERBOSE)
Output:
b^2*(d*ln(x)-1/x*c)+4*a*b*d*(-1/2*(d*x+c)^(1/2)/d/x-1/2/c^(1/2)*arctanh((d *x+c)^(1/2)/c^(1/2)))-a^2/x
Time = 0.11 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.67 \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^2} \, dx=\left [\frac {b^{2} c d x \log \left (x\right ) + a b \sqrt {c} d x \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) - b^{2} c^{2} - 2 \, \sqrt {d x + c} a b c - a^{2} c}{c x}, \frac {b^{2} c d x \log \left (x\right ) + 2 \, a b \sqrt {-c} d x \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x + c}}\right ) - b^{2} c^{2} - 2 \, \sqrt {d x + c} a b c - a^{2} c}{c x}\right ] \] Input:
integrate((a+b*(d*x+c)^(1/2))^2/x^2,x, algorithm="fricas")
Output:
[(b^2*c*d*x*log(x) + a*b*sqrt(c)*d*x*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) - b^2*c^2 - 2*sqrt(d*x + c)*a*b*c - a^2*c)/(c*x), (b^2*c*d*x*log(x ) + 2*a*b*sqrt(-c)*d*x*arctan(sqrt(-c)/sqrt(d*x + c)) - b^2*c^2 - 2*sqrt(d *x + c)*a*b*c - a^2*c)/(c*x)]
Time = 25.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.35 \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^2} \, dx=- \frac {a^{2}}{x} - \frac {2 a b \sqrt {d} \sqrt {\frac {c}{d x} + 1}}{\sqrt {x}} - \frac {2 a b d \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} \sqrt {x}} \right )}}{\sqrt {c}} - \frac {b^{2} c}{x} + b^{2} d \log {\left (x \right )} \] Input:
integrate((a+b*(d*x+c)**(1/2))**2/x**2,x)
Output:
-a**2/x - 2*a*b*sqrt(d)*sqrt(c/(d*x) + 1)/sqrt(x) - 2*a*b*d*asinh(sqrt(c)/ (sqrt(d)*sqrt(x)))/sqrt(c) - b**2*c/x + b**2*d*log(x)
Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.35 \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^2} \, dx={\left (b^{2} \log \left (d x\right ) + \frac {a b \log \left (\frac {\sqrt {d x + c} - \sqrt {c}}{\sqrt {d x + c} + \sqrt {c}}\right )}{\sqrt {c}} - \frac {b^{2} c + 2 \, \sqrt {d x + c} a b + a^{2}}{d x}\right )} d \] Input:
integrate((a+b*(d*x+c)^(1/2))^2/x^2,x, algorithm="maxima")
Output:
(b^2*log(d*x) + a*b*log((sqrt(d*x + c) - sqrt(c))/(sqrt(d*x + c) + sqrt(c) ))/sqrt(c) - (b^2*c + 2*sqrt(d*x + c)*a*b + a^2)/(d*x))*d
Time = 0.12 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.15 \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^2} \, dx={\left (b^{2} \log \left (d x\right ) + \frac {2 \, a b \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {b^{2} c + 2 \, \sqrt {d x + c} a b + a^{2}}{d x}\right )} d \] Input:
integrate((a+b*(d*x+c)^(1/2))^2/x^2,x, algorithm="giac")
Output:
(b^2*log(d*x) + 2*a*b*arctan(sqrt(d*x + c)/sqrt(-c))/sqrt(-c) - (b^2*c + 2 *sqrt(d*x + c)*a*b + a^2)/(d*x))*d
Time = 0.12 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.43 \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^2} \, dx=b\,d\,\ln \left (2\,b\,d\,\left (b+\frac {a}{\sqrt {c}}\right )\,\sqrt {c+d\,x}-2\,b^2\,d\,\sqrt {c+d\,x}-2\,a\,b\,d\right )\,\left (b+\frac {a}{\sqrt {c}}\right )-\frac {a^2\,d+b^2\,c\,d+2\,a\,b\,d\,\sqrt {c+d\,x}}{d\,x}+b\,d\,\ln \left (2\,b\,d\,\left (b-\frac {a}{\sqrt {c}}\right )\,\sqrt {c+d\,x}-2\,b^2\,d\,\sqrt {c+d\,x}-2\,a\,b\,d\right )\,\left (b-\frac {a}{\sqrt {c}}\right ) \] Input:
int((a + b*(c + d*x)^(1/2))^2/x^2,x)
Output:
b*d*log(2*b*d*(b + a/c^(1/2))*(c + d*x)^(1/2) - 2*b^2*d*(c + d*x)^(1/2) - 2*a*b*d)*(b + a/c^(1/2)) - (a^2*d + b^2*c*d + 2*a*b*d*(c + d*x)^(1/2))/(d* x) + b*d*log(2*b*d*(b - a/c^(1/2))*(c + d*x)^(1/2) - 2*b^2*d*(c + d*x)^(1/ 2) - 2*a*b*d)*(b - a/c^(1/2))
Time = 0.26 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.46 \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^2} \, dx=\frac {-2 \sqrt {d x +c}\, a b c +\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a b d x -\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a b d x +\mathrm {log}\left (x \right ) b^{2} c d x -a^{2} c -b^{2} c^{2}}{c x} \] Input:
int((a+b*(d*x+c)^(1/2))^2/x^2,x)
Output:
( - 2*sqrt(c + d*x)*a*b*c + sqrt(c)*log(sqrt(c + d*x) - sqrt(c))*a*b*d*x - sqrt(c)*log(sqrt(c + d*x) + sqrt(c))*a*b*d*x + log(x)*b**2*c*d*x - a**2*c - b**2*c**2)/(c*x)