Integrand size = 19, antiderivative size = 80 \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^3} \, dx=-\frac {b d \left (b c+a \sqrt {c+d x}\right )}{2 c x}-\frac {\left (a+b \sqrt {c+d x}\right )^2}{2 x^2}+\frac {a b d^2 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{2 c^{3/2}} \] Output:
-1/2*b*d*(b*c+a*(d*x+c)^(1/2))/c/x-1/2*(a+b*(d*x+c)^(1/2))^2/x^2+1/2*a*b*d ^2*arctanh((d*x+c)^(1/2)/c^(1/2))/c^(3/2)
Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^3} \, dx=-\frac {a^2 c+a b \sqrt {c+d x} (2 c+d x)+b^2 c (c+2 d x)}{2 c x^2}+\frac {a b d^2 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{2 c^{3/2}} \] Input:
Integrate[(a + b*Sqrt[c + d*x])^2/x^3,x]
Output:
-1/2*(a^2*c + a*b*Sqrt[c + d*x]*(2*c + d*x) + b^2*c*(c + 2*d*x))/(c*x^2) + (a*b*d^2*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(2*c^(3/2))
Time = 0.38 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {896, 25, 1732, 531, 27, 454, 219}
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.
\(\displaystyle \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^3} \, dx\) |
\(\Big \downarrow \) 896 |
\(\displaystyle d^2 \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{d^3 x^3}d(c+d x)\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -d^2 \int -\frac {\left (a+b \sqrt {c+d x}\right )^2}{d^3 x^3}d(c+d x)\) |
\(\Big \downarrow \) 1732 |
\(\displaystyle -2 d^2 \int -\frac {\sqrt {c+d x} \left (a+b \sqrt {c+d x}\right )^2}{d^3 x^3}d\sqrt {c+d x}\) |
\(\Big \downarrow \) 531 |
\(\displaystyle -2 d^2 \left (\frac {\int -\frac {2 b c \left (a+b \sqrt {c+d x}\right )}{d^2 x^2}d\sqrt {c+d x}}{4 c}+\frac {\left (a+b \sqrt {c+d x}\right )^2}{4 d^2 x^2}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -2 d^2 \left (\frac {\left (a+b \sqrt {c+d x}\right )^2}{4 d^2 x^2}-\frac {1}{2} b \int \frac {a+b \sqrt {c+d x}}{d^2 x^2}d\sqrt {c+d x}\right )\) |
\(\Big \downarrow \) 454 |
\(\displaystyle -2 d^2 \left (\frac {\left (a+b \sqrt {c+d x}\right )^2}{4 d^2 x^2}-\frac {1}{2} b \left (\frac {a \int -\frac {1}{d x}d\sqrt {c+d x}}{2 c}-\frac {a \sqrt {c+d x}+b c}{2 c d x}\right )\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -2 d^2 \left (\frac {\left (a+b \sqrt {c+d x}\right )^2}{4 d^2 x^2}-\frac {1}{2} b \left (\frac {a \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{2 c^{3/2}}-\frac {a \sqrt {c+d x}+b c}{2 c d x}\right )\right )\) |
Input:
Int[(a + b*Sqrt[c + d*x])^2/x^3,x]
Output:
-2*d^2*((a + b*Sqrt[c + d*x])^2/(4*d^2*x^2) - (b*(-1/2*(b*c + a*Sqrt[c + d *x])/(c*d*x) + (a*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(2*c^(3/2))))/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[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*d - b*c*x)/(2*a*b*(p + 1)))*(a + b*x^2)^(p + 1), x] + Simp[c*((2*p + 3)/(2*a *(p + 1))) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && L tQ[p, -1] && NeQ[p, -3/2]
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 = 0.37 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.01
method | result | size |
derivativedivides | \(2 d^{2} \left (-\frac {\frac {a b \left (d x +c \right )^{\frac {3}{2}}}{4 c}+\frac {b^{2} \left (d x +c \right )}{2}+\frac {a b \sqrt {d x +c}}{4}-\frac {b^{2} c}{4}+\frac {a^{2}}{4}}{d^{2} x^{2}}+\frac {a b \,\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{4 c^{\frac {3}{2}}}\right )\) | \(81\) |
default | \(b^{2} \left (-\frac {c}{2 x^{2}}-\frac {d}{x}\right )+4 a b \,d^{2} \left (-\frac {\frac {\left (d x +c \right )^{\frac {3}{2}}}{8 c}+\frac {\sqrt {d x +c}}{8}}{d^{2} x^{2}}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{8 c^{\frac {3}{2}}}\right )-\frac {a^{2}}{2 x^{2}}\) | \(82\) |
Input:
int((a+b*(d*x+c)^(1/2))^2/x^3,x,method=_RETURNVERBOSE)
Output:
2*d^2*(-(1/4*a*b/c*(d*x+c)^(3/2)+1/2*b^2*(d*x+c)+1/4*a*b*(d*x+c)^(1/2)-1/4 *b^2*c+1/4*a^2)/d^2/x^2+1/4*a*b/c^(3/2)*arctanh((d*x+c)^(1/2)/c^(1/2)))
Time = 0.14 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.22 \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^3} \, dx=\left [\frac {a b \sqrt {c} d^{2} x^{2} \log \left (\frac {d x + 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) - 4 \, b^{2} c^{2} d x - 2 \, b^{2} c^{3} - 2 \, a^{2} c^{2} - 2 \, {\left (a b c d x + 2 \, a b c^{2}\right )} \sqrt {d x + c}}{4 \, c^{2} x^{2}}, -\frac {a b \sqrt {-c} d^{2} x^{2} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x + c}}\right ) + 2 \, b^{2} c^{2} d x + b^{2} c^{3} + a^{2} c^{2} + {\left (a b c d x + 2 \, a b c^{2}\right )} \sqrt {d x + c}}{2 \, c^{2} x^{2}}\right ] \] Input:
integrate((a+b*(d*x+c)^(1/2))^2/x^3,x, algorithm="fricas")
Output:
[1/4*(a*b*sqrt(c)*d^2*x^2*log((d*x + 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) - 4 *b^2*c^2*d*x - 2*b^2*c^3 - 2*a^2*c^2 - 2*(a*b*c*d*x + 2*a*b*c^2)*sqrt(d*x + c))/(c^2*x^2), -1/2*(a*b*sqrt(-c)*d^2*x^2*arctan(sqrt(-c)/sqrt(d*x + c)) + 2*b^2*c^2*d*x + b^2*c^3 + a^2*c^2 + (a*b*c*d*x + 2*a*b*c^2)*sqrt(d*x + c))/(c^2*x^2)]
Time = 97.42 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.68 \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^3} \, dx=- \frac {a^{2}}{2 x^{2}} - \frac {a b c}{\sqrt {d} x^{\frac {5}{2}} \sqrt {\frac {c}{d x} + 1}} - \frac {3 a b \sqrt {d}}{2 x^{\frac {3}{2}} \sqrt {\frac {c}{d x} + 1}} - \frac {a b d^{\frac {3}{2}}}{2 c \sqrt {x} \sqrt {\frac {c}{d x} + 1}} + \frac {a b d^{2} \operatorname {asinh}{\left (\frac {\sqrt {c}}{\sqrt {d} \sqrt {x}} \right )}}{2 c^{\frac {3}{2}}} - \frac {b^{2} c}{2 x^{2}} - \frac {b^{2} d}{x} \] Input:
integrate((a+b*(d*x+c)**(1/2))**2/x**3,x)
Output:
-a**2/(2*x**2) - a*b*c/(sqrt(d)*x**(5/2)*sqrt(c/(d*x) + 1)) - 3*a*b*sqrt(d )/(2*x**(3/2)*sqrt(c/(d*x) + 1)) - a*b*d**(3/2)/(2*c*sqrt(x)*sqrt(c/(d*x) + 1)) + a*b*d**2*asinh(sqrt(c)/(sqrt(d)*sqrt(x)))/(2*c**(3/2)) - b**2*c/(2 *x**2) - b**2*d/x
Time = 0.11 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.41 \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^3} \, dx=-\frac {1}{4} \, {\left (\frac {a b \log \left (\frac {\sqrt {d x + c} - \sqrt {c}}{\sqrt {d x + c} + \sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {2 \, {\left (2 \, {\left (d x + c\right )} b^{2} c - b^{2} c^{2} + {\left (d x + c\right )}^{\frac {3}{2}} a b + \sqrt {d x + c} a b c + a^{2} c\right )}}{{\left (d x + c\right )}^{2} c - 2 \, {\left (d x + c\right )} c^{2} + c^{3}}\right )} d^{2} \] Input:
integrate((a+b*(d*x+c)^(1/2))^2/x^3,x, algorithm="maxima")
Output:
-1/4*(a*b*log((sqrt(d*x + c) - sqrt(c))/(sqrt(d*x + c) + sqrt(c)))/c^(3/2) + 2*(2*(d*x + c)*b^2*c - b^2*c^2 + (d*x + c)^(3/2)*a*b + sqrt(d*x + c)*a* b*c + a^2*c)/((d*x + c)^2*c - 2*(d*x + c)*c^2 + c^3))*d^2
Time = 0.11 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.31 \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^3} \, dx=-\frac {\frac {a b d^{3} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{\sqrt {-c} c} + \frac {2 \, {\left (d x + c\right )} b^{2} c d^{3} - b^{2} c^{2} d^{3} + {\left (d x + c\right )}^{\frac {3}{2}} a b d^{3} + \sqrt {d x + c} a b c d^{3} + a^{2} c d^{3}}{c d^{2} x^{2}}}{2 \, d} \] Input:
integrate((a+b*(d*x+c)^(1/2))^2/x^3,x, algorithm="giac")
Output:
-1/2*(a*b*d^3*arctan(sqrt(d*x + c)/sqrt(-c))/(sqrt(-c)*c) + (2*(d*x + c)*b ^2*c*d^3 - b^2*c^2*d^3 + (d*x + c)^(3/2)*a*b*d^3 + sqrt(d*x + c)*a*b*c*d^3 + a^2*c*d^3)/(c*d^2*x^2))/d
Time = 9.03 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^3} \, dx=\frac {a\,b\,d^2\,\mathrm {atanh}\left (\frac {\sqrt {c+d\,x}}{\sqrt {c}}\right )}{2\,c^{3/2}}-\frac {b^2\,c}{2\,x^2}-\frac {b^2\,d}{x}-\frac {a\,b\,\sqrt {c+d\,x}}{2\,x^2}-\frac {a\,b\,{\left (c+d\,x\right )}^{3/2}}{2\,c\,x^2}-\frac {a^2}{2\,x^2} \] Input:
int((a + b*(c + d*x)^(1/2))^2/x^3,x)
Output:
(a*b*d^2*atanh((c + d*x)^(1/2)/c^(1/2)))/(2*c^(3/2)) - (b^2*c)/(2*x^2) - ( b^2*d)/x - (a*b*(c + d*x)^(1/2))/(2*x^2) - (a*b*(c + d*x)^(3/2))/(2*c*x^2) - a^2/(2*x^2)
Time = 0.26 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.32 \[ \int \frac {\left (a+b \sqrt {c+d x}\right )^2}{x^3} \, dx=\frac {-4 \sqrt {d x +c}\, a b \,c^{2}-2 \sqrt {d x +c}\, a b c d x -\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}-\sqrt {c}\right ) a b \,d^{2} x^{2}+\sqrt {c}\, \mathrm {log}\left (\sqrt {d x +c}+\sqrt {c}\right ) a b \,d^{2} x^{2}-2 a^{2} c^{2}-2 b^{2} c^{3}-4 b^{2} c^{2} d x}{4 c^{2} x^{2}} \] Input:
int((a+b*(d*x+c)^(1/2))^2/x^3,x)
Output:
( - 4*sqrt(c + d*x)*a*b*c**2 - 2*sqrt(c + d*x)*a*b*c*d*x - sqrt(c)*log(sqr t(c + d*x) - sqrt(c))*a*b*d**2*x**2 + sqrt(c)*log(sqrt(c + d*x) + sqrt(c)) *a*b*d**2*x**2 - 2*a**2*c**2 - 2*b**2*c**3 - 4*b**2*c**2*d*x)/(4*c**2*x**2 )