Integrand size = 20, antiderivative size = 69 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{3/2}} \, dx=-\frac {2 A \sqrt {a+b x}}{\sqrt {x}}+B \sqrt {x} \sqrt {a+b x}+\frac {(2 A b+a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{\sqrt {b}} \] Output:
-2*A*(b*x+a)^(1/2)/x^(1/2)+B*x^(1/2)*(b*x+a)^(1/2)+(2*A*b+B*a)*arctanh(b^( 1/2)*x^(1/2)/(b*x+a)^(1/2))/b^(1/2)
Time = 0.16 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{3/2}} \, dx=\frac {\sqrt {a+b x} (-2 A+B x)}{\sqrt {x}}+\frac {2 (2 A b+a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )}{\sqrt {b}} \] Input:
Integrate[(Sqrt[a + b*x]*(A + B*x))/x^(3/2),x]
Output:
(Sqrt[a + b*x]*(-2*A + B*x))/Sqrt[x] + (2*(2*A*b + a*B)*ArcTanh[(Sqrt[b]*S qrt[x])/(-Sqrt[a] + Sqrt[a + b*x])])/Sqrt[b]
Time = 0.18 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {87, 60, 65, 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.
\(\displaystyle \int \frac {\sqrt {a+b x} (A+B x)}{x^{3/2}} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {(a B+2 A b) \int \frac {\sqrt {a+b x}}{\sqrt {x}}dx}{a}-\frac {2 A (a+b x)^{3/2}}{a \sqrt {x}}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {(a B+2 A b) \left (\frac {1}{2} a \int \frac {1}{\sqrt {x} \sqrt {a+b x}}dx+\sqrt {x} \sqrt {a+b x}\right )}{a}-\frac {2 A (a+b x)^{3/2}}{a \sqrt {x}}\) |
\(\Big \downarrow \) 65 |
\(\displaystyle \frac {(a B+2 A b) \left (a \int \frac {1}{1-\frac {b x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}+\sqrt {x} \sqrt {a+b x}\right )}{a}-\frac {2 A (a+b x)^{3/2}}{a \sqrt {x}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {(a B+2 A b) \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{\sqrt {b}}+\sqrt {x} \sqrt {a+b x}\right )}{a}-\frac {2 A (a+b x)^{3/2}}{a \sqrt {x}}\) |
Input:
Int[(Sqrt[a + b*x]*(A + B*x))/x^(3/2),x]
Output:
(-2*A*(a + b*x)^(3/2))/(a*Sqrt[x]) + ((2*A*b + a*B)*(Sqrt[x]*Sqrt[a + b*x] + (a*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/Sqrt[b]))/a
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2 Sub st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d }, x] && !GtQ[c, 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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])
Time = 0.15 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.12
method | result | size |
risch | \(-\frac {\sqrt {b x +a}\, \left (-B x +2 A \right )}{\sqrt {x}}+\frac {\left (A b +\frac {B a}{2}\right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{\sqrt {b}\, \sqrt {x}\, \sqrt {b x +a}}\) | \(77\) |
default | \(\frac {\sqrt {b x +a}\, \left (2 A \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) b x +B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a x +2 B x \sqrt {x \left (b x +a \right )}\, \sqrt {b}-4 A \sqrt {x \left (b x +a \right )}\, \sqrt {b}\right )}{2 \sqrt {x}\, \sqrt {x \left (b x +a \right )}\, \sqrt {b}}\) | \(118\) |
Input:
int((b*x+a)^(1/2)*(B*x+A)/x^(3/2),x,method=_RETURNVERBOSE)
Output:
-(b*x+a)^(1/2)*(-B*x+2*A)/x^(1/2)+(A*b+1/2*B*a)*ln((1/2*a+b*x)/b^(1/2)+(b* x^2+a*x)^(1/2))/b^(1/2)*(x*(b*x+a))^(1/2)/x^(1/2)/(b*x+a)^(1/2)
Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.86 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{3/2}} \, dx=\left [\frac {{\left (B a + 2 \, A b\right )} \sqrt {b} x \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (B b x - 2 \, A b\right )} \sqrt {b x + a} \sqrt {x}}{2 \, b x}, -\frac {{\left (B a + 2 \, A b\right )} \sqrt {-b} x \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {b x + a}}\right ) - {\left (B b x - 2 \, A b\right )} \sqrt {b x + a} \sqrt {x}}{b x}\right ] \] Input:
integrate((b*x+a)^(1/2)*(B*x+A)/x^(3/2),x, algorithm="fricas")
Output:
[1/2*((B*a + 2*A*b)*sqrt(b)*x*log(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) + 2*(B*b*x - 2*A*b)*sqrt(b*x + a)*sqrt(x))/(b*x), -((B*a + 2*A*b)*sqr t(-b)*x*arctan(sqrt(-b)*sqrt(x)/sqrt(b*x + a)) - (B*b*x - 2*A*b)*sqrt(b*x + a)*sqrt(x))/(b*x)]
Time = 2.21 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.75 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{3/2}} \, dx=- \frac {2 A \sqrt {a}}{\sqrt {x} \sqrt {1 + \frac {b x}{a}}} + 2 A \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} - \frac {2 A b \sqrt {x}}{\sqrt {a} \sqrt {1 + \frac {b x}{a}}} + B \sqrt {a} \sqrt {x} \sqrt {1 + \frac {b x}{a}} + \frac {B a \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{\sqrt {b}} \] Input:
integrate((b*x+a)**(1/2)*(B*x+A)/x**(3/2),x)
Output:
-2*A*sqrt(a)/(sqrt(x)*sqrt(1 + b*x/a)) + 2*A*sqrt(b)*asinh(sqrt(b)*sqrt(x) /sqrt(a)) - 2*A*b*sqrt(x)/(sqrt(a)*sqrt(1 + b*x/a)) + B*sqrt(a)*sqrt(x)*sq rt(1 + b*x/a) + B*a*asinh(sqrt(b)*sqrt(x)/sqrt(a))/sqrt(b)
Time = 0.04 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{3/2}} \, dx=\frac {B a \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{2 \, \sqrt {b}} + A \sqrt {b} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + \sqrt {b x^{2} + a x} B - \frac {2 \, \sqrt {b x^{2} + a x} A}{x} \] Input:
integrate((b*x+a)^(1/2)*(B*x+A)/x^(3/2),x, algorithm="maxima")
Output:
1/2*B*a*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/sqrt(b) + A*sqrt(b)*l og(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) + sqrt(b*x^2 + a*x)*B - 2*sqrt (b*x^2 + a*x)*A/x
Time = 75.89 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.45 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{3/2}} \, dx=\frac {{\left (\frac {\sqrt {b x + a} {\left (\frac {{\left (b x + a\right )} B}{b} - \frac {B a b + 2 \, A b^{2}}{b^{2}}\right )}}{\sqrt {{\left (b x + a\right )} b - a b}} - \frac {{\left (B a + 2 \, A b\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} + \sqrt {{\left (b x + a\right )} b - a b} \right |}\right )}{b^{\frac {3}{2}}}\right )} b^{2}}{{\left | b \right |}} \] Input:
integrate((b*x+a)^(1/2)*(B*x+A)/x^(3/2),x, algorithm="giac")
Output:
(sqrt(b*x + a)*((b*x + a)*B/b - (B*a*b + 2*A*b^2)/b^2)/sqrt((b*x + a)*b - a*b) - (B*a + 2*A*b)*log(abs(-sqrt(b*x + a)*sqrt(b) + sqrt((b*x + a)*b - a *b)))/b^(3/2))*b^2/abs(b)
Timed out. \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{3/2}} \, dx=\int \frac {\left (A+B\,x\right )\,\sqrt {a+b\,x}}{x^{3/2}} \,d x \] Input:
int(((A + B*x)*(a + b*x)^(1/2))/x^(3/2),x)
Output:
int(((A + B*x)*(a + b*x)^(1/2))/x^(3/2), x)
Time = 0.17 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{3/2}} \, dx=\frac {-8 \sqrt {x}\, \sqrt {b x +a}\, a +4 \sqrt {x}\, \sqrt {b x +a}\, b x +12 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a x -9 \sqrt {b}\, a x}{4 x} \] Input:
int((b*x+a)^(1/2)*(B*x+A)/x^(3/2),x)
Output:
( - 8*sqrt(x)*sqrt(a + b*x)*a + 4*sqrt(x)*sqrt(a + b*x)*b*x + 12*sqrt(b)*l og((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a*x - 9*sqrt(b)*a*x)/(4*x)