Integrand size = 20, antiderivative size = 69 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{5/2}} \, dx=-\frac {2 B \sqrt {a+b x}}{\sqrt {x}}-\frac {2 A (a+b x)^{3/2}}{3 a x^{3/2}}+2 \sqrt {b} B \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \] Output:
-2*B*(b*x+a)^(1/2)/x^(1/2)-2/3*A*(b*x+a)^(3/2)/a/x^(3/2)+2*b^(1/2)*B*arcta nh(b^(1/2)*x^(1/2)/(b*x+a)^(1/2))
Time = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{5/2}} \, dx=-\frac {2 \sqrt {a+b x} (a A+A b x+3 a B x)}{3 a x^{3/2}}-2 \sqrt {b} B \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right ) \] Input:
Integrate[(Sqrt[a + b*x]*(A + B*x))/x^(5/2),x]
Output:
(-2*Sqrt[a + b*x]*(a*A + A*b*x + 3*a*B*x))/(3*a*x^(3/2)) - 2*Sqrt[b]*B*Log [-(Sqrt[b]*Sqrt[x]) + Sqrt[a + b*x]]
Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {87, 57, 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.
\(\displaystyle \int \frac {\sqrt {a+b x} (A+B x)}{x^{5/2}} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle B \int \frac {\sqrt {a+b x}}{x^{3/2}}dx-\frac {2 A (a+b x)^{3/2}}{3 a x^{3/2}}\) |
\(\Big \downarrow \) 57 |
\(\displaystyle B \left (b \int \frac {1}{\sqrt {x} \sqrt {a+b x}}dx-\frac {2 \sqrt {a+b x}}{\sqrt {x}}\right )-\frac {2 A (a+b x)^{3/2}}{3 a x^{3/2}}\) |
\(\Big \downarrow \) 65 |
\(\displaystyle B \left (2 b \int \frac {1}{1-\frac {b x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}-\frac {2 \sqrt {a+b x}}{\sqrt {x}}\right )-\frac {2 A (a+b x)^{3/2}}{3 a x^{3/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle B \left (2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )-\frac {2 \sqrt {a+b x}}{\sqrt {x}}\right )-\frac {2 A (a+b x)^{3/2}}{3 a x^{3/2}}\) |
Input:
Int[(Sqrt[a + b*x]*(A + B*x))/x^(5/2),x]
Output:
(-2*A*(a + b*x)^(3/2))/(3*a*x^(3/2)) + B*((-2*Sqrt[a + b*x])/Sqrt[x] + 2*S qrt[b]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[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 = 78, normalized size of antiderivative = 1.13
method | result | size |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (A b x +3 B a x +A a \right )}{3 x^{\frac {3}{2}} a}+\frac {B \sqrt {b}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{\sqrt {x}\, \sqrt {b x +a}}\) | \(78\) |
default | \(-\frac {\sqrt {b x +a}\, \left (-3 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a b \,x^{2}+2 A \,b^{\frac {3}{2}} x \sqrt {x \left (b x +a \right )}+6 B a x \sqrt {b}\, \sqrt {x \left (b x +a \right )}+2 A a \sqrt {b}\, \sqrt {x \left (b x +a \right )}\right )}{3 x^{\frac {3}{2}} \sqrt {x \left (b x +a \right )}\, a \sqrt {b}}\) | \(112\) |
Input:
int((b*x+a)^(1/2)*(B*x+A)/x^(5/2),x,method=_RETURNVERBOSE)
Output:
-2/3*(b*x+a)^(1/2)*(A*b*x+3*B*a*x+A*a)/x^(3/2)/a+B*b^(1/2)*ln((1/2*a+b*x)/ b^(1/2)+(b*x^2+a*x)^(1/2))*(x*(b*x+a))^(1/2)/x^(1/2)/(b*x+a)^(1/2)
Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.90 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{5/2}} \, dx=\left [\frac {3 \, B a \sqrt {b} x^{2} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (A a + {\left (3 \, B a + A b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{3 \, a x^{2}}, -\frac {2 \, {\left (3 \, B a \sqrt {-b} x^{2} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {b x + a}}\right ) + {\left (A a + {\left (3 \, B a + A b\right )} x\right )} \sqrt {b x + a} \sqrt {x}\right )}}{3 \, a x^{2}}\right ] \] Input:
integrate((b*x+a)^(1/2)*(B*x+A)/x^(5/2),x, algorithm="fricas")
Output:
[1/3*(3*B*a*sqrt(b)*x^2*log(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) - 2*(A*a + (3*B*a + A*b)*x)*sqrt(b*x + a)*sqrt(x))/(a*x^2), -2/3*(3*B*a*sqr t(-b)*x^2*arctan(sqrt(-b)*sqrt(x)/sqrt(b*x + a)) + (A*a + (3*B*a + A*b)*x) *sqrt(b*x + a)*sqrt(x))/(a*x^2)]
Time = 1.88 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.70 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{5/2}} \, dx=- \frac {2 A \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{3 x} - \frac {2 A b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{3 a} - \frac {2 B \sqrt {a}}{\sqrt {x} \sqrt {1 + \frac {b x}{a}}} + 2 B \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} - \frac {2 B b \sqrt {x}}{\sqrt {a} \sqrt {1 + \frac {b x}{a}}} \] Input:
integrate((b*x+a)**(1/2)*(B*x+A)/x**(5/2),x)
Output:
-2*A*sqrt(b)*sqrt(a/(b*x) + 1)/(3*x) - 2*A*b**(3/2)*sqrt(a/(b*x) + 1)/(3*a ) - 2*B*sqrt(a)/(sqrt(x)*sqrt(1 + b*x/a)) + 2*B*sqrt(b)*asinh(sqrt(b)*sqrt (x)/sqrt(a)) - 2*B*b*sqrt(x)/(sqrt(a)*sqrt(1 + b*x/a))
Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{5/2}} \, dx=-{\left (\sqrt {b} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right ) + \frac {2 \, \sqrt {b x + a}}{\sqrt {x}}\right )} B - \frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}} A}{3 \, a x^{\frac {3}{2}}} \] Input:
integrate((b*x+a)^(1/2)*(B*x+A)/x^(5/2),x, algorithm="maxima")
Output:
-(sqrt(b)*log(-(sqrt(b) - sqrt(b*x + a)/sqrt(x))/(sqrt(b) + sqrt(b*x + a)/ sqrt(x))) + 2*sqrt(b*x + a)/sqrt(x))*B - 2/3*(b*x + a)^(3/2)*A/(a*x^(3/2))
Time = 75.20 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{5/2}} \, dx=-\frac {2 \, b^{3} {\left (\frac {3 \, B \log \left ({\left | -\sqrt {b x + a} \sqrt {b} + \sqrt {{\left (b x + a\right )} b - a b} \right |}\right )}{b^{\frac {3}{2}}} - \frac {{\left (3 \, B a - \frac {{\left (3 \, B a b + A b^{2}\right )} {\left (b x + a\right )}}{a b}\right )} \sqrt {b x + a}}{{\left ({\left (b x + a\right )} b - a b\right )}^{\frac {3}{2}}}\right )}}{3 \, {\left | b \right |}} \] Input:
integrate((b*x+a)^(1/2)*(B*x+A)/x^(5/2),x, algorithm="giac")
Output:
-2/3*b^3*(3*B*log(abs(-sqrt(b*x + a)*sqrt(b) + sqrt((b*x + a)*b - a*b)))/b ^(3/2) - (3*B*a - (3*B*a*b + A*b^2)*(b*x + a)/(a*b))*sqrt(b*x + a)/((b*x + a)*b - a*b)^(3/2))/abs(b)
Timed out. \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,\sqrt {a+b\,x}}{x^{5/2}} \,d x \] Input:
int(((A + B*x)*(a + b*x)^(1/2))/x^(5/2),x)
Output:
int(((A + B*x)*(a + b*x)^(1/2))/x^(5/2), x)
Time = 0.16 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^{5/2}} \, dx=\frac {-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a}{3}-\frac {8 \sqrt {x}\, \sqrt {b x +a}\, b x}{3}+2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) b \,x^{2}}{x^{2}} \] Input:
int((b*x+a)^(1/2)*(B*x+A)/x^(5/2),x)
Output:
(2*( - sqrt(x)*sqrt(a + b*x)*a - 4*sqrt(x)*sqrt(a + b*x)*b*x + 3*sqrt(b)*l og((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*b*x**2))/(3*x**2)