Integrand size = 20, antiderivative size = 82 \[ \int \frac {\sqrt {x} (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {2 (A b-a B) x^{3/2}}{3 a b (a+b x)^{3/2}}-\frac {2 B \sqrt {x}}{b^2 \sqrt {a+b x}}+\frac {2 B \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{5/2}} \] Output:
2/3*(A*b-B*a)*x^(3/2)/a/b/(b*x+a)^(3/2)-2*B*x^(1/2)/b^2/(b*x+a)^(1/2)+2*B* arctanh(b^(1/2)*x^(1/2)/(b*x+a)^(1/2))/b^(5/2)
Time = 0.16 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {x} (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {2 \sqrt {x} \left (-3 a^2 B+A b^2 x-4 a b B x\right )}{3 a b^2 (a+b x)^{3/2}}-\frac {2 B \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )}{b^{5/2}} \] Input:
Integrate[(Sqrt[x]*(A + B*x))/(a + b*x)^(5/2),x]
Output:
(2*Sqrt[x]*(-3*a^2*B + A*b^2*x - 4*a*b*B*x))/(3*a*b^2*(a + b*x)^(3/2)) - ( 2*B*Log[-(Sqrt[b]*Sqrt[x]) + Sqrt[a + b*x]])/b^(5/2)
Time = 0.18 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.05, 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.
\(\displaystyle \int \frac {\sqrt {x} (A+B x)}{(a+b x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {B \int \frac {\sqrt {x}}{(a+b x)^{3/2}}dx}{b}+\frac {2 x^{3/2} (A b-a B)}{3 a b (a+b x)^{3/2}}\) |
\(\Big \downarrow \) 57 |
\(\displaystyle \frac {B \left (\frac {\int \frac {1}{\sqrt {x} \sqrt {a+b x}}dx}{b}-\frac {2 \sqrt {x}}{b \sqrt {a+b x}}\right )}{b}+\frac {2 x^{3/2} (A b-a B)}{3 a b (a+b x)^{3/2}}\) |
\(\Big \downarrow \) 65 |
\(\displaystyle \frac {B \left (\frac {2 \int \frac {1}{1-\frac {b x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{b}-\frac {2 \sqrt {x}}{b \sqrt {a+b x}}\right )}{b}+\frac {2 x^{3/2} (A b-a B)}{3 a b (a+b x)^{3/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2 x^{3/2} (A b-a B)}{3 a b (a+b x)^{3/2}}+\frac {B \left (\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{b^{3/2}}-\frac {2 \sqrt {x}}{b \sqrt {a+b x}}\right )}{b}\) |
Input:
Int[(Sqrt[x]*(A + B*x))/(a + b*x)^(5/2),x]
Output:
(2*(A*b - a*B)*x^(3/2))/(3*a*b*(a + b*x)^(3/2)) + (B*((-2*Sqrt[x])/(b*Sqrt [a + b*x]) + (2*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/b^(3/2)))/b
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])
Leaf count of result is larger than twice the leaf count of optimal. \(181\) vs. \(2(64)=128\).
Time = 0.15 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.22
method | result | size |
default | \(\frac {\left (3 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a \,b^{2} x^{2}+2 A \sqrt {x \left (b x +a \right )}\, b^{\frac {5}{2}} x +6 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{2} b x -8 B \sqrt {x \left (b x +a \right )}\, b^{\frac {3}{2}} a x +3 B \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{3}-6 B \sqrt {x \left (b x +a \right )}\, \sqrt {b}\, a^{2}\right ) \sqrt {x}}{3 a \sqrt {x \left (b x +a \right )}\, b^{\frac {5}{2}} \left (b x +a \right )^{\frac {3}{2}}}\) | \(182\) |
Input:
int(x^(1/2)*(B*x+A)/(b*x+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/3*(3*B*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*a*b^2*x^2+2 *A*(x*(b*x+a))^(1/2)*b^(5/2)*x+6*B*ln(1/2*(2*(x*(b*x+a))^(1/2)*b^(1/2)+2*b *x+a)/b^(1/2))*a^2*b*x-8*B*(x*(b*x+a))^(1/2)*b^(3/2)*a*x+3*B*ln(1/2*(2*(x* (b*x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*a^3-6*B*(x*(b*x+a))^(1/2)*b^(1/2) *a^2)*x^(1/2)/a/(x*(b*x+a))^(1/2)/b^(5/2)/(b*x+a)^(3/2)
Time = 0.10 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.79 \[ \int \frac {\sqrt {x} (A+B x)}{(a+b x)^{5/2}} \, dx=\left [\frac {3 \, {\left (B a b^{2} x^{2} + 2 \, B a^{2} b x + B a^{3}\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (3 \, B a^{2} b + {\left (4 \, B a b^{2} - A b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{3 \, {\left (a b^{5} x^{2} + 2 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}, -\frac {2 \, {\left (3 \, {\left (B a b^{2} x^{2} + 2 \, B a^{2} b x + B a^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {b x + a}}\right ) + {\left (3 \, B a^{2} b + {\left (4 \, B a b^{2} - A b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}\right )}}{3 \, {\left (a b^{5} x^{2} + 2 \, a^{2} b^{4} x + a^{3} b^{3}\right )}}\right ] \] Input:
integrate(x^(1/2)*(B*x+A)/(b*x+a)^(5/2),x, algorithm="fricas")
Output:
[1/3*(3*(B*a*b^2*x^2 + 2*B*a^2*b*x + B*a^3)*sqrt(b)*log(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) - 2*(3*B*a^2*b + (4*B*a*b^2 - A*b^3)*x)*sqrt(b* x + a)*sqrt(x))/(a*b^5*x^2 + 2*a^2*b^4*x + a^3*b^3), -2/3*(3*(B*a*b^2*x^2 + 2*B*a^2*b*x + B*a^3)*sqrt(-b)*arctan(sqrt(-b)*sqrt(x)/sqrt(b*x + a)) + ( 3*B*a^2*b + (4*B*a*b^2 - A*b^3)*x)*sqrt(b*x + a)*sqrt(x))/(a*b^5*x^2 + 2*a ^2*b^4*x + a^3*b^3)]
Leaf count of result is larger than twice the leaf count of optimal. 376 vs. \(2 (78) = 156\).
Time = 24.54 (sec) , antiderivative size = 376, normalized size of antiderivative = 4.59 \[ \int \frac {\sqrt {x} (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {2 A x^{\frac {3}{2}}}{3 a^{\frac {5}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {3}{2}} b x \sqrt {1 + \frac {b x}{a}}} + B \left (\frac {6 a^{\frac {39}{2}} b^{11} x^{\frac {27}{2}} \sqrt {1 + \frac {b x}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {1 + \frac {b x}{a}}} + \frac {6 a^{\frac {37}{2}} b^{12} x^{\frac {29}{2}} \sqrt {1 + \frac {b x}{a}} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {1 + \frac {b x}{a}}} - \frac {6 a^{19} b^{\frac {23}{2}} x^{14}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {1 + \frac {b x}{a}}} - \frac {8 a^{18} b^{\frac {25}{2}} x^{15}}{3 a^{\frac {39}{2}} b^{\frac {27}{2}} x^{\frac {27}{2}} \sqrt {1 + \frac {b x}{a}} + 3 a^{\frac {37}{2}} b^{\frac {29}{2}} x^{\frac {29}{2}} \sqrt {1 + \frac {b x}{a}}}\right ) \] Input:
integrate(x**(1/2)*(B*x+A)/(b*x+a)**(5/2),x)
Output:
2*A*x**(3/2)/(3*a**(5/2)*sqrt(1 + b*x/a) + 3*a**(3/2)*b*x*sqrt(1 + b*x/a)) + B*(6*a**(39/2)*b**11*x**(27/2)*sqrt(1 + b*x/a)*asinh(sqrt(b)*sqrt(x)/sq rt(a))/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 + b*x/a) + 3*a**(37/2)*b**( 29/2)*x**(29/2)*sqrt(1 + b*x/a)) + 6*a**(37/2)*b**12*x**(29/2)*sqrt(1 + b* x/a)*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt( 1 + b*x/a) + 3*a**(37/2)*b**(29/2)*x**(29/2)*sqrt(1 + b*x/a)) - 6*a**19*b* *(23/2)*x**14/(3*a**(39/2)*b**(27/2)*x**(27/2)*sqrt(1 + b*x/a) + 3*a**(37/ 2)*b**(29/2)*x**(29/2)*sqrt(1 + b*x/a)) - 8*a**18*b**(25/2)*x**15/(3*a**(3 9/2)*b**(27/2)*x**(27/2)*sqrt(1 + b*x/a) + 3*a**(37/2)*b**(29/2)*x**(29/2) *sqrt(1 + b*x/a)))
Time = 0.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {x} (A+B x)}{(a+b x)^{5/2}} \, dx=-\frac {1}{3} \, B {\left (\frac {2 \, {\left (b + \frac {3 \, {\left (b x + a\right )}}{x}\right )} x^{\frac {3}{2}}}{{\left (b x + a\right )}^{\frac {3}{2}} b^{2}} + \frac {3 \, \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{b^{\frac {5}{2}}}\right )} + \frac {2 \, A x^{\frac {3}{2}}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a} \] Input:
integrate(x^(1/2)*(B*x+A)/(b*x+a)^(5/2),x, algorithm="maxima")
Output:
-1/3*B*(2*(b + 3*(b*x + a)/x)*x^(3/2)/((b*x + a)^(3/2)*b^2) + 3*log(-(sqrt (b) - sqrt(b*x + a)/sqrt(x))/(sqrt(b) + sqrt(b*x + a)/sqrt(x)))/b^(5/2)) + 2/3*A*x^(3/2)/((b*x + a)^(3/2)*a)
Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (64) = 128\).
Time = 15.33 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.62 \[ \int \frac {\sqrt {x} (A+B x)}{(a+b x)^{5/2}} \, dx=-\frac {B {\left | b \right |} \log \left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{b^{\frac {7}{2}}} - \frac {4 \, {\left (6 \, B a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} {\left | b \right |} + 6 \, B a^{2} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b {\left | b \right |} - 3 \, A {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b {\left | b \right |} + 4 \, B a^{3} b^{2} {\left | b \right |} - A a^{2} b^{3} {\left | b \right |}\right )}}{3 \, {\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} b^{\frac {5}{2}}} \] Input:
integrate(x^(1/2)*(B*x+A)/(b*x+a)^(5/2),x, algorithm="giac")
Output:
-B*abs(b)*log((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2)/b^(7/2) - 4/3*(6*B*a*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*abs(b) + 6*B*a^2*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b*abs(b) - 3* A*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*b*abs(b) + 4*B*a^3*b ^2*abs(b) - A*a^2*b^3*abs(b))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)^3*b^(5/2))
Timed out. \[ \int \frac {\sqrt {x} (A+B x)}{(a+b x)^{5/2}} \, dx=\int \frac {\sqrt {x}\,\left (A+B\,x\right )}{{\left (a+b\,x\right )}^{5/2}} \,d x \] Input:
int((x^(1/2)*(A + B*x))/(a + b*x)^(5/2),x)
Output:
int((x^(1/2)*(A + B*x))/(a + b*x)^(5/2), x)
Time = 0.16 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {x} (A+B x)}{(a+b x)^{5/2}} \, dx=\frac {2 \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right )-2 \sqrt {b}\, \sqrt {b x +a}-2 \sqrt {x}\, b}{\sqrt {b x +a}\, b^{2}} \] Input:
int(x^(1/2)*(B*x+A)/(b*x+a)^(5/2),x)
Output:
(2*(sqrt(b)*sqrt(a + b*x)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a)) - sqrt(b)*sqrt(a + b*x) - sqrt(x)*b))/(sqrt(a + b*x)*b**2)