Integrand size = 15, antiderivative size = 86 \[ \int \frac {x^{5/2}}{(-a+b x)^3} \, dx=\frac {2 \sqrt {x}}{b^3}-\frac {a^2 \sqrt {x}}{2 b^3 (a-b x)^2}+\frac {9 a \sqrt {x}}{4 b^3 (a-b x)}-\frac {15 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{7/2}} \] Output:
2*x^(1/2)/b^3-1/2*a^2*x^(1/2)/b^3/(-b*x+a)^2+9/4*a*x^(1/2)/b^3/(-b*x+a)-15 /4*a^(1/2)*arctanh(b^(1/2)*x^(1/2)/a^(1/2))/b^(7/2)
Time = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.83 \[ \int \frac {x^{5/2}}{(-a+b x)^3} \, dx=\frac {\sqrt {x} \left (15 a^2-25 a b x+8 b^2 x^2\right )}{4 b^3 (a-b x)^2}-\frac {15 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{7/2}} \] Input:
Integrate[x^(5/2)/(-a + b*x)^3,x]
Output:
(Sqrt[x]*(15*a^2 - 25*a*b*x + 8*b^2*x^2))/(4*b^3*(a - b*x)^2) - (15*Sqrt[a ]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*b^(7/2))
Time = 0.17 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {51, 51, 60, 73, 221}
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.
\(\displaystyle \int \frac {x^{5/2}}{(b x-a)^3} \, dx\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {5 \int \frac {x^{3/2}}{(a-b x)^2}dx}{4 b}-\frac {x^{5/2}}{2 b (a-b x)^2}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {5 \left (\frac {x^{3/2}}{b (a-b x)}-\frac {3 \int \frac {\sqrt {x}}{a-b x}dx}{2 b}\right )}{4 b}-\frac {x^{5/2}}{2 b (a-b x)^2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {5 \left (\frac {x^{3/2}}{b (a-b x)}-\frac {3 \left (\frac {a \int \frac {1}{\sqrt {x} (a-b x)}dx}{b}-\frac {2 \sqrt {x}}{b}\right )}{2 b}\right )}{4 b}-\frac {x^{5/2}}{2 b (a-b x)^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {5 \left (\frac {x^{3/2}}{b (a-b x)}-\frac {3 \left (\frac {2 a \int \frac {1}{a-b x}d\sqrt {x}}{b}-\frac {2 \sqrt {x}}{b}\right )}{2 b}\right )}{4 b}-\frac {x^{5/2}}{2 b (a-b x)^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {5 \left (\frac {x^{3/2}}{b (a-b x)}-\frac {3 \left (\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{3/2}}-\frac {2 \sqrt {x}}{b}\right )}{2 b}\right )}{4 b}-\frac {x^{5/2}}{2 b (a-b x)^2}\) |
Input:
Int[x^(5/2)/(-a + b*x)^3,x]
Output:
-1/2*x^(5/2)/(b*(a - b*x)^2) + (5*(x^(3/2)/(b*(a - b*x)) - (3*((-2*Sqrt[x] )/b + (2*Sqrt[a]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^(3/2)))/(2*b)))/(4* 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, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Time = 0.16 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.66
method | result | size |
derivativedivides | \(\frac {2 \sqrt {x}}{b^{3}}-\frac {2 a \left (\frac {\frac {9 b \,x^{\frac {3}{2}}}{8}-\frac {7 a \sqrt {x}}{8}}{\left (-b x +a \right )^{2}}+\frac {15 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{3}}\) | \(57\) |
default | \(\frac {2 \sqrt {x}}{b^{3}}-\frac {2 a \left (\frac {\frac {9 b \,x^{\frac {3}{2}}}{8}-\frac {7 a \sqrt {x}}{8}}{\left (-b x +a \right )^{2}}+\frac {15 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{3}}\) | \(57\) |
risch | \(\frac {2 \sqrt {x}}{b^{3}}+\frac {a \left (\frac {-\frac {9 b \,x^{\frac {3}{2}}}{4}+\frac {7 a \sqrt {x}}{4}}{\left (b x -a \right )^{2}}-\frac {15 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}}\right )}{b^{3}}\) | \(58\) |
Input:
int(x^(5/2)/(b*x-a)^3,x,method=_RETURNVERBOSE)
Output:
2*x^(1/2)/b^3-2/b^3*a*((9/8*b*x^(3/2)-7/8*a*x^(1/2))/(-b*x+a)^2+15/8/(a*b) ^(1/2)*arctanh(b*x^(1/2)/(a*b)^(1/2)))
Time = 0.09 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.31 \[ \int \frac {x^{5/2}}{(-a+b x)^3} \, dx=\left [\frac {15 \, {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt {\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {\frac {a}{b}} + a}{b x - a}\right ) + 2 \, {\left (8 \, b^{2} x^{2} - 25 \, a b x + 15 \, a^{2}\right )} \sqrt {x}}{8 \, {\left (b^{5} x^{2} - 2 \, a b^{4} x + a^{2} b^{3}\right )}}, \frac {15 \, {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt {-\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {-\frac {a}{b}}}{a}\right ) + {\left (8 \, b^{2} x^{2} - 25 \, a b x + 15 \, a^{2}\right )} \sqrt {x}}{4 \, {\left (b^{5} x^{2} - 2 \, a b^{4} x + a^{2} b^{3}\right )}}\right ] \] Input:
integrate(x^(5/2)/(b*x-a)^3,x, algorithm="fricas")
Output:
[1/8*(15*(b^2*x^2 - 2*a*b*x + a^2)*sqrt(a/b)*log((b*x - 2*b*sqrt(x)*sqrt(a /b) + a)/(b*x - a)) + 2*(8*b^2*x^2 - 25*a*b*x + 15*a^2)*sqrt(x))/(b^5*x^2 - 2*a*b^4*x + a^2*b^3), 1/4*(15*(b^2*x^2 - 2*a*b*x + a^2)*sqrt(-a/b)*arcta n(b*sqrt(x)*sqrt(-a/b)/a) + (8*b^2*x^2 - 25*a*b*x + 15*a^2)*sqrt(x))/(b^5* x^2 - 2*a*b^4*x + a^2*b^3)]
Leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (78) = 156\).
Time = 16.35 (sec) , antiderivative size = 624, normalized size of antiderivative = 7.26 \[ \int \frac {x^{5/2}}{(-a+b x)^3} \, dx=\begin {cases} \tilde {\infty } \sqrt {x} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2 x^{\frac {7}{2}}}{7 a^{3}} & \text {for}\: b = 0 \\\frac {2 \sqrt {x}}{b^{3}} & \text {for}\: a = 0 \\\frac {15 a^{3} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {\frac {a}{b}} - 16 a b^{5} x \sqrt {\frac {a}{b}} + 8 b^{6} x^{2} \sqrt {\frac {a}{b}}} - \frac {15 a^{3} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {\frac {a}{b}} - 16 a b^{5} x \sqrt {\frac {a}{b}} + 8 b^{6} x^{2} \sqrt {\frac {a}{b}}} + \frac {30 a^{2} b \sqrt {x} \sqrt {\frac {a}{b}}}{8 a^{2} b^{4} \sqrt {\frac {a}{b}} - 16 a b^{5} x \sqrt {\frac {a}{b}} + 8 b^{6} x^{2} \sqrt {\frac {a}{b}}} - \frac {30 a^{2} b x \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {\frac {a}{b}} - 16 a b^{5} x \sqrt {\frac {a}{b}} + 8 b^{6} x^{2} \sqrt {\frac {a}{b}}} + \frac {30 a^{2} b x \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {\frac {a}{b}} - 16 a b^{5} x \sqrt {\frac {a}{b}} + 8 b^{6} x^{2} \sqrt {\frac {a}{b}}} - \frac {50 a b^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b}}}{8 a^{2} b^{4} \sqrt {\frac {a}{b}} - 16 a b^{5} x \sqrt {\frac {a}{b}} + 8 b^{6} x^{2} \sqrt {\frac {a}{b}}} + \frac {15 a b^{2} x^{2} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {\frac {a}{b}} - 16 a b^{5} x \sqrt {\frac {a}{b}} + 8 b^{6} x^{2} \sqrt {\frac {a}{b}}} - \frac {15 a b^{2} x^{2} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {\frac {a}{b}} - 16 a b^{5} x \sqrt {\frac {a}{b}} + 8 b^{6} x^{2} \sqrt {\frac {a}{b}}} + \frac {16 b^{3} x^{\frac {5}{2}} \sqrt {\frac {a}{b}}}{8 a^{2} b^{4} \sqrt {\frac {a}{b}} - 16 a b^{5} x \sqrt {\frac {a}{b}} + 8 b^{6} x^{2} \sqrt {\frac {a}{b}}} & \text {otherwise} \end {cases} \] Input:
integrate(x**(5/2)/(b*x-a)**3,x)
Output:
Piecewise((zoo*sqrt(x), Eq(a, 0) & Eq(b, 0)), (-2*x**(7/2)/(7*a**3), Eq(b, 0)), (2*sqrt(x)/b**3, Eq(a, 0)), (15*a**3*log(sqrt(x) - sqrt(a/b))/(8*a** 2*b**4*sqrt(a/b) - 16*a*b**5*x*sqrt(a/b) + 8*b**6*x**2*sqrt(a/b)) - 15*a** 3*log(sqrt(x) + sqrt(a/b))/(8*a**2*b**4*sqrt(a/b) - 16*a*b**5*x*sqrt(a/b) + 8*b**6*x**2*sqrt(a/b)) + 30*a**2*b*sqrt(x)*sqrt(a/b)/(8*a**2*b**4*sqrt(a /b) - 16*a*b**5*x*sqrt(a/b) + 8*b**6*x**2*sqrt(a/b)) - 30*a**2*b*x*log(sqr t(x) - sqrt(a/b))/(8*a**2*b**4*sqrt(a/b) - 16*a*b**5*x*sqrt(a/b) + 8*b**6* x**2*sqrt(a/b)) + 30*a**2*b*x*log(sqrt(x) + sqrt(a/b))/(8*a**2*b**4*sqrt(a /b) - 16*a*b**5*x*sqrt(a/b) + 8*b**6*x**2*sqrt(a/b)) - 50*a*b**2*x**(3/2)* sqrt(a/b)/(8*a**2*b**4*sqrt(a/b) - 16*a*b**5*x*sqrt(a/b) + 8*b**6*x**2*sqr t(a/b)) + 15*a*b**2*x**2*log(sqrt(x) - sqrt(a/b))/(8*a**2*b**4*sqrt(a/b) - 16*a*b**5*x*sqrt(a/b) + 8*b**6*x**2*sqrt(a/b)) - 15*a*b**2*x**2*log(sqrt( x) + sqrt(a/b))/(8*a**2*b**4*sqrt(a/b) - 16*a*b**5*x*sqrt(a/b) + 8*b**6*x* *2*sqrt(a/b)) + 16*b**3*x**(5/2)*sqrt(a/b)/(8*a**2*b**4*sqrt(a/b) - 16*a*b **5*x*sqrt(a/b) + 8*b**6*x**2*sqrt(a/b)), True))
Time = 0.13 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.05 \[ \int \frac {x^{5/2}}{(-a+b x)^3} \, dx=-\frac {9 \, a b x^{\frac {3}{2}} - 7 \, a^{2} \sqrt {x}}{4 \, {\left (b^{5} x^{2} - 2 \, a b^{4} x + a^{2} b^{3}\right )}} + \frac {15 \, a \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{3}} + \frac {2 \, \sqrt {x}}{b^{3}} \] Input:
integrate(x^(5/2)/(b*x-a)^3,x, algorithm="maxima")
Output:
-1/4*(9*a*b*x^(3/2) - 7*a^2*sqrt(x))/(b^5*x^2 - 2*a*b^4*x + a^2*b^3) + 15/ 8*a*log((b*sqrt(x) - sqrt(a*b))/(b*sqrt(x) + sqrt(a*b)))/(sqrt(a*b)*b^3) + 2*sqrt(x)/b^3
Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.73 \[ \int \frac {x^{5/2}}{(-a+b x)^3} \, dx=\frac {15 \, a \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{4 \, \sqrt {-a b} b^{3}} + \frac {2 \, \sqrt {x}}{b^{3}} - \frac {9 \, a b x^{\frac {3}{2}} - 7 \, a^{2} \sqrt {x}}{4 \, {\left (b x - a\right )}^{2} b^{3}} \] Input:
integrate(x^(5/2)/(b*x-a)^3,x, algorithm="giac")
Output:
15/4*a*arctan(b*sqrt(x)/sqrt(-a*b))/(sqrt(-a*b)*b^3) + 2*sqrt(x)/b^3 - 1/4 *(9*a*b*x^(3/2) - 7*a^2*sqrt(x))/((b*x - a)^2*b^3)
Time = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.80 \[ \int \frac {x^{5/2}}{(-a+b x)^3} \, dx=\frac {\frac {7\,a^2\,\sqrt {x}}{4}-\frac {9\,a\,b\,x^{3/2}}{4}}{a^2\,b^3-2\,a\,b^4\,x+b^5\,x^2}+\frac {2\,\sqrt {x}}{b^3}-\frac {15\,\sqrt {a}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,b^{7/2}} \] Input:
int(-x^(5/2)/(a - b*x)^3,x)
Output:
((7*a^2*x^(1/2))/4 - (9*a*b*x^(3/2))/4)/(a^2*b^3 + b^5*x^2 - 2*a*b^4*x) + (2*x^(1/2))/b^3 - (15*a^(1/2)*atanh((b^(1/2)*x^(1/2))/a^(1/2)))/(4*b^(7/2) )
Time = 0.16 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.09 \[ \int \frac {x^{5/2}}{(-a+b x)^3} \, dx=\frac {15 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) a^{2}-30 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) a b x +15 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) b^{2} x^{2}-15 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) a^{2}+30 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) a b x -15 \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) b^{2} x^{2}+30 \sqrt {x}\, a^{2} b -50 \sqrt {x}\, a \,b^{2} x +16 \sqrt {x}\, b^{3} x^{2}}{8 b^{4} \left (b^{2} x^{2}-2 a b x +a^{2}\right )} \] Input:
int(x^(5/2)/(b*x-a)^3,x)
Output:
(15*sqrt(b)*sqrt(a)*log( - sqrt(b)*sqrt(a) + sqrt(x)*b)*a**2 - 30*sqrt(b)* sqrt(a)*log( - sqrt(b)*sqrt(a) + sqrt(x)*b)*a*b*x + 15*sqrt(b)*sqrt(a)*log ( - sqrt(b)*sqrt(a) + sqrt(x)*b)*b**2*x**2 - 15*sqrt(b)*sqrt(a)*log(sqrt(b )*sqrt(a) + sqrt(x)*b)*a**2 + 30*sqrt(b)*sqrt(a)*log(sqrt(b)*sqrt(a) + sqr t(x)*b)*a*b*x - 15*sqrt(b)*sqrt(a)*log(sqrt(b)*sqrt(a) + sqrt(x)*b)*b**2*x **2 + 30*sqrt(x)*a**2*b - 50*sqrt(x)*a*b**2*x + 16*sqrt(x)*b**3*x**2)/(8*b **4*(a**2 - 2*a*b*x + b**2*x**2))