Integrand size = 15, antiderivative size = 97 \[ \int \frac {1}{x^{5/2} (-a+b x)^3} \, dx=\frac {35}{12 a^3 x^{3/2}}+\frac {35 b}{4 a^4 \sqrt {x}}-\frac {1}{2 a x^{3/2} (a-b x)^2}-\frac {7}{4 a^2 x^{3/2} (a-b x)}-\frac {35 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2}} \] Output:
35/12/a^3/x^(3/2)+35/4*b/a^4/x^(1/2)-1/2/a/x^(3/2)/(-b*x+a)^2-7/4/a^2/x^(3 /2)/(-b*x+a)-35/4*b^(3/2)*arctanh(b^(1/2)*x^(1/2)/a^(1/2))/a^(9/2)
Time = 0.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^{5/2} (-a+b x)^3} \, dx=\frac {8 a^3+56 a^2 b x-175 a b^2 x^2+105 b^3 x^3}{12 a^4 x^{3/2} (a-b x)^2}-\frac {35 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 a^{9/2}} \] Input:
Integrate[1/(x^(5/2)*(-a + b*x)^3),x]
Output:
(8*a^3 + 56*a^2*b*x - 175*a*b^2*x^2 + 105*b^3*x^3)/(12*a^4*x^(3/2)*(a - b* x)^2) - (35*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(4*a^(9/2))
Time = 0.18 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {52, 52, 61, 61, 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.
\(\displaystyle \int \frac {1}{x^{5/2} (b x-a)^3} \, dx\) |
\(\Big \downarrow \) 52 |
\(\displaystyle -\frac {7 \int \frac {1}{x^{5/2} (a-b x)^2}dx}{4 a}-\frac {1}{2 a x^{3/2} (a-b x)^2}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle -\frac {7 \left (\frac {5 \int \frac {1}{x^{5/2} (a-b x)}dx}{2 a}+\frac {1}{a x^{3/2} (a-b x)}\right )}{4 a}-\frac {1}{2 a x^{3/2} (a-b x)^2}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {7 \left (\frac {5 \left (\frac {b \int \frac {1}{x^{3/2} (a-b x)}dx}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a-b x)}\right )}{4 a}-\frac {1}{2 a x^{3/2} (a-b x)^2}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {7 \left (\frac {5 \left (\frac {b \left (\frac {b \int \frac {1}{\sqrt {x} (a-b x)}dx}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a-b x)}\right )}{4 a}-\frac {1}{2 a x^{3/2} (a-b x)^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {7 \left (\frac {5 \left (\frac {b \left (\frac {2 b \int \frac {1}{a-b x}d\sqrt {x}}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a-b x)}\right )}{4 a}-\frac {1}{2 a x^{3/2} (a-b x)^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {7 \left (\frac {5 \left (\frac {b \left (\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a}+\frac {1}{a x^{3/2} (a-b x)}\right )}{4 a}-\frac {1}{2 a x^{3/2} (a-b x)^2}\) |
Input:
Int[1/(x^(5/2)*(-a + b*x)^3),x]
Output:
-1/2*1/(a*x^(3/2)*(a - b*x)^2) - (7*(1/(a*x^(3/2)*(a - b*x)) + (5*(-2/(3*a *x^(3/2)) + (b*(-2/(a*Sqrt[x]) + (2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt [a]])/a^(3/2)))/a))/(2*a)))/(4*a)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[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 = 0.15 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.68
method | result | size |
risch | \(\frac {6 b x +\frac {2 a}{3}}{a^{4} x^{\frac {3}{2}}}+\frac {b^{2} \left (\frac {\frac {11 b \,x^{\frac {3}{2}}}{4}-\frac {13 a \sqrt {x}}{4}}{\left (b x -a \right )^{2}}-\frac {35 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}}\right )}{a^{4}}\) | \(66\) |
derivativedivides | \(\frac {2}{3 a^{3} x^{\frac {3}{2}}}+\frac {6 b}{a^{4} \sqrt {x}}-\frac {2 b^{2} \left (\frac {-\frac {11 b \,x^{\frac {3}{2}}}{8}+\frac {13 a \sqrt {x}}{8}}{\left (-b x +a \right )^{2}}+\frac {35 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{4}}\) | \(68\) |
default | \(\frac {2}{3 a^{3} x^{\frac {3}{2}}}+\frac {6 b}{a^{4} \sqrt {x}}-\frac {2 b^{2} \left (\frac {-\frac {11 b \,x^{\frac {3}{2}}}{8}+\frac {13 a \sqrt {x}}{8}}{\left (-b x +a \right )^{2}}+\frac {35 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{a^{4}}\) | \(68\) |
Input:
int(1/x^(5/2)/(b*x-a)^3,x,method=_RETURNVERBOSE)
Output:
2/3*(9*b*x+a)/a^4/x^(3/2)+b^2/a^4*(2*(11/8*b*x^(3/2)-13/8*a*x^(1/2))/(b*x- a)^2-35/4/(a*b)^(1/2)*arctanh(b*x^(1/2)/(a*b)^(1/2)))
Time = 0.12 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.53 \[ \int \frac {1}{x^{5/2} (-a+b x)^3} \, dx=\left [\frac {105 \, {\left (b^{3} x^{4} - 2 \, a b^{2} x^{3} + a^{2} b x^{2}\right )} \sqrt {\frac {b}{a}} \log \left (\frac {b x - 2 \, a \sqrt {x} \sqrt {\frac {b}{a}} + a}{b x - a}\right ) + 2 \, {\left (105 \, b^{3} x^{3} - 175 \, a b^{2} x^{2} + 56 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt {x}}{24 \, {\left (a^{4} b^{2} x^{4} - 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}, \frac {105 \, {\left (b^{3} x^{4} - 2 \, a b^{2} x^{3} + a^{2} b x^{2}\right )} \sqrt {-\frac {b}{a}} \arctan \left (\sqrt {x} \sqrt {-\frac {b}{a}}\right ) + {\left (105 \, b^{3} x^{3} - 175 \, a b^{2} x^{2} + 56 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt {x}}{12 \, {\left (a^{4} b^{2} x^{4} - 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}\right ] \] Input:
integrate(1/x^(5/2)/(b*x-a)^3,x, algorithm="fricas")
Output:
[1/24*(105*(b^3*x^4 - 2*a*b^2*x^3 + a^2*b*x^2)*sqrt(b/a)*log((b*x - 2*a*sq rt(x)*sqrt(b/a) + a)/(b*x - a)) + 2*(105*b^3*x^3 - 175*a*b^2*x^2 + 56*a^2* b*x + 8*a^3)*sqrt(x))/(a^4*b^2*x^4 - 2*a^5*b*x^3 + a^6*x^2), 1/12*(105*(b^ 3*x^4 - 2*a*b^2*x^3 + a^2*b*x^2)*sqrt(-b/a)*arctan(sqrt(x)*sqrt(-b/a)) + ( 105*b^3*x^3 - 175*a*b^2*x^2 + 56*a^2*b*x + 8*a^3)*sqrt(x))/(a^4*b^2*x^4 - 2*a^5*b*x^3 + a^6*x^2)]
Leaf count of result is larger than twice the leaf count of optimal. 799 vs. \(2 (88) = 176\).
Time = 39.19 (sec) , antiderivative size = 799, normalized size of antiderivative = 8.24 \[ \int \frac {1}{x^{5/2} (-a+b x)^3} \, dx =\text {Too large to display} \] Input:
integrate(1/x**(5/2)/(b*x-a)**3,x)
Output:
Piecewise((zoo/x**(9/2), Eq(a, 0) & Eq(b, 0)), (2/(3*a**3*x**(3/2)), Eq(b, 0)), (-2/(9*b**3*x**(9/2)), Eq(a, 0)), (16*a**3*sqrt(a/b)/(24*a**6*x**(3/ 2)*sqrt(a/b) - 48*a**5*b*x**(5/2)*sqrt(a/b) + 24*a**4*b**2*x**(7/2)*sqrt(a /b)) + 105*a**2*b*x**(3/2)*log(sqrt(x) - sqrt(a/b))/(24*a**6*x**(3/2)*sqrt (a/b) - 48*a**5*b*x**(5/2)*sqrt(a/b) + 24*a**4*b**2*x**(7/2)*sqrt(a/b)) - 105*a**2*b*x**(3/2)*log(sqrt(x) + sqrt(a/b))/(24*a**6*x**(3/2)*sqrt(a/b) - 48*a**5*b*x**(5/2)*sqrt(a/b) + 24*a**4*b**2*x**(7/2)*sqrt(a/b)) + 112*a** 2*b*x*sqrt(a/b)/(24*a**6*x**(3/2)*sqrt(a/b) - 48*a**5*b*x**(5/2)*sqrt(a/b) + 24*a**4*b**2*x**(7/2)*sqrt(a/b)) - 210*a*b**2*x**(5/2)*log(sqrt(x) - sq rt(a/b))/(24*a**6*x**(3/2)*sqrt(a/b) - 48*a**5*b*x**(5/2)*sqrt(a/b) + 24*a **4*b**2*x**(7/2)*sqrt(a/b)) + 210*a*b**2*x**(5/2)*log(sqrt(x) + sqrt(a/b) )/(24*a**6*x**(3/2)*sqrt(a/b) - 48*a**5*b*x**(5/2)*sqrt(a/b) + 24*a**4*b** 2*x**(7/2)*sqrt(a/b)) - 350*a*b**2*x**2*sqrt(a/b)/(24*a**6*x**(3/2)*sqrt(a /b) - 48*a**5*b*x**(5/2)*sqrt(a/b) + 24*a**4*b**2*x**(7/2)*sqrt(a/b)) + 10 5*b**3*x**(7/2)*log(sqrt(x) - sqrt(a/b))/(24*a**6*x**(3/2)*sqrt(a/b) - 48* a**5*b*x**(5/2)*sqrt(a/b) + 24*a**4*b**2*x**(7/2)*sqrt(a/b)) - 105*b**3*x* *(7/2)*log(sqrt(x) + sqrt(a/b))/(24*a**6*x**(3/2)*sqrt(a/b) - 48*a**5*b*x* *(5/2)*sqrt(a/b) + 24*a**4*b**2*x**(7/2)*sqrt(a/b)) + 210*b**3*x**3*sqrt(a /b)/(24*a**6*x**(3/2)*sqrt(a/b) - 48*a**5*b*x**(5/2)*sqrt(a/b) + 24*a**4*b **2*x**(7/2)*sqrt(a/b)), True))
Time = 0.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^{5/2} (-a+b x)^3} \, dx=\frac {105 \, b^{3} x^{3} - 175 \, a b^{2} x^{2} + 56 \, a^{2} b x + 8 \, a^{3}}{12 \, {\left (a^{4} b^{2} x^{\frac {7}{2}} - 2 \, a^{5} b x^{\frac {5}{2}} + a^{6} x^{\frac {3}{2}}\right )}} + \frac {35 \, b^{2} \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{4}} \] Input:
integrate(1/x^(5/2)/(b*x-a)^3,x, algorithm="maxima")
Output:
1/12*(105*b^3*x^3 - 175*a*b^2*x^2 + 56*a^2*b*x + 8*a^3)/(a^4*b^2*x^(7/2) - 2*a^5*b*x^(5/2) + a^6*x^(3/2)) + 35/8*b^2*log((b*sqrt(x) - sqrt(a*b))/(b* sqrt(x) + sqrt(a*b)))/(sqrt(a*b)*a^4)
Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^{5/2} (-a+b x)^3} \, dx=\frac {35 \, b^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{4 \, \sqrt {-a b} a^{4}} + \frac {2 \, {\left (9 \, b x + a\right )}}{3 \, a^{4} x^{\frac {3}{2}}} + \frac {11 \, b^{3} x^{\frac {3}{2}} - 13 \, a b^{2} \sqrt {x}}{4 \, {\left (b x - a\right )}^{2} a^{4}} \] Input:
integrate(1/x^(5/2)/(b*x-a)^3,x, algorithm="giac")
Output:
35/4*b^2*arctan(b*sqrt(x)/sqrt(-a*b))/(sqrt(-a*b)*a^4) + 2/3*(9*b*x + a)/( a^4*x^(3/2)) + 1/4*(11*b^3*x^(3/2) - 13*a*b^2*sqrt(x))/((b*x - a)^2*a^4)
Time = 0.10 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^{5/2} (-a+b x)^3} \, dx=\frac {\frac {2}{3\,a}-\frac {175\,b^2\,x^2}{12\,a^3}+\frac {35\,b^3\,x^3}{4\,a^4}+\frac {14\,b\,x}{3\,a^2}}{a^2\,x^{3/2}+b^2\,x^{7/2}-2\,a\,b\,x^{5/2}}-\frac {35\,b^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,a^{9/2}} \] Input:
int(-1/(x^(5/2)*(a - b*x)^3),x)
Output:
(2/(3*a) - (175*b^2*x^2)/(12*a^3) + (35*b^3*x^3)/(4*a^4) + (14*b*x)/(3*a^2 ))/(a^2*x^(3/2) + b^2*x^(7/2) - 2*a*b*x^(5/2)) - (35*b^(3/2)*atanh((b^(1/2 )*x^(1/2))/a^(1/2)))/(4*a^(9/2))
Time = 0.15 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.23 \[ \int \frac {1}{x^{5/2} (-a+b x)^3} \, dx=\frac {105 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) a^{2} b x -210 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) a \,b^{2} x^{2}+105 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) b^{3} x^{3}-105 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) a^{2} b x +210 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) a \,b^{2} x^{2}-105 \sqrt {x}\, \sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}+\sqrt {x}\, b \right ) b^{3} x^{3}+16 a^{4}+112 a^{3} b x -350 a^{2} b^{2} x^{2}+210 a \,b^{3} x^{3}}{24 \sqrt {x}\, a^{5} x \left (b^{2} x^{2}-2 a b x +a^{2}\right )} \] Input:
int(1/x^(5/2)/(b*x-a)^3,x)
Output:
(105*sqrt(x)*sqrt(b)*sqrt(a)*log( - sqrt(b)*sqrt(a) + sqrt(x)*b)*a**2*b*x - 210*sqrt(x)*sqrt(b)*sqrt(a)*log( - sqrt(b)*sqrt(a) + sqrt(x)*b)*a*b**2*x **2 + 105*sqrt(x)*sqrt(b)*sqrt(a)*log( - sqrt(b)*sqrt(a) + sqrt(x)*b)*b**3 *x**3 - 105*sqrt(x)*sqrt(b)*sqrt(a)*log(sqrt(b)*sqrt(a) + sqrt(x)*b)*a**2* b*x + 210*sqrt(x)*sqrt(b)*sqrt(a)*log(sqrt(b)*sqrt(a) + sqrt(x)*b)*a*b**2* x**2 - 105*sqrt(x)*sqrt(b)*sqrt(a)*log(sqrt(b)*sqrt(a) + sqrt(x)*b)*b**3*x **3 + 16*a**4 + 112*a**3*b*x - 350*a**2*b**2*x**2 + 210*a*b**3*x**3)/(24*s qrt(x)*a**5*x*(a**2 - 2*a*b*x + b**2*x**2))