Integrand size = 15, antiderivative size = 116 \[ \int \frac {1}{x^3 (-a+b x)^{5/2}} \, dx=-\frac {35 b^2}{12 a^3 (-a+b x)^{3/2}}+\frac {1}{2 a x^2 (-a+b x)^{3/2}}+\frac {7 b}{4 a^2 x (-a+b x)^{3/2}}+\frac {35 b^2}{4 a^4 \sqrt {-a+b x}}+\frac {35 b^2 \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{4 a^{9/2}} \]
-35/12*b^2/a^3/(b*x-a)^(3/2)+1/2/a/x^2/(b*x-a)^(3/2)+7/4*b/a^2/x/(b*x-a)^( 3/2)+35/4*b^2*arctan((b*x-a)^(1/2)/a^(1/2))/a^(9/2)+35/4*b^2/a^4/(b*x-a)^( 1/2)
Time = 0.12 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x^3 (-a+b x)^{5/2}} \, dx=\frac {6 a^3+21 a^2 b x-140 a b^2 x^2+105 b^3 x^3}{12 a^4 x^2 (-a+b x)^{3/2}}+\frac {35 b^2 \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{4 a^{9/2}} \]
(6*a^3 + 21*a^2*b*x - 140*a*b^2*x^2 + 105*b^3*x^3)/(12*a^4*x^2*(-a + b*x)^ (3/2)) + (35*b^2*ArcTan[Sqrt[-a + b*x]/Sqrt[a]])/(4*a^(9/2))
Time = 0.19 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.06, 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, 218}
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.
\(\displaystyle \int \frac {1}{x^3 (b x-a)^{5/2}} \, dx\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {7 b \int \frac {1}{x^2 (b x-a)^{5/2}}dx}{4 a}+\frac {1}{2 a x^2 (b x-a)^{3/2}}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {7 b \left (\frac {5 b \int \frac {1}{x (b x-a)^{5/2}}dx}{2 a}+\frac {1}{a x (b x-a)^{3/2}}\right )}{4 a}+\frac {1}{2 a x^2 (b x-a)^{3/2}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {7 b \left (\frac {5 b \left (-\frac {\int \frac {1}{x (b x-a)^{3/2}}dx}{a}-\frac {2}{3 a (b x-a)^{3/2}}\right )}{2 a}+\frac {1}{a x (b x-a)^{3/2}}\right )}{4 a}+\frac {1}{2 a x^2 (b x-a)^{3/2}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {7 b \left (\frac {5 b \left (-\frac {-\frac {\int \frac {1}{x \sqrt {b x-a}}dx}{a}-\frac {2}{a \sqrt {b x-a}}}{a}-\frac {2}{3 a (b x-a)^{3/2}}\right )}{2 a}+\frac {1}{a x (b x-a)^{3/2}}\right )}{4 a}+\frac {1}{2 a x^2 (b x-a)^{3/2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {7 b \left (\frac {5 b \left (-\frac {-\frac {2 \int \frac {1}{\frac {a}{b}+\frac {b x-a}{b}}d\sqrt {b x-a}}{a b}-\frac {2}{a \sqrt {b x-a}}}{a}-\frac {2}{3 a (b x-a)^{3/2}}\right )}{2 a}+\frac {1}{a x (b x-a)^{3/2}}\right )}{4 a}+\frac {1}{2 a x^2 (b x-a)^{3/2}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {7 b \left (\frac {5 b \left (-\frac {-\frac {2 \arctan \left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {b x-a}}}{a}-\frac {2}{3 a (b x-a)^{3/2}}\right )}{2 a}+\frac {1}{a x (b x-a)^{3/2}}\right )}{4 a}+\frac {1}{2 a x^2 (b x-a)^{3/2}}\) |
1/(2*a*x^2*(-a + b*x)^(3/2)) + (7*b*(1/(a*x*(-a + b*x)^(3/2)) + (5*b*(-2/( 3*a*(-a + b*x)^(3/2)) - (-2/(a*Sqrt[-a + b*x]) - (2*ArcTan[Sqrt[-a + b*x]/ Sqrt[a]])/a^(3/2))/a))/(2*a)))/(4*a)
3.4.67.3.1 Defintions of rubi rules used
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)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Time = 0.14 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.77
method | result | size |
risch | \(-\frac {\left (-b x +a \right ) \left (11 b x +2 a \right )}{4 a^{4} x^{2} \sqrt {b x -a}}+\frac {35 b^{2} \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{4 a^{\frac {9}{2}}}+\frac {6 b^{2}}{a^{4} \sqrt {b x -a}}-\frac {2 b^{2}}{3 a^{3} \left (b x -a \right )^{\frac {3}{2}}}\) | \(89\) |
pseudoelliptic | \(-\frac {35 \left (\sqrt {b x -a}\, b^{2} x^{2} \left (-b x +a \right ) \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )-\sqrt {a}\, b^{3} x^{3}+\frac {4 a^{\frac {3}{2}} b^{2} x^{2}}{3}-\frac {a^{\frac {5}{2}} b x}{5}-\frac {2 a^{\frac {7}{2}}}{35}\right )}{4 \left (b x -a \right )^{\frac {3}{2}} a^{\frac {9}{2}} x^{2}}\) | \(89\) |
derivativedivides | \(2 b^{2} \left (-\frac {1}{3 a^{3} \left (b x -a \right )^{\frac {3}{2}}}+\frac {3}{a^{4} \sqrt {b x -a}}+\frac {\frac {\frac {11 \left (b x -a \right )^{\frac {3}{2}}}{8}+\frac {13 a \sqrt {b x -a}}{8}}{b^{2} x^{2}}+\frac {35 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a^{4}}\right )\) | \(90\) |
default | \(2 b^{2} \left (-\frac {1}{3 a^{3} \left (b x -a \right )^{\frac {3}{2}}}+\frac {3}{a^{4} \sqrt {b x -a}}+\frac {\frac {\frac {11 \left (b x -a \right )^{\frac {3}{2}}}{8}+\frac {13 a \sqrt {b x -a}}{8}}{b^{2} x^{2}}+\frac {35 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a^{4}}\right )\) | \(90\) |
-1/4*(-b*x+a)*(11*b*x+2*a)/a^4/x^2/(b*x-a)^(1/2)+35/4*b^2*arctan((b*x-a)^( 1/2)/a^(1/2))/a^(9/2)+6*b^2/a^4/(b*x-a)^(1/2)-2/3*b^2/a^3/(b*x-a)^(3/2)
Time = 0.24 (sec) , antiderivative size = 260, normalized size of antiderivative = 2.24 \[ \int \frac {1}{x^3 (-a+b x)^{5/2}} \, dx=\left [-\frac {105 \, {\left (b^{4} x^{4} - 2 \, a b^{3} x^{3} + a^{2} b^{2} x^{2}\right )} \sqrt {-a} \log \left (\frac {b x - 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) - 2 \, {\left (105 \, a b^{3} x^{3} - 140 \, a^{2} b^{2} x^{2} + 21 \, a^{3} b x + 6 \, a^{4}\right )} \sqrt {b x - a}}{24 \, {\left (a^{5} b^{2} x^{4} - 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}}, \frac {105 \, {\left (b^{4} x^{4} - 2 \, a b^{3} x^{3} + a^{2} b^{2} x^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + {\left (105 \, a b^{3} x^{3} - 140 \, a^{2} b^{2} x^{2} + 21 \, a^{3} b x + 6 \, a^{4}\right )} \sqrt {b x - a}}{12 \, {\left (a^{5} b^{2} x^{4} - 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}}\right ] \]
[-1/24*(105*(b^4*x^4 - 2*a*b^3*x^3 + a^2*b^2*x^2)*sqrt(-a)*log((b*x - 2*sq rt(b*x - a)*sqrt(-a) - 2*a)/x) - 2*(105*a*b^3*x^3 - 140*a^2*b^2*x^2 + 21*a ^3*b*x + 6*a^4)*sqrt(b*x - a))/(a^5*b^2*x^4 - 2*a^6*b*x^3 + a^7*x^2), 1/12 *(105*(b^4*x^4 - 2*a*b^3*x^3 + a^2*b^2*x^2)*sqrt(a)*arctan(sqrt(b*x - a)/s qrt(a)) + (105*a*b^3*x^3 - 140*a^2*b^2*x^2 + 21*a^3*b*x + 6*a^4)*sqrt(b*x - a))/(a^5*b^2*x^4 - 2*a^6*b*x^3 + a^7*x^2)]
Result contains complex when optimal does not.
Time = 17.58 (sec) , antiderivative size = 1108, normalized size of antiderivative = 9.55 \[ \int \frac {1}{x^3 (-a+b x)^{5/2}} \, dx=\text {Too large to display} \]
Piecewise((12*I*a**(89/2)*b**75*x**75/(24*a**(93/2)*b**(151/2)*x**(155/2)* sqrt(a/(b*x) - 1) - 24*a**(91/2)*b**(153/2)*x**(157/2)*sqrt(a/(b*x) - 1)) + 42*I*a**(87/2)*b**76*x**76/(24*a**(93/2)*b**(151/2)*x**(155/2)*sqrt(a/(b *x) - 1) - 24*a**(91/2)*b**(153/2)*x**(157/2)*sqrt(a/(b*x) - 1)) - 280*I*a **(85/2)*b**77*x**77/(24*a**(93/2)*b**(151/2)*x**(155/2)*sqrt(a/(b*x) - 1) - 24*a**(91/2)*b**(153/2)*x**(157/2)*sqrt(a/(b*x) - 1)) + 210*I*a**(83/2) *b**78*x**78/(24*a**(93/2)*b**(151/2)*x**(155/2)*sqrt(a/(b*x) - 1) - 24*a* *(91/2)*b**(153/2)*x**(157/2)*sqrt(a/(b*x) - 1)) + 210*I*a**42*b**(155/2)* x**(155/2)*sqrt(a/(b*x) - 1)*acosh(sqrt(a)/(sqrt(b)*sqrt(x)))/(24*a**(93/2 )*b**(151/2)*x**(155/2)*sqrt(a/(b*x) - 1) - 24*a**(91/2)*b**(153/2)*x**(15 7/2)*sqrt(a/(b*x) - 1)) - 105*pi*a**42*b**(155/2)*x**(155/2)*sqrt(a/(b*x) - 1)/(24*a**(93/2)*b**(151/2)*x**(155/2)*sqrt(a/(b*x) - 1) - 24*a**(91/2)* b**(153/2)*x**(157/2)*sqrt(a/(b*x) - 1)) - 210*I*a**41*b**(157/2)*x**(157/ 2)*sqrt(a/(b*x) - 1)*acosh(sqrt(a)/(sqrt(b)*sqrt(x)))/(24*a**(93/2)*b**(15 1/2)*x**(155/2)*sqrt(a/(b*x) - 1) - 24*a**(91/2)*b**(153/2)*x**(157/2)*sqr t(a/(b*x) - 1)) + 105*pi*a**41*b**(157/2)*x**(157/2)*sqrt(a/(b*x) - 1)/(24 *a**(93/2)*b**(151/2)*x**(155/2)*sqrt(a/(b*x) - 1) - 24*a**(91/2)*b**(153/ 2)*x**(157/2)*sqrt(a/(b*x) - 1)), Abs(a/(b*x)) > 1), (-6*a**(89/2)*b**75*x **75/(12*a**(93/2)*b**(151/2)*x**(155/2)*sqrt(-a/(b*x) + 1) - 12*a**(91/2) *b**(153/2)*x**(157/2)*sqrt(-a/(b*x) + 1)) - 21*a**(87/2)*b**76*x**76/(...
Time = 0.29 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^3 (-a+b x)^{5/2}} \, dx=\frac {105 \, {\left (b x - a\right )}^{3} b^{2} + 175 \, {\left (b x - a\right )}^{2} a b^{2} + 56 \, {\left (b x - a\right )} a^{2} b^{2} - 8 \, a^{3} b^{2}}{12 \, {\left ({\left (b x - a\right )}^{\frac {7}{2}} a^{4} + 2 \, {\left (b x - a\right )}^{\frac {5}{2}} a^{5} + {\left (b x - a\right )}^{\frac {3}{2}} a^{6}\right )}} + \frac {35 \, b^{2} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{4 \, a^{\frac {9}{2}}} \]
1/12*(105*(b*x - a)^3*b^2 + 175*(b*x - a)^2*a*b^2 + 56*(b*x - a)*a^2*b^2 - 8*a^3*b^2)/((b*x - a)^(7/2)*a^4 + 2*(b*x - a)^(5/2)*a^5 + (b*x - a)^(3/2) *a^6) + 35/4*b^2*arctan(sqrt(b*x - a)/sqrt(a))/a^(9/2)
Time = 0.29 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^3 (-a+b x)^{5/2}} \, dx=\frac {35 \, b^{2} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{4 \, a^{\frac {9}{2}}} + \frac {2 \, {\left (9 \, {\left (b x - a\right )} b^{2} - a b^{2}\right )}}{3 \, {\left (b x - a\right )}^{\frac {3}{2}} a^{4}} + \frac {11 \, {\left (b x - a\right )}^{\frac {3}{2}} b^{2} + 13 \, \sqrt {b x - a} a b^{2}}{4 \, a^{4} b^{2} x^{2}} \]
35/4*b^2*arctan(sqrt(b*x - a)/sqrt(a))/a^(9/2) + 2/3*(9*(b*x - a)*b^2 - a* b^2)/((b*x - a)^(3/2)*a^4) + 1/4*(11*(b*x - a)^(3/2)*b^2 + 13*sqrt(b*x - a )*a*b^2)/(a^4*b^2*x^2)
Time = 0.09 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^3 (-a+b x)^{5/2}} \, dx=\frac {35\,b^2\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )}{4\,a^{9/2}}-\frac {\frac {2\,b^2}{3\,a}-\frac {175\,b^2\,{\left (a-b\,x\right )}^2}{12\,a^3}+\frac {35\,b^2\,{\left (a-b\,x\right )}^3}{4\,a^4}+\frac {14\,b^2\,\left (a-b\,x\right )}{3\,a^2}}{2\,a\,{\left (b\,x-a\right )}^{5/2}+{\left (b\,x-a\right )}^{7/2}+a^2\,{\left (b\,x-a\right )}^{3/2}} \]
(35*b^2*atan((b*x - a)^(1/2)/a^(1/2)))/(4*a^(9/2)) - ((2*b^2)/(3*a) - (175 *b^2*(a - b*x)^2)/(12*a^3) + (35*b^2*(a - b*x)^3)/(4*a^4) + (14*b^2*(a - b *x))/(3*a^2))/(2*a*(b*x - a)^(5/2) + (b*x - a)^(7/2) + a^2*(b*x - a)^(3/2) )
Time = 0.00 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^3 (-a+b x)^{5/2}} \, dx=\frac {105 \sqrt {a}\, \sqrt {b x -a}\, \mathit {atan} \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right ) a \,b^{2} x^{2}-105 \sqrt {a}\, \sqrt {b x -a}\, \mathit {atan} \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right ) b^{3} x^{3}-6 a^{4}-21 a^{3} b x +140 a^{2} b^{2} x^{2}-105 a \,b^{3} x^{3}}{12 \sqrt {b x -a}\, a^{5} x^{2} \left (-b x +a \right )} \]