Integrand size = 13, antiderivative size = 106 \[ \int \frac {1}{x^3 (a+b x)^{5/2}} \, dx=\frac {35 b^2}{12 a^3 (a+b x)^{3/2}}-\frac {1}{2 a x^2 (a+b x)^{3/2}}+\frac {7 b}{4 a^2 x (a+b x)^{3/2}}+\frac {35 b^2}{4 a^4 \sqrt {a+b x}}-\frac {35 b^2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{9/2}} \]
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*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(9/2)+35/4*b^2/a^4/(b*x+a)^( 1/2)
Time = 0.11 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^3 (a+b x)^{5/2}} \, dx=\frac {-6 a^3+21 a^2 b x+140 a b^2 x^2+105 b^3 x^3}{12 a^4 x^2 (a+b x)^{3/2}}-\frac {35 b^2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{9/2}} \]
(-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*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(4*a^(9/2))
Time = 0.19 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, 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.
\(\displaystyle \int \frac {1}{x^3 (a+b x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 52 |
\(\displaystyle -\frac {7 b \int \frac {1}{x^2 (a+b x)^{5/2}}dx}{4 a}-\frac {1}{2 a x^2 (a+b x)^{3/2}}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle -\frac {7 b \left (-\frac {5 b \int \frac {1}{x (a+b x)^{5/2}}dx}{2 a}-\frac {1}{a x (a+b x)^{3/2}}\right )}{4 a}-\frac {1}{2 a x^2 (a+b x)^{3/2}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {7 b \left (-\frac {5 b \left (\frac {\int \frac {1}{x (a+b x)^{3/2}}dx}{a}+\frac {2}{3 a (a+b x)^{3/2}}\right )}{2 a}-\frac {1}{a x (a+b x)^{3/2}}\right )}{4 a}-\frac {1}{2 a x^2 (a+b x)^{3/2}}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {7 b \left (-\frac {5 b \left (\frac {\frac {\int \frac {1}{x \sqrt {a+b x}}dx}{a}+\frac {2}{a \sqrt {a+b x}}}{a}+\frac {2}{3 a (a+b x)^{3/2}}\right )}{2 a}-\frac {1}{a x (a+b x)^{3/2}}\right )}{4 a}-\frac {1}{2 a x^2 (a+b x)^{3/2}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {7 b \left (-\frac {5 b \left (\frac {\frac {2 \int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}}{a b}+\frac {2}{a \sqrt {a+b x}}}{a}+\frac {2}{3 a (a+b x)^{3/2}}\right )}{2 a}-\frac {1}{a x (a+b x)^{3/2}}\right )}{4 a}-\frac {1}{2 a x^2 (a+b x)^{3/2}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {7 b \left (-\frac {5 b \left (\frac {\frac {2}{a \sqrt {a+b x}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}}}{a}+\frac {2}{3 a (a+b x)^{3/2}}\right )}{2 a}-\frac {1}{a x (a+b x)^{3/2}}\right )}{4 a}-\frac {1}{2 a x^2 (a+b x)^{3/2}}\) |
-1/2*1/(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*ArcTanh[Sqrt[a + b*x]/ Sqrt[a]])/a^(3/2))/a))/(2*a)))/(4*a)
3.4.58.3.1 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)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Time = 0.14 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.66
method | result | size |
risch | \(-\frac {\sqrt {b x +a}\, \left (-11 b x +2 a \right )}{4 a^{4} x^{2}}+\frac {b^{2} \left (-\frac {70 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {48}{\sqrt {b x +a}}+\frac {16 a}{3 \left (b x +a \right )^{\frac {3}{2}}}\right )}{8 a^{4}}\) | \(70\) |
pseudoelliptic | \(-\frac {35 \left (\left (b x +a \right )^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) b^{2} x^{2}-\sqrt {a}\, b^{3} x^{3}-\frac {4 a^{\frac {3}{2}} b^{2} x^{2}}{3}-\frac {a^{\frac {5}{2}} b x}{5}+\frac {2 a^{\frac {7}{2}}}{35}\right )}{4 \left (b x +a \right )^{\frac {3}{2}} a^{\frac {9}{2}} x^{2}}\) | \(77\) |
derivativedivides | \(2 b^{2} \left (\frac {3}{a^{4} \sqrt {b x +a}}+\frac {1}{3 a^{3} \left (b x +a \right )^{\frac {3}{2}}}-\frac {\frac {-\frac {11 \left (b x +a \right )^{\frac {3}{2}}}{8}+\frac {13 a \sqrt {b x +a}}{8}}{b^{2} x^{2}}+\frac {35 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a^{4}}\right )\) | \(81\) |
default | \(2 b^{2} \left (\frac {3}{a^{4} \sqrt {b x +a}}+\frac {1}{3 a^{3} \left (b x +a \right )^{\frac {3}{2}}}-\frac {\frac {-\frac {11 \left (b x +a \right )^{\frac {3}{2}}}{8}+\frac {13 a \sqrt {b x +a}}{8}}{b^{2} x^{2}}+\frac {35 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 \sqrt {a}}}{a^{4}}\right )\) | \(81\) |
-1/4*(b*x+a)^(1/2)*(-11*b*x+2*a)/a^4/x^2+1/8/a^4*b^2*(-70*arctanh((b*x+a)^ (1/2)/a^(1/2))/a^(1/2)+48/(b*x+a)^(1/2)+16/3*a/(b*x+a)^(3/2))
Time = 0.24 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.41 \[ \int \frac {1}{x^3 (a+b x)^{5/2}} \, dx=\left [\frac {105 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{3} + a^{2} b^{2} x^{2}\right )} \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (105 \, a b^{3} x^{3} + 140 \, a^{2} b^{2} x^{2} + 21 \, a^{3} b x - 6 \, a^{4}\right )} \sqrt {b x + a}}{24 \, {\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}}, \frac {105 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{3} + a^{2} b^{2} x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (105 \, a b^{3} x^{3} + 140 \, a^{2} b^{2} x^{2} + 21 \, a^{3} b x - 6 \, a^{4}\right )} \sqrt {b x + a}}{12 \, {\left (a^{5} b^{2} x^{4} + 2 \, a^{6} b x^{3} + a^{7} x^{2}\right )}}\right ] \]
[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*sqrt (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*(1 05*(b^4*x^4 + 2*a*b^3*x^3 + a^2*b^2*x^2)*sqrt(-a)*arctan(sqrt(b*x + a)*sqr t(-a)/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)]
Leaf count of result is larger than twice the leaf count of optimal. 464 vs. \(2 (99) = 198\).
Time = 14.86 (sec) , antiderivative size = 464, normalized size of antiderivative = 4.38 \[ \int \frac {1}{x^3 (a+b x)^{5/2}} \, dx=- \frac {6 a^{\frac {89}{2}} b^{75} x^{75}}{12 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{\frac {155}{2}} \sqrt {\frac {a}{b x} + 1} + 12 a^{\frac {91}{2}} b^{\frac {153}{2}} x^{\frac {157}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {21 a^{\frac {87}{2}} b^{76} x^{76}}{12 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{\frac {155}{2}} \sqrt {\frac {a}{b x} + 1} + 12 a^{\frac {91}{2}} b^{\frac {153}{2}} x^{\frac {157}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {140 a^{\frac {85}{2}} b^{77} x^{77}}{12 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{\frac {155}{2}} \sqrt {\frac {a}{b x} + 1} + 12 a^{\frac {91}{2}} b^{\frac {153}{2}} x^{\frac {157}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {105 a^{\frac {83}{2}} b^{78} x^{78}}{12 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{\frac {155}{2}} \sqrt {\frac {a}{b x} + 1} + 12 a^{\frac {91}{2}} b^{\frac {153}{2}} x^{\frac {157}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {105 a^{42} b^{\frac {155}{2}} x^{\frac {155}{2}} \sqrt {\frac {a}{b x} + 1} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{12 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{\frac {155}{2}} \sqrt {\frac {a}{b x} + 1} + 12 a^{\frac {91}{2}} b^{\frac {153}{2}} x^{\frac {157}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {105 a^{41} b^{\frac {157}{2}} x^{\frac {157}{2}} \sqrt {\frac {a}{b x} + 1} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{12 a^{\frac {93}{2}} b^{\frac {151}{2}} x^{\frac {155}{2}} \sqrt {\frac {a}{b x} + 1} + 12 a^{\frac {91}{2}} b^{\frac {153}{2}} x^{\frac {157}{2}} \sqrt {\frac {a}{b x} + 1}} \]
-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/(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)) + 140*a**(85/2)*b**77*x* *77/(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)) + 105*a**(83/2)*b**78*x**78/(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)) - 105*a**42*b**(155/2)*x**(155/2)*sqrt(a/(b* x) + 1)*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(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 )) - 105*a**41*b**(157/2)*x**(157/2)*sqrt(a/(b*x) + 1)*asinh(sqrt(a)/(sqrt (b)*sqrt(x)))/(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))
Time = 0.29 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.16 \[ \int \frac {1}{x^3 (a+b x)^{5/2}} \, dx=\frac {105 \, {\left (b x + a\right )}^{3} b^{2} - 175 \, {\left (b x + a\right )}^{2} a b^{2} + 56 \, {\left (b x + a\right )} a^{2} b^{2} + 8 \, a^{3} b^{2}}{12 \, {\left ({\left (b x + a\right )}^{\frac {7}{2}} a^{4} - 2 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{5} + {\left (b x + a\right )}^{\frac {3}{2}} a^{6}\right )}} + \frac {35 \, b^{2} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{8 \, a^{\frac {9}{2}}} \]
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) + 35/8*b^2*log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a)))/ a^(9/2)
Time = 0.31 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^3 (a+b x)^{5/2}} \, dx=\frac {35 \, b^{2} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a^{4}} + \frac {2 \, {\left (9 \, {\left (b x + a\right )} b^{2} + a b^{2}\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4}} + \frac {11 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{2} - 13 \, \sqrt {b x + a} a b^{2}}{4 \, a^{4} b^{2} x^{2}} \]
35/4*b^2*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^4) + 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 = 0.15 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x^3 (a+b x)^{5/2}} \, dx=\frac {\frac {2\,b^2}{3\,a}-\frac {175\,b^2\,{\left (a+b\,x\right )}^2}{12\,a^3}+\frac {35\,b^2\,{\left (a+b\,x\right )}^3}{4\,a^4}+\frac {14\,b^2\,\left (a+b\,x\right )}{3\,a^2}}{{\left (a+b\,x\right )}^{7/2}-2\,a\,{\left (a+b\,x\right )}^{5/2}+a^2\,{\left (a+b\,x\right )}^{3/2}}-\frac {35\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )}{4\,a^{9/2}} \]
((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))/((a + b*x)^(7/2) - 2*a*(a + b*x)^(5/2) + a^2*(a + b*x)^(3/2)) - (35*b^2*atanh((a + b*x)^(1/2)/a^(1/2)))/(4*a^(9/2 ))
Time = 0.00 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.57 \[ \int \frac {1}{x^3 (a+b x)^{5/2}} \, dx=\frac {105 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a \,b^{2} x^{2}+105 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) b^{3} x^{3}-105 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a \,b^{2} x^{2}-105 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) b^{3} x^{3}-12 a^{4}+42 a^{3} b x +280 a^{2} b^{2} x^{2}+210 a \,b^{3} x^{3}}{24 \sqrt {b x +a}\, a^{5} x^{2} \left (b x +a \right )} \]
(105*sqrt(a)*sqrt(a + b*x)*log(sqrt(a + b*x) - sqrt(a))*a*b**2*x**2 + 105* sqrt(a)*sqrt(a + b*x)*log(sqrt(a + b*x) - sqrt(a))*b**3*x**3 - 105*sqrt(a) *sqrt(a + b*x)*log(sqrt(a + b*x) + sqrt(a))*a*b**2*x**2 - 105*sqrt(a)*sqrt (a + b*x)*log(sqrt(a + b*x) + sqrt(a))*b**3*x**3 - 12*a**4 + 42*a**3*b*x + 280*a**2*b**2*x**2 + 210*a*b**3*x**3)/(24*sqrt(a + b*x)*a**5*x**2*(a + b* x))