Integrand size = 19, antiderivative size = 126 \[ \int \frac {x^{5/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\frac {x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac {8 x^{3/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac {16 \sqrt {x}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac {64}{35 a^4 \sqrt {x} \sqrt {a x+b x^3}}-\frac {128 \sqrt {a x+b x^3}}{35 a^5 x^{3/2}} \]
1/7*x^(5/2)/a/(b*x^3+a*x)^(7/2)+8/35*x^(3/2)/a^2/(b*x^3+a*x)^(5/2)+16/35*x ^(1/2)/a^3/(b*x^3+a*x)^(3/2)+64/35/a^4/x^(1/2)/(b*x^3+a*x)^(1/2)-128/35*(b *x^3+a*x)^(1/2)/a^5/x^(3/2)
Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.54 \[ \int \frac {x^{5/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\frac {x^{5/2} \left (-35 a^4-280 a^3 b x^2-560 a^2 b^2 x^4-448 a b^3 x^6-128 b^4 x^8\right )}{35 a^5 \left (x \left (a+b x^2\right )\right )^{7/2}} \]
(x^(5/2)*(-35*a^4 - 280*a^3*b*x^2 - 560*a^2*b^2*x^4 - 448*a*b^3*x^6 - 128* b^4*x^8))/(35*a^5*(x*(a + b*x^2))^(7/2))
Time = 0.33 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.15, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {1921, 1921, 1921, 1921, 1920}
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.
\(\displaystyle \int \frac {x^{5/2}}{\left (a x+b x^3\right )^{9/2}} \, dx\) |
\(\Big \downarrow \) 1921 |
\(\displaystyle \frac {8 \int \frac {x^{3/2}}{\left (b x^3+a x\right )^{7/2}}dx}{7 a}+\frac {x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}}\) |
\(\Big \downarrow \) 1921 |
\(\displaystyle \frac {8 \left (\frac {6 \int \frac {\sqrt {x}}{\left (b x^3+a x\right )^{5/2}}dx}{5 a}+\frac {x^{3/2}}{5 a \left (a x+b x^3\right )^{5/2}}\right )}{7 a}+\frac {x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}}\) |
\(\Big \downarrow \) 1921 |
\(\displaystyle \frac {8 \left (\frac {6 \left (\frac {4 \int \frac {1}{\sqrt {x} \left (b x^3+a x\right )^{3/2}}dx}{3 a}+\frac {\sqrt {x}}{3 a \left (a x+b x^3\right )^{3/2}}\right )}{5 a}+\frac {x^{3/2}}{5 a \left (a x+b x^3\right )^{5/2}}\right )}{7 a}+\frac {x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}}\) |
\(\Big \downarrow \) 1921 |
\(\displaystyle \frac {8 \left (\frac {6 \left (\frac {4 \left (\frac {2 \int \frac {1}{x^{3/2} \sqrt {b x^3+a x}}dx}{a}+\frac {1}{a \sqrt {x} \sqrt {a x+b x^3}}\right )}{3 a}+\frac {\sqrt {x}}{3 a \left (a x+b x^3\right )^{3/2}}\right )}{5 a}+\frac {x^{3/2}}{5 a \left (a x+b x^3\right )^{5/2}}\right )}{7 a}+\frac {x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}}\) |
\(\Big \downarrow \) 1920 |
\(\displaystyle \frac {8 \left (\frac {6 \left (\frac {4 \left (\frac {1}{a \sqrt {x} \sqrt {a x+b x^3}}-\frac {2 \sqrt {a x+b x^3}}{a^2 x^{3/2}}\right )}{3 a}+\frac {\sqrt {x}}{3 a \left (a x+b x^3\right )^{3/2}}\right )}{5 a}+\frac {x^{3/2}}{5 a \left (a x+b x^3\right )^{5/2}}\right )}{7 a}+\frac {x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}}\) |
x^(5/2)/(7*a*(a*x + b*x^3)^(7/2)) + (8*(x^(3/2)/(5*a*(a*x + b*x^3)^(5/2)) + (6*(Sqrt[x]/(3*a*(a*x + b*x^3)^(3/2)) + (4*(1/(a*Sqrt[x]*Sqrt[a*x + b*x^ 3]) - (2*Sqrt[a*x + b*x^3])/(a^2*x^(3/2))))/(3*a)))/(5*a)))/(7*a)
3.1.87.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[(-c^(j - 1))*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(n - j )*(p + 1))), x] /; FreeQ[{a, b, c, j, m, n, p}, x] && !IntegerQ[p] && NeQ[ n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[(-c^(j - 1))*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(n - j )*(p + 1))), x] + Simp[c^j*((m + n*p + n - j + 1)/(a*(n - j)*(p + 1))) In t[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, j, m, n} , x] && !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(m + n*p + n - j + 1)/( n - j)], 0] && LtQ[p, -1] && (IntegerQ[j] || GtQ[c, 0])
Time = 2.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.56
method | result | size |
gosper | \(-\frac {x^{\frac {7}{2}} \left (b \,x^{2}+a \right ) \left (128 x^{8} b^{4}+448 a \,b^{3} x^{6}+560 a^{2} x^{4} b^{2}+280 a^{3} b \,x^{2}+35 a^{4}\right )}{35 a^{5} \left (b \,x^{3}+a x \right )^{\frac {9}{2}}}\) | \(70\) |
default | \(-\frac {\sqrt {x \left (b \,x^{2}+a \right )}\, \left (128 x^{8} b^{4}+448 a \,b^{3} x^{6}+560 a^{2} x^{4} b^{2}+280 a^{3} b \,x^{2}+35 a^{4}\right )}{35 x^{\frac {3}{2}} \left (b \,x^{2}+a \right )^{4} a^{5}}\) | \(72\) |
risch | \(-\frac {b \,x^{2}+a}{a^{5} \sqrt {x}\, \sqrt {x \left (b \,x^{2}+a \right )}}-\frac {\left (b \,x^{2}+a \right ) x^{\frac {3}{2}} \left (93 b^{3} x^{6}+308 a \,b^{2} x^{4}+350 a^{2} b \,x^{2}+140 a^{3}\right ) b}{35 \left (x^{8} b^{4}+4 a \,b^{3} x^{6}+6 a^{2} x^{4} b^{2}+4 a^{3} b \,x^{2}+a^{4}\right ) a^{5} \sqrt {x \left (b \,x^{2}+a \right )}}\) | \(129\) |
-1/35*x^(7/2)*(b*x^2+a)*(128*b^4*x^8+448*a*b^3*x^6+560*a^2*b^2*x^4+280*a^3 *b*x^2+35*a^4)/a^5/(b*x^3+a*x)^(9/2)
Time = 0.42 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.87 \[ \int \frac {x^{5/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=-\frac {{\left (128 \, b^{4} x^{8} + 448 \, a b^{3} x^{6} + 560 \, a^{2} b^{2} x^{4} + 280 \, a^{3} b x^{2} + 35 \, a^{4}\right )} \sqrt {b x^{3} + a x} \sqrt {x}}{35 \, {\left (a^{5} b^{4} x^{10} + 4 \, a^{6} b^{3} x^{8} + 6 \, a^{7} b^{2} x^{6} + 4 \, a^{8} b x^{4} + a^{9} x^{2}\right )}} \]
-1/35*(128*b^4*x^8 + 448*a*b^3*x^6 + 560*a^2*b^2*x^4 + 280*a^3*b*x^2 + 35* a^4)*sqrt(b*x^3 + a*x)*sqrt(x)/(a^5*b^4*x^10 + 4*a^6*b^3*x^8 + 6*a^7*b^2*x ^6 + 4*a^8*b*x^4 + a^9*x^2)
\[ \int \frac {x^{5/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\int \frac {x^{\frac {5}{2}}}{\left (x \left (a + b x^{2}\right )\right )^{\frac {9}{2}}}\, dx \]
\[ \int \frac {x^{5/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\int { \frac {x^{\frac {5}{2}}}{{\left (b x^{3} + a x\right )}^{\frac {9}{2}}} \,d x } \]
Time = 0.31 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.71 \[ \int \frac {x^{5/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=-\frac {{\left ({\left (x^{2} {\left (\frac {93 \, b^{4} x^{2}}{a^{5}} + \frac {308 \, b^{3}}{a^{4}}\right )} + \frac {350 \, b^{2}}{a^{3}}\right )} x^{2} + \frac {140 \, b}{a^{2}}\right )} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} + \frac {2 \, \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )} a^{4}} \]
-1/35*((x^2*(93*b^4*x^2/a^5 + 308*b^3/a^4) + 350*b^2/a^3)*x^2 + 140*b/a^2) *x/(b*x^2 + a)^(7/2) + 2*sqrt(b)/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)*a^ 4)
Timed out. \[ \int \frac {x^{5/2}}{\left (a x+b x^3\right )^{9/2}} \, dx=\int \frac {x^{5/2}}{{\left (b\,x^3+a\,x\right )}^{9/2}} \,d x \]