Integrand size = 17, antiderivative size = 100 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^{10}} \, dx=-\frac {2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}+\frac {12 b \left (a x+b x^2\right )^{7/2}}{143 a^2 x^9}-\frac {16 b^2 \left (a x+b x^2\right )^{7/2}}{429 a^3 x^8}+\frac {32 b^3 \left (a x+b x^2\right )^{7/2}}{3003 a^4 x^7} \]
-2/13*(b*x^2+a*x)^(7/2)/a/x^10+12/143*b*(b*x^2+a*x)^(7/2)/a^2/x^9-16/429*b ^2*(b*x^2+a*x)^(7/2)/a^3/x^8+32/3003*b^3*(b*x^2+a*x)^(7/2)/a^4/x^7
Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.51 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^{10}} \, dx=-\frac {2 (x (a+b x))^{7/2} \left (231 a^3-126 a^2 b x+56 a b^2 x^2-16 b^3 x^3\right )}{3003 a^4 x^{10}} \]
(-2*(x*(a + b*x))^(7/2)*(231*a^3 - 126*a^2*b*x + 56*a*b^2*x^2 - 16*b^3*x^3 ))/(3003*a^4*x^10)
Time = 0.24 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {1129, 1129, 1129, 1123}
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.
\(\displaystyle \int \frac {\left (a x+b x^2\right )^{5/2}}{x^{10}} \, dx\) |
\(\Big \downarrow \) 1129 |
\(\displaystyle -\frac {6 b \int \frac {\left (b x^2+a x\right )^{5/2}}{x^9}dx}{13 a}-\frac {2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}\) |
\(\Big \downarrow \) 1129 |
\(\displaystyle -\frac {6 b \left (-\frac {4 b \int \frac {\left (b x^2+a x\right )^{5/2}}{x^8}dx}{11 a}-\frac {2 \left (a x+b x^2\right )^{7/2}}{11 a x^9}\right )}{13 a}-\frac {2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}\) |
\(\Big \downarrow \) 1129 |
\(\displaystyle -\frac {6 b \left (-\frac {4 b \left (-\frac {2 b \int \frac {\left (b x^2+a x\right )^{5/2}}{x^7}dx}{9 a}-\frac {2 \left (a x+b x^2\right )^{7/2}}{9 a x^8}\right )}{11 a}-\frac {2 \left (a x+b x^2\right )^{7/2}}{11 a x^9}\right )}{13 a}-\frac {2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}\) |
\(\Big \downarrow \) 1123 |
\(\displaystyle -\frac {6 b \left (-\frac {4 b \left (\frac {4 b \left (a x+b x^2\right )^{7/2}}{63 a^2 x^7}-\frac {2 \left (a x+b x^2\right )^{7/2}}{9 a x^8}\right )}{11 a}-\frac {2 \left (a x+b x^2\right )^{7/2}}{11 a x^9}\right )}{13 a}-\frac {2 \left (a x+b x^2\right )^{7/2}}{13 a x^{10}}\) |
(-2*(a*x + b*x^2)^(7/2))/(13*a*x^10) - (6*b*((-2*(a*x + b*x^2)^(7/2))/(11* a*x^9) - (4*b*((-2*(a*x + b*x^2)^(7/2))/(9*a*x^8) + (4*b*(a*x + b*x^2)^(7/ 2))/(63*a^2*x^7)))/(11*a)))/(13*a)
3.1.36.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(2*c*d - b *e))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + 2*p + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* c*d - b*e))), x] + Simp[c*(Simplify[m + 2*p + 2]/((m + p + 1)*(2*c*d - b*e) )) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d , e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 2], 0]
Time = 2.30 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.55
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right ) \left (-16 b^{3} x^{3}+56 a \,b^{2} x^{2}-126 a^{2} b x +231 a^{3}\right ) \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{3003 a^{4} x^{9}}\) | \(55\) |
pseudoelliptic | \(-\frac {2 \left (b x +a \right )^{3} \sqrt {x \left (b x +a \right )}\, \left (-16 b^{3} x^{3}+56 a \,b^{2} x^{2}-126 a^{2} b x +231 a^{3}\right )}{3003 x^{7} a^{4}}\) | \(55\) |
trager | \(-\frac {2 \left (-16 b^{6} x^{6}+8 a \,x^{5} b^{5}-6 a^{2} x^{4} b^{4}+5 a^{3} x^{3} b^{3}+371 a^{4} x^{2} b^{2}+567 a^{5} x b +231 a^{6}\right ) \sqrt {b \,x^{2}+a x}}{3003 a^{4} x^{7}}\) | \(83\) |
risch | \(-\frac {2 \left (b x +a \right ) \left (-16 b^{6} x^{6}+8 a \,x^{5} b^{5}-6 a^{2} x^{4} b^{4}+5 a^{3} x^{3} b^{3}+371 a^{4} x^{2} b^{2}+567 a^{5} x b +231 a^{6}\right )}{3003 x^{6} \sqrt {x \left (b x +a \right )}\, a^{4}}\) | \(86\) |
default | \(-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{13 a \,x^{10}}-\frac {6 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{11 a \,x^{9}}-\frac {4 b \left (-\frac {2 \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{9 a \,x^{8}}+\frac {4 b \left (b \,x^{2}+a x \right )^{\frac {7}{2}}}{63 a^{2} x^{7}}\right )}{11 a}\right )}{13 a}\) | \(93\) |
Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^{10}} \, dx=\frac {2 \, {\left (16 \, b^{6} x^{6} - 8 \, a b^{5} x^{5} + 6 \, a^{2} b^{4} x^{4} - 5 \, a^{3} b^{3} x^{3} - 371 \, a^{4} b^{2} x^{2} - 567 \, a^{5} b x - 231 \, a^{6}\right )} \sqrt {b x^{2} + a x}}{3003 \, a^{4} x^{7}} \]
2/3003*(16*b^6*x^6 - 8*a*b^5*x^5 + 6*a^2*b^4*x^4 - 5*a^3*b^3*x^3 - 371*a^4 *b^2*x^2 - 567*a^5*b*x - 231*a^6)*sqrt(b*x^2 + a*x)/(a^4*x^7)
\[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^{10}} \, dx=\int \frac {\left (x \left (a + b x\right )\right )^{\frac {5}{2}}}{x^{10}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (84) = 168\).
Time = 0.19 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.78 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^{10}} \, dx=\frac {32 \, \sqrt {b x^{2} + a x} b^{6}}{3003 \, a^{4} x} - \frac {16 \, \sqrt {b x^{2} + a x} b^{5}}{3003 \, a^{3} x^{2}} + \frac {4 \, \sqrt {b x^{2} + a x} b^{4}}{1001 \, a^{2} x^{3}} - \frac {10 \, \sqrt {b x^{2} + a x} b^{3}}{3003 \, a x^{4}} + \frac {5 \, \sqrt {b x^{2} + a x} b^{2}}{1716 \, x^{5}} - \frac {3 \, \sqrt {b x^{2} + a x} a b}{1144 \, x^{6}} - \frac {3 \, \sqrt {b x^{2} + a x} a^{2}}{104 \, x^{7}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} a}{8 \, x^{8}} - \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}}}{4 \, x^{9}} \]
32/3003*sqrt(b*x^2 + a*x)*b^6/(a^4*x) - 16/3003*sqrt(b*x^2 + a*x)*b^5/(a^3 *x^2) + 4/1001*sqrt(b*x^2 + a*x)*b^4/(a^2*x^3) - 10/3003*sqrt(b*x^2 + a*x) *b^3/(a*x^4) + 5/1716*sqrt(b*x^2 + a*x)*b^2/x^5 - 3/1144*sqrt(b*x^2 + a*x) *a*b/x^6 - 3/104*sqrt(b*x^2 + a*x)*a^2/x^7 + 1/8*(b*x^2 + a*x)^(3/2)*a/x^8 - 1/4*(b*x^2 + a*x)^(5/2)/x^9
Leaf count of result is larger than twice the leaf count of optimal. 281 vs. \(2 (84) = 168\).
Time = 0.29 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.81 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^{10}} \, dx=\frac {2 \, {\left (6006 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{9} b^{\frac {9}{2}} + 36036 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{8} a b^{4} + 99099 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{7} a^{2} b^{\frac {7}{2}} + 161733 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} a^{3} b^{3} + 171171 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} a^{4} b^{\frac {5}{2}} + 121121 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a^{5} b^{2} + 57057 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{6} b^{\frac {3}{2}} + 17199 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{7} b + 3003 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{8} \sqrt {b} + 231 \, a^{9}\right )}}{3003 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{13}} \]
2/3003*(6006*(sqrt(b)*x - sqrt(b*x^2 + a*x))^9*b^(9/2) + 36036*(sqrt(b)*x - sqrt(b*x^2 + a*x))^8*a*b^4 + 99099*(sqrt(b)*x - sqrt(b*x^2 + a*x))^7*a^2 *b^(7/2) + 161733*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*a^3*b^3 + 171171*(sqrt (b)*x - sqrt(b*x^2 + a*x))^5*a^4*b^(5/2) + 121121*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a^5*b^2 + 57057*(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*a^6*b^(3/2) + 17199*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a^7*b + 3003*(sqrt(b)*x - sqrt(b*x ^2 + a*x))*a^8*sqrt(b) + 231*a^9)/(sqrt(b)*x - sqrt(b*x^2 + a*x))^13
Time = 10.38 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^{10}} \, dx=\frac {4\,b^4\,\sqrt {b\,x^2+a\,x}}{1001\,a^2\,x^3}-\frac {106\,b^2\,\sqrt {b\,x^2+a\,x}}{429\,x^5}-\frac {10\,b^3\,\sqrt {b\,x^2+a\,x}}{3003\,a\,x^4}-\frac {2\,a^2\,\sqrt {b\,x^2+a\,x}}{13\,x^7}-\frac {16\,b^5\,\sqrt {b\,x^2+a\,x}}{3003\,a^3\,x^2}+\frac {32\,b^6\,\sqrt {b\,x^2+a\,x}}{3003\,a^4\,x}-\frac {54\,a\,b\,\sqrt {b\,x^2+a\,x}}{143\,x^6} \]
(4*b^4*(a*x + b*x^2)^(1/2))/(1001*a^2*x^3) - (106*b^2*(a*x + b*x^2)^(1/2)) /(429*x^5) - (10*b^3*(a*x + b*x^2)^(1/2))/(3003*a*x^4) - (2*a^2*(a*x + b*x ^2)^(1/2))/(13*x^7) - (16*b^5*(a*x + b*x^2)^(1/2))/(3003*a^3*x^2) + (32*b^ 6*(a*x + b*x^2)^(1/2))/(3003*a^4*x) - (54*a*b*(a*x + b*x^2)^(1/2))/(143*x^ 6)