Integrand size = 33, antiderivative size = 167 \[ \int \frac {c x^3+d x^5+e x^7+f x^9}{\sqrt {a+b x^2}} \, dx=-\frac {a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \sqrt {a+b x^2}}{b^5}+\frac {\left (b^3 c-2 a b^2 d+3 a^2 b e-4 a^3 f\right ) \left (a+b x^2\right )^{3/2}}{3 b^5}+\frac {\left (b^2 d-3 a b e+6 a^2 f\right ) \left (a+b x^2\right )^{5/2}}{5 b^5}+\frac {(b e-4 a f) \left (a+b x^2\right )^{7/2}}{7 b^5}+\frac {f \left (a+b x^2\right )^{9/2}}{9 b^5} \]
1/3*(-4*a^3*f+3*a^2*b*e-2*a*b^2*d+b^3*c)*(b*x^2+a)^(3/2)/b^5+1/5*(6*a^2*f- 3*a*b*e+b^2*d)*(b*x^2+a)^(5/2)/b^5+1/7*(-4*a*f+b*e)*(b*x^2+a)^(7/2)/b^5+1/ 9*f*(b*x^2+a)^(9/2)/b^5-a*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*(b*x^2+a)^(1/2)/b ^5
Time = 0.02 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.73 \[ \int \frac {c x^3+d x^5+e x^7+f x^9}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {a+b x^2} \left (128 a^4 f-16 a^3 b \left (9 e+4 f x^2\right )+24 a^2 b^2 \left (7 d+3 e x^2+2 f x^4\right )-2 a b^3 \left (105 c+42 d x^2+27 e x^4+20 f x^6\right )+b^4 x^2 \left (105 c+63 d x^2+45 e x^4+35 f x^6\right )\right )}{315 b^5} \]
(Sqrt[a + b*x^2]*(128*a^4*f - 16*a^3*b*(9*e + 4*f*x^2) + 24*a^2*b^2*(7*d + 3*e*x^2 + 2*f*x^4) - 2*a*b^3*(105*c + 42*d*x^2 + 27*e*x^4 + 20*f*x^6) + b ^4*x^2*(105*c + 63*d*x^2 + 45*e*x^4 + 35*f*x^6)))/(315*b^5)
Time = 0.45 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2029, 2331, 2123, 2009}
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.
| \(\displaystyle \int \frac {c x^3+d x^5+e x^7+f x^9}{\sqrt {a+b x^2}} \, dx\) |
| \(\Big \downarrow \) 2029 |
| \(\displaystyle \int \frac {x^3 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt {a+b x^2}}dx\) |
| \(\Big \downarrow \) 2331 |
| \(\displaystyle \frac {1}{2} \int \frac {x^2 \left (f x^6+e x^4+d x^2+c\right )}{\sqrt {b x^2+a}}dx^2\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {f \left (b x^2+a\right )^{7/2}}{b^4}+\frac {(b e-4 a f) \left (b x^2+a\right )^{5/2}}{b^4}+\frac {\left (6 f a^2-3 b e a+b^2 d\right ) \left (b x^2+a\right )^{3/2}}{b^4}+\frac {\left (-4 f a^3+3 b e a^2-2 b^2 d a+b^3 c\right ) \sqrt {b x^2+a}}{b^4}+\frac {a \left (f a^3-b e a^2+b^2 d a-b^3 c\right )}{b^4 \sqrt {b x^2+a}}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \left (a+b x^2\right )^{5/2} \left (6 a^2 f-3 a b e+b^2 d\right )}{5 b^5}+\frac {2 \left (a+b x^2\right )^{3/2} \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{3 b^5}-\frac {2 a \sqrt {a+b x^2} \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^5}+\frac {2 \left (a+b x^2\right )^{7/2} (b e-4 a f)}{7 b^5}+\frac {2 f \left (a+b x^2\right )^{9/2}}{9 b^5}\right )\) |
((-2*a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Sqrt[a + b*x^2])/b^5 + (2*(b^3* c - 2*a*b^2*d + 3*a^2*b*e - 4*a^3*f)*(a + b*x^2)^(3/2))/(3*b^5) + (2*(b^2* d - 3*a*b*e + 6*a^2*f)*(a + b*x^2)^(5/2))/(5*b^5) + (2*(b*e - 4*a*f)*(a + b*x^2)^(7/2))/(7*b^5) + (2*f*(a + b*x^2)^(9/2))/(9*b^5))/2
3.2.70.3.1 Defintions of rubi rules used
Int[(Fx_.)*((d_.)*(x_)^(q_.) + (a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.) + (c_.)* (x_)^(t_.))^(p_.), x_Symbol] :> Int[x^(p*r)*(a + b*x^(s - r) + c*x^(t - r) + d*x^(q - r))^p*Fx, x] /; FreeQ[{a, b, c, d, r, s, t, q}, x] && IntegerQ[p ] && PosQ[s - r] && PosQ[t - r] && PosQ[q - r] && !(EqQ[p, 1] && EqQ[u, 1] )
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
Time = 3.51 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.66
| method | result | size |
| pseudoelliptic | \(\frac {128 \left (\frac {105 \left (\frac {1}{3} f \,x^{6}+\frac {3}{7} e \,x^{4}+\frac {3}{5} d \,x^{2}+c \right ) x^{2} b^{4}}{128}-\frac {105 \left (\frac {4}{21} f \,x^{6}+\frac {9}{35} e \,x^{4}+\frac {2}{5} d \,x^{2}+c \right ) a \,b^{3}}{64}+\frac {21 \left (\frac {2}{7} f \,x^{4}+\frac {3}{7} e \,x^{2}+d \right ) a^{2} b^{2}}{16}-\frac {9 \left (\frac {4 f \,x^{2}}{9}+e \right ) a^{3} b}{8}+a^{4} f \right ) \sqrt {b \,x^{2}+a}}{315 b^{5}}\) | \(111\) |
| gosper | \(\frac {\sqrt {b \,x^{2}+a}\, \left (35 f \,x^{8} b^{4}-40 a \,b^{3} f \,x^{6}+45 b^{4} e \,x^{6}+48 a^{2} b^{2} f \,x^{4}-54 a \,b^{3} e \,x^{4}+63 b^{4} d \,x^{4}-64 a^{3} b f \,x^{2}+72 a^{2} b^{2} e \,x^{2}-84 a \,b^{3} d \,x^{2}+105 b^{4} c \,x^{2}+128 a^{4} f -144 a^{3} b e +168 a^{2} b^{2} d -210 a \,b^{3} c \right )}{315 b^{5}}\) | \(145\) |
| trager | \(\frac {\sqrt {b \,x^{2}+a}\, \left (35 f \,x^{8} b^{4}-40 a \,b^{3} f \,x^{6}+45 b^{4} e \,x^{6}+48 a^{2} b^{2} f \,x^{4}-54 a \,b^{3} e \,x^{4}+63 b^{4} d \,x^{4}-64 a^{3} b f \,x^{2}+72 a^{2} b^{2} e \,x^{2}-84 a \,b^{3} d \,x^{2}+105 b^{4} c \,x^{2}+128 a^{4} f -144 a^{3} b e +168 a^{2} b^{2} d -210 a \,b^{3} c \right )}{315 b^{5}}\) | \(145\) |
| risch | \(\frac {\sqrt {b \,x^{2}+a}\, \left (35 f \,x^{8} b^{4}-40 a \,b^{3} f \,x^{6}+45 b^{4} e \,x^{6}+48 a^{2} b^{2} f \,x^{4}-54 a \,b^{3} e \,x^{4}+63 b^{4} d \,x^{4}-64 a^{3} b f \,x^{2}+72 a^{2} b^{2} e \,x^{2}-84 a \,b^{3} d \,x^{2}+105 b^{4} c \,x^{2}+128 a^{4} f -144 a^{3} b e +168 a^{2} b^{2} d -210 a \,b^{3} c \right )}{315 b^{5}}\) | \(145\) |
| default | \(f \left (\frac {x^{8} \sqrt {b \,x^{2}+a}}{9 b}-\frac {8 a \left (\frac {x^{6} \sqrt {b \,x^{2}+a}}{7 b}-\frac {6 a \left (\frac {x^{4} \sqrt {b \,x^{2}+a}}{5 b}-\frac {4 a \left (\frac {x^{2} \sqrt {b \,x^{2}+a}}{3 b}-\frac {2 a \sqrt {b \,x^{2}+a}}{3 b^{2}}\right )}{5 b}\right )}{7 b}\right )}{9 b}\right )+e \left (\frac {x^{6} \sqrt {b \,x^{2}+a}}{7 b}-\frac {6 a \left (\frac {x^{4} \sqrt {b \,x^{2}+a}}{5 b}-\frac {4 a \left (\frac {x^{2} \sqrt {b \,x^{2}+a}}{3 b}-\frac {2 a \sqrt {b \,x^{2}+a}}{3 b^{2}}\right )}{5 b}\right )}{7 b}\right )+d \left (\frac {x^{4} \sqrt {b \,x^{2}+a}}{5 b}-\frac {4 a \left (\frac {x^{2} \sqrt {b \,x^{2}+a}}{3 b}-\frac {2 a \sqrt {b \,x^{2}+a}}{3 b^{2}}\right )}{5 b}\right )+c \left (\frac {x^{2} \sqrt {b \,x^{2}+a}}{3 b}-\frac {2 a \sqrt {b \,x^{2}+a}}{3 b^{2}}\right )\) | \(286\) |
128/315*(105/128*(1/3*f*x^6+3/7*e*x^4+3/5*d*x^2+c)*x^2*b^4-105/64*(4/21*f* x^6+9/35*e*x^4+2/5*d*x^2+c)*a*b^3+21/16*(2/7*f*x^4+3/7*e*x^2+d)*a^2*b^2-9/ 8*(4/9*f*x^2+e)*a^3*b+a^4*f)*(b*x^2+a)^(1/2)/b^5
Time = 0.29 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.80 \[ \int \frac {c x^3+d x^5+e x^7+f x^9}{\sqrt {a+b x^2}} \, dx=\frac {{\left (35 \, b^{4} f x^{8} + 5 \, {\left (9 \, b^{4} e - 8 \, a b^{3} f\right )} x^{6} - 210 \, a b^{3} c + 168 \, a^{2} b^{2} d - 144 \, a^{3} b e + 128 \, a^{4} f + 3 \, {\left (21 \, b^{4} d - 18 \, a b^{3} e + 16 \, a^{2} b^{2} f\right )} x^{4} + {\left (105 \, b^{4} c - 84 \, a b^{3} d + 72 \, a^{2} b^{2} e - 64 \, a^{3} b f\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{315 \, b^{5}} \]
1/315*(35*b^4*f*x^8 + 5*(9*b^4*e - 8*a*b^3*f)*x^6 - 210*a*b^3*c + 168*a^2* b^2*d - 144*a^3*b*e + 128*a^4*f + 3*(21*b^4*d - 18*a*b^3*e + 16*a^2*b^2*f) *x^4 + (105*b^4*c - 84*a*b^3*d + 72*a^2*b^2*e - 64*a^3*b*f)*x^2)*sqrt(b*x^ 2 + a)/b^5
Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (163) = 326\).
Time = 0.37 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.04 \[ \int \frac {c x^3+d x^5+e x^7+f x^9}{\sqrt {a+b x^2}} \, dx=\begin {cases} \frac {128 a^{4} f \sqrt {a + b x^{2}}}{315 b^{5}} - \frac {16 a^{3} e \sqrt {a + b x^{2}}}{35 b^{4}} - \frac {64 a^{3} f x^{2} \sqrt {a + b x^{2}}}{315 b^{4}} + \frac {8 a^{2} d \sqrt {a + b x^{2}}}{15 b^{3}} + \frac {8 a^{2} e x^{2} \sqrt {a + b x^{2}}}{35 b^{3}} + \frac {16 a^{2} f x^{4} \sqrt {a + b x^{2}}}{105 b^{3}} - \frac {2 a c \sqrt {a + b x^{2}}}{3 b^{2}} - \frac {4 a d x^{2} \sqrt {a + b x^{2}}}{15 b^{2}} - \frac {6 a e x^{4} \sqrt {a + b x^{2}}}{35 b^{2}} - \frac {8 a f x^{6} \sqrt {a + b x^{2}}}{63 b^{2}} + \frac {c x^{2} \sqrt {a + b x^{2}}}{3 b} + \frac {d x^{4} \sqrt {a + b x^{2}}}{5 b} + \frac {e x^{6} \sqrt {a + b x^{2}}}{7 b} + \frac {f x^{8} \sqrt {a + b x^{2}}}{9 b} & \text {for}\: b \neq 0 \\\frac {\frac {c x^{4}}{4} + \frac {d x^{6}}{6} + \frac {e x^{8}}{8} + \frac {f x^{10}}{10}}{\sqrt {a}} & \text {otherwise} \end {cases} \]
Piecewise((128*a**4*f*sqrt(a + b*x**2)/(315*b**5) - 16*a**3*e*sqrt(a + b*x **2)/(35*b**4) - 64*a**3*f*x**2*sqrt(a + b*x**2)/(315*b**4) + 8*a**2*d*sqr t(a + b*x**2)/(15*b**3) + 8*a**2*e*x**2*sqrt(a + b*x**2)/(35*b**3) + 16*a* *2*f*x**4*sqrt(a + b*x**2)/(105*b**3) - 2*a*c*sqrt(a + b*x**2)/(3*b**2) - 4*a*d*x**2*sqrt(a + b*x**2)/(15*b**2) - 6*a*e*x**4*sqrt(a + b*x**2)/(35*b* *2) - 8*a*f*x**6*sqrt(a + b*x**2)/(63*b**2) + c*x**2*sqrt(a + b*x**2)/(3*b ) + d*x**4*sqrt(a + b*x**2)/(5*b) + e*x**6*sqrt(a + b*x**2)/(7*b) + f*x**8 *sqrt(a + b*x**2)/(9*b), Ne(b, 0)), ((c*x**4/4 + d*x**6/6 + e*x**8/8 + f*x **10/10)/sqrt(a), True))
Time = 0.21 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.57 \[ \int \frac {c x^3+d x^5+e x^7+f x^9}{\sqrt {a+b x^2}} \, dx=\frac {\sqrt {b x^{2} + a} f x^{8}}{9 \, b} + \frac {\sqrt {b x^{2} + a} e x^{6}}{7 \, b} - \frac {8 \, \sqrt {b x^{2} + a} a f x^{6}}{63 \, b^{2}} + \frac {\sqrt {b x^{2} + a} d x^{4}}{5 \, b} - \frac {6 \, \sqrt {b x^{2} + a} a e x^{4}}{35 \, b^{2}} + \frac {16 \, \sqrt {b x^{2} + a} a^{2} f x^{4}}{105 \, b^{3}} + \frac {\sqrt {b x^{2} + a} c x^{2}}{3 \, b} - \frac {4 \, \sqrt {b x^{2} + a} a d x^{2}}{15 \, b^{2}} + \frac {8 \, \sqrt {b x^{2} + a} a^{2} e x^{2}}{35 \, b^{3}} - \frac {64 \, \sqrt {b x^{2} + a} a^{3} f x^{2}}{315 \, b^{4}} - \frac {2 \, \sqrt {b x^{2} + a} a c}{3 \, b^{2}} + \frac {8 \, \sqrt {b x^{2} + a} a^{2} d}{15 \, b^{3}} - \frac {16 \, \sqrt {b x^{2} + a} a^{3} e}{35 \, b^{4}} + \frac {128 \, \sqrt {b x^{2} + a} a^{4} f}{315 \, b^{5}} \]
1/9*sqrt(b*x^2 + a)*f*x^8/b + 1/7*sqrt(b*x^2 + a)*e*x^6/b - 8/63*sqrt(b*x^ 2 + a)*a*f*x^6/b^2 + 1/5*sqrt(b*x^2 + a)*d*x^4/b - 6/35*sqrt(b*x^2 + a)*a* e*x^4/b^2 + 16/105*sqrt(b*x^2 + a)*a^2*f*x^4/b^3 + 1/3*sqrt(b*x^2 + a)*c*x ^2/b - 4/15*sqrt(b*x^2 + a)*a*d*x^2/b^2 + 8/35*sqrt(b*x^2 + a)*a^2*e*x^2/b ^3 - 64/315*sqrt(b*x^2 + a)*a^3*f*x^2/b^4 - 2/3*sqrt(b*x^2 + a)*a*c/b^2 + 8/15*sqrt(b*x^2 + a)*a^2*d/b^3 - 16/35*sqrt(b*x^2 + a)*a^3*e/b^4 + 128/315 *sqrt(b*x^2 + a)*a^4*f/b^5
Time = 0.29 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.16 \[ \int \frac {c x^3+d x^5+e x^7+f x^9}{\sqrt {a+b x^2}} \, dx=-\frac {{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \sqrt {b x^{2} + a}}{b^{5}} + \frac {105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3} c + 63 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2} d - 210 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a b^{2} d + 45 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b e - 189 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b e + 315 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b e + 35 \, {\left (b x^{2} + a\right )}^{\frac {9}{2}} f - 180 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a f + 378 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{2} f - 420 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{3} f}{315 \, b^{5}} \]
-(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*sqrt(b*x^2 + a)/b^5 + 1/315*(105* (b*x^2 + a)^(3/2)*b^3*c + 63*(b*x^2 + a)^(5/2)*b^2*d - 210*(b*x^2 + a)^(3/ 2)*a*b^2*d + 45*(b*x^2 + a)^(7/2)*b*e - 189*(b*x^2 + a)^(5/2)*a*b*e + 315* (b*x^2 + a)^(3/2)*a^2*b*e + 35*(b*x^2 + a)^(9/2)*f - 180*(b*x^2 + a)^(7/2) *a*f + 378*(b*x^2 + a)^(5/2)*a^2*f - 420*(b*x^2 + a)^(3/2)*a^3*f)/b^5
Time = 5.89 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.87 \[ \int \frac {c x^3+d x^5+e x^7+f x^9}{\sqrt {a+b x^2}} \, dx=\sqrt {b\,x^2+a}\,\left (\frac {128\,f\,a^4-144\,e\,a^3\,b+168\,d\,a^2\,b^2-210\,c\,a\,b^3}{315\,b^5}+\frac {x^4\,\left (48\,f\,a^2\,b^2-54\,e\,a\,b^3+63\,d\,b^4\right )}{315\,b^5}+\frac {f\,x^8}{9\,b}+\frac {x^6\,\left (45\,b^4\,e-40\,a\,b^3\,f\right )}{315\,b^5}+\frac {x^2\,\left (-64\,f\,a^3\,b+72\,e\,a^2\,b^2-84\,d\,a\,b^3+105\,c\,b^4\right )}{315\,b^5}\right ) \]