Integrand size = 23, antiderivative size = 119 \[ \int \frac {x \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {x (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {5 A b+2 a C-b B x}{35 a b^2 \left (a+b x^2\right )^{5/2}}+\frac {4 B x}{105 a^2 b \left (a+b x^2\right )^{3/2}}+\frac {8 B x}{105 a^3 b \sqrt {a+b x^2}} \]
-1/7*x*(B*a-(A*b-C*a)*x)/a/b/(b*x^2+a)^(7/2)+1/35*(B*b*x-5*A*b-2*C*a)/a/b^ 2/(b*x^2+a)^(5/2)+4/105*B*x/a^2/b/(b*x^2+a)^(3/2)+8/105*B*x/a^3/b/(b*x^2+a )^(1/2)
Time = 0.79 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.64 \[ \int \frac {x \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {-15 a^3 A b-6 a^4 C-21 a^3 b C x^2+35 a^2 b^2 B x^3+28 a b^3 B x^5+8 b^4 B x^7}{105 a^3 b^2 \left (a+b x^2\right )^{7/2}} \]
(-15*a^3*A*b - 6*a^4*C - 21*a^3*b*C*x^2 + 35*a^2*b^2*B*x^3 + 28*a*b^3*B*x^ 5 + 8*b^4*B*x^7)/(105*a^3*b^2*(a + b*x^2)^(7/2))
Time = 0.28 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2335, 25, 454, 209, 208}
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.
| \(\displaystyle \int \frac {x \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2335 |
| \(\displaystyle -\frac {\int -\frac {a B+(5 A b+2 a C) x}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}-\frac {x (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {a B+(5 A b+2 a C) x}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}-\frac {x (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 454 |
| \(\displaystyle \frac {\frac {4}{5} B \int \frac {1}{\left (b x^2+a\right )^{5/2}}dx-\frac {2 a C+5 A b-b B x}{5 b \left (a+b x^2\right )^{5/2}}}{7 a b}-\frac {x (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {\frac {4}{5} B \left (\frac {2 \int \frac {1}{\left (b x^2+a\right )^{3/2}}dx}{3 a}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )-\frac {2 a C+5 A b-b B x}{5 b \left (a+b x^2\right )^{5/2}}}{7 a b}-\frac {x (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {\frac {4}{5} B \left (\frac {2 x}{3 a^2 \sqrt {a+b x^2}}+\frac {x}{3 a \left (a+b x^2\right )^{3/2}}\right )-\frac {2 a C+5 A b-b B x}{5 b \left (a+b x^2\right )^{5/2}}}{7 a b}-\frac {x (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
-1/7*(x*(a*B - (A*b - a*C)*x))/(a*b*(a + b*x^2)^(7/2)) + (-1/5*(5*A*b + 2* a*C - b*B*x)/(b*(a + b*x^2)^(5/2)) + (4*B*(x/(3*a*(a + b*x^2)^(3/2)) + (2* x)/(3*a^2*Sqrt[a + b*x^2])))/5)/(7*a*b)
3.1.53.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*d - b*c*x)/(2*a*b*(p + 1)))*(a + b*x^2)^(p + 1), x] + Simp[c*((2*p + 3)/(2*a *(p + 1))) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && L tQ[p, -1] && NeQ[p, -3/2]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq , a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(c*x)^m*(a + b*x^2)^(p + 1)*((a*g - b*f*x)/(2*a*b*(p + 1))), x] + Simp[c/(2*a*b*(p + 1)) Int[(c*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSu m[2*a*b*(p + 1)*x*Q - a*g*m + b*f*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && GtQ[m, 0]
Time = 3.59 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.61
| method | result | size |
| gosper | \(-\frac {-8 x^{7} B \,b^{4}-28 x^{5} B a \,b^{3}-35 B \,a^{2} b^{2} x^{3}+21 C \,a^{3} b \,x^{2}+15 A \,a^{3} b +6 C \,a^{4}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{3} b^{2}}\) | \(73\) |
| trager | \(-\frac {-8 x^{7} B \,b^{4}-28 x^{5} B a \,b^{3}-35 B \,a^{2} b^{2} x^{3}+21 C \,a^{3} b \,x^{2}+15 A \,a^{3} b +6 C \,a^{4}}{105 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{3} b^{2}}\) | \(73\) |
| default | \(C \left (-\frac {x^{2}}{5 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {2 a}{35 b^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}\right )+B \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )-\frac {A}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}\) | \(149\) |
-1/105*(-8*B*b^4*x^7-28*B*a*b^3*x^5-35*B*a^2*b^2*x^3+21*C*a^3*b*x^2+15*A*a ^3*b+6*C*a^4)/(b*x^2+a)^(7/2)/a^3/b^2
Time = 0.40 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left (8 \, B b^{4} x^{7} + 28 \, B a b^{3} x^{5} + 35 \, B a^{2} b^{2} x^{3} - 21 \, C a^{3} b x^{2} - 6 \, C a^{4} - 15 \, A a^{3} b\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a^{3} b^{6} x^{8} + 4 \, a^{4} b^{5} x^{6} + 6 \, a^{5} b^{4} x^{4} + 4 \, a^{6} b^{3} x^{2} + a^{7} b^{2}\right )}} \]
1/105*(8*B*b^4*x^7 + 28*B*a*b^3*x^5 + 35*B*a^2*b^2*x^3 - 21*C*a^3*b*x^2 - 6*C*a^4 - 15*A*a^3*b)*sqrt(b*x^2 + a)/(a^3*b^6*x^8 + 4*a^4*b^5*x^6 + 6*a^5 *b^4*x^4 + 4*a^6*b^3*x^2 + a^7*b^2)
Time = 19.63 (sec) , antiderivative size = 796, normalized size of antiderivative = 6.69 \[ \int \frac {x \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=A \left (\begin {cases} - \frac {1}{7 a^{3} b \sqrt {a + b x^{2}} + 21 a^{2} b^{2} x^{2} \sqrt {a + b x^{2}} + 21 a b^{3} x^{4} \sqrt {a + b x^{2}} + 7 b^{4} x^{6} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {9}{2}}} & \text {otherwise} \end {cases}\right ) + B \left (\frac {35 a^{5} x^{3}}{105 a^{\frac {19}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {17}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 630 a^{\frac {15}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {13}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {11}{2}} b^{4} x^{8} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {63 a^{4} b x^{5}}{105 a^{\frac {19}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {17}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 630 a^{\frac {15}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {13}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {11}{2}} b^{4} x^{8} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {36 a^{3} b^{2} x^{7}}{105 a^{\frac {19}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {17}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 630 a^{\frac {15}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {13}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {11}{2}} b^{4} x^{8} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {8 a^{2} b^{3} x^{9}}{105 a^{\frac {19}{2}} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {17}{2}} b x^{2} \sqrt {1 + \frac {b x^{2}}{a}} + 630 a^{\frac {15}{2}} b^{2} x^{4} \sqrt {1 + \frac {b x^{2}}{a}} + 420 a^{\frac {13}{2}} b^{3} x^{6} \sqrt {1 + \frac {b x^{2}}{a}} + 105 a^{\frac {11}{2}} b^{4} x^{8} \sqrt {1 + \frac {b x^{2}}{a}}}\right ) + C \left (\begin {cases} - \frac {2 a}{35 a^{3} b^{2} \sqrt {a + b x^{2}} + 105 a^{2} b^{3} x^{2} \sqrt {a + b x^{2}} + 105 a b^{4} x^{4} \sqrt {a + b x^{2}} + 35 b^{5} x^{6} \sqrt {a + b x^{2}}} - \frac {7 b x^{2}}{35 a^{3} b^{2} \sqrt {a + b x^{2}} + 105 a^{2} b^{3} x^{2} \sqrt {a + b x^{2}} + 105 a b^{4} x^{4} \sqrt {a + b x^{2}} + 35 b^{5} x^{6} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {9}{2}}} & \text {otherwise} \end {cases}\right ) \]
A*Piecewise((-1/(7*a**3*b*sqrt(a + b*x**2) + 21*a**2*b**2*x**2*sqrt(a + b* x**2) + 21*a*b**3*x**4*sqrt(a + b*x**2) + 7*b**4*x**6*sqrt(a + b*x**2)), N e(b, 0)), (x**2/(2*a**(9/2)), True)) + B*(35*a**5*x**3/(105*a**(19/2)*sqrt (1 + b*x**2/a) + 420*a**(17/2)*b*x**2*sqrt(1 + b*x**2/a) + 630*a**(15/2)*b **2*x**4*sqrt(1 + b*x**2/a) + 420*a**(13/2)*b**3*x**6*sqrt(1 + b*x**2/a) + 105*a**(11/2)*b**4*x**8*sqrt(1 + b*x**2/a)) + 63*a**4*b*x**5/(105*a**(19/ 2)*sqrt(1 + b*x**2/a) + 420*a**(17/2)*b*x**2*sqrt(1 + b*x**2/a) + 630*a**( 15/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 420*a**(13/2)*b**3*x**6*sqrt(1 + b*x* *2/a) + 105*a**(11/2)*b**4*x**8*sqrt(1 + b*x**2/a)) + 36*a**3*b**2*x**7/(1 05*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a**(17/2)*b*x**2*sqrt(1 + b*x**2/a) + 630*a**(15/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 420*a**(13/2)*b**3*x**6*sqr t(1 + b*x**2/a) + 105*a**(11/2)*b**4*x**8*sqrt(1 + b*x**2/a)) + 8*a**2*b** 3*x**9/(105*a**(19/2)*sqrt(1 + b*x**2/a) + 420*a**(17/2)*b*x**2*sqrt(1 + b *x**2/a) + 630*a**(15/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 420*a**(13/2)*b**3 *x**6*sqrt(1 + b*x**2/a) + 105*a**(11/2)*b**4*x**8*sqrt(1 + b*x**2/a))) + C*Piecewise((-2*a/(35*a**3*b**2*sqrt(a + b*x**2) + 105*a**2*b**3*x**2*sqrt (a + b*x**2) + 105*a*b**4*x**4*sqrt(a + b*x**2) + 35*b**5*x**6*sqrt(a + b* x**2)) - 7*b*x**2/(35*a**3*b**2*sqrt(a + b*x**2) + 105*a**2*b**3*x**2*sqrt (a + b*x**2) + 105*a*b**4*x**4*sqrt(a + b*x**2) + 35*b**5*x**6*sqrt(a + b* x**2)), Ne(b, 0)), (x**4/(4*a**(9/2)), True))
Time = 0.19 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.03 \[ \int \frac {x \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {C x^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {B x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {8 \, B x}{105 \, \sqrt {b x^{2} + a} a^{3} b} + \frac {4 \, B x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} b} + \frac {B x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a b} - \frac {2 \, C a}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {A}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} \]
-1/5*C*x^2/((b*x^2 + a)^(7/2)*b) - 1/7*B*x/((b*x^2 + a)^(7/2)*b) + 8/105*B *x/(sqrt(b*x^2 + a)*a^3*b) + 4/105*B*x/((b*x^2 + a)^(3/2)*a^2*b) + 1/35*B* x/((b*x^2 + a)^(5/2)*a*b) - 2/35*C*a/((b*x^2 + a)^(7/2)*b^2) - 1/7*A/((b*x ^2 + a)^(7/2)*b)
Time = 0.32 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.69 \[ \int \frac {x \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left ({\left (4 \, {\left (\frac {2 \, B b^{2} x^{2}}{a^{3}} + \frac {7 \, B b}{a^{2}}\right )} x^{2} + \frac {35 \, B}{a}\right )} x - \frac {21 \, C}{b}\right )} x^{2} - \frac {3 \, {\left (2 \, C a^{4} b + 5 \, A a^{3} b^{2}\right )}}{a^{3} b^{3}}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} \]
1/105*(((4*(2*B*b^2*x^2/a^3 + 7*B*b/a^2)*x^2 + 35*B/a)*x - 21*C/b)*x^2 - 3 *(2*C*a^4*b + 5*A*a^3*b^2)/(a^3*b^3))/(b*x^2 + a)^(7/2)
Time = 5.76 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.83 \[ \int \frac {x \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {8\,B\,x}{105\,a^3\,b\,\sqrt {b\,x^2+a}}-\frac {\frac {A}{7\,b}-\frac {C\,a}{7\,b^2}+\frac {B\,x}{7\,b}}{{\left (b\,x^2+a\right )}^{7/2}}-\frac {\frac {C}{5\,b^2}-\frac {B\,x}{35\,a\,b}}{{\left (b\,x^2+a\right )}^{5/2}}+\frac {4\,B\,x}{105\,a^2\,b\,{\left (b\,x^2+a\right )}^{3/2}} \]