Integrand size = 25, antiderivative size = 213 \[ \int \frac {x^7 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {x^7 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^5 (7 a B-(A b-8 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x^3 (35 a B-6 (A b-8 a C) x)}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac {x (35 a B-8 (A b-8 a C) x)}{35 a b^4 \sqrt {a+b x^2}}-\frac {16 (A b-8 a C) \sqrt {a+b x^2}}{35 a b^5}+\frac {B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{9/2}} \]
-1/7*x^7*(B*a-(A*b-C*a)*x)/a/b/(b*x^2+a)^(7/2)-1/35*x^5*(7*B*a-(A*b-8*C*a) *x)/a/b^2/(b*x^2+a)^(5/2)-1/105*x^3*(35*B*a-6*(A*b-8*C*a)*x)/a/b^3/(b*x^2+ a)^(3/2)+B*arctanh(x*b^(1/2)/(b*x^2+a)^(1/2))/b^(9/2)-1/35*x*(35*B*a-8*(A* b-8*C*a)*x)/a/b^4/(b*x^2+a)^(1/2)-16/35*(A*b-8*C*a)*(b*x^2+a)^(1/2)/a/b^5
Time = 1.05 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.73 \[ \int \frac {x^7 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {384 a^4 C-3 a^3 b (16 A+7 x (5 B-64 C x))+14 a^2 b^2 x^2 (-12 A+5 x (-5 B+24 C x))+14 a b^3 x^4 (-15 A+x (-29 B+60 C x))+b^4 x^6 (-105 A+x (-176 B+105 C x))-105 \sqrt {b} B \left (a+b x^2\right )^{7/2} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{105 b^5 \left (a+b x^2\right )^{7/2}} \]
(384*a^4*C - 3*a^3*b*(16*A + 7*x*(5*B - 64*C*x)) + 14*a^2*b^2*x^2*(-12*A + 5*x*(-5*B + 24*C*x)) + 14*a*b^3*x^4*(-15*A + x*(-29*B + 60*C*x)) + b^4*x^ 6*(-105*A + x*(-176*B + 105*C*x)) - 105*Sqrt[b]*B*(a + b*x^2)^(7/2)*Log[-( Sqrt[b]*x) + Sqrt[a + b*x^2]])/(105*b^5*(a + b*x^2)^(7/2))
Time = 0.82 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2335, 25, 530, 25, 2345, 2345, 27, 455, 224, 219}
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.
| \(\displaystyle \int \frac {x^7 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2335 |
| \(\displaystyle -\frac {\int -\frac {x^6 (7 a B-(A b-8 a C) x)}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}-\frac {x^7 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {x^6 (7 a B-(A b-8 a C) x)}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}-\frac {x^7 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 530 |
| \(\displaystyle \frac {-\frac {\int -\frac {-5 a \left (A-\frac {8 a C}{b}\right ) x^5+\frac {35 a^2 B x^4}{b}+\frac {5 a^2 (A b-8 a C) x^3}{b^2}-\frac {35 a^3 B x^2}{b^2}-\frac {5 a^3 (A b-8 a C) x}{b^3}+\frac {7 a^4 B}{b^3}}{\left (b x^2+a\right )^{5/2}}dx}{5 a}-\frac {a^3 (-8 a C+A b+7 b B x)}{5 b^4 \left (a+b x^2\right )^{5/2}}}{7 a b}-\frac {x^7 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {-5 a \left (A-\frac {8 a C}{b}\right ) x^5+\frac {35 a^2 B x^4}{b}+\frac {5 a^2 (A b-8 a C) x^3}{b^2}-\frac {35 a^3 B x^2}{b^2}-\frac {5 a^3 (A b-8 a C) x}{b^3}+\frac {7 a^4 B}{b^3}}{\left (b x^2+a\right )^{5/2}}dx}{5 a}-\frac {a^3 (-8 a C+A b+7 b B x)}{5 b^4 \left (a+b x^2\right )^{5/2}}}{7 a b}-\frac {x^7 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {\frac {\frac {a^3 (15 (A b-8 a C)+77 b B x)}{3 b^4 \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {\frac {56 B a^4}{b^3}-\frac {105 B x^2 a^3}{b^2}-\frac {30 (A b-8 a C) x a^3}{b^3}+\frac {15 (A b-8 a C) x^3 a^2}{b^2}}{\left (b x^2+a\right )^{3/2}}dx}{3 a}}{5 a}-\frac {a^3 (-8 a C+A b+7 b B x)}{5 b^4 \left (a+b x^2\right )^{5/2}}}{7 a b}-\frac {x^7 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {\frac {\frac {a^3 (15 (A b-8 a C)+77 b B x)}{3 b^4 \left (a+b x^2\right )^{3/2}}-\frac {\frac {a^3 (45 (A b-8 a C)+161 b B x)}{b^4 \sqrt {a+b x^2}}-\frac {\int \frac {15 a^3 (7 a B-(A b-8 a C) x)}{b^3 \sqrt {b x^2+a}}dx}{a}}{3 a}}{5 a}-\frac {a^3 (-8 a C+A b+7 b B x)}{5 b^4 \left (a+b x^2\right )^{5/2}}}{7 a b}-\frac {x^7 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {a^3 (15 (A b-8 a C)+77 b B x)}{3 b^4 \left (a+b x^2\right )^{3/2}}-\frac {\frac {a^3 (45 (A b-8 a C)+161 b B x)}{b^4 \sqrt {a+b x^2}}-\frac {15 a^2 \int \frac {7 a B-(A b-8 a C) x}{\sqrt {b x^2+a}}dx}{b^3}}{3 a}}{5 a}-\frac {a^3 (-8 a C+A b+7 b B x)}{5 b^4 \left (a+b x^2\right )^{5/2}}}{7 a b}-\frac {x^7 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {\frac {\frac {a^3 (15 (A b-8 a C)+77 b B x)}{3 b^4 \left (a+b x^2\right )^{3/2}}-\frac {\frac {a^3 (45 (A b-8 a C)+161 b B x)}{b^4 \sqrt {a+b x^2}}-\frac {15 a^2 \left (7 a B \int \frac {1}{\sqrt {b x^2+a}}dx-\frac {\sqrt {a+b x^2} (A b-8 a C)}{b}\right )}{b^3}}{3 a}}{5 a}-\frac {a^3 (-8 a C+A b+7 b B x)}{5 b^4 \left (a+b x^2\right )^{5/2}}}{7 a b}-\frac {x^7 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {\frac {a^3 (15 (A b-8 a C)+77 b B x)}{3 b^4 \left (a+b x^2\right )^{3/2}}-\frac {\frac {a^3 (45 (A b-8 a C)+161 b B x)}{b^4 \sqrt {a+b x^2}}-\frac {15 a^2 \left (7 a B \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}-\frac {\sqrt {a+b x^2} (A b-8 a C)}{b}\right )}{b^3}}{3 a}}{5 a}-\frac {a^3 (-8 a C+A b+7 b B x)}{5 b^4 \left (a+b x^2\right )^{5/2}}}{7 a b}-\frac {x^7 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {a^3 (15 (A b-8 a C)+77 b B x)}{3 b^4 \left (a+b x^2\right )^{3/2}}-\frac {\frac {a^3 (45 (A b-8 a C)+161 b B x)}{b^4 \sqrt {a+b x^2}}-\frac {15 a^2 \left (\frac {7 a B \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{\sqrt {b}}-\frac {\sqrt {a+b x^2} (A b-8 a C)}{b}\right )}{b^3}}{3 a}}{5 a}-\frac {a^3 (-8 a C+A b+7 b B x)}{5 b^4 \left (a+b x^2\right )^{5/2}}}{7 a b}-\frac {x^7 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
-1/7*(x^7*(a*B - (A*b - a*C)*x))/(a*b*(a + b*x^2)^(7/2)) + (-1/5*(a^3*(A*b - 8*a*C + 7*b*B*x))/(b^4*(a + b*x^2)^(5/2)) + ((a^3*(15*(A*b - 8*a*C) + 7 7*b*B*x))/(3*b^4*(a + b*x^2)^(3/2)) - ((a^3*(45*(A*b - 8*a*C) + 161*b*B*x) )/(b^4*Sqrt[a + b*x^2]) - (15*a^2*(-(((A*b - 8*a*C)*Sqrt[a + b*x^2])/b) + (7*a*B*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/Sqrt[b]))/b^3)/(3*a))/(5*a))/ (7*a*b)
3.1.47.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symb ol] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Co eff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Po lynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x )*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Qx + e*(2*p + 3), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 0] && LtQ[p, -1] && EqQ[n, 1] && IntegerQ[2*p]
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]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[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[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Time = 3.71 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.38
| method | result | size |
| default | \(C \left (\frac {x^{8}}{b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {8 a \left (-\frac {x^{6}}{b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {6 a \left (-\frac {x^{4}}{3 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {4 a \left (-\frac {x^{2}}{5 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {2 a}{35 b^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}\right )}{3 b}\right )}{b}\right )}{b}\right )+B \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )+A \left (-\frac {x^{6}}{b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {6 a \left (-\frac {x^{4}}{3 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {4 a \left (-\frac {x^{2}}{5 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {2 a}{35 b^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}\right )}{3 b}\right )}{b}\right )\) | \(295\) |
| risch | \(\text {Expression too large to display}\) | \(2131\) |
C*(x^8/b/(b*x^2+a)^(7/2)-8*a/b*(-x^6/b/(b*x^2+a)^(7/2)+6*a/b*(-1/3*x^4/b/( b*x^2+a)^(7/2)+4/3*a/b*(-1/5*x^2/b/(b*x^2+a)^(7/2)-2/35*a/b^2/(b*x^2+a)^(7 /2)))))+B*(-1/7*x^7/b/(b*x^2+a)^(7/2)+1/b*(-1/5*x^5/b/(b*x^2+a)^(5/2)+1/b* (-1/3*x^3/b/(b*x^2+a)^(3/2)+1/b*(-x/b/(b*x^2+a)^(1/2)+1/b^(3/2)*ln(x*b^(1/ 2)+(b*x^2+a)^(1/2))))))+A*(-x^6/b/(b*x^2+a)^(7/2)+6*a/b*(-1/3*x^4/b/(b*x^2 +a)^(7/2)+4/3*a/b*(-1/5*x^2/b/(b*x^2+a)^(7/2)-2/35*a/b^2/(b*x^2+a)^(7/2))) )
Time = 0.31 (sec) , antiderivative size = 522, normalized size of antiderivative = 2.45 \[ \int \frac {x^7 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\left [\frac {105 \, {\left (B b^{4} x^{8} + 4 \, B a b^{3} x^{6} + 6 \, B a^{2} b^{2} x^{4} + 4 \, B a^{3} b x^{2} + B a^{4}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (105 \, C b^{4} x^{8} - 176 \, B b^{4} x^{7} - 406 \, B a b^{3} x^{5} - 350 \, B a^{2} b^{2} x^{3} + 105 \, {\left (8 \, C a b^{3} - A b^{4}\right )} x^{6} - 105 \, B a^{3} b x + 384 \, C a^{4} - 48 \, A a^{3} b + 210 \, {\left (8 \, C a^{2} b^{2} - A a b^{3}\right )} x^{4} + 168 \, {\left (8 \, C a^{3} b - A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{210 \, {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}}, -\frac {105 \, {\left (B b^{4} x^{8} + 4 \, B a b^{3} x^{6} + 6 \, B a^{2} b^{2} x^{4} + 4 \, B a^{3} b x^{2} + B a^{4}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (105 \, C b^{4} x^{8} - 176 \, B b^{4} x^{7} - 406 \, B a b^{3} x^{5} - 350 \, B a^{2} b^{2} x^{3} + 105 \, {\left (8 \, C a b^{3} - A b^{4}\right )} x^{6} - 105 \, B a^{3} b x + 384 \, C a^{4} - 48 \, A a^{3} b + 210 \, {\left (8 \, C a^{2} b^{2} - A a b^{3}\right )} x^{4} + 168 \, {\left (8 \, C a^{3} b - A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}}\right ] \]
[1/210*(105*(B*b^4*x^8 + 4*B*a*b^3*x^6 + 6*B*a^2*b^2*x^4 + 4*B*a^3*b*x^2 + B*a^4)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(105*C *b^4*x^8 - 176*B*b^4*x^7 - 406*B*a*b^3*x^5 - 350*B*a^2*b^2*x^3 + 105*(8*C* a*b^3 - A*b^4)*x^6 - 105*B*a^3*b*x + 384*C*a^4 - 48*A*a^3*b + 210*(8*C*a^2 *b^2 - A*a*b^3)*x^4 + 168*(8*C*a^3*b - A*a^2*b^2)*x^2)*sqrt(b*x^2 + a))/(b ^9*x^8 + 4*a*b^8*x^6 + 6*a^2*b^7*x^4 + 4*a^3*b^6*x^2 + a^4*b^5), -1/105*(1 05*(B*b^4*x^8 + 4*B*a*b^3*x^6 + 6*B*a^2*b^2*x^4 + 4*B*a^3*b*x^2 + B*a^4)*s qrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (105*C*b^4*x^8 - 176*B*b^4*x^ 7 - 406*B*a*b^3*x^5 - 350*B*a^2*b^2*x^3 + 105*(8*C*a*b^3 - A*b^4)*x^6 - 10 5*B*a^3*b*x + 384*C*a^4 - 48*A*a^3*b + 210*(8*C*a^2*b^2 - A*a*b^3)*x^4 + 1 68*(8*C*a^3*b - A*a^2*b^2)*x^2)*sqrt(b*x^2 + a))/(b^9*x^8 + 4*a*b^8*x^6 + 6*a^2*b^7*x^4 + 4*a^3*b^6*x^2 + a^4*b^5)]
Time = 34.30 (sec) , antiderivative size = 3806, normalized size of antiderivative = 17.87 \[ \int \frac {x^7 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \]
A*Piecewise((-16*a**3/(35*a**3*b**4*sqrt(a + b*x**2) + 105*a**2*b**5*x**2* sqrt(a + b*x**2) + 105*a*b**6*x**4*sqrt(a + b*x**2) + 35*b**7*x**6*sqrt(a + b*x**2)) - 56*a**2*b*x**2/(35*a**3*b**4*sqrt(a + b*x**2) + 105*a**2*b**5 *x**2*sqrt(a + b*x**2) + 105*a*b**6*x**4*sqrt(a + b*x**2) + 35*b**7*x**6*s qrt(a + b*x**2)) - 70*a*b**2*x**4/(35*a**3*b**4*sqrt(a + b*x**2) + 105*a** 2*b**5*x**2*sqrt(a + b*x**2) + 105*a*b**6*x**4*sqrt(a + b*x**2) + 35*b**7* x**6*sqrt(a + b*x**2)) - 35*b**3*x**6/(35*a**3*b**4*sqrt(a + b*x**2) + 105 *a**2*b**5*x**2*sqrt(a + b*x**2) + 105*a*b**6*x**4*sqrt(a + b*x**2) + 35*b **7*x**6*sqrt(a + b*x**2)), Ne(b, 0)), (x**8/(8*a**(9/2)), True)) + B*(105 *a**(205/2)*b**45*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(105*a**(205 /2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a** (199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x* *8*sqrt(1 + b*x**2/a) + 630*a**(195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) + 630*a**(203/2)*b* *46*x**2*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(105*a**(205/2)*b**(9 9/2)*sqrt(1 + b*x**2/a) + 630*a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a ) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a**(199/2)*b **(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105...
Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (189) = 378\).
Time = 0.22 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.04 \[ \int \frac {x^7 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {C x^{8}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {1}{35} \, {\left (\frac {35 \, x^{6}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {70 \, a x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} + \frac {56 \, a^{2} x^{2}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {16 \, a^{3}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}}\right )} B x + \frac {8 \, C a x^{6}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {A x^{6}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {B x {\left (\frac {15 \, x^{4}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b} + \frac {20 \, a x^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {8 \, a^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}}\right )}}{15 \, b} - \frac {B x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{3 \, b^{2}} + \frac {16 \, C a^{2} x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} - \frac {2 \, A a x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {B a x^{3}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} + \frac {64 \, C a^{3} x^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}} - \frac {8 \, A a^{2} x^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {139 \, B x}{105 \, \sqrt {b x^{2} + a} b^{4}} + \frac {17 \, B a x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{4}} - \frac {29 \, B a^{2} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{4}} + \frac {B \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {9}{2}}} + \frac {128 \, C a^{4}}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{5}} - \frac {16 \, A a^{3}}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}} \]
C*x^8/((b*x^2 + a)^(7/2)*b) - 1/35*(35*x^6/((b*x^2 + a)^(7/2)*b) + 70*a*x^ 4/((b*x^2 + a)^(7/2)*b^2) + 56*a^2*x^2/((b*x^2 + a)^(7/2)*b^3) + 16*a^3/(( b*x^2 + a)^(7/2)*b^4))*B*x + 8*C*a*x^6/((b*x^2 + a)^(7/2)*b^2) - A*x^6/((b *x^2 + a)^(7/2)*b) - 1/15*B*x*(15*x^4/((b*x^2 + a)^(5/2)*b) + 20*a*x^2/((b *x^2 + a)^(5/2)*b^2) + 8*a^2/((b*x^2 + a)^(5/2)*b^3))/b - 1/3*B*x*(3*x^2/( (b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b^2 + 16*C*a^2*x^4/((b *x^2 + a)^(7/2)*b^3) - 2*A*a*x^4/((b*x^2 + a)^(7/2)*b^2) - B*a*x^3/((b*x^2 + a)^(5/2)*b^3) + 64/5*C*a^3*x^2/((b*x^2 + a)^(7/2)*b^4) - 8/5*A*a^2*x^2/ ((b*x^2 + a)^(7/2)*b^3) + 139/105*B*x/(sqrt(b*x^2 + a)*b^4) + 17/105*B*a*x /((b*x^2 + a)^(3/2)*b^4) - 29/35*B*a^2*x/((b*x^2 + a)^(5/2)*b^4) + B*arcsi nh(b*x/sqrt(a*b))/b^(9/2) + 128/35*C*a^4/((b*x^2 + a)^(7/2)*b^5) - 16/35*A *a^3/((b*x^2 + a)^(7/2)*b^4)
Time = 0.35 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.96 \[ \int \frac {x^7 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {{\left ({\left ({\left ({\left ({\left ({\left ({\left (\frac {105 \, C x}{b} - \frac {176 \, B}{b}\right )} x + \frac {105 \, {\left (8 \, C a^{4} b^{7} - A a^{3} b^{8}\right )}}{a^{3} b^{9}}\right )} x - \frac {406 \, B a}{b^{2}}\right )} x + \frac {210 \, {\left (8 \, C a^{5} b^{6} - A a^{4} b^{7}\right )}}{a^{3} b^{9}}\right )} x - \frac {350 \, B a^{2}}{b^{3}}\right )} x + \frac {168 \, {\left (8 \, C a^{6} b^{5} - A a^{5} b^{6}\right )}}{a^{3} b^{9}}\right )} x - \frac {105 \, B a^{3}}{b^{4}}\right )} x + \frac {48 \, {\left (8 \, C a^{7} b^{4} - A a^{6} b^{5}\right )}}{a^{3} b^{9}}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} - \frac {B \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{b^{\frac {9}{2}}} \]
1/105*((((((((105*C*x/b - 176*B/b)*x + 105*(8*C*a^4*b^7 - A*a^3*b^8)/(a^3* b^9))*x - 406*B*a/b^2)*x + 210*(8*C*a^5*b^6 - A*a^4*b^7)/(a^3*b^9))*x - 35 0*B*a^2/b^3)*x + 168*(8*C*a^6*b^5 - A*a^5*b^6)/(a^3*b^9))*x - 105*B*a^3/b^ 4)*x + 48*(8*C*a^7*b^4 - A*a^6*b^5)/(a^3*b^9))/(b*x^2 + a)^(7/2) - B*log(a bs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)
Timed out. \[ \int \frac {x^7 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\int \frac {x^7\,\left (C\,x^2+B\,x+A\right )}{{\left (b\,x^2+a\right )}^{9/2}} \,d x \]