Integrand size = 25, antiderivative size = 172 \[ \int \frac {x^6 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {a^2 (a B-(A b-a C) x)}{7 b^4 \left (a+b x^2\right )^{7/2}}-\frac {a (21 a B-(15 A b-22 a C) x)}{35 b^4 \left (a+b x^2\right )^{5/2}}+\frac {105 a B-(45 A b-122 a C) x}{105 b^4 \left (a+b x^2\right )^{3/2}}-\frac {105 a B-(15 A b-176 a C) x}{105 a b^4 \sqrt {a+b x^2}}+\frac {C \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{9/2}} \] Output:
1/7*a^2*(B*a-(A*b-C*a)*x)/b^4/(b*x^2+a)^(7/2)-1/35*a*(21*B*a-(15*A*b-22*C* a)*x)/b^4/(b*x^2+a)^(5/2)+1/105*(105*B*a-(45*A*b-122*C*a)*x)/b^4/(b*x^2+a) ^(3/2)-1/105*(105*B*a-(15*A*b-176*C*a)*x)/a/b^4/(b*x^2+a)^(1/2)+C*arctanh( b^(1/2)*x/(b*x^2+a)^(1/2))/b^(9/2)
Time = 1.24 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.74 \[ \int \frac {x^6 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {15 A b^4 x^7-14 a^3 b x^2 (12 B+25 C x)-14 a^2 b^2 x^4 (15 B+29 C x)-3 a^4 (16 B+35 C x)-a b^3 x^6 (105 B+176 C x)}{105 a b^4 \left (a+b x^2\right )^{7/2}}-\frac {C \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{b^{9/2}} \] Input:
Integrate[(x^6*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]
Output:
(15*A*b^4*x^7 - 14*a^3*b*x^2*(12*B + 25*C*x) - 14*a^2*b^2*x^4*(15*B + 29*C *x) - 3*a^4*(16*B + 35*C*x) - a*b^3*x^6*(105*B + 176*C*x))/(105*a*b^4*(a + b*x^2)^(7/2)) - (C*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/b^(9/2)
Time = 0.60 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2335, 25, 27, 530, 25, 2345, 2345, 27, 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.
| \(\displaystyle \int \frac {x^6 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx\) |
| \(\Big \downarrow \) 2335 |
| \(\displaystyle -\frac {\int -\frac {a x^5 (6 B+7 C x)}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}-\frac {x^6 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {a x^5 (6 B+7 C x)}{\left (b x^2+a\right )^{7/2}}dx}{7 a b}-\frac {x^6 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {x^5 (6 B+7 C x)}{\left (b x^2+a\right )^{7/2}}dx}{7 b}-\frac {x^6 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 530 |
| \(\displaystyle \frac {-\frac {\int -\frac {\frac {35 a C x^4}{b}+\frac {30 a B x^3}{b}-\frac {35 a^2 C x^2}{b^2}-\frac {30 a^2 B x}{b^2}+\frac {7 a^3 C}{b^3}}{\left (b x^2+a\right )^{5/2}}dx}{5 a}-\frac {a^2 (6 B+7 C x)}{5 b^3 \left (a+b x^2\right )^{5/2}}}{7 b}-\frac {x^6 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\frac {35 a C x^4}{b}+\frac {30 a B x^3}{b}-\frac {35 a^2 C x^2}{b^2}-\frac {30 a^2 B x}{b^2}+\frac {7 a^3 C}{b^3}}{\left (b x^2+a\right )^{5/2}}dx}{5 a}-\frac {a^2 (6 B+7 C x)}{5 b^3 \left (a+b x^2\right )^{5/2}}}{7 b}-\frac {x^6 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {\frac {\frac {a^2 (60 B+77 C x)}{3 b^3 \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {\frac {56 C a^3}{b^3}-\frac {105 C x^2 a^2}{b^2}-\frac {90 B x a^2}{b^2}}{\left (b x^2+a\right )^{3/2}}dx}{3 a}}{5 a}-\frac {a^2 (6 B+7 C x)}{5 b^3 \left (a+b x^2\right )^{5/2}}}{7 b}-\frac {x^6 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {\frac {\frac {a^2 (60 B+77 C x)}{3 b^3 \left (a+b x^2\right )^{3/2}}-\frac {\frac {a^2 (90 B+161 C x)}{b^3 \sqrt {a+b x^2}}-\frac {\int \frac {105 a^3 C}{b^3 \sqrt {b x^2+a}}dx}{a}}{3 a}}{5 a}-\frac {a^2 (6 B+7 C x)}{5 b^3 \left (a+b x^2\right )^{5/2}}}{7 b}-\frac {x^6 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {a^2 (60 B+77 C x)}{3 b^3 \left (a+b x^2\right )^{3/2}}-\frac {\frac {a^2 (90 B+161 C x)}{b^3 \sqrt {a+b x^2}}-\frac {105 a^2 C \int \frac {1}{\sqrt {b x^2+a}}dx}{b^3}}{3 a}}{5 a}-\frac {a^2 (6 B+7 C x)}{5 b^3 \left (a+b x^2\right )^{5/2}}}{7 b}-\frac {x^6 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {\frac {a^2 (60 B+77 C x)}{3 b^3 \left (a+b x^2\right )^{3/2}}-\frac {\frac {a^2 (90 B+161 C x)}{b^3 \sqrt {a+b x^2}}-\frac {105 a^2 C \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{b^3}}{3 a}}{5 a}-\frac {a^2 (6 B+7 C x)}{5 b^3 \left (a+b x^2\right )^{5/2}}}{7 b}-\frac {x^6 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {a^2 (60 B+77 C x)}{3 b^3 \left (a+b x^2\right )^{3/2}}-\frac {\frac {a^2 (90 B+161 C x)}{b^3 \sqrt {a+b x^2}}-\frac {105 a^2 C \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{7/2}}}{3 a}}{5 a}-\frac {a^2 (6 B+7 C x)}{5 b^3 \left (a+b x^2\right )^{5/2}}}{7 b}-\frac {x^6 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}\) |
Input:
Int[(x^6*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]
Output:
-1/7*(x^6*(a*B - (A*b - a*C)*x))/(a*b*(a + b*x^2)^(7/2)) + (-1/5*(a^2*(6*B + 7*C*x))/(b^3*(a + b*x^2)^(5/2)) + ((a^2*(60*B + 77*C*x))/(3*b^3*(a + b* x^2)^(3/2)) - ((a^2*(90*B + 161*C*x))/(b^3*Sqrt[a + b*x^2]) - (105*a^2*C*A rcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/b^(7/2))/(3*a))/(5*a))/(7*b)
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[(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]
Leaf count of result is larger than twice the leaf count of optimal. \(333\) vs. \(2(150)=300\).
Time = 0.57 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.94
| method | result | size |
| default | \(A \left (-\frac {x^{5}}{2 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {5 a \left (-\frac {x^{3}}{4 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {3 a \left (-\frac {x}{6 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {a \left (\frac {x}{7 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 a \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {8 x}{15 a^{2} \sqrt {b \,x^{2}+a}}\right )}{7 a}}{a}\right )}{6 b}\right )}{4 b}\right )}{2 b}\right )+B \left (-\frac {x^{6}}{b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {6 a \left (-\frac {x^{4}}{3 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {4 a \left (-\frac {x^{2}}{5 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {2 a}{35 b^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}\right )}{3 b}\right )}{b}\right )+C \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )\) | \(334\) |
Input:
int(x^6*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x,method=_RETURNVERBOSE)
Output:
A*(-1/2*x^5/b/(b*x^2+a)^(7/2)+5/2*a/b*(-1/4*x^3/b/(b*x^2+a)^(7/2)+3/4*a/b* (-1/6*x/b/(b*x^2+a)^(7/2)+1/6*a/b*(1/7*x/a/(b*x^2+a)^(7/2)+6/7/a*(1/5*x/a/ (b*x^2+a)^(5/2)+4/5/a*(1/3*x/a/(b*x^2+a)^(3/2)+2/3*x/a^2/(b*x^2+a)^(1/2))) ))))+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))))+C*(-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(b^(1/2)*x+(b*x^2+a)^(1/2))))))
Time = 0.11 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.72 \[ \int \frac {x^6 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\left [\frac {105 \, {\left (C a b^{4} x^{8} + 4 \, C a^{2} b^{3} x^{6} + 6 \, C a^{3} b^{2} x^{4} + 4 \, C a^{4} b x^{2} + C a^{5}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - 2 \, {\left (105 \, B a b^{4} x^{6} + 406 \, C a^{2} b^{3} x^{5} + 210 \, B a^{2} b^{3} x^{4} + 350 \, C a^{3} b^{2} x^{3} + 168 \, B a^{3} b^{2} x^{2} + {\left (176 \, C a b^{4} - 15 \, A b^{5}\right )} x^{7} + 105 \, C a^{4} b x + 48 \, B a^{4} b\right )} \sqrt {b x^{2} + a}}{210 \, {\left (a b^{9} x^{8} + 4 \, a^{2} b^{8} x^{6} + 6 \, a^{3} b^{7} x^{4} + 4 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}}, -\frac {105 \, {\left (C a b^{4} x^{8} + 4 \, C a^{2} b^{3} x^{6} + 6 \, C a^{3} b^{2} x^{4} + 4 \, C a^{4} b x^{2} + C a^{5}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) + {\left (105 \, B a b^{4} x^{6} + 406 \, C a^{2} b^{3} x^{5} + 210 \, B a^{2} b^{3} x^{4} + 350 \, C a^{3} b^{2} x^{3} + 168 \, B a^{3} b^{2} x^{2} + {\left (176 \, C a b^{4} - 15 \, A b^{5}\right )} x^{7} + 105 \, C a^{4} b x + 48 \, B a^{4} b\right )} \sqrt {b x^{2} + a}}{105 \, {\left (a b^{9} x^{8} + 4 \, a^{2} b^{8} x^{6} + 6 \, a^{3} b^{7} x^{4} + 4 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}}\right ] \] Input:
integrate(x^6*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x, algorithm="fricas")
Output:
[1/210*(105*(C*a*b^4*x^8 + 4*C*a^2*b^3*x^6 + 6*C*a^3*b^2*x^4 + 4*C*a^4*b*x ^2 + C*a^5)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(1 05*B*a*b^4*x^6 + 406*C*a^2*b^3*x^5 + 210*B*a^2*b^3*x^4 + 350*C*a^3*b^2*x^3 + 168*B*a^3*b^2*x^2 + (176*C*a*b^4 - 15*A*b^5)*x^7 + 105*C*a^4*b*x + 48*B *a^4*b)*sqrt(b*x^2 + a))/(a*b^9*x^8 + 4*a^2*b^8*x^6 + 6*a^3*b^7*x^4 + 4*a^ 4*b^6*x^2 + a^5*b^5), -1/105*(105*(C*a*b^4*x^8 + 4*C*a^2*b^3*x^6 + 6*C*a^3 *b^2*x^4 + 4*C*a^4*b*x^2 + C*a^5)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (105*B*a*b^4*x^6 + 406*C*a^2*b^3*x^5 + 210*B*a^2*b^3*x^4 + 350*C*a^3 *b^2*x^3 + 168*B*a^3*b^2*x^2 + (176*C*a*b^4 - 15*A*b^5)*x^7 + 105*C*a^4*b* x + 48*B*a^4*b)*sqrt(b*x^2 + a))/(a*b^9*x^8 + 4*a^2*b^8*x^6 + 6*a^3*b^7*x^ 4 + 4*a^4*b^6*x^2 + a^5*b^5)]
Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (151) = 302\).
Time = 38.26 (sec) , antiderivative size = 3448, normalized size of antiderivative = 20.05 \[ \int \frac {x^6 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\text {Too large to display} \] Input:
integrate(x**6*(C*x**2+B*x+A)/(b*x**2+a)**(9/2),x)
Output:
A*x**7/(7*a**(9/2)*sqrt(1 + b*x**2/a) + 21*a**(7/2)*b*x**2*sqrt(1 + b*x**2 /a) + 21*a**(5/2)*b**2*x**4*sqrt(1 + b*x**2/a) + 7*a**(3/2)*b**3*x**6*sqrt (1 + b*x**2/a)) + B*Piecewise((-16*a**3/(35*a**3*b**4*sqrt(a + b*x**2) + 1 05*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*sqrt(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)) + C*(105*a**(205/2)*b**45*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqr t(a))/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(203/2)*b**(10 1/2)*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**(19 7/2)*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))/(10 5*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...
Leaf count of result is larger than twice the leaf count of optimal. 447 vs. \(2 (146) = 292\).
Time = 0.06 (sec) , antiderivative size = 447, normalized size of antiderivative = 2.60 \[ \int \frac {x^6 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {1}{35} \, {\left (\frac {35 \, x^{6}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {70 \, a x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} + \frac {56 \, a^{2} x^{2}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {16 \, a^{3}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}}\right )} C x - \frac {B x^{6}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {C x {\left (\frac {15 \, x^{4}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b} + \frac {20 \, a x^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {8 \, a^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}}\right )}}{15 \, b} - \frac {A x^{5}}{2 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {C x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{3 \, b^{2}} - \frac {2 \, B a x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {C a x^{3}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} - \frac {5 \, A a x^{3}}{8 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {8 \, B a^{2} x^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {139 \, C x}{105 \, \sqrt {b x^{2} + a} b^{4}} + \frac {17 \, C a x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{4}} - \frac {29 \, C a^{2} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{4}} + \frac {A x}{14 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3}} + \frac {A x}{7 \, \sqrt {b x^{2} + a} a b^{3}} + \frac {3 \, A a x}{56 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} - \frac {15 \, A a^{2} x}{56 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {C \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {9}{2}}} - \frac {16 \, B a^{3}}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}} \] Input:
integrate(x^6*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x, algorithm="maxima")
Output:
-1/35*(35*x^6/((b*x^2 + a)^(7/2)*b) + 70*a*x^4/((b*x^2 + a)^(7/2)*b^2) + 5 6*a^2*x^2/((b*x^2 + a)^(7/2)*b^3) + 16*a^3/((b*x^2 + a)^(7/2)*b^4))*C*x - B*x^6/((b*x^2 + a)^(7/2)*b) - 1/15*C*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 - 1/2*A*x ^5/((b*x^2 + a)^(7/2)*b) - 1/3*C*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b* x^2 + a)^(3/2)*b^2))/b^2 - 2*B*a*x^4/((b*x^2 + a)^(7/2)*b^2) - C*a*x^3/((b *x^2 + a)^(5/2)*b^3) - 5/8*A*a*x^3/((b*x^2 + a)^(7/2)*b^2) - 8/5*B*a^2*x^2 /((b*x^2 + a)^(7/2)*b^3) + 139/105*C*x/(sqrt(b*x^2 + a)*b^4) + 17/105*C*a* x/((b*x^2 + a)^(3/2)*b^4) - 29/35*C*a^2*x/((b*x^2 + a)^(5/2)*b^4) + 1/14*A *x/((b*x^2 + a)^(3/2)*b^3) + 1/7*A*x/(sqrt(b*x^2 + a)*a*b^3) + 3/56*A*a*x/ ((b*x^2 + a)^(5/2)*b^3) - 15/56*A*a^2*x/((b*x^2 + a)^(7/2)*b^3) + C*arcsin h(b*x/sqrt(a*b))/b^(9/2) - 16/35*B*a^3/((b*x^2 + a)^(7/2)*b^4)
Time = 0.14 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.80 \[ \int \frac {x^6 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=-\frac {{\left ({\left ({\left ({\left ({\left (x {\left (\frac {105 \, B}{b} + \frac {{\left (176 \, C a^{3} b^{7} - 15 \, A a^{2} b^{8}\right )} x}{a^{3} b^{8}}\right )} + \frac {406 \, C a}{b^{2}}\right )} x + \frac {210 \, B a}{b^{2}}\right )} x + \frac {350 \, C a^{2}}{b^{3}}\right )} x + \frac {168 \, B a^{2}}{b^{3}}\right )} x + \frac {105 \, C a^{3}}{b^{4}}\right )} x + \frac {48 \, B a^{3}}{b^{4}}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} - \frac {C \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{b^{\frac {9}{2}}} \] Input:
integrate(x^6*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x, algorithm="giac")
Output:
-1/105*((((((x*(105*B/b + (176*C*a^3*b^7 - 15*A*a^2*b^8)*x/(a^3*b^8)) + 40 6*C*a/b^2)*x + 210*B*a/b^2)*x + 350*C*a^2/b^3)*x + 168*B*a^2/b^3)*x + 105* C*a^3/b^4)*x + 48*B*a^3/b^4)/(b*x^2 + a)^(7/2) - C*log(abs(-sqrt(b)*x + sq rt(b*x^2 + a)))/b^(9/2)
Timed out. \[ \int \frac {x^6 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\int \frac {x^6\,\left (C\,x^2+B\,x+A\right )}{{\left (b\,x^2+a\right )}^{9/2}} \,d x \] Input:
int((x^6*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x)
Output:
int((x^6*(A + B*x + C*x^2))/(a + b*x^2)^(9/2), x)
Time = 0.20 (sec) , antiderivative size = 467, normalized size of antiderivative = 2.72 \[ \int \frac {x^6 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx=\frac {-48 \sqrt {b \,x^{2}+a}\, a^{3} b^{2}-105 \sqrt {b \,x^{2}+a}\, a^{3} b c x -168 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} x^{2}-350 \sqrt {b \,x^{2}+a}\, a^{2} b^{2} c \,x^{3}-210 \sqrt {b \,x^{2}+a}\, a \,b^{4} x^{4}-406 \sqrt {b \,x^{2}+a}\, a \,b^{3} c \,x^{5}+15 \sqrt {b \,x^{2}+a}\, b^{5} x^{7}-105 \sqrt {b \,x^{2}+a}\, b^{5} x^{6}-176 \sqrt {b \,x^{2}+a}\, b^{4} c \,x^{7}+105 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} c +420 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{3} b c \,x^{2}+630 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{2} b^{2} c \,x^{4}+420 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a \,b^{3} c \,x^{6}+105 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b^{4} c \,x^{8}+15 \sqrt {b}\, a^{4} b +56 \sqrt {b}\, a^{4} c +60 \sqrt {b}\, a^{3} b^{2} x^{2}+224 \sqrt {b}\, a^{3} b c \,x^{2}+90 \sqrt {b}\, a^{2} b^{3} x^{4}+336 \sqrt {b}\, a^{2} b^{2} c \,x^{4}+60 \sqrt {b}\, a \,b^{4} x^{6}+224 \sqrt {b}\, a \,b^{3} c \,x^{6}+15 \sqrt {b}\, b^{5} x^{8}+56 \sqrt {b}\, b^{4} c \,x^{8}}{105 b^{5} \left (b^{4} x^{8}+4 a \,b^{3} x^{6}+6 a^{2} b^{2} x^{4}+4 a^{3} b \,x^{2}+a^{4}\right )} \] Input:
int(x^6*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x)
Output:
( - 48*sqrt(a + b*x**2)*a**3*b**2 - 105*sqrt(a + b*x**2)*a**3*b*c*x - 168* sqrt(a + b*x**2)*a**2*b**3*x**2 - 350*sqrt(a + b*x**2)*a**2*b**2*c*x**3 - 210*sqrt(a + b*x**2)*a*b**4*x**4 - 406*sqrt(a + b*x**2)*a*b**3*c*x**5 + 15 *sqrt(a + b*x**2)*b**5*x**7 - 105*sqrt(a + b*x**2)*b**5*x**6 - 176*sqrt(a + b*x**2)*b**4*c*x**7 + 105*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqr t(a))*a**4*c + 420*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a** 3*b*c*x**2 + 630*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**2* b**2*c*x**4 + 420*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a*b* *3*c*x**6 + 105*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*b**4*c *x**8 + 15*sqrt(b)*a**4*b + 56*sqrt(b)*a**4*c + 60*sqrt(b)*a**3*b**2*x**2 + 224*sqrt(b)*a**3*b*c*x**2 + 90*sqrt(b)*a**2*b**3*x**4 + 336*sqrt(b)*a**2 *b**2*c*x**4 + 60*sqrt(b)*a*b**4*x**6 + 224*sqrt(b)*a*b**3*c*x**6 + 15*sqr t(b)*b**5*x**8 + 56*sqrt(b)*b**4*c*x**8)/(105*b**5*(a**4 + 4*a**3*b*x**2 + 6*a**2*b**2*x**4 + 4*a*b**3*x**6 + b**4*x**8))