3.168 \(\int \frac {A+B x^2+C x^4+D x^6}{x^{10} (a+b x^2)^{9/2}} \, dx\)

Optimal. Leaf size=392 \[ -\frac {32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac {16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac {256 b^2 x \left (128 A b^3-3 a \left (5 a^2 D-12 a b C+24 b^2 B\right )\right )}{315 a^9 \sqrt {a+b x^2}}-\frac {128 b^2 x \left (128 A b^3-3 a \left (5 a^2 D-12 a b C+24 b^2 B\right )\right )}{315 a^8 \left (a+b x^2\right )^{3/2}}-\frac {32 b^2 x \left (128 A b^3-3 a \left (5 a^2 D-12 a b C+24 b^2 B\right )\right )}{105 a^7 \left (a+b x^2\right )^{5/2}}-\frac {16 b^2 x \left (128 A b^3-3 a \left (5 a^2 D-12 a b C+24 b^2 B\right )\right )}{63 a^6 \left (a+b x^2\right )^{7/2}}-\frac {2 b \left (128 A b^3-3 a \left (5 a^2 D-12 a b C+24 b^2 B\right )\right )}{9 a^5 x \left (a+b x^2\right )^{7/2}}+\frac {128 A b^3-3 a \left (5 a^2 D-12 a b C+24 b^2 B\right )}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}} \]

[Out]

-1/9*A/a/x^9/(b*x^2+a)^(7/2)+1/63*(16*A*b-9*B*a)/a^2/x^7/(b*x^2+a)^(7/2)+1/45*(-32*A*b^2+9*a*(2*B*b-C*a))/a^3/
x^5/(b*x^2+a)^(7/2)+1/45*(128*A*b^3-3*a*(24*B*b^2-12*C*a*b+5*D*a^2))/a^4/x^3/(b*x^2+a)^(7/2)-2/9*b*(128*A*b^3-
3*a*(24*B*b^2-12*C*a*b+5*D*a^2))/a^5/x/(b*x^2+a)^(7/2)-16/63*b^2*(128*A*b^3-3*a*(24*B*b^2-12*C*a*b+5*D*a^2))*x
/a^6/(b*x^2+a)^(7/2)-32/105*b^2*(128*A*b^3-3*a*(24*B*b^2-12*C*a*b+5*D*a^2))*x/a^7/(b*x^2+a)^(5/2)-128/315*b^2*
(128*A*b^3-3*a*(24*B*b^2-12*C*a*b+5*D*a^2))*x/a^8/(b*x^2+a)^(3/2)-256/315*b^2*(128*A*b^3-3*a*(24*B*b^2-12*C*a*
b+5*D*a^2))*x/a^9/(b*x^2+a)^(1/2)

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Rubi [A]  time = 0.55, antiderivative size = 380, normalized size of antiderivative = 0.97, number of steps used = 10, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {1803, 12, 271, 192, 191} \[ -\frac {256 b^2 x \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right )}{315 a^9 \sqrt {a+b x^2}}-\frac {128 b^2 x \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right )}{315 a^8 \left (a+b x^2\right )^{3/2}}-\frac {32 b^2 x \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right )}{105 a^7 \left (a+b x^2\right )^{5/2}}-\frac {16 b^2 x \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right )}{63 a^6 \left (a+b x^2\right )^{7/2}}-\frac {2 b \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right )}{9 a^5 x \left (a+b x^2\right )^{7/2}}+\frac {-15 a^3 D-36 a b (2 b B-a C)+128 A b^3}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}-\frac {32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac {16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x^2 + C*x^4 + D*x^6)/(x^10*(a + b*x^2)^(9/2)),x]

[Out]

-A/(9*a*x^9*(a + b*x^2)^(7/2)) + (16*A*b - 9*a*B)/(63*a^2*x^7*(a + b*x^2)^(7/2)) - (32*A*b^2 - 9*a*(2*b*B - a*
C))/(45*a^3*x^5*(a + b*x^2)^(7/2)) + (128*A*b^3 - 36*a*b*(2*b*B - a*C) - 15*a^3*D)/(45*a^4*x^3*(a + b*x^2)^(7/
2)) - (2*b*(128*A*b^3 - 36*a*b*(2*b*B - a*C) - 15*a^3*D))/(9*a^5*x*(a + b*x^2)^(7/2)) - (16*b^2*(128*A*b^3 - 3
6*a*b*(2*b*B - a*C) - 15*a^3*D)*x)/(63*a^6*(a + b*x^2)^(7/2)) - (32*b^2*(128*A*b^3 - 36*a*b*(2*b*B - a*C) - 15
*a^3*D)*x)/(105*a^7*(a + b*x^2)^(5/2)) - (128*b^2*(128*A*b^3 - 36*a*b*(2*b*B - a*C) - 15*a^3*D)*x)/(315*a^8*(a
+ b*x^2)^(3/2)) - (256*b^2*(128*A*b^3 - 36*a*b*(2*b*B - a*C) - 15*a^3*D)*x)/(315*a^9*Sqrt[a + b*x^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 1803

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{A = Coeff[Pq, x, 0], Q = PolynomialQuotient
[Pq - Coeff[Pq, x, 0], x^2, x]}, Simp[(A*x^(m + 1)*(a + b*x^2)^(p + 1))/(a*(m + 1)), x] + Dist[1/(a*(m + 1)),
Int[x^(m + 2)*(a + b*x^2)^p*(a*(m + 1)*Q - A*b*(m + 2*(p + 1) + 1)), x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq,
x^2] && IntegerQ[m/2] && ILtQ[(m + 1)/2 + p, 0] && LtQ[m + Expon[Pq, x] + 2*p + 1, 0]

Rubi steps

\begin {align*} \int \frac {A+B x^2+C x^4+D x^6}{x^{10} \left (a+b x^2\right )^{9/2}} \, dx &=-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {16 A b-9 a \left (B+C x^2+D x^4\right )}{x^8 \left (a+b x^2\right )^{9/2}} \, dx}{9 a}\\ &=-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac {16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}+\frac {\int \frac {14 b (16 A b-9 a B)-7 a \left (-9 a C-9 a D x^2\right )}{x^6 \left (a+b x^2\right )^{9/2}} \, dx}{63 a^2}\\ &=-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac {16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac {32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {12 b \left (224 A b^2-126 a b B+63 a^2 C\right )-315 a^3 D}{x^4 \left (a+b x^2\right )^{9/2}} \, dx}{315 a^3}\\ &=-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac {16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac {32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}-\frac {\left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) \int \frac {1}{x^4 \left (a+b x^2\right )^{9/2}} \, dx}{15 a^3}\\ &=-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac {16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac {32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac {128 A b^3-36 a b (2 b B-a C)-15 a^3 D}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}+\frac {\left (2 b \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )\right ) \int \frac {1}{x^2 \left (a+b x^2\right )^{9/2}} \, dx}{9 a^4}\\ &=-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac {16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac {32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac {128 A b^3-36 a b (2 b B-a C)-15 a^3 D}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}-\frac {2 b \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )}{9 a^5 x \left (a+b x^2\right )^{7/2}}-\frac {\left (16 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )\right ) \int \frac {1}{\left (a+b x^2\right )^{9/2}} \, dx}{9 a^5}\\ &=-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac {16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac {32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac {128 A b^3-36 a b (2 b B-a C)-15 a^3 D}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}-\frac {2 b \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )}{9 a^5 x \left (a+b x^2\right )^{7/2}}-\frac {16 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{63 a^6 \left (a+b x^2\right )^{7/2}}-\frac {\left (32 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )\right ) \int \frac {1}{\left (a+b x^2\right )^{7/2}} \, dx}{21 a^6}\\ &=-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac {16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac {32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac {128 A b^3-36 a b (2 b B-a C)-15 a^3 D}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}-\frac {2 b \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )}{9 a^5 x \left (a+b x^2\right )^{7/2}}-\frac {16 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{63 a^6 \left (a+b x^2\right )^{7/2}}-\frac {32 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{105 a^7 \left (a+b x^2\right )^{5/2}}-\frac {\left (128 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )\right ) \int \frac {1}{\left (a+b x^2\right )^{5/2}} \, dx}{105 a^7}\\ &=-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac {16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac {32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac {128 A b^3-36 a b (2 b B-a C)-15 a^3 D}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}-\frac {2 b \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )}{9 a^5 x \left (a+b x^2\right )^{7/2}}-\frac {16 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{63 a^6 \left (a+b x^2\right )^{7/2}}-\frac {32 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{105 a^7 \left (a+b x^2\right )^{5/2}}-\frac {128 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{315 a^8 \left (a+b x^2\right )^{3/2}}-\frac {\left (256 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )\right ) \int \frac {1}{\left (a+b x^2\right )^{3/2}} \, dx}{315 a^8}\\ &=-\frac {A}{9 a x^9 \left (a+b x^2\right )^{7/2}}+\frac {16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac {32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac {128 A b^3-36 a b (2 b B-a C)-15 a^3 D}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}-\frac {2 b \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right )}{9 a^5 x \left (a+b x^2\right )^{7/2}}-\frac {16 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{63 a^6 \left (a+b x^2\right )^{7/2}}-\frac {32 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{105 a^7 \left (a+b x^2\right )^{5/2}}-\frac {128 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{315 a^8 \left (a+b x^2\right )^{3/2}}-\frac {256 b^2 \left (128 A b^3-36 a b (2 b B-a C)-15 a^3 D\right ) x}{315 a^9 \sqrt {a+b x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 270, normalized size = 0.69 \[ \frac {-a^8 \left (35 A+45 B x^2+63 C x^4+105 D x^6\right )+2 a^7 b x^2 \left (40 A+21 \left (3 B x^2+6 C x^4+25 D x^6\right )\right )-56 a^6 b^2 x^4 \left (4 A+9 B x^2+45 C x^4-150 D x^6\right )+112 a^5 b^3 x^6 \left (8 A+45 B x^2-180 C x^4+150 D x^6\right )+4480 a^4 b^4 x^8 \left (-2 A+9 B x^2-9 C x^4+3 D x^6\right )+256 a^3 b^5 x^{10} \left (-280 A+315 B x^2-126 C x^4+15 D x^6\right )-1024 a^2 b^6 x^{12} \left (140 A-63 B x^2+9 C x^4\right )+2048 a b^7 x^{14} \left (9 B x^2-56 A\right )-32768 A b^8 x^{16}}{315 a^9 x^9 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x^2 + C*x^4 + D*x^6)/(x^10*(a + b*x^2)^(9/2)),x]

[Out]

(-32768*A*b^8*x^16 + 2048*a*b^7*x^14*(-56*A + 9*B*x^2) - 1024*a^2*b^6*x^12*(140*A - 63*B*x^2 + 9*C*x^4) - 56*a
^6*b^2*x^4*(4*A + 9*B*x^2 + 45*C*x^4 - 150*D*x^6) + 4480*a^4*b^4*x^8*(-2*A + 9*B*x^2 - 9*C*x^4 + 3*D*x^6) + 25
6*a^3*b^5*x^10*(-280*A + 315*B*x^2 - 126*C*x^4 + 15*D*x^6) - a^8*(35*A + 45*B*x^2 + 63*C*x^4 + 105*D*x^6) + 11
2*a^5*b^3*x^6*(8*A + 45*B*x^2 - 180*C*x^4 + 150*D*x^6) + 2*a^7*b*x^2*(40*A + 21*(3*B*x^2 + 6*C*x^4 + 25*D*x^6)
))/(315*a^9*x^9*(a + b*x^2)^(7/2))

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fricas [A]  time = 2.31, size = 354, normalized size = 0.90 \[ \frac {{\left (256 \, {\left (15 \, D a^{3} b^{5} - 36 \, C a^{2} b^{6} + 72 \, B a b^{7} - 128 \, A b^{8}\right )} x^{16} + 896 \, {\left (15 \, D a^{4} b^{4} - 36 \, C a^{3} b^{5} + 72 \, B a^{2} b^{6} - 128 \, A a b^{7}\right )} x^{14} + 1120 \, {\left (15 \, D a^{5} b^{3} - 36 \, C a^{4} b^{4} + 72 \, B a^{3} b^{5} - 128 \, A a^{2} b^{6}\right )} x^{12} + 560 \, {\left (15 \, D a^{6} b^{2} - 36 \, C a^{5} b^{3} + 72 \, B a^{4} b^{4} - 128 \, A a^{3} b^{5}\right )} x^{10} - 35 \, A a^{8} + 70 \, {\left (15 \, D a^{7} b - 36 \, C a^{6} b^{2} + 72 \, B a^{5} b^{3} - 128 \, A a^{4} b^{4}\right )} x^{8} - 7 \, {\left (15 \, D a^{8} - 36 \, C a^{7} b + 72 \, B a^{6} b^{2} - 128 \, A a^{5} b^{3}\right )} x^{6} - 7 \, {\left (9 \, C a^{8} - 18 \, B a^{7} b + 32 \, A a^{6} b^{2}\right )} x^{4} - 5 \, {\left (9 \, B a^{8} - 16 \, A a^{7} b\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{315 \, {\left (a^{9} b^{4} x^{17} + 4 \, a^{10} b^{3} x^{15} + 6 \, a^{11} b^{2} x^{13} + 4 \, a^{12} b x^{11} + a^{13} x^{9}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^6+C*x^4+B*x^2+A)/x^10/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

1/315*(256*(15*D*a^3*b^5 - 36*C*a^2*b^6 + 72*B*a*b^7 - 128*A*b^8)*x^16 + 896*(15*D*a^4*b^4 - 36*C*a^3*b^5 + 72
*B*a^2*b^6 - 128*A*a*b^7)*x^14 + 1120*(15*D*a^5*b^3 - 36*C*a^4*b^4 + 72*B*a^3*b^5 - 128*A*a^2*b^6)*x^12 + 560*
(15*D*a^6*b^2 - 36*C*a^5*b^3 + 72*B*a^4*b^4 - 128*A*a^3*b^5)*x^10 - 35*A*a^8 + 70*(15*D*a^7*b - 36*C*a^6*b^2 +
72*B*a^5*b^3 - 128*A*a^4*b^4)*x^8 - 7*(15*D*a^8 - 36*C*a^7*b + 72*B*a^6*b^2 - 128*A*a^5*b^3)*x^6 - 7*(9*C*a^8
- 18*B*a^7*b + 32*A*a^6*b^2)*x^4 - 5*(9*B*a^8 - 16*A*a^7*b)*x^2)*sqrt(b*x^2 + a)/(a^9*b^4*x^17 + 4*a^10*b^3*x
^15 + 6*a^11*b^2*x^13 + 4*a^12*b*x^11 + a^13*x^9)

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giac [B]  time = 0.74, size = 1162, normalized size = 2.96 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^6+C*x^4+B*x^2+A)/x^10/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

1/105*((x^2*((790*D*a^24*b^8 - 1686*C*a^23*b^9 + 3072*B*a^22*b^10 - 5053*A*a^21*b^11)*x^2/(a^30*b^3) + 7*(365*
D*a^25*b^7 - 768*C*a^24*b^8 + 1386*B*a^23*b^9 - 2264*A*a^22*b^10)/(a^30*b^3)) + 35*(80*D*a^26*b^6 - 165*C*a^25
*b^7 + 294*B*a^24*b^8 - 476*A*a^23*b^9)/(a^30*b^3))*x^2 + 105*(10*D*a^27*b^5 - 20*C*a^26*b^6 + 35*B*a^25*b^7 -
56*A*a^24*b^8)/(a^30*b^3))*x/(b*x^2 + a)^(7/2) - 2/315*(1260*(sqrt(b)*x - sqrt(b*x^2 + a))^16*D*a^3*b^(3/2) -
3150*(sqrt(b)*x - sqrt(b*x^2 + a))^16*C*a^2*b^(5/2) + 6300*(sqrt(b)*x - sqrt(b*x^2 + a))^16*B*a*b^(7/2) - 110
25*(sqrt(b)*x - sqrt(b*x^2 + a))^16*A*b^(9/2) - 10710*(sqrt(b)*x - sqrt(b*x^2 + a))^14*D*a^4*b^(3/2) + 27720*(
sqrt(b)*x - sqrt(b*x^2 + a))^14*C*a^3*b^(5/2) - 56700*(sqrt(b)*x - sqrt(b*x^2 + a))^14*B*a^2*b^(7/2) + 100800*
(sqrt(b)*x - sqrt(b*x^2 + a))^14*A*a*b^(9/2) + 39270*(sqrt(b)*x - sqrt(b*x^2 + a))^12*D*a^5*b^(3/2) - 105840*(
sqrt(b)*x - sqrt(b*x^2 + a))^12*C*a^4*b^(5/2) + 223020*(sqrt(b)*x - sqrt(b*x^2 + a))^12*B*a^3*b^(7/2) - 405300
*(sqrt(b)*x - sqrt(b*x^2 + a))^12*A*a^2*b^(9/2) - 81270*(sqrt(b)*x - sqrt(b*x^2 + a))^10*D*a^6*b^(3/2) + 22680
0*(sqrt(b)*x - sqrt(b*x^2 + a))^10*C*a^5*b^(5/2) - 495180*(sqrt(b)*x - sqrt(b*x^2 + a))^10*B*a^4*b^(7/2) + 927
360*(sqrt(b)*x - sqrt(b*x^2 + a))^10*A*a^3*b^(9/2) + 103950*(sqrt(b)*x - sqrt(b*x^2 + a))^8*D*a^7*b^(3/2) - 29
7108*(sqrt(b)*x - sqrt(b*x^2 + a))^8*C*a^6*b^(5/2) + 666036*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a^5*b^(7/2) - 12
91374*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*a^4*b^(9/2) - 84210*(sqrt(b)*x - sqrt(b*x^2 + a))^6*D*a^8*b^(3/2) + 24
3432*(sqrt(b)*x - sqrt(b*x^2 + a))^6*C*a^7*b^(5/2) - 551124*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^6*b^(7/2) + 10
73856*(sqrt(b)*x - sqrt(b*x^2 + a))^6*A*a^5*b^(9/2) + 42210*(sqrt(b)*x - sqrt(b*x^2 + a))^4*D*a^9*b^(3/2) - 12
1968*(sqrt(b)*x - sqrt(b*x^2 + a))^4*C*a^8*b^(5/2) + 275076*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^7*b^(7/2) - 53
3124*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a^6*b^(9/2) - 11970*(sqrt(b)*x - sqrt(b*x^2 + a))^2*D*a^10*b^(3/2) + 34
272*(sqrt(b)*x - sqrt(b*x^2 + a))^2*C*a^9*b^(5/2) - 76644*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^8*b^(7/2) + 1474
56*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a^7*b^(9/2) + 1470*D*a^11*b^(3/2) - 4158*C*a^10*b^(5/2) + 9216*B*a^9*b^(7
/2) - 17609*A*a^8*b^(9/2))/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^9*a^8)

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maple [A]  time = 0.01, size = 349, normalized size = 0.89 \[ -\frac {32768 A \,b^{8} x^{16}-18432 B a \,b^{7} x^{16}+9216 C \,a^{2} b^{6} x^{16}-3840 D a^{3} b^{5} x^{16}+114688 A a \,b^{7} x^{14}-64512 B \,a^{2} b^{6} x^{14}+32256 C \,a^{3} b^{5} x^{14}-13440 D a^{4} b^{4} x^{14}+143360 A \,a^{2} b^{6} x^{12}-80640 B \,a^{3} b^{5} x^{12}+40320 C \,a^{4} b^{4} x^{12}-16800 D a^{5} b^{3} x^{12}+71680 A \,a^{3} b^{5} x^{10}-40320 B \,a^{4} b^{4} x^{10}+20160 C \,a^{5} b^{3} x^{10}-8400 D a^{6} b^{2} x^{10}+8960 A \,a^{4} b^{4} x^{8}-5040 B \,a^{5} b^{3} x^{8}+2520 C \,a^{6} b^{2} x^{8}-1050 D a^{7} b \,x^{8}-896 A \,a^{5} b^{3} x^{6}+504 B \,a^{6} b^{2} x^{6}-252 C \,a^{7} b \,x^{6}+105 D a^{8} x^{6}+224 A \,a^{6} b^{2} x^{4}-126 B \,a^{7} b \,x^{4}+63 C \,a^{8} x^{4}-80 A \,a^{7} b \,x^{2}+45 B \,a^{8} x^{2}+35 A \,a^{8}}{315 \left (b \,x^{2}+a \right )^{\frac {7}{2}} a^{9} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((D*x^6+C*x^4+B*x^2+A)/x^10/(b*x^2+a)^(9/2),x)

[Out]

-1/315*(32768*A*b^8*x^16-18432*B*a*b^7*x^16+9216*C*a^2*b^6*x^16-3840*D*a^3*b^5*x^16+114688*A*a*b^7*x^14-64512*
B*a^2*b^6*x^14+32256*C*a^3*b^5*x^14-13440*D*a^4*b^4*x^14+143360*A*a^2*b^6*x^12-80640*B*a^3*b^5*x^12+40320*C*a^
4*b^4*x^12-16800*D*a^5*b^3*x^12+71680*A*a^3*b^5*x^10-40320*B*a^4*b^4*x^10+20160*C*a^5*b^3*x^10-8400*D*a^6*b^2*
x^10+8960*A*a^4*b^4*x^8-5040*B*a^5*b^3*x^8+2520*C*a^6*b^2*x^8-1050*D*a^7*b*x^8-896*A*a^5*b^3*x^6+504*B*a^6*b^2
*x^6-252*C*a^7*b*x^6+105*D*a^8*x^6+224*A*a^6*b^2*x^4-126*B*a^7*b*x^4+63*C*a^8*x^4-80*A*a^7*b*x^2+45*B*a^8*x^2+
35*A*a^8)/(b*x^2+a)^(7/2)/x^9/a^9

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maxima [A]  time = 1.57, size = 579, normalized size = 1.48 \[ \frac {256 \, D b^{2} x}{21 \, \sqrt {b x^{2} + a} a^{6}} + \frac {128 \, D b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{5}} + \frac {32 \, D b^{2} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{4}} + \frac {80 \, D b^{2} x}{21 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3}} - \frac {1024 \, C b^{3} x}{35 \, \sqrt {b x^{2} + a} a^{7}} - \frac {512 \, C b^{3} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{6}} - \frac {384 \, C b^{3} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{5}} - \frac {64 \, C b^{3} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{4}} + \frac {2048 \, B b^{4} x}{35 \, \sqrt {b x^{2} + a} a^{8}} + \frac {1024 \, B b^{4} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{7}} + \frac {768 \, B b^{4} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{6}} + \frac {128 \, B b^{4} x}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{5}} - \frac {32768 \, A b^{5} x}{315 \, \sqrt {b x^{2} + a} a^{9}} - \frac {16384 \, A b^{5} x}{315 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{8}} - \frac {4096 \, A b^{5} x}{105 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} a^{7}} - \frac {2048 \, A b^{5} x}{63 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{6}} + \frac {10 \, D b}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x} - \frac {8 \, C b^{2}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3} x} + \frac {16 \, B b^{3}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{4} x} - \frac {256 \, A b^{4}}{9 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{5} x} - \frac {D}{3 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{3}} + \frac {4 \, C b}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x^{3}} - \frac {8 \, B b^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3} x^{3}} + \frac {128 \, A b^{3}}{45 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{4} x^{3}} - \frac {C}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{5}} + \frac {2 \, B b}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x^{5}} - \frac {32 \, A b^{2}}{45 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{3} x^{5}} - \frac {B}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{7}} + \frac {16 \, A b}{63 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a^{2} x^{7}} - \frac {A}{9 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} a x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x^6+C*x^4+B*x^2+A)/x^10/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

256/21*D*b^2*x/(sqrt(b*x^2 + a)*a^6) + 128/21*D*b^2*x/((b*x^2 + a)^(3/2)*a^5) + 32/7*D*b^2*x/((b*x^2 + a)^(5/2
)*a^4) + 80/21*D*b^2*x/((b*x^2 + a)^(7/2)*a^3) - 1024/35*C*b^3*x/(sqrt(b*x^2 + a)*a^7) - 512/35*C*b^3*x/((b*x^
2 + a)^(3/2)*a^6) - 384/35*C*b^3*x/((b*x^2 + a)^(5/2)*a^5) - 64/7*C*b^3*x/((b*x^2 + a)^(7/2)*a^4) + 2048/35*B*
b^4*x/(sqrt(b*x^2 + a)*a^8) + 1024/35*B*b^4*x/((b*x^2 + a)^(3/2)*a^7) + 768/35*B*b^4*x/((b*x^2 + a)^(5/2)*a^6)
+ 128/7*B*b^4*x/((b*x^2 + a)^(7/2)*a^5) - 32768/315*A*b^5*x/(sqrt(b*x^2 + a)*a^9) - 16384/315*A*b^5*x/((b*x^2
+ a)^(3/2)*a^8) - 4096/105*A*b^5*x/((b*x^2 + a)^(5/2)*a^7) - 2048/63*A*b^5*x/((b*x^2 + a)^(7/2)*a^6) + 10/3*D
*b/((b*x^2 + a)^(7/2)*a^2*x) - 8*C*b^2/((b*x^2 + a)^(7/2)*a^3*x) + 16*B*b^3/((b*x^2 + a)^(7/2)*a^4*x) - 256/9*
A*b^4/((b*x^2 + a)^(7/2)*a^5*x) - 1/3*D/((b*x^2 + a)^(7/2)*a*x^3) + 4/5*C*b/((b*x^2 + a)^(7/2)*a^2*x^3) - 8/5*
B*b^2/((b*x^2 + a)^(7/2)*a^3*x^3) + 128/45*A*b^3/((b*x^2 + a)^(7/2)*a^4*x^3) - 1/5*C/((b*x^2 + a)^(7/2)*a*x^5)
+ 2/5*B*b/((b*x^2 + a)^(7/2)*a^2*x^5) - 32/45*A*b^2/((b*x^2 + a)^(7/2)*a^3*x^5) - 1/7*B/((b*x^2 + a)^(7/2)*a*
x^7) + 16/63*A*b/((b*x^2 + a)^(7/2)*a^2*x^7) - 1/9*A/((b*x^2 + a)^(7/2)*a*x^9)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {A+B\,x^2+C\,x^4+x^6\,D}{x^{10}\,{\left (b\,x^2+a\right )}^{9/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^2 + C*x^4 + x^6*D)/(x^10*(a + b*x^2)^(9/2)),x)

[Out]

int((A + B*x^2 + C*x^4 + x^6*D)/(x^10*(a + b*x^2)^(9/2)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((D*x**6+C*x**4+B*x**2+A)/x**10/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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