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

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]

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

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Rubi [A]  time = 1.13509, 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 \[ -\frac{32 A b^2-9 a (2 b B-a C)}{45 a^3 x^5 \left (a+b x^2\right )^{7/2}}+\frac{16 A b-9 a B}{63 a^2 x^7 \left (a+b x^2\right )^{7/2}}-\frac{256 b^2 x \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right )}{315 a^9 \sqrt{a+b x^2}}-\frac{128 b^2 x \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right )}{315 a^8 \left (a+b x^2\right )^{3/2}}-\frac{32 b^2 x \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right )}{105 a^7 \left (a+b x^2\right )^{5/2}}-\frac{16 b^2 x \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right )}{63 a^6 \left (a+b x^2\right )^{7/2}}-\frac{2 b \left (-15 a^3 D-36 a b (2 b B-a C)+128 A b^3\right )}{9 a^5 x \left (a+b x^2\right )^{7/2}}+\frac{-15 a^3 D-36 a b (2 b B-a C)+128 A b^3}{45 a^4 x^3 \left (a+b x^2\right )^{7/2}}-\frac{A}{9 a x^9 \left (a+b x^2\right )^{7/2}} \]

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 - 36*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])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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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Mathematica [A]  time = 0.475293, 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) + 256*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) + 112*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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Maple [A]  time = 0.013, size = 349, normalized size = 0.9 \[ -{\frac{32768\,A{b}^{8}{x}^{16}-18432\,Ba{b}^{7}{x}^{16}+9216\,C{a}^{2}{b}^{6}{x}^{16}-3840\,D{a}^{3}{b}^{5}{x}^{16}+114688\,Aa{b}^{7}{x}^{14}-64512\,B{a}^{2}{b}^{6}{x}^{14}+32256\,C{a}^{3}{b}^{5}{x}^{14}-13440\,D{a}^{4}{b}^{4}{x}^{14}+143360\,A{a}^{2}{b}^{6}{x}^{12}-80640\,B{a}^{3}{b}^{5}{x}^{12}+40320\,C{a}^{4}{b}^{4}{x}^{12}-16800\,D{a}^{5}{b}^{3}{x}^{12}+71680\,A{a}^{3}{b}^{5}{x}^{10}-40320\,B{a}^{4}{b}^{4}{x}^{10}+20160\,C{a}^{5}{b}^{3}{x}^{10}-8400\,D{a}^{6}{b}^{2}{x}^{10}+8960\,A{a}^{4}{b}^{4}{x}^{8}-5040\,B{a}^{5}{b}^{3}{x}^{8}+2520\,C{a}^{6}{b}^{2}{x}^{8}-1050\,D{a}^{7}b{x}^{8}-896\,A{a}^{5}{b}^{3}{x}^{6}+504\,B{a}^{6}{b}^{2}{x}^{6}-252\,C{a}^{7}b{x}^{6}+105\,D{a}^{8}{x}^{6}+224\,A{a}^{6}{b}^{2}{x}^{4}-126\,B{a}^{7}b{x}^{4}+63\,C{a}^{8}{x}^{4}-80\,A{a}^{7}b{x}^{2}+45\,B{a}^{8}{x}^{2}+35\,A{a}^{8}}{315\,{x}^{9}{a}^{9}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \]

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)/
x^9/(b*x^2+a)^(7/2)/a^9

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.1156, size = 478, normalized size = 1.22 \[ \frac{{\left (256 \,{\left (15 \, D a^{3} b^{5} - 36 \, C a^{2} b^{6} + 72 \, B a b^{7} - 128 \, A b^{8}\right )} x^{16} + 896 \,{\left (15 \, D a^{4} b^{4} - 36 \, C a^{3} b^{5} + 72 \, B a^{2} b^{6} - 128 \, A a b^{7}\right )} x^{14} + 1120 \,{\left (15 \, D a^{5} b^{3} - 36 \, C a^{4} b^{4} + 72 \, B a^{3} b^{5} - 128 \, A a^{2} b^{6}\right )} x^{12} + 560 \,{\left (15 \, D a^{6} b^{2} - 36 \, C a^{5} b^{3} + 72 \, B a^{4} b^{4} - 128 \, A a^{3} b^{5}\right )} x^{10} - 35 \, A a^{8} + 70 \,{\left (15 \, D a^{7} b - 36 \, C a^{6} b^{2} + 72 \, B a^{5} b^{3} - 128 \, A a^{4} b^{4}\right )} x^{8} - 7 \,{\left (15 \, D a^{8} - 36 \, C a^{7} b + 72 \, B a^{6} b^{2} - 128 \, A a^{5} b^{3}\right )} x^{6} - 7 \,{\left (9 \, C a^{8} - 18 \, B a^{7} b + 32 \, A a^{6} b^{2}\right )} x^{4} - 5 \,{\left (9 \, B a^{8} - 16 \, A a^{7} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{315 \,{\left (a^{9} b^{4} x^{17} + 4 \, a^{10} b^{3} x^{15} + 6 \, a^{11} b^{2} x^{13} + 4 \, a^{12} b x^{11} + a^{13} x^{9}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^6 + C*x^4 + B*x^2 + A)/((b*x^2 + a)^(9/2)*x^10),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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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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GIAC/XCAS [A]  time = 0.259431, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done