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

Optimal. Leaf size=281 \[ -\frac{24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}+\frac{12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac{16 x \left (192 A b^3-a \left (3 a^2 D-24 a b C+80 b^2 B\right )\right )}{105 a^7 \sqrt{a+b x^2}}-\frac{8 x \left (192 A b^3-a \left (3 a^2 D-24 a b C+80 b^2 B\right )\right )}{105 a^6 \left (a+b x^2\right )^{3/2}}-\frac{2 x \left (192 A b^3-a \left (3 a^2 D-24 a b C+80 b^2 B\right )\right )}{35 a^5 \left (a+b x^2\right )^{5/2}}-\frac{x \left (192 A b^3-a \left (3 a^2 D-24 a b C+80 b^2 B\right )\right )}{21 a^4 \left (a+b x^2\right )^{7/2}}-\frac{A}{5 a x^5 \left (a+b x^2\right )^{7/2}} \]

[Out]

-A/(5*a*x^5*(a + b*x^2)^(7/2)) + (12*A*b - 5*a*B)/(15*a^2*x^3*(a + b*x^2)^(7/2))
- (24*A*b^2 - a*(10*b*B - 3*a*C))/(3*a^3*x*(a + b*x^2)^(7/2)) - ((192*A*b^3 - a
*(80*b^2*B - 24*a*b*C + 3*a^2*D))*x)/(21*a^4*(a + b*x^2)^(7/2)) - (2*(192*A*b^3
- a*(80*b^2*B - 24*a*b*C + 3*a^2*D))*x)/(35*a^5*(a + b*x^2)^(5/2)) - (8*(192*A*b
^3 - a*(80*b^2*B - 24*a*b*C + 3*a^2*D))*x)/(105*a^6*(a + b*x^2)^(3/2)) - (16*(19
2*A*b^3 - a*(80*b^2*B - 24*a*b*C + 3*a^2*D))*x)/(105*a^7*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.843296, antiderivative size = 275, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{24 A b^2-a (10 b B-3 a C)}{3 a^3 x \left (a+b x^2\right )^{7/2}}+\frac{12 A b-5 a B}{15 a^2 x^3 \left (a+b x^2\right )^{7/2}}-\frac{16 x \left (-3 a^3 D-8 a b (10 b B-3 a C)+192 A b^3\right )}{105 a^7 \sqrt{a+b x^2}}-\frac{8 x \left (192 A b^3-a \left (3 a^2 D-24 a b C+80 b^2 B\right )\right )}{105 a^6 \left (a+b x^2\right )^{3/2}}-\frac{2 x \left (-3 a^3 D-8 a b (10 b B-3 a C)+192 A b^3\right )}{35 a^5 \left (a+b x^2\right )^{5/2}}-\frac{x \left (-3 a^3 D-8 a b (10 b B-3 a C)+192 A b^3\right )}{21 a^4 \left (a+b x^2\right )^{7/2}}-\frac{A}{5 a x^5 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

-A/(5*a*x^5*(a + b*x^2)^(7/2)) + (12*A*b - 5*a*B)/(15*a^2*x^3*(a + b*x^2)^(7/2))
- (24*A*b^2 - a*(10*b*B - 3*a*C))/(3*a^3*x*(a + b*x^2)^(7/2)) - ((192*A*b^3 - 8
*a*b*(10*b*B - 3*a*C) - 3*a^3*D)*x)/(21*a^4*(a + b*x^2)^(7/2)) - (2*(192*A*b^3 -
8*a*b*(10*b*B - 3*a*C) - 3*a^3*D)*x)/(35*a^5*(a + b*x^2)^(5/2)) - (8*(192*A*b^3
- a*(80*b^2*B - 24*a*b*C + 3*a^2*D))*x)/(105*a^6*(a + b*x^2)^(3/2)) - (16*(192*
A*b^3 - 8*a*b*(10*b*B - 3*a*C) - 3*a^3*D)*x)/(105*a^7*Sqrt[a + b*x^2])

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Rubi in Sympy [A]  time = 171.121, size = 309, normalized size = 1.1 \[ - \frac{D}{8 b^{2} x^{3} \left (a + b x^{2}\right )^{\frac{5}{2}}} + \frac{x \left (\frac{A b^{3}}{x^{6}} - \frac{B a b^{2}}{x^{6}} + \frac{C a^{2} b}{x^{6}} - \frac{D a^{3}}{x^{6}}\right )}{7 a b^{3} \left (a + b x^{2}\right )^{\frac{7}{2}}} - \frac{B b^{2} - C a b + D a^{2}}{5 a b^{3} x^{5} \left (a + b x^{2}\right )^{\frac{5}{2}}} + \frac{16 B b^{2} - 24 C a b + 27 D a^{2}}{24 a^{2} b^{2} x^{3} \left (a + b x^{2}\right )^{\frac{5}{2}}} - \frac{16 B b^{2} - 24 C a b + 27 D a^{2}}{3 a^{3} b x \left (a + b x^{2}\right )^{\frac{5}{2}}} - \frac{2 x \left (16 B b^{2} - 24 C a b + 27 D a^{2}\right )}{5 a^{4} \left (a + b x^{2}\right )^{\frac{5}{2}}} - \frac{8 x \left (16 B b^{2} - 24 C a b + 27 D a^{2}\right )}{15 a^{5} \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{16 x \left (16 B b^{2} - 24 C a b + 27 D a^{2}\right )}{15 a^{6} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((D*x**6+C*x**4+B*x**2+A)/x**6/(b*x**2+a)**(9/2),x)

[Out]

-D/(8*b**2*x**3*(a + b*x**2)**(5/2)) + x*(A*b**3/x**6 - B*a*b**2/x**6 + C*a**2*b
/x**6 - D*a**3/x**6)/(7*a*b**3*(a + b*x**2)**(7/2)) - (B*b**2 - C*a*b + D*a**2)/
(5*a*b**3*x**5*(a + b*x**2)**(5/2)) + (16*B*b**2 - 24*C*a*b + 27*D*a**2)/(24*a**
2*b**2*x**3*(a + b*x**2)**(5/2)) - (16*B*b**2 - 24*C*a*b + 27*D*a**2)/(3*a**3*b*
x*(a + b*x**2)**(5/2)) - 2*x*(16*B*b**2 - 24*C*a*b + 27*D*a**2)/(5*a**4*(a + b*x
**2)**(5/2)) - 8*x*(16*B*b**2 - 24*C*a*b + 27*D*a**2)/(15*a**5*(a + b*x**2)**(3/
2)) - 16*x*(16*B*b**2 - 24*C*a*b + 27*D*a**2)/(15*a**6*sqrt(a + b*x**2))

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Mathematica [A]  time = 0.293513, size = 202, normalized size = 0.72 \[ \frac{-7 a^6 \left (3 A+5 x^2 \left (B+3 C x^2-3 D x^4\right )\right )+14 a^5 b x^2 \left (6 A+25 B x^2-60 C x^4+15 D x^6\right )+56 a^4 b^2 x^4 \left (-15 A+50 B x^2-30 C x^4+3 D x^6\right )+16 a^3 b^3 x^6 \left (-420 A+350 B x^2-84 C x^4+3 D x^6\right )-128 a^2 b^4 x^8 \left (105 A-35 B x^2+3 C x^4\right )+256 a b^5 x^{10} \left (5 B x^2-42 A\right )-3072 A b^6 x^{12}}{105 a^7 x^5 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-3072*A*b^6*x^12 + 256*a*b^5*x^10*(-42*A + 5*B*x^2) - 128*a^2*b^4*x^8*(105*A -
35*B*x^2 + 3*C*x^4) + 16*a^3*b^3*x^6*(-420*A + 350*B*x^2 - 84*C*x^4 + 3*D*x^6) +
56*a^4*b^2*x^4*(-15*A + 50*B*x^2 - 30*C*x^4 + 3*D*x^6) + 14*a^5*b*x^2*(6*A + 25
*B*x^2 - 60*C*x^4 + 15*D*x^6) - 7*a^6*(3*A + 5*x^2*(B + 3*C*x^2 - 3*D*x^4)))/(10
5*a^7*x^5*(a + b*x^2)^(7/2))

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Maple [A]  time = 0.012, size = 253, normalized size = 0.9 \[ -{\frac{3072\,A{b}^{6}{x}^{12}-1280\,Ba{b}^{5}{x}^{12}+384\,C{a}^{2}{b}^{4}{x}^{12}-48\,D{a}^{3}{b}^{3}{x}^{12}+10752\,Aa{b}^{5}{x}^{10}-4480\,B{a}^{2}{b}^{4}{x}^{10}+1344\,C{a}^{3}{b}^{3}{x}^{10}-168\,D{a}^{4}{b}^{2}{x}^{10}+13440\,A{a}^{2}{b}^{4}{x}^{8}-5600\,B{a}^{3}{b}^{3}{x}^{8}+1680\,C{a}^{4}{b}^{2}{x}^{8}-210\,D{a}^{5}b{x}^{8}+6720\,A{a}^{3}{b}^{3}{x}^{6}-2800\,B{a}^{4}{b}^{2}{x}^{6}+840\,C{a}^{5}b{x}^{6}-105\,D{a}^{6}{x}^{6}+840\,A{a}^{4}{b}^{2}{x}^{4}-350\,B{a}^{5}b{x}^{4}+105\,C{a}^{6}{x}^{4}-84\,A{a}^{5}b{x}^{2}+35\,B{a}^{6}{x}^{2}+21\,A{a}^{6}}{105\,{x}^{5}{a}^{7}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/105*(3072*A*b^6*x^12-1280*B*a*b^5*x^12+384*C*a^2*b^4*x^12-48*D*a^3*b^3*x^12+1
0752*A*a*b^5*x^10-4480*B*a^2*b^4*x^10+1344*C*a^3*b^3*x^10-168*D*a^4*b^2*x^10+134
40*A*a^2*b^4*x^8-5600*B*a^3*b^3*x^8+1680*C*a^4*b^2*x^8-210*D*a^5*b*x^8+6720*A*a^
3*b^3*x^6-2800*B*a^4*b^2*x^6+840*C*a^5*b*x^6-105*D*a^6*x^6+840*A*a^4*b^2*x^4-350
*B*a^5*b*x^4+105*C*a^6*x^4-84*A*a^5*b*x^2+35*B*a^6*x^2+21*A*a^6)/x^5/(b*x^2+a)^(
7/2)/a^7

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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^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.12548, size = 365, normalized size = 1.3 \[ \frac{{\left (16 \,{\left (3 \, D a^{3} b^{3} - 24 \, C a^{2} b^{4} + 80 \, B a b^{5} - 192 \, A b^{6}\right )} x^{12} + 56 \,{\left (3 \, D a^{4} b^{2} - 24 \, C a^{3} b^{3} + 80 \, B a^{2} b^{4} - 192 \, A a b^{5}\right )} x^{10} + 70 \,{\left (3 \, D a^{5} b - 24 \, C a^{4} b^{2} + 80 \, B a^{3} b^{3} - 192 \, A a^{2} b^{4}\right )} x^{8} - 21 \, A a^{6} + 35 \,{\left (3 \, D a^{6} - 24 \, C a^{5} b + 80 \, B a^{4} b^{2} - 192 \, A a^{3} b^{3}\right )} x^{6} - 35 \,{\left (3 \, C a^{6} - 10 \, B a^{5} b + 24 \, A a^{4} b^{2}\right )} x^{4} - 7 \,{\left (5 \, B a^{6} - 12 \, A a^{5} b\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{105 \,{\left (a^{7} b^{4} x^{13} + 4 \, a^{8} b^{3} x^{11} + 6 \, a^{9} b^{2} x^{9} + 4 \, a^{10} b x^{7} + a^{11} x^{5}\right )}} \]

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^6),x, algorithm="fricas")

[Out]

1/105*(16*(3*D*a^3*b^3 - 24*C*a^2*b^4 + 80*B*a*b^5 - 192*A*b^6)*x^12 + 56*(3*D*a
^4*b^2 - 24*C*a^3*b^3 + 80*B*a^2*b^4 - 192*A*a*b^5)*x^10 + 70*(3*D*a^5*b - 24*C*
a^4*b^2 + 80*B*a^3*b^3 - 192*A*a^2*b^4)*x^8 - 21*A*a^6 + 35*(3*D*a^6 - 24*C*a^5*
b + 80*B*a^4*b^2 - 192*A*a^3*b^3)*x^6 - 35*(3*C*a^6 - 10*B*a^5*b + 24*A*a^4*b^2)
*x^4 - 7*(5*B*a^6 - 12*A*a^5*b)*x^2)*sqrt(b*x^2 + a)/(a^7*b^4*x^13 + 4*a^8*b^3*x
^11 + 6*a^9*b^2*x^9 + 4*a^10*b*x^7 + a^11*x^5)

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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**6/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.233374, size = 799, normalized size = 2.84 \[ \frac{{\left ({\left (x^{2}{\left (\frac{{\left (48 \, D a^{18} b^{6} - 279 \, C a^{17} b^{7} + 790 \, B a^{16} b^{8} - 1686 \, A a^{15} b^{9}\right )} x^{2}}{a^{22} b^{3}} + \frac{7 \,{\left (24 \, D a^{19} b^{5} - 132 \, C a^{18} b^{6} + 365 \, B a^{17} b^{7} - 768 \, A a^{16} b^{8}\right )}}{a^{22} b^{3}}\right )} + \frac{35 \,{\left (6 \, D a^{20} b^{4} - 30 \, C a^{19} b^{5} + 80 \, B a^{18} b^{6} - 165 \, A a^{17} b^{7}\right )}}{a^{22} b^{3}}\right )} x^{2} + \frac{105 \,{\left (D a^{21} b^{3} - 4 \, C a^{20} b^{4} + 10 \, B a^{19} b^{5} - 20 \, A a^{18} b^{6}\right )}}{a^{22} b^{3}}\right )} x}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} + \frac{2 \,{\left (15 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} C a^{2} \sqrt{b} - 60 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} B a b^{\frac{3}{2}} + 150 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} A b^{\frac{5}{2}} - 60 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} C a^{3} \sqrt{b} + 270 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} B a^{2} b^{\frac{3}{2}} - 720 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} A a b^{\frac{5}{2}} + 90 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} C a^{4} \sqrt{b} - 430 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} B a^{3} b^{\frac{3}{2}} + 1260 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} A a^{2} b^{\frac{5}{2}} - 60 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} C a^{5} \sqrt{b} + 290 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} B a^{4} b^{\frac{3}{2}} - 840 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} A a^{3} b^{\frac{5}{2}} + 15 \, C a^{6} \sqrt{b} - 70 \, B a^{5} b^{\frac{3}{2}} + 198 \, A a^{4} b^{\frac{5}{2}}\right )}}{15 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{5} a^{6}} \]

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^6),x, algorithm="giac")

[Out]

1/105*((x^2*((48*D*a^18*b^6 - 279*C*a^17*b^7 + 790*B*a^16*b^8 - 1686*A*a^15*b^9)
*x^2/(a^22*b^3) + 7*(24*D*a^19*b^5 - 132*C*a^18*b^6 + 365*B*a^17*b^7 - 768*A*a^1
6*b^8)/(a^22*b^3)) + 35*(6*D*a^20*b^4 - 30*C*a^19*b^5 + 80*B*a^18*b^6 - 165*A*a^
17*b^7)/(a^22*b^3))*x^2 + 105*(D*a^21*b^3 - 4*C*a^20*b^4 + 10*B*a^19*b^5 - 20*A*
a^18*b^6)/(a^22*b^3))*x/(b*x^2 + a)^(7/2) + 2/15*(15*(sqrt(b)*x - sqrt(b*x^2 + a
))^8*C*a^2*sqrt(b) - 60*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a*b^(3/2) + 150*(sqrt(
b)*x - sqrt(b*x^2 + a))^8*A*b^(5/2) - 60*(sqrt(b)*x - sqrt(b*x^2 + a))^6*C*a^3*s
qrt(b) + 270*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^2*b^(3/2) - 720*(sqrt(b)*x - sq
rt(b*x^2 + a))^6*A*a*b^(5/2) + 90*(sqrt(b)*x - sqrt(b*x^2 + a))^4*C*a^4*sqrt(b)
- 430*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^3*b^(3/2) + 1260*(sqrt(b)*x - sqrt(b*x
^2 + a))^4*A*a^2*b^(5/2) - 60*(sqrt(b)*x - sqrt(b*x^2 + a))^2*C*a^5*sqrt(b) + 29
0*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^4*b^(3/2) - 840*(sqrt(b)*x - sqrt(b*x^2 +
a))^2*A*a^3*b^(5/2) + 15*C*a^6*sqrt(b) - 70*B*a^5*b^(3/2) + 198*A*a^4*b^(5/2))/(
((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^5*a^6)