3.159 \(\int \frac{x^8 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx\)

Optimal. Leaf size=381 \[ -\frac{x^9 \left (2 A b^3-a \left (23 a^2 D-16 a b C+9 b^2 B\right )\right )}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac{x^9 \left (A-\frac{a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac{x \sqrt{a+b x^2} \left (16 A b^3-3 a \left (143 a^2 D-66 a b C+24 b^2 B\right )\right )}{16 a b^7}+\frac{x^3 \sqrt{a+b x^2} \left (16 A b^3-3 a \left (143 a^2 D-66 a b C+24 b^2 B\right )\right )}{24 a^2 b^6}-\frac{x^5 \left (16 A b^3-3 a \left (143 a^2 D-66 a b C+24 b^2 B\right )\right )}{30 a^2 b^5 \sqrt{a+b x^2}}-\frac{x^7 \left (16 A b^3-3 a \left (143 a^2 D-66 a b C+24 b^2 B\right )\right )}{210 a^2 b^4 \left (a+b x^2\right )^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \left (-429 a^3 D+198 a^2 b C-72 a b^2 B+16 A b^3\right )}{16 b^{15/2}}+\frac{D x^9}{6 b^3 \left (a+b x^2\right )^{3/2}} \]

[Out]

((A - (a*(b^2*B - a*b*C + a^2*D))/b^3)*x^9)/(7*a*(a + b*x^2)^(7/2)) - ((2*A*b^3
- a*(9*b^2*B - 16*a*b*C + 23*a^2*D))*x^9)/(35*a^2*b^3*(a + b*x^2)^(5/2)) - ((16*
A*b^3 - 3*a*(24*b^2*B - 66*a*b*C + 143*a^2*D))*x^7)/(210*a^2*b^4*(a + b*x^2)^(3/
2)) + (D*x^9)/(6*b^3*(a + b*x^2)^(3/2)) - ((16*A*b^3 - 3*a*(24*b^2*B - 66*a*b*C
+ 143*a^2*D))*x^5)/(30*a^2*b^5*Sqrt[a + b*x^2]) - ((16*A*b^3 - 3*a*(24*b^2*B - 6
6*a*b*C + 143*a^2*D))*x*Sqrt[a + b*x^2])/(16*a*b^7) + ((16*A*b^3 - 3*a*(24*b^2*B
- 66*a*b*C + 143*a^2*D))*x^3*Sqrt[a + b*x^2])/(24*a^2*b^6) + ((16*A*b^3 - 72*a*
b^2*B + 198*a^2*b*C - 429*a^3*D)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(15
/2))

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Rubi [A]  time = 1.62045, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{x^9 \left (2 A b^3-a \left (23 a^2 D-16 a b C+9 b^2 B\right )\right )}{35 a^2 b^3 \left (a+b x^2\right )^{5/2}}+\frac{x^9 \left (A-\frac{a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}-\frac{x \sqrt{a+b x^2} \left (16 A b^3-3 a \left (143 a^2 D-66 a b C+24 b^2 B\right )\right )}{16 a b^7}+\frac{x^3 \sqrt{a+b x^2} \left (16 A b^3-3 a \left (143 a^2 D-66 a b C+24 b^2 B\right )\right )}{24 a^2 b^6}-\frac{x^5 \left (16 A b^3-3 a \left (143 a^2 D-66 a b C+24 b^2 B\right )\right )}{30 a^2 b^5 \sqrt{a+b x^2}}-\frac{x^7 \left (16 A b^3-3 a \left (143 a^2 D-66 a b C+24 b^2 B\right )\right )}{210 a^2 b^4 \left (a+b x^2\right )^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \left (-429 a^3 D+198 a^2 b C-72 a b^2 B+16 A b^3\right )}{16 b^{15/2}}+\frac{D x^9}{6 b^3 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^8*(A + B*x^2 + C*x^4 + D*x^6))/(a + b*x^2)^(9/2),x]

[Out]

((A - (a*(b^2*B - a*b*C + a^2*D))/b^3)*x^9)/(7*a*(a + b*x^2)^(7/2)) - ((2*A*b^3
- a*(9*b^2*B - 16*a*b*C + 23*a^2*D))*x^9)/(35*a^2*b^3*(a + b*x^2)^(5/2)) - ((16*
A*b^3 - 3*a*(24*b^2*B - 66*a*b*C + 143*a^2*D))*x^7)/(210*a^2*b^4*(a + b*x^2)^(3/
2)) + (D*x^9)/(6*b^3*(a + b*x^2)^(3/2)) - ((16*A*b^3 - 3*a*(24*b^2*B - 66*a*b*C
+ 143*a^2*D))*x^5)/(30*a^2*b^5*Sqrt[a + b*x^2]) - ((16*A*b^3 - 3*a*(24*b^2*B - 6
6*a*b*C + 143*a^2*D))*x*Sqrt[a + b*x^2])/(16*a*b^7) + ((16*A*b^3 - 3*a*(24*b^2*B
- 66*a*b*C + 143*a^2*D))*x^3*Sqrt[a + b*x^2])/(24*a^2*b^6) + ((16*A*b^3 - 72*a*
b^2*B + 198*a^2*b*C - 429*a^3*D)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(15
/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(x**8*(D*x**6+C*x**4+B*x**2+A)/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.524869, size = 252, normalized size = 0.66 \[ \frac{\log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right ) \left (16 A b^3-3 a \left (143 a^2 D-66 a b C+24 b^2 B\right )\right )}{16 b^{15/2}}+\frac{x \left (45045 a^6 D-2310 a^5 b \left (9 C-65 D x^2\right )+42 a^4 b^2 \left (180 B-1650 C x^2+4147 D x^4\right )-12 a^3 b^3 \left (140 A-2100 B x^2+6699 C x^4-6292 D x^6\right )+a^2 b^4 x^2 \left (-5600 A+29232 B x^2-34848 C x^4+5005 D x^6\right )-2 a b^5 x^4 \left (3248 A-6336 B x^2+1155 C x^4+455 D x^6\right )+4 b^6 x^6 \left (35 \left (6 B x^2+3 C x^4+2 D x^6\right )-704 A\right )\right )}{1680 b^7 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^8*(A + B*x^2 + C*x^4 + D*x^6))/(a + b*x^2)^(9/2),x]

[Out]

(x*(45045*a^6*D - 2310*a^5*b*(9*C - 65*D*x^2) + 42*a^4*b^2*(180*B - 1650*C*x^2 +
4147*D*x^4) - 12*a^3*b^3*(140*A - 2100*B*x^2 + 6699*C*x^4 - 6292*D*x^6) - 2*a*b
^5*x^4*(3248*A - 6336*B*x^2 + 1155*C*x^4 + 455*D*x^6) + a^2*b^4*x^2*(-5600*A + 2
9232*B*x^2 - 34848*C*x^4 + 5005*D*x^6) + 4*b^6*x^6*(-704*A + 35*(6*B*x^2 + 3*C*x
^4 + 2*D*x^6))))/(1680*b^7*(a + b*x^2)^(7/2)) + ((16*A*b^3 - 3*a*(24*b^2*B - 66*
a*b*C + 143*a^2*D))*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(16*b^(15/2))

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Maple [A]  time = 0.402, size = 517, normalized size = 1.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(D*x^6+C*x^4+B*x^2+A)/(b*x^2+a)^(9/2),x)

[Out]

9/2*B*a/b^5*x/(b*x^2+a)^(1/2)-11/8*C*a/b^2*x^9/(b*x^2+a)^(7/2)-99/56*C*a^2/b^3*x
^7/(b*x^2+a)^(7/2)-13/24*D*a/b^2*x^11/(b*x^2+a)^(7/2)+143/48*D*a^2/b^3*x^9/(b*x^
2+a)^(7/2)+429/112*D*a^3/b^4*x^7/(b*x^2+a)^(7/2)+9/14*B*a/b^2*x^7/(b*x^2+a)^(7/2
)+9/10*B*a/b^3*x^5/(b*x^2+a)^(5/2)+3/2*B*a/b^4*x^3/(b*x^2+a)^(3/2)+429/80*D*a^3/
b^5*x^5/(b*x^2+a)^(5/2)+143/16*D*a^3/b^6*x^3/(b*x^2+a)^(3/2)+429/16*D*a^3/b^7*x/
(b*x^2+a)^(1/2)-99/40*C*a^2/b^4*x^5/(b*x^2+a)^(5/2)-33/8*C*a^2/b^5*x^3/(b*x^2+a)
^(3/2)-99/8*C*a^2/b^6*x/(b*x^2+a)^(1/2)+A/b^(9/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))-
1/7*A*x^7/b/(b*x^2+a)^(7/2)-1/5*A/b^2*x^5/(b*x^2+a)^(5/2)-1/3*A/b^3*x^3/(b*x^2+a
)^(3/2)-A/b^4*x/(b*x^2+a)^(1/2)+1/4*C*x^11/b/(b*x^2+a)^(7/2)+99/8*C*a^2/b^(13/2)
*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/2*B*x^9/b/(b*x^2+a)^(7/2)-9/2*B*a/b^(11/2)*ln(x
*b^(1/2)+(b*x^2+a)^(1/2))+1/6*D*x^13/b/(b*x^2+a)^(7/2)-429/16*D*a^3/b^(15/2)*ln(
x*b^(1/2)+(b*x^2+a)^(1/2))

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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)*x^8/(b*x^2 + a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.11271, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^6 + C*x^4 + B*x^2 + A)*x^8/(b*x^2 + a)^(9/2),x, algorithm="fricas")

[Out]

[1/3360*(2*(280*D*b^6*x^13 - 70*(13*D*a*b^5 - 6*C*b^6)*x^11 + 35*(143*D*a^2*b^4
- 66*C*a*b^5 + 24*B*b^6)*x^9 + 176*(429*D*a^3*b^3 - 198*C*a^2*b^4 + 72*B*a*b^5 -
16*A*b^6)*x^7 + 406*(429*D*a^4*b^2 - 198*C*a^3*b^3 + 72*B*a^2*b^4 - 16*A*a*b^5)
*x^5 + 350*(429*D*a^5*b - 198*C*a^4*b^2 + 72*B*a^3*b^3 - 16*A*a^2*b^4)*x^3 + 105
*(429*D*a^6 - 198*C*a^5*b + 72*B*a^4*b^2 - 16*A*a^3*b^3)*x)*sqrt(b*x^2 + a)*sqrt
(b) - 105*((429*D*a^3*b^4 - 198*C*a^2*b^5 + 72*B*a*b^6 - 16*A*b^7)*x^8 + 429*D*a
^7 - 198*C*a^6*b + 72*B*a^5*b^2 - 16*A*a^4*b^3 + 4*(429*D*a^4*b^3 - 198*C*a^3*b^
4 + 72*B*a^2*b^5 - 16*A*a*b^6)*x^6 + 6*(429*D*a^5*b^2 - 198*C*a^4*b^3 + 72*B*a^3
*b^4 - 16*A*a^2*b^5)*x^4 + 4*(429*D*a^6*b - 198*C*a^5*b^2 + 72*B*a^4*b^3 - 16*A*
a^3*b^4)*x^2)*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)))/((b^11*x^8 +
4*a*b^10*x^6 + 6*a^2*b^9*x^4 + 4*a^3*b^8*x^2 + a^4*b^7)*sqrt(b)), 1/1680*((280*D
*b^6*x^13 - 70*(13*D*a*b^5 - 6*C*b^6)*x^11 + 35*(143*D*a^2*b^4 - 66*C*a*b^5 + 24
*B*b^6)*x^9 + 176*(429*D*a^3*b^3 - 198*C*a^2*b^4 + 72*B*a*b^5 - 16*A*b^6)*x^7 +
406*(429*D*a^4*b^2 - 198*C*a^3*b^3 + 72*B*a^2*b^4 - 16*A*a*b^5)*x^5 + 350*(429*D
*a^5*b - 198*C*a^4*b^2 + 72*B*a^3*b^3 - 16*A*a^2*b^4)*x^3 + 105*(429*D*a^6 - 198
*C*a^5*b + 72*B*a^4*b^2 - 16*A*a^3*b^3)*x)*sqrt(b*x^2 + a)*sqrt(-b) - 105*((429*
D*a^3*b^4 - 198*C*a^2*b^5 + 72*B*a*b^6 - 16*A*b^7)*x^8 + 429*D*a^7 - 198*C*a^6*b
+ 72*B*a^5*b^2 - 16*A*a^4*b^3 + 4*(429*D*a^4*b^3 - 198*C*a^3*b^4 + 72*B*a^2*b^5
- 16*A*a*b^6)*x^6 + 6*(429*D*a^5*b^2 - 198*C*a^4*b^3 + 72*B*a^3*b^4 - 16*A*a^2*
b^5)*x^4 + 4*(429*D*a^6*b - 198*C*a^5*b^2 + 72*B*a^4*b^3 - 16*A*a^3*b^4)*x^2)*ar
ctan(sqrt(-b)*x/sqrt(b*x^2 + a)))/((b^11*x^8 + 4*a*b^10*x^6 + 6*a^2*b^9*x^4 + 4*
a^3*b^8*x^2 + a^4*b^7)*sqrt(-b))]

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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(x**8*(D*x**6+C*x**4+B*x**2+A)/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.231411, size = 462, normalized size = 1.21 \[ \frac{{\left ({\left ({\left ({\left (35 \,{\left (2 \,{\left (\frac{4 \, D x^{2}}{b} - \frac{13 \, D a^{4} b^{11} - 6 \, C a^{3} b^{12}}{a^{3} b^{13}}\right )} x^{2} + \frac{143 \, D a^{5} b^{10} - 66 \, C a^{4} b^{11} + 24 \, B a^{3} b^{12}}{a^{3} b^{13}}\right )} x^{2} + \frac{176 \,{\left (429 \, D a^{6} b^{9} - 198 \, C a^{5} b^{10} + 72 \, B a^{4} b^{11} - 16 \, A a^{3} b^{12}\right )}}{a^{3} b^{13}}\right )} x^{2} + \frac{406 \,{\left (429 \, D a^{7} b^{8} - 198 \, C a^{6} b^{9} + 72 \, B a^{5} b^{10} - 16 \, A a^{4} b^{11}\right )}}{a^{3} b^{13}}\right )} x^{2} + \frac{350 \,{\left (429 \, D a^{8} b^{7} - 198 \, C a^{7} b^{8} + 72 \, B a^{6} b^{9} - 16 \, A a^{5} b^{10}\right )}}{a^{3} b^{13}}\right )} x^{2} + \frac{105 \,{\left (429 \, D a^{9} b^{6} - 198 \, C a^{8} b^{7} + 72 \, B a^{7} b^{8} - 16 \, A a^{6} b^{9}\right )}}{a^{3} b^{13}}\right )} x}{1680 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} + \frac{{\left (429 \, D a^{3} - 198 \, C a^{2} b + 72 \, B a b^{2} - 16 \, A b^{3}\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{16 \, b^{\frac{15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^6 + C*x^4 + B*x^2 + A)*x^8/(b*x^2 + a)^(9/2),x, algorithm="giac")

[Out]

1/1680*((((35*(2*(4*D*x^2/b - (13*D*a^4*b^11 - 6*C*a^3*b^12)/(a^3*b^13))*x^2 + (
143*D*a^5*b^10 - 66*C*a^4*b^11 + 24*B*a^3*b^12)/(a^3*b^13))*x^2 + 176*(429*D*a^6
*b^9 - 198*C*a^5*b^10 + 72*B*a^4*b^11 - 16*A*a^3*b^12)/(a^3*b^13))*x^2 + 406*(42
9*D*a^7*b^8 - 198*C*a^6*b^9 + 72*B*a^5*b^10 - 16*A*a^4*b^11)/(a^3*b^13))*x^2 + 3
50*(429*D*a^8*b^7 - 198*C*a^7*b^8 + 72*B*a^6*b^9 - 16*A*a^5*b^10)/(a^3*b^13))*x^
2 + 105*(429*D*a^9*b^6 - 198*C*a^8*b^7 + 72*B*a^7*b^8 - 16*A*a^6*b^9)/(a^3*b^13)
)*x/(b*x^2 + a)^(7/2) + 1/16*(429*D*a^3 - 198*C*a^2*b + 72*B*a*b^2 - 16*A*b^3)*l
n(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(15/2)