3.787 \(\int \frac{\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^8} \, dx\)

Optimal. Leaf size=133 \[ -\frac{\left (a^2-b^2 x^2\right )^{5/2}}{33 a^2 b (a+b x)^7}-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{11 a b (a+b x)^8}-\frac{2 \left (a^2-b^2 x^2\right )^{5/2}}{1155 a^4 b (a+b x)^5}-\frac{2 \left (a^2-b^2 x^2\right )^{5/2}}{231 a^3 b (a+b x)^6} \]

[Out]

-(a^2 - b^2*x^2)^(5/2)/(11*a*b*(a + b*x)^8) - (a^2 - b^2*x^2)^(5/2)/(33*a^2*b*(a
+ b*x)^7) - (2*(a^2 - b^2*x^2)^(5/2))/(231*a^3*b*(a + b*x)^6) - (2*(a^2 - b^2*x
^2)^(5/2))/(1155*a^4*b*(a + b*x)^5)

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Rubi [A]  time = 0.166231, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{\left (a^2-b^2 x^2\right )^{5/2}}{33 a^2 b (a+b x)^7}-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{11 a b (a+b x)^8}-\frac{2 \left (a^2-b^2 x^2\right )^{5/2}}{1155 a^4 b (a+b x)^5}-\frac{2 \left (a^2-b^2 x^2\right )^{5/2}}{231 a^3 b (a+b x)^6} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^8,x]

[Out]

-(a^2 - b^2*x^2)^(5/2)/(11*a*b*(a + b*x)^8) - (a^2 - b^2*x^2)^(5/2)/(33*a^2*b*(a
+ b*x)^7) - (2*(a^2 - b^2*x^2)^(5/2))/(231*a^3*b*(a + b*x)^6) - (2*(a^2 - b^2*x
^2)^(5/2))/(1155*a^4*b*(a + b*x)^5)

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Rubi in Sympy [A]  time = 17.8784, size = 110, normalized size = 0.83 \[ - \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{5}{2}}}{11 a b \left (a + b x\right )^{8}} - \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{5}{2}}}{33 a^{2} b \left (a + b x\right )^{7}} - \frac{2 \left (a^{2} - b^{2} x^{2}\right )^{\frac{5}{2}}}{231 a^{3} b \left (a + b x\right )^{6}} - \frac{2 \left (a^{2} - b^{2} x^{2}\right )^{\frac{5}{2}}}{1155 a^{4} b \left (a + b x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b**2*x**2+a**2)**(3/2)/(b*x+a)**8,x)

[Out]

-(a**2 - b**2*x**2)**(5/2)/(11*a*b*(a + b*x)**8) - (a**2 - b**2*x**2)**(5/2)/(33
*a**2*b*(a + b*x)**7) - 2*(a**2 - b**2*x**2)**(5/2)/(231*a**3*b*(a + b*x)**6) -
2*(a**2 - b**2*x**2)**(5/2)/(1155*a**4*b*(a + b*x)**5)

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Mathematica [A]  time = 0.0605853, size = 71, normalized size = 0.53 \[ -\frac{(a-b x)^2 \sqrt{a^2-b^2 x^2} \left (152 a^3+61 a^2 b x+16 a b^2 x^2+2 b^3 x^3\right )}{1155 a^4 b (a+b x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^8,x]

[Out]

-((a - b*x)^2*Sqrt[a^2 - b^2*x^2]*(152*a^3 + 61*a^2*b*x + 16*a*b^2*x^2 + 2*b^3*x
^3))/(1155*a^4*b*(a + b*x)^6)

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Maple [A]  time = 0.01, size = 66, normalized size = 0.5 \[ -{\frac{ \left ( 2\,{b}^{3}{x}^{3}+16\,a{b}^{2}{x}^{2}+61\,{a}^{2}bx+152\,{a}^{3} \right ) \left ( -bx+a \right ) }{1155\, \left ( bx+a \right ) ^{7}{a}^{4}b} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b^2*x^2+a^2)^(3/2)/(b*x+a)^8,x)

[Out]

-1/1155*(-b*x+a)*(2*b^3*x^3+16*a*b^2*x^2+61*a^2*b*x+152*a^3)*(-b^2*x^2+a^2)^(3/2
)/(b*x+a)^7/a^4/b

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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((-b^2*x^2 + a^2)^(3/2)/(b*x + a)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25472, size = 629, normalized size = 4.73 \[ -\frac{150 \, b^{10} x^{11} - 22 \, a b^{9} x^{10} - 5071 \, a^{2} b^{8} x^{9} - 16665 \, a^{3} b^{7} x^{8} - 10989 \, a^{4} b^{6} x^{7} + 35343 \, a^{5} b^{5} x^{6} + 66066 \, a^{6} b^{4} x^{5} + 32340 \, a^{7} b^{3} x^{4} - 18480 \, a^{8} b^{2} x^{3} - 55440 \, a^{9} b x^{2} - 36960 \, a^{10} x + 11 \,{\left (14 \, b^{9} x^{10} + 152 \, a b^{8} x^{9} + 381 \, a^{2} b^{7} x^{8} - 324 \, a^{3} b^{6} x^{7} - 2793 \, a^{4} b^{5} x^{6} - 3906 \, a^{5} b^{4} x^{5} - 420 \, a^{6} b^{3} x^{4} + 3360 \, a^{7} b^{2} x^{3} + 5040 \, a^{8} b x^{2} + 3360 \, a^{9} x\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{1155 \,{\left (a^{4} b^{11} x^{11} - 33 \, a^{6} b^{9} x^{9} - 110 \, a^{7} b^{8} x^{8} - 77 \, a^{8} b^{7} x^{7} + 220 \, a^{9} b^{6} x^{6} + 473 \, a^{10} b^{5} x^{5} + 242 \, a^{11} b^{4} x^{4} - 220 \, a^{12} b^{3} x^{3} - 352 \, a^{13} b^{2} x^{2} - 176 \, a^{14} b x - 32 \, a^{15} +{\left (a^{4} b^{10} x^{10} + 11 \, a^{5} b^{9} x^{9} + 28 \, a^{6} b^{8} x^{8} - 22 \, a^{7} b^{7} x^{7} - 199 \, a^{8} b^{6} x^{6} - 297 \, a^{9} b^{5} x^{5} - 54 \, a^{10} b^{4} x^{4} + 308 \, a^{11} b^{3} x^{3} + 368 \, a^{12} b^{2} x^{2} + 176 \, a^{13} b x + 32 \, a^{14}\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b^2*x^2 + a^2)^(3/2)/(b*x + a)^8,x, algorithm="fricas")

[Out]

-1/1155*(150*b^10*x^11 - 22*a*b^9*x^10 - 5071*a^2*b^8*x^9 - 16665*a^3*b^7*x^8 -
10989*a^4*b^6*x^7 + 35343*a^5*b^5*x^6 + 66066*a^6*b^4*x^5 + 32340*a^7*b^3*x^4 -
18480*a^8*b^2*x^3 - 55440*a^9*b*x^2 - 36960*a^10*x + 11*(14*b^9*x^10 + 152*a*b^8
*x^9 + 381*a^2*b^7*x^8 - 324*a^3*b^6*x^7 - 2793*a^4*b^5*x^6 - 3906*a^5*b^4*x^5 -
420*a^6*b^3*x^4 + 3360*a^7*b^2*x^3 + 5040*a^8*b*x^2 + 3360*a^9*x)*sqrt(-b^2*x^2
+ a^2))/(a^4*b^11*x^11 - 33*a^6*b^9*x^9 - 110*a^7*b^8*x^8 - 77*a^8*b^7*x^7 + 22
0*a^9*b^6*x^6 + 473*a^10*b^5*x^5 + 242*a^11*b^4*x^4 - 220*a^12*b^3*x^3 - 352*a^1
3*b^2*x^2 - 176*a^14*b*x - 32*a^15 + (a^4*b^10*x^10 + 11*a^5*b^9*x^9 + 28*a^6*b^
8*x^8 - 22*a^7*b^7*x^7 - 199*a^8*b^6*x^6 - 297*a^9*b^5*x^5 - 54*a^10*b^4*x^4 + 3
08*a^11*b^3*x^3 + 368*a^12*b^2*x^2 + 176*a^13*b*x + 32*a^14)*sqrt(-b^2*x^2 + a^2
))

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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((-b**2*x**2+a**2)**(3/2)/(b*x+a)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.239866, size = 474, normalized size = 3.56 \[ \frac{2 \,{\left (\frac{517 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}}{b^{2} x} + \frac{4895 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + \frac{11220 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{3}}{b^{6} x^{3}} + \frac{27060 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{4}}{b^{8} x^{4}} + \frac{32802 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{5}}{b^{10} x^{5}} + \frac{37422 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{6}}{b^{12} x^{6}} + \frac{23100 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{7}}{b^{14} x^{7}} + \frac{13860 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{8}}{b^{16} x^{8}} + \frac{3465 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{9}}{b^{18} x^{9}} + \frac{1155 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{10}}{b^{20} x^{10}} + 152\right )}}{1155 \, a^{4}{\left (\frac{a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}}{b^{2} x} + 1\right )}^{11}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b^2*x^2 + a^2)^(3/2)/(b*x + a)^8,x, algorithm="giac")

[Out]

2/1155*(517*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 4895*(a*b + sqrt(-b^2*
x^2 + a^2)*abs(b))^2/(b^4*x^2) + 11220*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^3/(b^
6*x^3) + 27060*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^4/(b^8*x^4) + 32802*(a*b + sq
rt(-b^2*x^2 + a^2)*abs(b))^5/(b^10*x^5) + 37422*(a*b + sqrt(-b^2*x^2 + a^2)*abs(
b))^6/(b^12*x^6) + 23100*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^7/(b^14*x^7) + 1386
0*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^8/(b^16*x^8) + 3465*(a*b + sqrt(-b^2*x^2 +
a^2)*abs(b))^9/(b^18*x^9) + 1155*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^10/(b^20*x
^10) + 152)/(a^4*((a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 1)^11*abs(b))