3.776 \(\int \frac{\sqrt{a^2-b^2 x^2}}{(a+b x)^7} \, dx\)

Optimal. Leaf size=166 \[ -\frac{4 \left (a^2-b^2 x^2\right )^{3/2}}{99 a^2 b (a+b x)^6}-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{11 a b (a+b x)^7}-\frac{8 \left (a^2-b^2 x^2\right )^{3/2}}{3465 a^5 b (a+b x)^3}-\frac{8 \left (a^2-b^2 x^2\right )^{3/2}}{1155 a^4 b (a+b x)^4}-\frac{4 \left (a^2-b^2 x^2\right )^{3/2}}{231 a^3 b (a+b x)^5} \]

[Out]

-(a^2 - b^2*x^2)^(3/2)/(11*a*b*(a + b*x)^7) - (4*(a^2 - b^2*x^2)^(3/2))/(99*a^2*
b*(a + b*x)^6) - (4*(a^2 - b^2*x^2)^(3/2))/(231*a^3*b*(a + b*x)^5) - (8*(a^2 - b
^2*x^2)^(3/2))/(1155*a^4*b*(a + b*x)^4) - (8*(a^2 - b^2*x^2)^(3/2))/(3465*a^5*b*
(a + b*x)^3)

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Rubi [A]  time = 0.217013, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{4 \left (a^2-b^2 x^2\right )^{3/2}}{99 a^2 b (a+b x)^6}-\frac{\left (a^2-b^2 x^2\right )^{3/2}}{11 a b (a+b x)^7}-\frac{8 \left (a^2-b^2 x^2\right )^{3/2}}{3465 a^5 b (a+b x)^3}-\frac{8 \left (a^2-b^2 x^2\right )^{3/2}}{1155 a^4 b (a+b x)^4}-\frac{4 \left (a^2-b^2 x^2\right )^{3/2}}{231 a^3 b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a^2 - b^2*x^2]/(a + b*x)^7,x]

[Out]

-(a^2 - b^2*x^2)^(3/2)/(11*a*b*(a + b*x)^7) - (4*(a^2 - b^2*x^2)^(3/2))/(99*a^2*
b*(a + b*x)^6) - (4*(a^2 - b^2*x^2)^(3/2))/(231*a^3*b*(a + b*x)^5) - (8*(a^2 - b
^2*x^2)^(3/2))/(1155*a^4*b*(a + b*x)^4) - (8*(a^2 - b^2*x^2)^(3/2))/(3465*a^5*b*
(a + b*x)^3)

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Rubi in Sympy [A]  time = 24.8251, size = 141, normalized size = 0.85 \[ - \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{11 a b \left (a + b x\right )^{7}} - \frac{4 \left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{99 a^{2} b \left (a + b x\right )^{6}} - \frac{4 \left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{231 a^{3} b \left (a + b x\right )^{5}} - \frac{8 \left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{1155 a^{4} b \left (a + b x\right )^{4}} - \frac{8 \left (a^{2} - b^{2} x^{2}\right )^{\frac{3}{2}}}{3465 a^{5} b \left (a + b x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b**2*x**2+a**2)**(1/2)/(b*x+a)**7,x)

[Out]

-(a**2 - b**2*x**2)**(3/2)/(11*a*b*(a + b*x)**7) - 4*(a**2 - b**2*x**2)**(3/2)/(
99*a**2*b*(a + b*x)**6) - 4*(a**2 - b**2*x**2)**(3/2)/(231*a**3*b*(a + b*x)**5)
- 8*(a**2 - b**2*x**2)**(3/2)/(1155*a**4*b*(a + b*x)**4) - 8*(a**2 - b**2*x**2)*
*(3/2)/(3465*a**5*b*(a + b*x)**3)

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Mathematica [A]  time = 0.0522724, size = 85, normalized size = 0.51 \[ \frac{\sqrt{a^2-b^2 x^2} \left (-547 a^5+183 a^4 b x+184 a^3 b^2 x^2+124 a^2 b^3 x^3+48 a b^4 x^4+8 b^5 x^5\right )}{3465 a^5 b (a+b x)^6} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a^2 - b^2*x^2]/(a + b*x)^7,x]

[Out]

(Sqrt[a^2 - b^2*x^2]*(-547*a^5 + 183*a^4*b*x + 184*a^3*b^2*x^2 + 124*a^2*b^3*x^3
+ 48*a*b^4*x^4 + 8*b^5*x^5))/(3465*a^5*b*(a + b*x)^6)

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Maple [A]  time = 0.01, size = 77, normalized size = 0.5 \[ -{\frac{ \left ( 8\,{b}^{4}{x}^{4}+56\,{x}^{3}a{b}^{3}+180\,{x}^{2}{a}^{2}{b}^{2}+364\,x{a}^{3}b+547\,{a}^{4} \right ) \left ( -bx+a \right ) }{3465\, \left ( bx+a \right ) ^{6}{a}^{5}b}\sqrt{-{b}^{2}{x}^{2}+{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b^2*x^2+a^2)^(1/2)/(b*x+a)^7,x)

[Out]

-1/3465*(-b*x+a)*(8*b^4*x^4+56*a*b^3*x^3+180*a^2*b^2*x^2+364*a^3*b*x+547*a^4)*(-
b^2*x^2+a^2)^(1/2)/(b*x+a)^6/a^5/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(sqrt(-b^2*x^2 + a^2)/(b*x + a)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.267049, size = 629, normalized size = 3.79 \[ -\frac{555 \, b^{10} x^{11} + 88 \, a b^{9} x^{10} - 17831 \, a^{2} b^{8} x^{9} - 60390 \, a^{3} b^{7} x^{8} - 43824 \, a^{4} b^{6} x^{7} + 117348 \, a^{5} b^{5} x^{6} + 255486 \, a^{6} b^{4} x^{5} + 147840 \, a^{7} b^{3} x^{4} - 101640 \, a^{8} b^{2} x^{3} - 221760 \, a^{9} b x^{2} - 110880 \, a^{10} x + 11 \,{\left (49 \, b^{9} x^{10} + 547 \, a b^{8} x^{9} + 1416 \, a^{2} b^{7} x^{8} - 1014 \, a^{3} b^{6} x^{7} - 9828 \, a^{4} b^{5} x^{6} - 14826 \, a^{5} b^{4} x^{5} - 3360 \, a^{6} b^{3} x^{4} + 14280 \, a^{7} b^{2} x^{3} + 20160 \, a^{8} b x^{2} + 10080 \, a^{9} x\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{3465 \,{\left (a^{5} b^{11} x^{11} - 33 \, a^{7} b^{9} x^{9} - 110 \, a^{8} b^{8} x^{8} - 77 \, a^{9} b^{7} x^{7} + 220 \, a^{10} b^{6} x^{6} + 473 \, a^{11} b^{5} x^{5} + 242 \, a^{12} b^{4} x^{4} - 220 \, a^{13} b^{3} x^{3} - 352 \, a^{14} b^{2} x^{2} - 176 \, a^{15} b x - 32 \, a^{16} +{\left (a^{5} b^{10} x^{10} + 11 \, a^{6} b^{9} x^{9} + 28 \, a^{7} b^{8} x^{8} - 22 \, a^{8} b^{7} x^{7} - 199 \, a^{9} b^{6} x^{6} - 297 \, a^{10} b^{5} x^{5} - 54 \, a^{11} b^{4} x^{4} + 308 \, a^{12} b^{3} x^{3} + 368 \, a^{13} b^{2} x^{2} + 176 \, a^{14} b x + 32 \, a^{15}\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b^2*x^2 + a^2)/(b*x + a)^7,x, algorithm="fricas")

[Out]

-1/3465*(555*b^10*x^11 + 88*a*b^9*x^10 - 17831*a^2*b^8*x^9 - 60390*a^3*b^7*x^8 -
43824*a^4*b^6*x^7 + 117348*a^5*b^5*x^6 + 255486*a^6*b^4*x^5 + 147840*a^7*b^3*x^
4 - 101640*a^8*b^2*x^3 - 221760*a^9*b*x^2 - 110880*a^10*x + 11*(49*b^9*x^10 + 54
7*a*b^8*x^9 + 1416*a^2*b^7*x^8 - 1014*a^3*b^6*x^7 - 9828*a^4*b^5*x^6 - 14826*a^5
*b^4*x^5 - 3360*a^6*b^3*x^4 + 14280*a^7*b^2*x^3 + 20160*a^8*b*x^2 + 10080*a^9*x)
*sqrt(-b^2*x^2 + a^2))/(a^5*b^11*x^11 - 33*a^7*b^9*x^9 - 110*a^8*b^8*x^8 - 77*a^
9*b^7*x^7 + 220*a^10*b^6*x^6 + 473*a^11*b^5*x^5 + 242*a^12*b^4*x^4 - 220*a^13*b^
3*x^3 - 352*a^14*b^2*x^2 - 176*a^15*b*x - 32*a^16 + (a^5*b^10*x^10 + 11*a^6*b^9*
x^9 + 28*a^7*b^8*x^8 - 22*a^8*b^7*x^7 - 199*a^9*b^6*x^6 - 297*a^10*b^5*x^5 - 54*
a^11*b^4*x^4 + 308*a^12*b^3*x^3 + 368*a^13*b^2*x^2 + 176*a^14*b*x + 32*a^15)*sqr
t(-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)**(1/2)/(b*x+a)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.239461, size = 474, normalized size = 2.86 \[ \frac{2 \,{\left (\frac{2552 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}}{b^{2} x} + \frac{16225 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + \frac{42900 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{3}}{b^{6} x^{3}} + \frac{92730 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{4}}{b^{8} x^{4}} + \frac{122892 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{5}}{b^{10} x^{5}} + \frac{129822 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{6}}{b^{12} x^{6}} + \frac{87780 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{7}}{b^{14} x^{7}} + \frac{47355 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{8}}{b^{16} x^{8}} + \frac{13860 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{9}}{b^{18} x^{9}} + \frac{3465 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{10}}{b^{20} x^{10}} + 547\right )}}{3465 \, a^{5}{\left (\frac{a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}}{b^{2} x} + 1\right )}^{11}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b^2*x^2 + a^2)/(b*x + a)^7,x, algorithm="giac")

[Out]

2/3465*(2552*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 16225*(a*b + sqrt(-b^
2*x^2 + a^2)*abs(b))^2/(b^4*x^2) + 42900*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^3/(
b^6*x^3) + 92730*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^4/(b^8*x^4) + 122892*(a*b +
sqrt(-b^2*x^2 + a^2)*abs(b))^5/(b^10*x^5) + 129822*(a*b + sqrt(-b^2*x^2 + a^2)*
abs(b))^6/(b^12*x^6) + 87780*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^7/(b^14*x^7) +
47355*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^8/(b^16*x^8) + 13860*(a*b + sqrt(-b^2*
x^2 + a^2)*abs(b))^9/(b^18*x^9) + 3465*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^10/(b
^20*x^10) + 547)/(a^5*((a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 1)^11*abs(b
))