3.786 \(\int \frac{\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^7} \, dx\)

Optimal. Leaf size=100 \[ -\frac{2 \left (a^2-b^2 x^2\right )^{5/2}}{63 a^2 b (a+b x)^6}-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{9 a b (a+b x)^7}-\frac{2 \left (a^2-b^2 x^2\right )^{5/2}}{315 a^3 b (a+b x)^5} \]

[Out]

-(a^2 - b^2*x^2)^(5/2)/(9*a*b*(a + b*x)^7) - (2*(a^2 - b^2*x^2)^(5/2))/(63*a^2*b
*(a + b*x)^6) - (2*(a^2 - b^2*x^2)^(5/2))/(315*a^3*b*(a + b*x)^5)

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Rubi [A]  time = 0.120342, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{2 \left (a^2-b^2 x^2\right )^{5/2}}{63 a^2 b (a+b x)^6}-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{9 a b (a+b x)^7}-\frac{2 \left (a^2-b^2 x^2\right )^{5/2}}{315 a^3 b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^7,x]

[Out]

-(a^2 - b^2*x^2)^(5/2)/(9*a*b*(a + b*x)^7) - (2*(a^2 - b^2*x^2)^(5/2))/(63*a^2*b
*(a + b*x)^6) - (2*(a^2 - b^2*x^2)^(5/2))/(315*a^3*b*(a + b*x)^5)

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Rubi in Sympy [A]  time = 12.7077, size = 83, normalized size = 0.83 \[ - \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{5}{2}}}{9 a b \left (a + b x\right )^{7}} - \frac{2 \left (a^{2} - b^{2} x^{2}\right )^{\frac{5}{2}}}{63 a^{2} b \left (a + b x\right )^{6}} - \frac{2 \left (a^{2} - b^{2} x^{2}\right )^{\frac{5}{2}}}{315 a^{3} b \left (a + b x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b**2*x**2+a**2)**(3/2)/(b*x+a)**7,x)

[Out]

-(a**2 - b**2*x**2)**(5/2)/(9*a*b*(a + b*x)**7) - 2*(a**2 - b**2*x**2)**(5/2)/(6
3*a**2*b*(a + b*x)**6) - 2*(a**2 - b**2*x**2)**(5/2)/(315*a**3*b*(a + b*x)**5)

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Mathematica [A]  time = 0.0538237, size = 60, normalized size = 0.6 \[ -\frac{(a-b x)^2 \sqrt{a^2-b^2 x^2} \left (47 a^2+14 a b x+2 b^2 x^2\right )}{315 a^3 b (a+b x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^7,x]

[Out]

-((a - b*x)^2*Sqrt[a^2 - b^2*x^2]*(47*a^2 + 14*a*b*x + 2*b^2*x^2))/(315*a^3*b*(a
+ b*x)^5)

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Maple [A]  time = 0.01, size = 55, normalized size = 0.6 \[ -{\frac{ \left ( 2\,{b}^{2}{x}^{2}+14\,abx+47\,{a}^{2} \right ) \left ( -bx+a \right ) }{315\, \left ( bx+a \right ) ^{6}{a}^{3}b} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b^2*x^2+a^2)^(3/2)/(b*x+a)^7,x)

[Out]

-1/315*(-b*x+a)*(2*b^2*x^2+14*a*b*x+47*a^2)*(-b^2*x^2+a^2)^(3/2)/(b*x+a)^6/a^3/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)^7,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236822, size = 512, normalized size = 5.12 \[ -\frac{49 \, b^{8} x^{9} + 423 \, a b^{7} x^{8} + 801 \, a^{2} b^{6} x^{7} - 1071 \, a^{3} b^{5} x^{6} - 4158 \, a^{4} b^{4} x^{5} - 3780 \, a^{5} b^{3} x^{4} - 840 \, a^{6} b^{2} x^{3} + 5040 \, a^{7} b x^{2} + 5040 \, a^{8} x - 3 \,{\left (15 \, b^{7} x^{8} - 6 \, a b^{6} x^{7} - 357 \, a^{2} b^{5} x^{6} - 896 \, a^{3} b^{4} x^{5} - 420 \, a^{4} b^{3} x^{4} + 560 \, a^{5} b^{2} x^{3} + 1680 \, a^{6} b x^{2} + 1680 \, a^{7} x\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{315 \,{\left (a^{3} b^{9} x^{9} + 9 \, a^{4} b^{8} x^{8} + 18 \, a^{5} b^{7} x^{7} - 18 \, a^{6} b^{6} x^{6} - 99 \, a^{7} b^{5} x^{5} - 99 \, a^{8} b^{4} x^{4} + 24 \, a^{9} b^{3} x^{3} + 108 \, a^{10} b^{2} x^{2} + 72 \, a^{11} b x + 16 \, a^{12} -{\left (a^{3} b^{8} x^{8} - 22 \, a^{5} b^{6} x^{6} - 60 \, a^{6} b^{5} x^{5} - 39 \, a^{7} b^{4} x^{4} + 60 \, a^{8} b^{3} x^{3} + 116 \, a^{9} b^{2} x^{2} + 72 \, a^{10} b x + 16 \, a^{11}\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b^2*x^2 + a^2)^(3/2)/(b*x + a)^7,x, algorithm="fricas")

[Out]

-1/315*(49*b^8*x^9 + 423*a*b^7*x^8 + 801*a^2*b^6*x^7 - 1071*a^3*b^5*x^6 - 4158*a
^4*b^4*x^5 - 3780*a^5*b^3*x^4 - 840*a^6*b^2*x^3 + 5040*a^7*b*x^2 + 5040*a^8*x -
3*(15*b^7*x^8 - 6*a*b^6*x^7 - 357*a^2*b^5*x^6 - 896*a^3*b^4*x^5 - 420*a^4*b^3*x^
4 + 560*a^5*b^2*x^3 + 1680*a^6*b*x^2 + 1680*a^7*x)*sqrt(-b^2*x^2 + a^2))/(a^3*b^
9*x^9 + 9*a^4*b^8*x^8 + 18*a^5*b^7*x^7 - 18*a^6*b^6*x^6 - 99*a^7*b^5*x^5 - 99*a^
8*b^4*x^4 + 24*a^9*b^3*x^3 + 108*a^10*b^2*x^2 + 72*a^11*b*x + 16*a^12 - (a^3*b^8
*x^8 - 22*a^5*b^6*x^6 - 60*a^6*b^5*x^5 - 39*a^7*b^4*x^4 + 60*a^8*b^3*x^3 + 116*a
^9*b^2*x^2 + 72*a^10*b*x + 16*a^11)*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)**7,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.237159, size = 390, normalized size = 3.9 \[ \frac{2 \,{\left (\frac{108 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}}{b^{2} x} + \frac{1062 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + \frac{1638 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{3}}{b^{6} x^{3}} + \frac{3402 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{4}}{b^{8} x^{4}} + \frac{2520 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{5}}{b^{10} x^{5}} + \frac{2310 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{6}}{b^{12} x^{6}} + \frac{630 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{7}}{b^{14} x^{7}} + \frac{315 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{8}}{b^{16} x^{8}} + 47\right )}}{315 \, a^{3}{\left (\frac{a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}}{b^{2} x} + 1\right )}^{9}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b^2*x^2 + a^2)^(3/2)/(b*x + a)^7,x, algorithm="giac")

[Out]

2/315*(108*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 1062*(a*b + sqrt(-b^2*x
^2 + a^2)*abs(b))^2/(b^4*x^2) + 1638*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^3/(b^6*
x^3) + 3402*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^4/(b^8*x^4) + 2520*(a*b + sqrt(-
b^2*x^2 + a^2)*abs(b))^5/(b^10*x^5) + 2310*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^6
/(b^12*x^6) + 630*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^7/(b^14*x^7) + 315*(a*b +
sqrt(-b^2*x^2 + a^2)*abs(b))^8/(b^16*x^8) + 47)/(a^3*((a*b + sqrt(-b^2*x^2 + a^2
)*abs(b))/(b^2*x) + 1)^9*abs(b))