3.788 \(\int \frac{\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^9} \, dx\)

Optimal. Leaf size=166 \[ -\frac{4 \left (a^2-b^2 x^2\right )^{5/2}}{143 a^2 b (a+b x)^8}-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{13 a b (a+b x)^9}-\frac{8 \left (a^2-b^2 x^2\right )^{5/2}}{15015 a^5 b (a+b x)^5}-\frac{8 \left (a^2-b^2 x^2\right )^{5/2}}{3003 a^4 b (a+b x)^6}-\frac{4 \left (a^2-b^2 x^2\right )^{5/2}}{429 a^3 b (a+b x)^7} \]

[Out]

-(a^2 - b^2*x^2)^(5/2)/(13*a*b*(a + b*x)^9) - (4*(a^2 - b^2*x^2)^(5/2))/(143*a^2
*b*(a + b*x)^8) - (4*(a^2 - b^2*x^2)^(5/2))/(429*a^3*b*(a + b*x)^7) - (8*(a^2 -
b^2*x^2)^(5/2))/(3003*a^4*b*(a + b*x)^6) - (8*(a^2 - b^2*x^2)^(5/2))/(15015*a^5*
b*(a + b*x)^5)

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Rubi [A]  time = 0.220715, 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 \[ -\frac{4 \left (a^2-b^2 x^2\right )^{5/2}}{143 a^2 b (a+b x)^8}-\frac{\left (a^2-b^2 x^2\right )^{5/2}}{13 a b (a+b x)^9}-\frac{8 \left (a^2-b^2 x^2\right )^{5/2}}{15015 a^5 b (a+b x)^5}-\frac{8 \left (a^2-b^2 x^2\right )^{5/2}}{3003 a^4 b (a+b x)^6}-\frac{4 \left (a^2-b^2 x^2\right )^{5/2}}{429 a^3 b (a+b x)^7} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^9,x]

[Out]

-(a^2 - b^2*x^2)^(5/2)/(13*a*b*(a + b*x)^9) - (4*(a^2 - b^2*x^2)^(5/2))/(143*a^2
*b*(a + b*x)^8) - (4*(a^2 - b^2*x^2)^(5/2))/(429*a^3*b*(a + b*x)^7) - (8*(a^2 -
b^2*x^2)^(5/2))/(3003*a^4*b*(a + b*x)^6) - (8*(a^2 - b^2*x^2)^(5/2))/(15015*a^5*
b*(a + b*x)^5)

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Rubi in Sympy [A]  time = 23.9761, size = 141, normalized size = 0.85 \[ - \frac{\left (a^{2} - b^{2} x^{2}\right )^{\frac{5}{2}}}{13 a b \left (a + b x\right )^{9}} - \frac{4 \left (a^{2} - b^{2} x^{2}\right )^{\frac{5}{2}}}{143 a^{2} b \left (a + b x\right )^{8}} - \frac{4 \left (a^{2} - b^{2} x^{2}\right )^{\frac{5}{2}}}{429 a^{3} b \left (a + b x\right )^{7}} - \frac{8 \left (a^{2} - b^{2} x^{2}\right )^{\frac{5}{2}}}{3003 a^{4} b \left (a + b x\right )^{6}} - \frac{8 \left (a^{2} - b^{2} x^{2}\right )^{\frac{5}{2}}}{15015 a^{5} b \left (a + b x\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b**2*x**2+a**2)**(3/2)/(b*x+a)**9,x)

[Out]

-(a**2 - b**2*x**2)**(5/2)/(13*a*b*(a + b*x)**9) - 4*(a**2 - b**2*x**2)**(5/2)/(
143*a**2*b*(a + b*x)**8) - 4*(a**2 - b**2*x**2)**(5/2)/(429*a**3*b*(a + b*x)**7)
- 8*(a**2 - b**2*x**2)**(5/2)/(3003*a**4*b*(a + b*x)**6) - 8*(a**2 - b**2*x**2)
**(5/2)/(15015*a**5*b*(a + b*x)**5)

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Mathematica [A]  time = 0.0640715, size = 82, normalized size = 0.49 \[ -\frac{(a-b x)^2 \sqrt{a^2-b^2 x^2} \left (1763 a^4+852 a^3 b x+308 a^2 b^2 x^2+72 a b^3 x^3+8 b^4 x^4\right )}{15015 a^5 b (a+b x)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 - b^2*x^2)^(3/2)/(a + b*x)^9,x]

[Out]

-((a - b*x)^2*Sqrt[a^2 - b^2*x^2]*(1763*a^4 + 852*a^3*b*x + 308*a^2*b^2*x^2 + 72
*a*b^3*x^3 + 8*b^4*x^4))/(15015*a^5*b*(a + b*x)^7)

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Maple [A]  time = 0.012, size = 77, normalized size = 0.5 \[ -{\frac{ \left ( 8\,{b}^{4}{x}^{4}+72\,{x}^{3}a{b}^{3}+308\,{x}^{2}{a}^{2}{b}^{2}+852\,x{a}^{3}b+1763\,{a}^{4} \right ) \left ( -bx+a \right ) }{15015\, \left ( bx+a \right ) ^{8}{a}^{5}b} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b^2*x^2+a^2)^(3/2)/(b*x+a)^9,x)

[Out]

-1/15015*(-b*x+a)*(8*b^4*x^4+72*a*b^3*x^3+308*a^2*b^2*x^2+852*a^3*b*x+1763*a^4)*
(-b^2*x^2+a^2)^(3/2)/(b*x+a)^8/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((-b^2*x^2 + a^2)^(3/2)/(b*x + a)^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.293954, size = 749, normalized size = 4.51 \[ -\frac{1771 \, b^{12} x^{13} + 22919 \, a b^{11} x^{12} + 68393 \, a^{2} b^{10} x^{11} - 70213 \, a^{3} b^{9} x^{10} - 711854 \, a^{4} b^{8} x^{9} - 1214070 \, a^{5} b^{7} x^{8} + 55770 \, a^{6} b^{6} x^{7} + 2594592 \, a^{7} b^{5} x^{6} + 2870868 \, a^{8} b^{4} x^{5} + 480480 \, a^{9} b^{3} x^{4} - 1441440 \, a^{10} b^{2} x^{3} - 1921920 \, a^{11} b x^{2} - 960960 \, a^{12} x - 13 \,{\left (135 \, b^{11} x^{12} - 8 \, a b^{10} x^{11} - 6127 \, a^{2} b^{9} x^{10} - 24662 \, a^{3} b^{8} x^{9} - 25938 \, a^{4} b^{7} x^{8} + 50028 \, a^{5} b^{6} x^{7} + 162624 \, a^{6} b^{5} x^{6} + 137676 \, a^{7} b^{4} x^{5} - 36960 \, a^{8} b^{3} x^{4} - 147840 \, a^{9} b^{2} x^{3} - 147840 \, a^{10} b x^{2} - 73920 \, a^{11} x\right )} \sqrt{-b^{2} x^{2} + a^{2}}}{15015 \,{\left (a^{5} b^{13} x^{13} + 13 \, a^{6} b^{12} x^{12} + 39 \, a^{7} b^{11} x^{11} - 39 \, a^{8} b^{10} x^{10} - 403 \, a^{9} b^{9} x^{9} - 689 \, a^{10} b^{8} x^{8} + 13 \, a^{11} b^{7} x^{7} + 1443 \, a^{12} b^{6} x^{6} + 1742 \, a^{13} b^{5} x^{5} + 312 \, a^{14} b^{4} x^{4} - 1040 \, a^{15} b^{3} x^{3} - 1040 \, a^{16} b^{2} x^{2} - 416 \, a^{17} b x - 64 \, a^{18} -{\left (a^{5} b^{12} x^{12} - 45 \, a^{7} b^{10} x^{10} - 182 \, a^{8} b^{9} x^{9} - 193 \, a^{9} b^{8} x^{8} + 364 \, a^{10} b^{7} x^{7} + 1189 \, a^{11} b^{6} x^{6} + 1066 \, a^{12} b^{5} x^{5} - 232 \, a^{13} b^{4} x^{4} - 1248 \, a^{14} b^{3} x^{3} - 1072 \, a^{15} b^{2} x^{2} - 416 \, a^{16} b x - 64 \, a^{17}\right )} \sqrt{-b^{2} x^{2} + a^{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b^2*x^2 + a^2)^(3/2)/(b*x + a)^9,x, algorithm="fricas")

[Out]

-1/15015*(1771*b^12*x^13 + 22919*a*b^11*x^12 + 68393*a^2*b^10*x^11 - 70213*a^3*b
^9*x^10 - 711854*a^4*b^8*x^9 - 1214070*a^5*b^7*x^8 + 55770*a^6*b^6*x^7 + 2594592
*a^7*b^5*x^6 + 2870868*a^8*b^4*x^5 + 480480*a^9*b^3*x^4 - 1441440*a^10*b^2*x^3 -
1921920*a^11*b*x^2 - 960960*a^12*x - 13*(135*b^11*x^12 - 8*a*b^10*x^11 - 6127*a
^2*b^9*x^10 - 24662*a^3*b^8*x^9 - 25938*a^4*b^7*x^8 + 50028*a^5*b^6*x^7 + 162624
*a^6*b^5*x^6 + 137676*a^7*b^4*x^5 - 36960*a^8*b^3*x^4 - 147840*a^9*b^2*x^3 - 147
840*a^10*b*x^2 - 73920*a^11*x)*sqrt(-b^2*x^2 + a^2))/(a^5*b^13*x^13 + 13*a^6*b^1
2*x^12 + 39*a^7*b^11*x^11 - 39*a^8*b^10*x^10 - 403*a^9*b^9*x^9 - 689*a^10*b^8*x^
8 + 13*a^11*b^7*x^7 + 1443*a^12*b^6*x^6 + 1742*a^13*b^5*x^5 + 312*a^14*b^4*x^4 -
1040*a^15*b^3*x^3 - 1040*a^16*b^2*x^2 - 416*a^17*b*x - 64*a^18 - (a^5*b^12*x^12
- 45*a^7*b^10*x^10 - 182*a^8*b^9*x^9 - 193*a^9*b^8*x^8 + 364*a^10*b^7*x^7 + 118
9*a^11*b^6*x^6 + 1066*a^12*b^5*x^5 - 232*a^13*b^4*x^4 - 1248*a^14*b^3*x^3 - 1072
*a^15*b^2*x^2 - 416*a^16*b*x - 64*a^17)*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)**9,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.239188, size = 558, normalized size = 3.36 \[ \frac{2 \,{\left (\frac{7904 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}}{b^{2} x} + \frac{77454 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + \frac{233948 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{3}}{b^{6} x^{3}} + \frac{659945 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{4}}{b^{8} x^{4}} + \frac{1094808 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{5}}{b^{10} x^{5}} + \frac{1559844 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{6}}{b^{12} x^{6}} + \frac{1465464 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{7}}{b^{14} x^{7}} + \frac{1174173 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{8}}{b^{16} x^{8}} + \frac{600600 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{9}}{b^{18} x^{9}} + \frac{270270 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{10}}{b^{20} x^{10}} + \frac{60060 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{11}}{b^{22} x^{11}} + \frac{15015 \,{\left (a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}\right )}^{12}}{b^{24} x^{12}} + 1763\right )}}{15015 \, a^{5}{\left (\frac{a b + \sqrt{-b^{2} x^{2} + a^{2}}{\left | b \right |}}{b^{2} x} + 1\right )}^{13}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b^2*x^2 + a^2)^(3/2)/(b*x + a)^9,x, algorithm="giac")

[Out]

2/15015*(7904*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 77454*(a*b + sqrt(-b
^2*x^2 + a^2)*abs(b))^2/(b^4*x^2) + 233948*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^3
/(b^6*x^3) + 659945*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^4/(b^8*x^4) + 1094808*(a
*b + sqrt(-b^2*x^2 + a^2)*abs(b))^5/(b^10*x^5) + 1559844*(a*b + sqrt(-b^2*x^2 +
a^2)*abs(b))^6/(b^12*x^6) + 1465464*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^7/(b^14*
x^7) + 1174173*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^8/(b^16*x^8) + 600600*(a*b +
sqrt(-b^2*x^2 + a^2)*abs(b))^9/(b^18*x^9) + 270270*(a*b + sqrt(-b^2*x^2 + a^2)*a
bs(b))^10/(b^20*x^10) + 60060*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^11/(b^22*x^11)
+ 15015*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^12/(b^24*x^12) + 1763)/(a^5*((a*b +
sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 1)^13*abs(b))