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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]