Optimal. Leaf size=88 \[ -\frac{64 \left (a-b x^2\right )^{13/4}}{585 a^3 c (c x)^{13/2}}+\frac{16 \left (a-b x^2\right )^{9/4}}{45 a^2 c (c x)^{13/2}}-\frac{2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{13/2}} \]
[Out]
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Rubi [A] time = 0.0907427, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{64 \left (a-b x^2\right )^{13/4}}{585 a^3 c (c x)^{13/2}}+\frac{16 \left (a-b x^2\right )^{9/4}}{45 a^2 c (c x)^{13/2}}-\frac{2 \left (a-b x^2\right )^{5/4}}{5 a c (c x)^{13/2}} \]
Antiderivative was successfully verified.
[In] Int[(a - b*x^2)^(1/4)/(c*x)^(15/2),x]
[Out]
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Rubi in Sympy [A] time = 11.5074, size = 73, normalized size = 0.83 \[ - \frac{2 \left (a - b x^{2}\right )^{\frac{5}{4}}}{5 a c \left (c x\right )^{\frac{13}{2}}} + \frac{16 \left (a - b x^{2}\right )^{\frac{9}{4}}}{45 a^{2} c \left (c x\right )^{\frac{13}{2}}} - \frac{64 \left (a - b x^{2}\right )^{\frac{13}{4}}}{585 a^{3} c \left (c x\right )^{\frac{13}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-b*x**2+a)**(1/4)/(c*x)**(15/2),x)
[Out]
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Mathematica [A] time = 0.0413316, size = 64, normalized size = 0.73 \[ \frac{2 \sqrt{c x} \sqrt [4]{a-b x^2} \left (-45 a^3+5 a^2 b x^2+8 a b^2 x^4+32 b^3 x^6\right )}{585 a^3 c^8 x^7} \]
Antiderivative was successfully verified.
[In] Integrate[(a - b*x^2)^(1/4)/(c*x)^(15/2),x]
[Out]
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Maple [A] time = 0.007, size = 43, normalized size = 0.5 \[ -{\frac{2\,x \left ( 32\,{b}^{2}{x}^{4}+40\,ab{x}^{2}+45\,{a}^{2} \right ) }{585\,{a}^{3}} \left ( -b{x}^{2}+a \right ) ^{{\frac{5}{4}}} \left ( cx \right ) ^{-{\frac{15}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-b*x^2+a)^(1/4)/(c*x)^(15/2),x)
[Out]
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Maxima [A] time = 1.3957, size = 78, normalized size = 0.89 \[ -\frac{2 \,{\left (\frac{117 \,{\left (-b x^{2} + a\right )}^{\frac{5}{4}} b^{2}}{x^{\frac{5}{2}}} + \frac{130 \,{\left (-b x^{2} + a\right )}^{\frac{9}{4}} b}{x^{\frac{9}{2}}} + \frac{45 \,{\left (-b x^{2} + a\right )}^{\frac{13}{4}}}{x^{\frac{13}{2}}}\right )}}{585 \, a^{3} c^{\frac{15}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x^2 + a)^(1/4)/(c*x)^(15/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.219121, size = 78, normalized size = 0.89 \[ \frac{2 \,{\left (32 \, b^{3} x^{6} + 8 \, a b^{2} x^{4} + 5 \, a^{2} b x^{2} - 45 \, a^{3}\right )}{\left (-b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x}}{585 \, a^{3} c^{8} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x^2 + a)^(1/4)/(c*x)^(15/2),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*x**2+a)**(1/4)/(c*x)**(15/2),x)
[Out]
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GIAC/XCAS [A] time = 0.236179, size = 251, normalized size = 2.85 \[ \frac{2 \,{\left (\frac{117 \,{\left (-b c^{4} x^{2} + a c^{4}\right )}^{\frac{1}{4}}{\left (b c^{2} - \frac{a c^{2}}{x^{2}}\right )} b^{2} c^{4}}{\sqrt{c x}} - \frac{130 \,{\left (b^{2} c^{8} x^{4} - 2 \, a b c^{8} x^{2} + a^{2} c^{8}\right )}{\left (-b c^{4} x^{2} + a c^{4}\right )}^{\frac{1}{4}} b}{\sqrt{c x} c^{2} x^{4}} + \frac{45 \,{\left (b^{3} c^{12} x^{6} - 3 \, a b^{2} c^{12} x^{4} + 3 \, a^{2} b c^{12} x^{2} - a^{3} c^{12}\right )}{\left (-b c^{4} x^{2} + a c^{4}\right )}^{\frac{1}{4}}}{\sqrt{c x} c^{6} x^{6}}\right )}}{585 \, a^{3} c^{14}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x^2 + a)^(1/4)/(c*x)^(15/2),x, algorithm="giac")
[Out]