Optimal. Leaf size=97 \[ \frac{4 b^{3/2} (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 a^{3/2} c^4 \left (a+b x^2\right )^{3/4}}-\frac{2 \sqrt [4]{a+b x^2}}{3 a c (c x)^{3/2}} \]
[Out]
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Rubi [A] time = 0.202989, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{4 b^{3/2} (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 a^{3/2} c^4 \left (a+b x^2\right )^{3/4}}-\frac{2 \sqrt [4]{a+b x^2}}{3 a c (c x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((c*x)^(5/2)*(a + b*x^2)^(3/4)),x]
[Out]
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Rubi in Sympy [A] time = 23.6327, size = 85, normalized size = 0.88 \[ - \frac{2 \sqrt [4]{a + b x^{2}}}{3 a c \left (c x\right )^{\frac{3}{2}}} + \frac{4 b^{\frac{3}{2}} \left (c x\right )^{\frac{3}{2}} \left (\frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{3 a^{\frac{3}{2}} c^{4} \left (a + b x^{2}\right )^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c*x)**(5/2)/(b*x**2+a)**(3/4),x)
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Mathematica [C] time = 0.0554217, size = 72, normalized size = 0.74 \[ -\frac{2 x \left (2 b x^2 \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )+a+b x^2\right )}{3 a (c x)^{5/2} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((c*x)^(5/2)*(a + b*x^2)^(3/4)),x]
[Out]
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Maple [F] time = 0.046, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-{\frac{5}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c*x)^(5/2)/(b*x^2+a)^(3/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \left (c x\right )^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(3/4)*(c*x)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x} c^{2} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(3/4)*(c*x)^(5/2)),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(1/(c*x)**(5/2)/(b*x**2+a)**(3/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \left (c x\right )^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(3/4)*(c*x)^(5/2)),x, algorithm="giac")
[Out]