Optimal. Leaf size=90 \[ \frac{2 \sqrt{b} \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} c^2 \sqrt [4]{a+b x^2}}-\frac{2}{c \sqrt{c x} \sqrt [4]{a+b x^2}} \]
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Rubi [A] time = 0.114408, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{2 \sqrt{b} \sqrt{c x} \sqrt [4]{\frac{a}{b x^2}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} c^2 \sqrt [4]{a+b x^2}}-\frac{2}{c \sqrt{c x} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Int[1/((c*x)^(3/2)*(a + b*x^2)^(1/4)),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{2}{c \sqrt{c x} \sqrt [4]{a + b x^{2}}} + \frac{\sqrt{c x} \sqrt [4]{\frac{a}{b x^{2}} + 1} \int ^{\frac{1}{x}} \frac{1}{\left (\frac{a x^{2}}{b} + 1\right )^{\frac{5}{4}}}\, dx}{c^{2} \sqrt [4]{a + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c*x)**(3/2)/(b*x**2+a)**(1/4),x)
[Out]
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Mathematica [C] time = 0.0577323, size = 75, normalized size = 0.83 \[ \frac{x \left (4 b x^2 \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )-6 \left (a+b x^2\right )\right )}{3 a (c x)^{3/2} \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((c*x)^(3/2)*(a + b*x^2)^(1/4)),x]
[Out]
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Maple [F] time = 0.053, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c*x)^(3/2)/(b*x^2+a)^(1/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{1}{4}} \left (c x\right )^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(1/4)*(c*x)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x} c x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(1/4)*(c*x)^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 14.2025, size = 31, normalized size = 0.34 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{\sqrt [4]{b} c^{\frac{3}{2}} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c*x)**(3/2)/(b*x**2+a)**(1/4),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{1}{4}} \left (c x\right )^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(1/4)*(c*x)^(3/2)),x, algorithm="giac")
[Out]