Optimal. Leaf size=118 \[ \frac{a^{3/2} (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{6 \sqrt{b} \left (a+b x^2\right )^{3/4}}+\frac{(c x)^{5/2} \sqrt [4]{a+b x^2}}{3 c}+\frac{a c \sqrt{c x} \sqrt [4]{a+b x^2}}{6 b} \]
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Rubi [A] time = 0.242395, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ \frac{a^{3/2} (c x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{6 \sqrt{b} \left (a+b x^2\right )^{3/4}}+\frac{(c x)^{5/2} \sqrt [4]{a+b x^2}}{3 c}+\frac{a c \sqrt{c x} \sqrt [4]{a+b x^2}}{6 b} \]
Antiderivative was successfully verified.
[In] Int[(c*x)^(3/2)*(a + b*x^2)^(1/4),x]
[Out]
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Rubi in Sympy [A] time = 26.628, size = 100, normalized size = 0.85 \[ \frac{a^{\frac{3}{2}} \left (c x\right )^{\frac{3}{2}} \left (\frac{a}{b x^{2}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )}}{2}\middle | 2\right )}{6 \sqrt{b} \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{a c \sqrt{c x} \sqrt [4]{a + b x^{2}}}{6 b} + \frac{\left (c x\right )^{\frac{5}{2}} \sqrt [4]{a + b x^{2}}}{3 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x)**(3/2)*(b*x**2+a)**(1/4),x)
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Mathematica [C] time = 0.0535818, size = 83, normalized size = 0.7 \[ \frac{c \sqrt{c x} \left (-a^2 \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )+a^2+3 a b x^2+2 b^2 x^4\right )}{6 b \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(c*x)^(3/2)*(a + b*x^2)^(1/4),x]
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Maple [F] time = 0.039, size = 0, normalized size = 0. \[ \int \left ( cx \right ) ^{{\frac{3}{2}}}\sqrt [4]{b{x}^{2}+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((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. \[ \int{\left (b x^{2} + a\right )}^{\frac{1}{4}} \left (c x\right )^{\frac{3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(1/4)*(c*x)^(3/2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x} c x, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(1/4)*(c*x)^(3/2),x, algorithm="fricas")
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Sympy [A] time = 22.9869, size = 46, normalized size = 0.39 \[ \frac{\sqrt [4]{a} c^{\frac{3}{2}} x^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((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. \[ \int{\left (b x^{2} + a\right )}^{\frac{1}{4}} \left (c x\right )^{\frac{3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(1/4)*(c*x)^(3/2),x, algorithm="giac")
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