Optimal. Leaf size=113 \[ -\frac{d \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}}-\frac{2 c \sqrt [4]{a+b x^2}}{a e \sqrt{e x}} \]
[Out]
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Rubi [A] time = 0.236559, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{d \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}}-\frac{2 c \sqrt [4]{a+b x^2}}{a e \sqrt{e x}} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^2)/((e*x)^(3/2)*(a + b*x^2)^(3/4)),x]
[Out]
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Rubi in Sympy [A] time = 28.3293, size = 104, normalized size = 0.92 \[ - \frac{d \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{b^{\frac{3}{4}} e^{\frac{3}{2}}} + \frac{d \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{b^{\frac{3}{4}} e^{\frac{3}{2}}} - \frac{2 c \sqrt [4]{a + b x^{2}}}{a e \sqrt{e x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)/(e*x)**(3/2)/(b*x**2+a)**(3/4),x)
[Out]
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Mathematica [C] time = 0.0712407, size = 77, normalized size = 0.68 \[ \frac{x \left (2 a d x^2 \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )-6 c \left (a+b x^2\right )\right )}{3 a (e x)^{3/2} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^2)/((e*x)^(3/2)*(a + b*x^2)^(3/4)),x]
[Out]
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Maple [F] time = 0.05, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{-{\frac{3}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2+c)/(e*x)^(3/2)/(b*x^2+a)^(3/4),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((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 40.7813, size = 85, normalized size = 0.75 \[ \frac{\sqrt [4]{b} c \sqrt [4]{\frac{a}{b x^{2}} + 1} \Gamma \left (- \frac{1}{4}\right )}{2 a e^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right )} + \frac{d x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{4}} e^{\frac{3}{2}} \Gamma \left (\frac{7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**2+c)/(e*x)**(3/2)/(b*x**2+a)**(3/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \left (e x\right )^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(3/2)),x, algorithm="giac")
[Out]