Optimal. Leaf size=125 \[ -\frac{d \sqrt{e} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{7/4}}+\frac{d \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{7/4}}+\frac{2 (e x)^{3/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}} \]
[Out]
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Rubi [A] time = 0.243671, antiderivative size = 125, 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 \[ -\frac{d \sqrt{e} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{7/4}}+\frac{d \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{7/4}}+\frac{2 (e x)^{3/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[e*x]*(c + d*x^2))/(a + b*x^2)^(7/4),x]
[Out]
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Rubi in Sympy [A] time = 29.3493, size = 112, normalized size = 0.9 \[ - \frac{d \sqrt{e} \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{b^{\frac{7}{4}}} + \frac{d \sqrt{e} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{b^{\frac{7}{4}}} - \frac{2 \left (e x\right )^{\frac{3}{2}} \left (a d - b c\right )}{3 a b e \left (a + b x^{2}\right )^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**(1/2)*(d*x**2+c)/(b*x**2+a)**(7/4),x)
[Out]
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Mathematica [C] time = 0.0663421, size = 73, normalized size = 0.58 \[ \frac{2 x \sqrt{e x} \left (a d \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^2}{a}\right )-a d+b c\right )}{3 a b \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[e*x]*(c + d*x^2))/(a + b*x^2)^(7/4),x]
[Out]
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Maple [F] time = 0.054, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c)\sqrt{ex} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(1/2)*(d*x^2+c)/(b*x^2+a)^(7/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ d \sqrt{e} \int \frac{x^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{\frac{7}{4}}}\,{d x} + \frac{2 \, c \sqrt{e} x^{\frac{3}{2}}}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{4}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*sqrt(e*x)/(b*x^2 + a)^(7/4),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)*sqrt(e*x)/(b*x^2 + a)^(7/4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 173.65, size = 87, normalized size = 0.7 \[ \frac{c \sqrt{e} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right )}{2 a^{\frac{7}{4}} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{3}{4}} \Gamma \left (\frac{7}{4}\right )} + \frac{d \sqrt{e} x^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{7}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{7}{4}} \Gamma \left (\frac{11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)**(1/2)*(d*x**2+c)/(b*x**2+a)**(7/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )} \sqrt{e x}}{{\left (b x^{2} + a\right )}^{\frac{7}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)*sqrt(e*x)/(b*x^2 + a)^(7/4),x, algorithm="giac")
[Out]