Optimal. Leaf size=113 \[ -\frac{d \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}}-\frac{2 c \sqrt [4]{a+b x^2}}{a e \sqrt{e x}} \]
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Rubi [A] time = 0.0800657, 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, Rules used = {451, 329, 331, 298, 205, 208} \[ -\frac{d \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{b^{3/4} e^{3/2}}-\frac{2 c \sqrt [4]{a+b x^2}}{a e \sqrt{e x}} \]
Antiderivative was successfully verified.
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Rule 451
Rule 329
Rule 331
Rule 298
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \frac{c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{3/4}} \, dx &=-\frac{2 c \sqrt [4]{a+b x^2}}{a e \sqrt{e x}}+\frac{d \int \frac{\sqrt{e x}}{\left (a+b x^2\right )^{3/4}} \, dx}{e^2}\\ &=-\frac{2 c \sqrt [4]{a+b x^2}}{a e \sqrt{e x}}+\frac{(2 d) \operatorname{Subst}\left (\int \frac{x^2}{\left (a+\frac{b x^4}{e^2}\right )^{3/4}} \, dx,x,\sqrt{e x}\right )}{e^3}\\ &=-\frac{2 c \sqrt [4]{a+b x^2}}{a e \sqrt{e x}}+\frac{(2 d) \operatorname{Subst}\left (\int \frac{x^2}{1-\frac{b x^4}{e^2}} \, dx,x,\frac{\sqrt{e x}}{\sqrt [4]{a+b x^2}}\right )}{e^3}\\ &=-\frac{2 c \sqrt [4]{a+b x^2}}{a e \sqrt{e x}}+\frac{d \operatorname{Subst}\left (\int \frac{1}{e-\sqrt{b} x^2} \, dx,x,\frac{\sqrt{e x}}{\sqrt [4]{a+b x^2}}\right )}{\sqrt{b} e}-\frac{d \operatorname{Subst}\left (\int \frac{1}{e+\sqrt{b} x^2} \, dx,x,\frac{\sqrt{e x}}{\sqrt [4]{a+b x^2}}\right )}{\sqrt{b} e}\\ &=-\frac{2 c \sqrt [4]{a+b x^2}}{a e \sqrt{e x}}-\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}}\\ \end{align*}
Mathematica [A] time = 0.0303367, size = 100, normalized size = 0.88 \[ \frac{x \left (-2 b^{3/4} c \sqrt [4]{a+b x^2}-a d \sqrt{x} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a+b x^2}}\right )+a d \sqrt{x} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a+b x^2}}\right )\right )}{a b^{3/4} (e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{(d{x}^{2}+c) \left ( ex \right ) ^{-{\frac{3}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \left (e x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 13.9976, size = 85, normalized size = 0.75 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \left (e x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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