Optimal. Leaf size=173 \[ \frac{3 a e^{5/2} (8 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{32 b^{11/4}}-\frac{3 a e^{5/2} (8 b c-7 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{32 b^{11/4}}+\frac{e (e x)^{3/2} \sqrt [4]{a+b x^2} (8 b c-7 a d)}{16 b^2}+\frac{d (e x)^{7/2} \sqrt [4]{a+b x^2}}{4 b e} \]
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Rubi [A] time = 0.126232, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {459, 321, 329, 331, 298, 205, 208} \[ \frac{3 a e^{5/2} (8 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{32 b^{11/4}}-\frac{3 a e^{5/2} (8 b c-7 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{32 b^{11/4}}+\frac{e (e x)^{3/2} \sqrt [4]{a+b x^2} (8 b c-7 a d)}{16 b^2}+\frac{d (e x)^{7/2} \sqrt [4]{a+b x^2}}{4 b e} \]
Antiderivative was successfully verified.
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Rule 459
Rule 321
Rule 329
Rule 331
Rule 298
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \frac{(e x)^{5/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{3/4}} \, dx &=\frac{d (e x)^{7/2} \sqrt [4]{a+b x^2}}{4 b e}-\frac{\left (-4 b c+\frac{7 a d}{2}\right ) \int \frac{(e x)^{5/2}}{\left (a+b x^2\right )^{3/4}} \, dx}{4 b}\\ &=\frac{(8 b c-7 a d) e (e x)^{3/2} \sqrt [4]{a+b x^2}}{16 b^2}+\frac{d (e x)^{7/2} \sqrt [4]{a+b x^2}}{4 b e}-\frac{\left (3 a (8 b c-7 a d) e^2\right ) \int \frac{\sqrt{e x}}{\left (a+b x^2\right )^{3/4}} \, dx}{32 b^2}\\ &=\frac{(8 b c-7 a d) e (e x)^{3/2} \sqrt [4]{a+b x^2}}{16 b^2}+\frac{d (e x)^{7/2} \sqrt [4]{a+b x^2}}{4 b e}-\frac{(3 a (8 b c-7 a d) e) \operatorname{Subst}\left (\int \frac{x^2}{\left (a+\frac{b x^4}{e^2}\right )^{3/4}} \, dx,x,\sqrt{e x}\right )}{16 b^2}\\ &=\frac{(8 b c-7 a d) e (e x)^{3/2} \sqrt [4]{a+b x^2}}{16 b^2}+\frac{d (e x)^{7/2} \sqrt [4]{a+b x^2}}{4 b e}-\frac{(3 a (8 b c-7 a d) e) \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 )}{16 b^2}\\ &=\frac{(8 b c-7 a d) e (e x)^{3/2} \sqrt [4]{a+b x^2}}{16 b^2}+\frac{d (e x)^{7/2} \sqrt [4]{a+b x^2}}{4 b e}-\frac{\left (3 a (8 b c-7 a d) e^3\right ) \operatorname{Subst}\left (\int \frac{1}{e-\sqrt{b} x^2} \, dx,x,\frac{\sqrt{e x}}{\sqrt [4]{a+b x^2}}\right )}{32 b^{5/2}}+\frac{\left (3 a (8 b c-7 a d) e^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+\sqrt{b} x^2} \, dx,x,\frac{\sqrt{e x}}{\sqrt [4]{a+b x^2}}\right )}{32 b^{5/2}}\\ &=\frac{(8 b c-7 a d) e (e x)^{3/2} \sqrt [4]{a+b x^2}}{16 b^2}+\frac{d (e x)^{7/2} \sqrt [4]{a+b x^2}}{4 b e}+\frac{3 a (8 b c-7 a d) e^{5/2} \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{32 b^{11/4}}-\frac{3 a (8 b c-7 a d) e^{5/2} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{32 b^{11/4}}\\ \end{align*}
Mathematica [A] time = 0.165653, size = 131, normalized size = 0.76 \[ \frac{(e x)^{5/2} \left (2 b^{3/4} x^{3/2} \sqrt [4]{a+b x^2} \left (-7 a d+8 b c+4 b d x^2\right )-3 a (7 a d-8 b c) \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a+b x^2}}\right )+3 a (7 a d-8 b c) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a+b x^2}}\right )\right )}{32 b^{11/4} x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.031, size = 0, normalized size = 0. \begin{align*} \int{(d{x}^{2}+c) \left ( ex \right ) ^{{\frac{5}{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{{\left (d x^{2} + c\right )} \left (e x\right )^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{\frac{3}{4}}}\,{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 = 142.568, size = 94, normalized size = 0.54 \begin{align*} \frac{c e^{\frac{5}{2}} x^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{4}} \Gamma \left (\frac{11}{4}\right )} + \frac{d e^{\frac{5}{2}} x^{\frac{11}{2}} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{4}} \Gamma \left (\frac{15}{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{{\left (d x^{2} + c\right )} \left (e x\right )^{\frac{5}{2}}}{{\left (b x^{2} + a\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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