Optimal. Leaf size=184 \[ -\frac{e^{5/2} (4 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{11/4}}+\frac{e^{5/2} (4 b c-7 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{11/4}}-\frac{e (e x)^{3/2} \sqrt [4]{a+b x^2} (4 b c-7 a d)}{6 a b^2}+\frac{2 (e x)^{7/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}} \]
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Rubi [A] time = 0.119237, antiderivative size = 184, 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 = {457, 321, 329, 331, 298, 205, 208} \[ -\frac{e^{5/2} (4 b c-7 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{11/4}}+\frac{e^{5/2} (4 b c-7 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{11/4}}-\frac{e (e x)^{3/2} \sqrt [4]{a+b x^2} (4 b c-7 a d)}{6 a b^2}+\frac{2 (e x)^{7/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 457
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 )^{7/4}} \, dx &=\frac{2 (b c-a d) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/4}}+\frac{\left (2 \left (-2 b c+\frac{7 a d}{2}\right )\right ) \int \frac{(e x)^{5/2}}{\left (a+b x^2\right )^{3/4}} \, dx}{3 a b}\\ &=\frac{2 (b c-a d) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac{(4 b c-7 a d) e (e x)^{3/2} \sqrt [4]{a+b x^2}}{6 a b^2}+\frac{\left ((4 b c-7 a d) e^2\right ) \int \frac{\sqrt{e x}}{\left (a+b x^2\right )^{3/4}} \, dx}{4 b^2}\\ &=\frac{2 (b c-a d) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac{(4 b c-7 a d) e (e x)^{3/2} \sqrt [4]{a+b x^2}}{6 a b^2}+\frac{((4 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 )}{2 b^2}\\ &=\frac{2 (b c-a d) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac{(4 b c-7 a d) e (e x)^{3/2} \sqrt [4]{a+b x^2}}{6 a b^2}+\frac{((4 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 )}{2 b^2}\\ &=\frac{2 (b c-a d) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac{(4 b c-7 a d) e (e x)^{3/2} \sqrt [4]{a+b x^2}}{6 a b^2}+\frac{\left ((4 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 )}{4 b^{5/2}}-\frac{\left ((4 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 )}{4 b^{5/2}}\\ &=\frac{2 (b c-a d) (e x)^{7/2}}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac{(4 b c-7 a d) e (e x)^{3/2} \sqrt [4]{a+b x^2}}{6 a b^2}-\frac{(4 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 )}{4 b^{11/4}}+\frac{(4 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 )}{4 b^{11/4}}\\ \end{align*}
Mathematica [C] time = 0.127221, size = 77, normalized size = 0.42 \[ \frac{x (e x)^{5/2} \left (\left (\frac{b x^2}{a}+1\right )^{3/4} (4 b c-7 a d) \, _2F_1\left (\frac{7}{4},\frac{7}{4};\frac{11}{4};-\frac{b x^2}{a}\right )+7 a d\right )}{14 a b \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.049, 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{7}{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{7}{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 [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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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{7}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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