Optimal. Leaf size=192 \[ \frac{5 \sqrt{a} e^2 (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (2 b c-3 a d) \text{EllipticF}\left (\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),2\right )}{6 b^{5/2} \left (a+b x^2\right )^{3/4}}+\frac{5 e^3 \sqrt{e x} \sqrt [4]{a+b x^2} (2 b c-3 a d)}{6 b^3}-\frac{e (e x)^{5/2} \sqrt [4]{a+b x^2} (2 b c-3 a d)}{3 a b^2}+\frac{2 (e x)^{9/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}} \]
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Rubi [A] time = 0.134827, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {457, 321, 329, 237, 335, 275, 231} \[ \frac{5 e^3 \sqrt{e x} \sqrt [4]{a+b x^2} (2 b c-3 a d)}{6 b^3}+\frac{5 \sqrt{a} e^2 (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (2 b c-3 a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{6 b^{5/2} \left (a+b x^2\right )^{3/4}}-\frac{e (e x)^{5/2} \sqrt [4]{a+b x^2} (2 b c-3 a d)}{3 a b^2}+\frac{2 (e x)^{9/2} (b c-a d)}{3 a b e \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 457
Rule 321
Rule 329
Rule 237
Rule 335
Rule 275
Rule 231
Rubi steps
\begin{align*} \int \frac{(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx &=\frac{2 (b c-a d) (e x)^{9/2}}{3 a b e \left (a+b x^2\right )^{3/4}}+\frac{\left (2 \left (-3 b c+\frac{9 a d}{2}\right )\right ) \int \frac{(e x)^{7/2}}{\left (a+b x^2\right )^{3/4}} \, dx}{3 a b}\\ &=\frac{2 (b c-a d) (e x)^{9/2}}{3 a b e \left (a+b x^2\right )^{3/4}}-\frac{(2 b c-3 a d) e (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 a b^2}+\frac{\left (5 (2 b c-3 a d) e^2\right ) \int \frac{(e x)^{3/2}}{\left (a+b x^2\right )^{3/4}} \, dx}{6 b^2}\\ &=\frac{2 (b c-a d) (e x)^{9/2}}{3 a b e \left (a+b x^2\right )^{3/4}}+\frac{5 (2 b c-3 a d) e^3 \sqrt{e x} \sqrt [4]{a+b x^2}}{6 b^3}-\frac{(2 b c-3 a d) e (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 a b^2}-\frac{\left (5 a (2 b c-3 a d) e^4\right ) \int \frac{1}{\sqrt{e x} \left (a+b x^2\right )^{3/4}} \, dx}{12 b^3}\\ &=\frac{2 (b c-a d) (e x)^{9/2}}{3 a b e \left (a+b x^2\right )^{3/4}}+\frac{5 (2 b c-3 a d) e^3 \sqrt{e x} \sqrt [4]{a+b x^2}}{6 b^3}-\frac{(2 b c-3 a d) e (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 a b^2}-\frac{\left (5 a (2 b c-3 a d) e^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a+\frac{b x^4}{e^2}\right )^{3/4}} \, dx,x,\sqrt{e x}\right )}{6 b^3}\\ &=\frac{2 (b c-a d) (e x)^{9/2}}{3 a b e \left (a+b x^2\right )^{3/4}}+\frac{5 (2 b c-3 a d) e^3 \sqrt{e x} \sqrt [4]{a+b x^2}}{6 b^3}-\frac{(2 b c-3 a d) e (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 a b^2}-\frac{\left (5 a (2 b c-3 a d) e^3 \left (1+\frac{a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{a e^2}{b x^4}\right )^{3/4} x^3} \, dx,x,\sqrt{e x}\right )}{6 b^3 \left (a+b x^2\right )^{3/4}}\\ &=\frac{2 (b c-a d) (e x)^{9/2}}{3 a b e \left (a+b x^2\right )^{3/4}}+\frac{5 (2 b c-3 a d) e^3 \sqrt{e x} \sqrt [4]{a+b x^2}}{6 b^3}-\frac{(2 b c-3 a d) e (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 a b^2}+\frac{\left (5 a (2 b c-3 a d) e^3 \left (1+\frac{a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{x}{\left (1+\frac{a e^2 x^4}{b}\right )^{3/4}} \, dx,x,\frac{1}{\sqrt{e x}}\right )}{6 b^3 \left (a+b x^2\right )^{3/4}}\\ &=\frac{2 (b c-a d) (e x)^{9/2}}{3 a b e \left (a+b x^2\right )^{3/4}}+\frac{5 (2 b c-3 a d) e^3 \sqrt{e x} \sqrt [4]{a+b x^2}}{6 b^3}-\frac{(2 b c-3 a d) e (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 a b^2}+\frac{\left (5 a (2 b c-3 a d) e^3 \left (1+\frac{a}{b x^2}\right )^{3/4} (e x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{a e^2 x^2}{b}\right )^{3/4}} \, dx,x,\frac{1}{e x}\right )}{12 b^3 \left (a+b x^2\right )^{3/4}}\\ &=\frac{2 (b c-a d) (e x)^{9/2}}{3 a b e \left (a+b x^2\right )^{3/4}}+\frac{5 (2 b c-3 a d) e^3 \sqrt{e x} \sqrt [4]{a+b x^2}}{6 b^3}-\frac{(2 b c-3 a d) e (e x)^{5/2} \sqrt [4]{a+b x^2}}{3 a b^2}+\frac{5 \sqrt{a} (2 b c-3 a d) e^2 \left (1+\frac{a}{b x^2}\right )^{3/4} (e x)^{3/2} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{6 b^{5/2} \left (a+b x^2\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.138387, size = 110, normalized size = 0.57 \[ \frac{e^3 \sqrt{e x} \left (-15 a^2 d+5 a \left (\frac{b x^2}{a}+1\right )^{3/4} (3 a d-2 b c) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )+a b \left (10 c-9 d x^2\right )+2 b^2 x^2 \left (3 c+d x^2\right )\right )}{6 b^3 \left (a+b x^2\right )^{3/4}} \]
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{7}{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{7}{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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (d e^{3} x^{5} + c e^{3} x^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{e x}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}, x\right ) \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{7}{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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