Optimal. Leaf size=144 \[ \frac{4 \sqrt{b} (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (2 b c-a d) \text{EllipticF}\left (\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),2\right )}{3 a^{5/2} e^4 \left (a+b x^2\right )^{3/4}}-\frac{2 \sqrt{e x} (2 b c-a d)}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}-\frac{2 c}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/4}} \]
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Rubi [A] time = 0.112312, antiderivative size = 144, 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 = {453, 290, 329, 237, 335, 275, 231} \[ -\frac{2 \sqrt{e x} (2 b c-a d)}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}+\frac{4 \sqrt{b} (e x)^{3/2} \left (\frac{a}{b x^2}+1\right )^{3/4} (2 b c-a d) F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 a^{5/2} e^4 \left (a+b x^2\right )^{3/4}}-\frac{2 c}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 453
Rule 290
Rule 329
Rule 237
Rule 335
Rule 275
Rule 231
Rubi steps
\begin{align*} \int \frac{c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{7/4}} \, dx &=-\frac{2 c}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/4}}-\frac{(2 b c-a d) \int \frac{1}{\sqrt{e x} \left (a+b x^2\right )^{7/4}} \, dx}{a e^2}\\ &=-\frac{2 c}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/4}}-\frac{2 (2 b c-a d) \sqrt{e x}}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}-\frac{(2 (2 b c-a d)) \int \frac{1}{\sqrt{e x} \left (a+b x^2\right )^{3/4}} \, dx}{3 a^2 e^2}\\ &=-\frac{2 c}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/4}}-\frac{2 (2 b c-a d) \sqrt{e x}}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}-\frac{(4 (2 b c-a d)) \operatorname{Subst}\left (\int \frac{1}{\left (a+\frac{b x^4}{e^2}\right )^{3/4}} \, dx,x,\sqrt{e x}\right )}{3 a^2 e^3}\\ &=-\frac{2 c}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/4}}-\frac{2 (2 b c-a d) \sqrt{e x}}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}-\frac{\left (4 (2 b c-a d) \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 )}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}\\ &=-\frac{2 c}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/4}}-\frac{2 (2 b c-a d) \sqrt{e x}}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}+\frac{\left (4 (2 b c-a d) \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 )}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}\\ &=-\frac{2 c}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/4}}-\frac{2 (2 b c-a d) \sqrt{e x}}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}+\frac{\left (2 (2 b c-a d) \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 )}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}\\ &=-\frac{2 c}{3 a e (e x)^{3/2} \left (a+b x^2\right )^{3/4}}-\frac{2 (2 b c-a d) \sqrt{e x}}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}+\frac{4 \sqrt{b} (2 b c-a d) \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 )}{3 a^{5/2} e^4 \left (a+b x^2\right )^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0494491, size = 91, normalized size = 0.63 \[ \frac{x \left (4 x^2 \left (\frac{b x^2}{a}+1\right )^{3/4} (a d-2 b c) \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^2}{a}\right )-2 a c+2 a d x^2-4 b c x^2\right )}{3 a^2 (e x)^{5/2} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.047, 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{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{7}{4}} \left (e x\right )^{\frac{5}{2}}}\,{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 (b x^{2} + a\right )}^{\frac{1}{4}}{\left (d x^{2} + c\right )} \sqrt{e x}}{b^{2} e^{3} x^{7} + 2 \, a b e^{3} x^{5} + a^{2} e^{3} x^{3}}, 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{d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac{7}{4}} \left (e x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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