Optimal. Leaf size=239 \[ \frac{2 \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \sqrt [4]{\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )}{\sqrt{d+e x} \sqrt [4]{a+b x+c x^2} \left (e \sqrt{b^2-4 a c}-b e+2 c d\right )} \]
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Rubi [A] time = 0.153645, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {726} \[ \frac{2 \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \sqrt [4]{\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (b-\sqrt{b^2-4 a c}\right )\right )}{\left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right )}} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b+2 c x-\sqrt{b^2-4 a c}\right )}\right )}{\sqrt{d+e x} \sqrt [4]{a+b x+c x^2} \left (e \sqrt{b^2-4 a c}-b e+2 c d\right )} \]
Antiderivative was successfully verified.
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Rule 726
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^{3/2} \sqrt [4]{a+b x+c x^2}} \, dx &=\frac{2 \left (b-\sqrt{b^2-4 a c}+2 c x\right ) \sqrt [4]{\frac{\left (2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e\right ) \left (b+\sqrt{b^2-4 a c}+2 c x\right )}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b-\sqrt{b^2-4 a c}+2 c x\right )}} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b-\sqrt{b^2-4 a c}+2 c x\right )}\right )}{\left (2 c d-b e+\sqrt{b^2-4 a c} e\right ) \sqrt{d+e x} \sqrt [4]{a+b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.264665, size = 235, normalized size = 0.98 \[ \frac{2 \left (-\sqrt{b^2-4 a c}+b+2 c x\right ) \sqrt [4]{\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) \left (e \left (\sqrt{b^2-4 a c}-b\right )+2 c d\right )}{\left (\sqrt{b^2-4 a c}-b-2 c x\right ) \left (e \left (\sqrt{b^2-4 a c}+b\right )-2 c d\right )}} \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{1}{2};-\frac{4 c \sqrt{b^2-4 a c} (d+e x)}{\left (\left (b+\sqrt{b^2-4 a c}\right ) e-2 c d\right ) \left (-b-2 c x+\sqrt{b^2-4 a c}\right )}\right )}{\sqrt{d+e x} \sqrt [4]{a+x (b+c x)} \left (e \left (\sqrt{b^2-4 a c}-b\right )+2 c d\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.284, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt [4]{c{x}^{2}+bx+a}}}}\, 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{1}{{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}{\left (e x + d\right )}^{\frac{3}{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 (c x^{2} + b x + a\right )}^{\frac{3}{4}} \sqrt{e x + d}}{c e^{2} x^{4} +{\left (2 \, c d e + b e^{2}\right )} x^{3} + a d^{2} +{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{2} +{\left (b d^{2} + 2 \, a d e\right )} x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d + e x\right )^{\frac{3}{2}} \sqrt [4]{a + b x + c x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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