Optimal. Leaf size=241 \[ -\frac{\left (b^2-4 a c\right )^{5/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right ),\frac{1}{2}\right )}{12 \sqrt{2} c^{9/4} (b+2 c x)}+\frac{(b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e)}{6 c^2}+\frac{2 e \left (a+b x+c x^2\right )^{5/4}}{5 c} \]
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Rubi [A] time = 0.183802, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {640, 612, 623, 220} \[ -\frac{\left (b^2-4 a c\right )^{5/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{12 \sqrt{2} c^{9/4} (b+2 c x)}+\frac{(b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e)}{6 c^2}+\frac{2 e \left (a+b x+c x^2\right )^{5/4}}{5 c} \]
Antiderivative was successfully verified.
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Rule 640
Rule 612
Rule 623
Rule 220
Rubi steps
\begin{align*} \int (d+e x) \sqrt [4]{a+b x+c x^2} \, dx &=\frac{2 e \left (a+b x+c x^2\right )^{5/4}}{5 c}+\frac{(2 c d-b e) \int \sqrt [4]{a+b x+c x^2} \, dx}{2 c}\\ &=\frac{(2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{6 c^2}+\frac{2 e \left (a+b x+c x^2\right )^{5/4}}{5 c}-\frac{\left (\left (b^2-4 a c\right ) (2 c d-b e)\right ) \int \frac{1}{\left (a+b x+c x^2\right )^{3/4}} \, dx}{24 c^2}\\ &=\frac{(2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{6 c^2}+\frac{2 e \left (a+b x+c x^2\right )^{5/4}}{5 c}-\frac{\left (\left (b^2-4 a c\right ) (2 c d-b e) \sqrt{(b+2 c x)^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{6 c^2 (b+2 c x)}\\ &=\frac{(2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{6 c^2}+\frac{2 e \left (a+b x+c x^2\right )^{5/4}}{5 c}-\frac{\left (b^2-4 a c\right )^{5/4} (2 c d-b e) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )^2}} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{12 \sqrt{2} c^{9/4} (b+2 c x)}\\ \end{align*}
Mathematica [A] time = 0.258339, size = 140, normalized size = 0.58 \[ \frac{5 (2 c d-b e) \left (2 c (b+2 c x) (a+x (b+c x))-\sqrt{2} \left (b^2-4 a c\right )^{3/2} \left (\frac{c (a+x (b+c x))}{4 a c-b^2}\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \sin ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ),2\right )\right )+24 c^2 e (a+x (b+c x))^2}{60 c^3 (a+x (b+c x))^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.99, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) \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{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}{\left (e x + d\right )}\,{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 ({\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}{\left (e x + d\right )}, 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 \left (d + e x\right ) \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}{\left (e x + d\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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