Optimal. Leaf size=285 \[ \frac{5 \left (b^2-4 a c\right )^{9/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right ),\frac{1}{2}\right )}{336 \sqrt{2} c^{13/4} (b+2 c x)}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e)}{168 c^3}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/4} (2 c d-b e)}{14 c^2}+\frac{2 e \left (a+b x+c x^2\right )^{9/4}}{9 c} \]
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Rubi [A] time = 0.212182, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {640, 612, 623, 220} \[ -\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e)}{168 c^3}+\frac{5 \left (b^2-4 a c\right )^{9/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{336 \sqrt{2} c^{13/4} (b+2 c x)}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/4} (2 c d-b e)}{14 c^2}+\frac{2 e \left (a+b x+c x^2\right )^{9/4}}{9 c} \]
Antiderivative was successfully verified.
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Rule 640
Rule 612
Rule 623
Rule 220
Rubi steps
\begin{align*} \int (d+e x) \left (a+b x+c x^2\right )^{5/4} \, dx &=\frac{2 e \left (a+b x+c x^2\right )^{9/4}}{9 c}+\frac{(2 c d-b e) \int \left (a+b x+c x^2\right )^{5/4} \, dx}{2 c}\\ &=\frac{(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{14 c^2}+\frac{2 e \left (a+b x+c x^2\right )^{9/4}}{9 c}-\frac{\left (5 \left (b^2-4 a c\right ) (2 c d-b e)\right ) \int \sqrt [4]{a+b x+c x^2} \, dx}{56 c^2}\\ &=-\frac{5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{168 c^3}+\frac{(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{14 c^2}+\frac{2 e \left (a+b x+c x^2\right )^{9/4}}{9 c}+\frac{\left (5 \left (b^2-4 a c\right )^2 (2 c d-b e)\right ) \int \frac{1}{\left (a+b x+c x^2\right )^{3/4}} \, dx}{672 c^3}\\ &=-\frac{5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{168 c^3}+\frac{(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{14 c^2}+\frac{2 e \left (a+b x+c x^2\right )^{9/4}}{9 c}+\frac{\left (5 \left (b^2-4 a c\right )^2 (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 )}{168 c^3 (b+2 c x)}\\ &=-\frac{5 \left (b^2-4 a c\right ) (2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{168 c^3}+\frac{(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{14 c^2}+\frac{2 e \left (a+b x+c x^2\right )^{9/4}}{9 c}+\frac{5 \left (b^2-4 a c\right )^{9/4} (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 )}{336 \sqrt{2} c^{13/4} (b+2 c x)}\\ \end{align*}
Mathematica [A] time = 0.348044, size = 175, normalized size = 0.61 \[ \frac{(2 c d-b e) \left (24 c^2 (b+2 c x) (a+x (b+c x))^2-5 \left (b^2-4 a c\right ) \left (2 c (b+2 c x) (a+x (b+c x))-\sqrt{2} \left (b^2-4 a c\right )^{3/2} \left (\frac{c (a+x (b+c x))}{4 a c-b^2}\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \sin ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ),2\right )\right )\right )}{336 c^4 (a+x (b+c x))^{3/4}}+\frac{2 e (a+x (b+c x))^{9/4}}{9 c} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.121, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{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{\left (c x^{2} + b x + a\right )}^{\frac{5}{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 e x^{3} +{\left (c d + b e\right )} x^{2} + a d +{\left (b d + a e\right )} x\right )}{\left (c x^{2} + b x + a\right )}^{\frac{1}{4}}, 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 ) \left (a + b x + c x^{2}\right )^{\frac{5}{4}}\, 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{5}{4}}{\left (e x + d\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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