Optimal. Leaf size=262 \[ \frac{\sqrt [4]{b^2-4 a c} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right ),\frac{1}{2}\right )}{6 \sqrt{2} c^{9/4} (b+2 c x)}+\frac{5 e \sqrt [4]{a+b x+c x^2} (2 c d-b e)}{3 c^2}+\frac{2 e (d+e x) \sqrt [4]{a+b x+c x^2}}{3 c} \]
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Rubi [A] time = 0.254795, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {742, 640, 623, 220} \[ \frac{\sqrt [4]{b^2-4 a c} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{6 \sqrt{2} c^{9/4} (b+2 c x)}+\frac{5 e \sqrt [4]{a+b x+c x^2} (2 c d-b e)}{3 c^2}+\frac{2 e (d+e x) \sqrt [4]{a+b x+c x^2}}{3 c} \]
Antiderivative was successfully verified.
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Rule 742
Rule 640
Rule 623
Rule 220
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\left (a+b x+c x^2\right )^{3/4}} \, dx &=\frac{2 e (d+e x) \sqrt [4]{a+b x+c x^2}}{3 c}+\frac{2 \int \frac{\frac{1}{4} \left (6 c d^2-e (b d+4 a e)\right )+\frac{5}{4} e (2 c d-b e) x}{\left (a+b x+c x^2\right )^{3/4}} \, dx}{3 c}\\ &=\frac{5 e (2 c d-b e) \sqrt [4]{a+b x+c x^2}}{3 c^2}+\frac{2 e (d+e x) \sqrt [4]{a+b x+c x^2}}{3 c}+\frac{\left (-\frac{5}{4} b e (2 c d-b e)+\frac{1}{2} c \left (6 c d^2-e (b d+4 a e)\right )\right ) \int \frac{1}{\left (a+b x+c x^2\right )^{3/4}} \, dx}{3 c^2}\\ &=\frac{5 e (2 c d-b e) \sqrt [4]{a+b x+c x^2}}{3 c^2}+\frac{2 e (d+e x) \sqrt [4]{a+b x+c x^2}}{3 c}+\frac{\left (4 \left (-\frac{5}{4} b e (2 c d-b e)+\frac{1}{2} c \left (6 c d^2-e (b d+4 a e)\right )\right ) \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 )}{3 c^2 (b+2 c x)}\\ &=\frac{5 e (2 c d-b e) \sqrt [4]{a+b x+c x^2}}{3 c^2}+\frac{2 e (d+e x) \sqrt [4]{a+b x+c x^2}}{3 c}+\frac{\sqrt [4]{b^2-4 a c} \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \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 )}{6 \sqrt{2} c^{9/4} (b+2 c x)}\\ \end{align*}
Mathematica [A] time = 0.265212, size = 150, normalized size = 0.57 \[ \frac{\sqrt{2} \sqrt{b^2-4 a c} \left (\frac{c (a+x (b+c x))}{4 a c-b^2}\right )^{3/4} \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right ) \text{EllipticF}\left (\frac{1}{2} \sin ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ),2\right )+2 c e (a+x (b+c x)) (2 c (6 d+e x)-5 b e)}{6 c^3 (a+x (b+c x))^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.944, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex+d \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{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 (e x + d\right )}^{2}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{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{e^{2} x^{2} + 2 \, d e x + d^{2}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{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 \frac{\left (d + e x\right )^{2}}{\left (a + b x + c x^{2}\right )^{\frac{3}{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 \frac{{\left (e x + d\right )}^{2}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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