Optimal. Leaf size=374 \[ -\frac{\left (b^2-4 a c\right )^{5/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (6 a e+7 b d)+13 b^2 e^2+28 c^2 d^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right ),\frac{1}{2}\right )}{336 \sqrt{2} c^{17/4} (b+2 c x)}+\frac{e \left (a+b x+c x^2\right )^{5/4} \left (-2 c e (56 a e+243 b d)+117 b^2 e^2+130 c e x (2 c d-b e)+616 c^2 d^2\right )}{630 c^3}+\frac{(b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e) \left (-4 c e (6 a e+7 b d)+13 b^2 e^2+28 c^2 d^2\right )}{168 c^4}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{5/4}}{9 c} \]
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Rubi [A] time = 0.518445, antiderivative size = 374, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {742, 779, 612, 623, 220} \[ \frac{e \left (a+b x+c x^2\right )^{5/4} \left (-2 c e (56 a e+243 b d)+117 b^2 e^2+130 c e x (2 c d-b e)+616 c^2 d^2\right )}{630 c^3}+\frac{(b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e) \left (-4 c e (6 a e+7 b d)+13 b^2 e^2+28 c^2 d^2\right )}{168 c^4}-\frac{\left (b^2-4 a c\right )^{5/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (6 a e+7 b d)+13 b^2 e^2+28 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{336 \sqrt{2} c^{17/4} (b+2 c x)}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{5/4}}{9 c} \]
Antiderivative was successfully verified.
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Rule 742
Rule 779
Rule 612
Rule 623
Rule 220
Rubi steps
\begin{align*} \int (d+e x)^3 \sqrt [4]{a+b x+c x^2} \, dx &=\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{5/4}}{9 c}+\frac{2 \int (d+e x) \left (\frac{1}{4} \left (18 c d^2-5 b d e-8 a e^2\right )+\frac{13}{4} e (2 c d-b e) x\right ) \sqrt [4]{a+b x+c x^2} \, dx}{9 c}\\ &=\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{5/4}}{9 c}+\frac{e \left (616 c^2 d^2+117 b^2 e^2-2 c e (243 b d+56 a e)+130 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/4}}{630 c^3}+\frac{\left ((2 c d-b e) \left (28 c^2 d^2+13 b^2 e^2-4 c e (7 b d+6 a e)\right )\right ) \int \sqrt [4]{a+b x+c x^2} \, dx}{56 c^3}\\ &=\frac{(2 c d-b e) \left (28 c^2 d^2+13 b^2 e^2-4 c e (7 b d+6 a e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{168 c^4}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{5/4}}{9 c}+\frac{e \left (616 c^2 d^2+117 b^2 e^2-2 c e (243 b d+56 a e)+130 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/4}}{630 c^3}-\frac{\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (28 c^2 d^2+13 b^2 e^2-4 c e (7 b d+6 a e)\right )\right ) \int \frac{1}{\left (a+b x+c x^2\right )^{3/4}} \, dx}{672 c^4}\\ &=\frac{(2 c d-b e) \left (28 c^2 d^2+13 b^2 e^2-4 c e (7 b d+6 a e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{168 c^4}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{5/4}}{9 c}+\frac{e \left (616 c^2 d^2+117 b^2 e^2-2 c e (243 b d+56 a e)+130 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/4}}{630 c^3}-\frac{\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (28 c^2 d^2+13 b^2 e^2-4 c e (7 b d+6 a e)\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 )}{168 c^4 (b+2 c x)}\\ &=\frac{(2 c d-b e) \left (28 c^2 d^2+13 b^2 e^2-4 c e (7 b d+6 a e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{168 c^4}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{5/4}}{9 c}+\frac{e \left (616 c^2 d^2+117 b^2 e^2-2 c e (243 b d+56 a e)+130 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/4}}{630 c^3}-\frac{\left (b^2-4 a c\right )^{5/4} (2 c d-b e) \left (28 c^2 d^2+13 b^2 e^2-4 c e (7 b d+6 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 )}{336 \sqrt{2} c^{17/4} (b+2 c x)}\\ \end{align*}
Mathematica [A] time = 0.924641, size = 235, normalized size = 0.63 \[ \frac{15 (2 c d-b e) \left (-4 c e (6 a e+7 b d)+13 b^2 e^2+28 c^2 d^2\right ) \left (2 c (b+2 c x) (a+x (b+c x))-\sqrt{2} \left (b^2-4 a c\right )^{3/2} \left (\frac{c (a+x (b+c x))}{4 a c-b^2}\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \sin ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ),2\right )\right )+8 c^2 e (a+x (b+c x))^2 \left (-2 c e (56 a e+243 b d+65 b e x)+117 b^2 e^2+4 c^2 d (154 d+65 e x)\right )+1120 c^4 e (d+e x)^2 (a+x (b+c x))^2}{5040 c^5 (a+x (b+c x))^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.073, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{3}\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 )}^{3}\,{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 (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\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 )^{3} \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 )}^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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