Optimal. Leaf size=448 \[ \frac{5 \left (b^2-4 a c\right )^{9/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (6 a e+11 b d)+17 b^2 e^2+44 c^2 d^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right ),\frac{1}{2}\right )}{14784 \sqrt{2} c^{21/4} (b+2 c x)}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e) \left (-4 c e (6 a e+11 b d)+17 b^2 e^2+44 c^2 d^2\right )}{7392 c^5}+\frac{e \left (a+b x+c x^2\right )^{9/4} \left (-2 c e (88 a e+507 b d)+221 b^2 e^2+306 c e x (2 c d-b e)+1320 c^2 d^2\right )}{2574 c^3}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/4} (2 c d-b e) \left (-4 c e (6 a e+11 b d)+17 b^2 e^2+44 c^2 d^2\right )}{616 c^4}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c} \]
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Rubi [A] time = 0.559006, antiderivative size = 448, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{5 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e) \left (-4 c e (6 a e+11 b d)+17 b^2 e^2+44 c^2 d^2\right )}{7392 c^5}+\frac{e \left (a+b x+c x^2\right )^{9/4} \left (-2 c e (88 a e+507 b d)+221 b^2 e^2+306 c e x (2 c d-b e)+1320 c^2 d^2\right )}{2574 c^3}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/4} (2 c d-b e) \left (-4 c e (6 a e+11 b d)+17 b^2 e^2+44 c^2 d^2\right )}{616 c^4}+\frac{5 \left (b^2-4 a c\right )^{9/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (6 a e+11 b d)+17 b^2 e^2+44 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{14784 \sqrt{2} c^{21/4} (b+2 c x)}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c} \]
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 \left (a+b x+c x^2\right )^{5/4} \, dx &=\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c}+\frac{2 \int (d+e x) \left (\frac{1}{4} \left (26 c d^2-9 b d e-8 a e^2\right )+\frac{17}{4} e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{5/4} \, dx}{13 c}\\ &=\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c}+\frac{e \left (1320 c^2 d^2+221 b^2 e^2-2 c e (507 b d+88 a e)+306 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{9/4}}{2574 c^3}+\frac{\left ((2 c d-b e) \left (44 c^2 d^2+17 b^2 e^2-4 c e (11 b d+6 a e)\right )\right ) \int \left (a+b x+c x^2\right )^{5/4} \, dx}{88 c^3}\\ &=\frac{(2 c d-b e) \left (44 c^2 d^2+17 b^2 e^2-4 c e (11 b d+6 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{616 c^4}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c}+\frac{e \left (1320 c^2 d^2+221 b^2 e^2-2 c e (507 b d+88 a e)+306 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{9/4}}{2574 c^3}-\frac{\left (5 \left (b^2-4 a c\right ) (2 c d-b e) \left (44 c^2 d^2+17 b^2 e^2-4 c e (11 b d+6 a e)\right )\right ) \int \sqrt [4]{a+b x+c x^2} \, dx}{2464 c^4}\\ &=-\frac{5 \left (b^2-4 a c\right ) (2 c d-b e) \left (44 c^2 d^2+17 b^2 e^2-4 c e (11 b d+6 a e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{7392 c^5}+\frac{(2 c d-b e) \left (44 c^2 d^2+17 b^2 e^2-4 c e (11 b d+6 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{616 c^4}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c}+\frac{e \left (1320 c^2 d^2+221 b^2 e^2-2 c e (507 b d+88 a e)+306 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{9/4}}{2574 c^3}+\frac{\left (5 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (44 c^2 d^2+17 b^2 e^2-4 c e (11 b d+6 a e)\right )\right ) \int \frac{1}{\left (a+b x+c x^2\right )^{3/4}} \, dx}{29568 c^5}\\ &=-\frac{5 \left (b^2-4 a c\right ) (2 c d-b e) \left (44 c^2 d^2+17 b^2 e^2-4 c e (11 b d+6 a e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{7392 c^5}+\frac{(2 c d-b e) \left (44 c^2 d^2+17 b^2 e^2-4 c e (11 b d+6 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{616 c^4}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c}+\frac{e \left (1320 c^2 d^2+221 b^2 e^2-2 c e (507 b d+88 a e)+306 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{9/4}}{2574 c^3}+\frac{\left (5 \left (b^2-4 a c\right )^2 (2 c d-b e) \left (44 c^2 d^2+17 b^2 e^2-4 c e (11 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 )}{7392 c^5 (b+2 c x)}\\ &=-\frac{5 \left (b^2-4 a c\right ) (2 c d-b e) \left (44 c^2 d^2+17 b^2 e^2-4 c e (11 b d+6 a e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{7392 c^5}+\frac{(2 c d-b e) \left (44 c^2 d^2+17 b^2 e^2-4 c e (11 b d+6 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/4}}{616 c^4}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{9/4}}{13 c}+\frac{e \left (1320 c^2 d^2+221 b^2 e^2-2 c e (507 b d+88 a e)+306 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{9/4}}{2574 c^3}+\frac{5 \left (b^2-4 a c\right )^{9/4} (2 c d-b e) \left (44 c^2 d^2+17 b^2 e^2-4 c e (11 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 )}{14784 \sqrt{2} c^{21/4} (b+2 c x)}\\ \end{align*}
Mathematica [A] time = 1.17966, size = 286, normalized size = 0.64 \[ \frac{39 (2 c d-b e) \left (-4 c e (6 a e+11 b d)+17 b^2 e^2+44 c^2 d^2\right ) \left (5 \sqrt{2} \left (b^2-4 a c\right )^{5/2} \left (\frac{c (a+x (b+c x))}{4 a c-b^2}\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \sin ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ),2\right )+2 c (b+2 c x) \left (32 a^2 c+a \left (-5 b^2+44 b c x+44 c^2 x^2\right )+x \left (7 b^2 c x-5 b^3+24 b c^2 x^2+12 c^3 x^3\right )\right )\right )+224 c^3 e (a+x (b+c x))^3 \left (-2 c e (88 a e+507 b d+153 b e x)+221 b^2 e^2+12 c^2 d (110 d+51 e x)\right )+88704 c^5 e (d+e x)^2 (a+x (b+c x))^3}{576576 c^6 (a+x (b+c x))^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.085, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{3} \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 )}^{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 (c e^{3} x^{5} +{\left (3 \, c d e^{2} + b e^{3}\right )} x^{4} + a d^{3} +{\left (3 \, c d^{2} e + 3 \, b d e^{2} + a e^{3}\right )} x^{3} +{\left (c d^{3} + 3 \, b d^{2} e + 3 \, a d e^{2}\right )} x^{2} +{\left (b d^{3} + 3 \, a d^{2} 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 )^{3} \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 )}^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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