Optimal. Leaf size=307 \[ \frac{\sqrt [4]{b^2-4 a c} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right ),\frac{1}{2}\right )}{4 \sqrt{2} c^{13/4} (b+2 c x)}+\frac{e \sqrt [4]{a+b x+c x^2} \left (-2 c e (8 a e+25 b d)+15 b^2 e^2+6 c e x (2 c d-b e)+56 c^2 d^2\right )}{10 c^3}+\frac{2 e (d+e x)^2 \sqrt [4]{a+b x+c x^2}}{5 c} \]
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Rubi [A] time = 0.329917, antiderivative size = 307, 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, 779, 623, 220} \[ \frac{e \sqrt [4]{a+b x+c x^2} \left (-2 c e (8 a e+25 b d)+15 b^2 e^2+6 c e x (2 c d-b e)+56 c^2 d^2\right )}{10 c^3}+\frac{\sqrt [4]{b^2-4 a c} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{4 \sqrt{2} c^{13/4} (b+2 c x)}+\frac{2 e (d+e x)^2 \sqrt [4]{a+b x+c x^2}}{5 c} \]
Antiderivative was successfully verified.
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Rule 742
Rule 779
Rule 623
Rule 220
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\left (a+b x+c x^2\right )^{3/4}} \, dx &=\frac{2 e (d+e x)^2 \sqrt [4]{a+b x+c x^2}}{5 c}+\frac{2 \int \frac{(d+e x) \left (\frac{1}{4} \left (10 c d^2-e (b d+8 a e)\right )+\frac{9}{4} e (2 c d-b e) x\right )}{\left (a+b x+c x^2\right )^{3/4}} \, dx}{5 c}\\ &=\frac{2 e (d+e x)^2 \sqrt [4]{a+b x+c x^2}}{5 c}+\frac{e \left (56 c^2 d^2+15 b^2 e^2-2 c e (25 b d+8 a e)+6 c e (2 c d-b e) x\right ) \sqrt [4]{a+b x+c x^2}}{10 c^3}+\frac{\left ((2 c d-b e) \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right )\right ) \int \frac{1}{\left (a+b x+c x^2\right )^{3/4}} \, dx}{8 c^3}\\ &=\frac{2 e (d+e x)^2 \sqrt [4]{a+b x+c x^2}}{5 c}+\frac{e \left (56 c^2 d^2+15 b^2 e^2-2 c e (25 b d+8 a e)+6 c e (2 c d-b e) x\right ) \sqrt [4]{a+b x+c x^2}}{10 c^3}+\frac{\left ((2 c d-b e) \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 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 )}{2 c^3 (b+2 c x)}\\ &=\frac{2 e (d+e x)^2 \sqrt [4]{a+b x+c x^2}}{5 c}+\frac{e \left (56 c^2 d^2+15 b^2 e^2-2 c e (25 b d+8 a e)+6 c e (2 c d-b e) x\right ) \sqrt [4]{a+b x+c x^2}}{10 c^3}+\frac{\sqrt [4]{b^2-4 a c} (2 c d-b e) \left (4 c^2 d^2+3 b^2 e^2-4 c e (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 )}{4 \sqrt{2} c^{13/4} (b+2 c x)}\\ \end{align*}
Mathematica [A] time = 0.424407, size = 244, normalized size = 0.79 \[ \frac{2 c e \left (-16 a^2 c e^2+a \left (15 b^2 e^2-2 b c e (25 d+11 e x)+4 c^2 \left (15 d^2+5 d e x-3 e^2 x^2\right )\right )+x (b+c x) \left (15 b^2 e^2-2 b c e (25 d+3 e x)+4 c^2 \left (15 d^2+5 d e x+e^2 x^2\right )\right )\right )-5 \sqrt{2} \sqrt{b^2-4 a c} \left (\frac{c (a+x (b+c x))}{4 a c-b^2}\right )^{3/4} (b e-2 c d) \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right ) \text{EllipticF}\left (\frac{1}{2} \sin ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ),2\right )}{20 c^4 (a+x (b+c x))^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.95, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ex+d \right ) ^{3} \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 )}^{3}}{{\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^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}{{\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 )^{3}}{\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 )}^{3}}{{\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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