Optimal. Leaf size=662 \[ \frac{\sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (6 a e+b d)+7 b^2 e^2+4 c^2 d^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right ),\frac{1}{2}\right )}{2 \sqrt{2} c^{11/4} \sqrt [4]{b^2-4 a c} (b+2 c x)}+\frac{2 e \left (a+b x+c x^2\right )^{3/4} \left (-2 c e (8 a e+9 b d)+7 b^2 e^2+6 c e x (2 c d-b e)+24 c^2 d^2\right )}{3 c^2 \left (b^2-4 a c\right )}+\frac{(b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e) \left (-4 c e (6 a e+b d)+7 b^2 e^2+4 c^2 d^2\right )}{c^{5/2} \left (b^2-4 a c\right )^{3/2} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}-\frac{\sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (6 a e+b d)+7 b^2 e^2+4 c^2 d^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} c^{11/4} \sqrt [4]{b^2-4 a c} (b+2 c x)}-\frac{4 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}} \]
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Rubi [A] time = 0.7245, antiderivative size = 662, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {738, 779, 623, 305, 220, 1196} \[ \frac{2 e \left (a+b x+c x^2\right )^{3/4} \left (-2 c e (8 a e+9 b d)+7 b^2 e^2+6 c e x (2 c d-b e)+24 c^2 d^2\right )}{3 c^2 \left (b^2-4 a c\right )}+\frac{(b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e) \left (-4 c e (6 a e+b d)+7 b^2 e^2+4 c^2 d^2\right )}{c^{5/2} \left (b^2-4 a c\right )^{3/2} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}+\frac{\sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (6 a e+b d)+7 b^2 e^2+4 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{2 \sqrt{2} c^{11/4} \sqrt [4]{b^2-4 a c} (b+2 c x)}-\frac{\sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (6 a e+b d)+7 b^2 e^2+4 c^2 d^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{\sqrt{2} c^{11/4} \sqrt [4]{b^2-4 a c} (b+2 c x)}-\frac{4 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 738
Rule 779
Rule 623
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\left (a+b x+c x^2\right )^{5/4}} \, dx &=-\frac{4 (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}}-\frac{4 \int \frac{(d+e x) \left (\frac{1}{2} \left (-2 c d^2-3 b d e+8 a e^2\right )-\frac{5}{2} e (2 c d-b e) x\right )}{\sqrt [4]{a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=-\frac{4 (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}}+\frac{2 e \left (24 c^2 d^2+7 b^2 e^2-2 c e (9 b d+8 a e)+6 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/4}}{3 c^2 \left (b^2-4 a c\right )}+\frac{\left ((2 c d-b e) \left (4 c^2 d^2+7 b^2 e^2-4 c e (b d+6 a e)\right )\right ) \int \frac{1}{\sqrt [4]{a+b x+c x^2}} \, dx}{2 c^2 \left (b^2-4 a c\right )}\\ &=-\frac{4 (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}}+\frac{2 e \left (24 c^2 d^2+7 b^2 e^2-2 c e (9 b d+8 a e)+6 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/4}}{3 c^2 \left (b^2-4 a c\right )}+\frac{\left (2 (2 c d-b e) \left (4 c^2 d^2+7 b^2 e^2-4 c e (b d+6 a e)\right ) \sqrt{(b+2 c x)^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{c^2 \left (b^2-4 a c\right ) (b+2 c x)}\\ &=-\frac{4 (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}}+\frac{2 e \left (24 c^2 d^2+7 b^2 e^2-2 c e (9 b d+8 a e)+6 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/4}}{3 c^2 \left (b^2-4 a c\right )}+\frac{\left ((2 c d-b e) \left (4 c^2 d^2+7 b^2 e^2-4 c e (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 )}{c^{5/2} \sqrt{b^2-4 a c} (b+2 c x)}-\frac{\left ((2 c d-b e) \left (4 c^2 d^2+7 b^2 e^2-4 c e (b d+6 a e)\right ) \sqrt{(b+2 c x)^2}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{2 \sqrt{c} x^2}{\sqrt{b^2-4 a c}}}{\sqrt{b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{c^{5/2} \sqrt{b^2-4 a c} (b+2 c x)}\\ &=-\frac{4 (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}}+\frac{2 e \left (24 c^2 d^2+7 b^2 e^2-2 c e (9 b d+8 a e)+6 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/4}}{3 c^2 \left (b^2-4 a c\right )}+\frac{(2 c d-b e) \left (4 c^2 d^2+7 b^2 e^2-4 c e (b d+6 a e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{c^{5/2} \left (b^2-4 a c\right )^{3/2} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}-\frac{(2 c d-b e) \left (4 c^2 d^2+7 b^2 e^2-4 c e (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 ) E\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 )}{\sqrt{2} c^{11/4} \sqrt [4]{b^2-4 a c} (b+2 c x)}+\frac{(2 c d-b e) \left (4 c^2 d^2+7 b^2 e^2-4 c e (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 )}{2 \sqrt{2} c^{11/4} \sqrt [4]{b^2-4 a c} (b+2 c x)}\\ \end{align*}
Mathematica [C] time = 0.570505, size = 247, normalized size = 0.37 \[ \frac{3 \sqrt{2} (b+2 c x) \sqrt [4]{\frac{c (a+x (b+c x))}{4 a c-b^2}} (2 c d-b e) \left (-4 c e (6 a e+b d)+7 b^2 e^2+4 c^2 d^2\right ) \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )-8 c \left (4 c \left (4 a^2 e^3+a c e \left (-9 d^2-9 d e x+e^2 x^2\right )+3 c^2 d^3 x\right )-b^2 e^2 (7 a e+c x (e x-18 d))+2 b c \left (a e^2 (9 d+11 e x)+3 c d^2 (d-3 e x)\right )-7 b^3 e^3 x\right )}{12 c^3 \left (b^2-4 a c\right ) \sqrt [4]{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 2.526, 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 \frac{{\left (e x + d\right )}^{3}}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{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{{\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{3}{4}}}{c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}}, 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{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 \frac{{\left (e x + d\right )}^{3}}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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