Optimal. Leaf size=510 \[ -\frac{3 \left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right ),\frac{1}{2}\right )}{40 \sqrt{2} c^{11/4} (b+2 c x)}-\frac{3 \sqrt{b^2-4 a c} (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e)}{20 c^{5/2} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}+\frac{3 \left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{20 \sqrt{2} c^{11/4} (b+2 c x)}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/4} (2 c d-b e)}{10 c^2}+\frac{2 e \left (a+b x+c x^2\right )^{7/4}}{7 c} \]
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Rubi [A] time = 0.438951, antiderivative size = 510, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {640, 612, 623, 305, 220, 1196} \[ -\frac{3 \sqrt{b^2-4 a c} (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e)}{20 c^{5/2} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}-\frac{3 \left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{40 \sqrt{2} c^{11/4} (b+2 c x)}+\frac{3 \left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{20 \sqrt{2} c^{11/4} (b+2 c x)}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/4} (2 c d-b e)}{10 c^2}+\frac{2 e \left (a+b x+c x^2\right )^{7/4}}{7 c} \]
Antiderivative was successfully verified.
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Rule 640
Rule 612
Rule 623
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int (d+e x) \left (a+b x+c x^2\right )^{3/4} \, dx &=\frac{2 e \left (a+b x+c x^2\right )^{7/4}}{7 c}+\frac{(2 c d-b e) \int \left (a+b x+c x^2\right )^{3/4} \, dx}{2 c}\\ &=\frac{(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{10 c^2}+\frac{2 e \left (a+b x+c x^2\right )^{7/4}}{7 c}-\frac{\left (3 \left (b^2-4 a c\right ) (2 c d-b e)\right ) \int \frac{1}{\sqrt [4]{a+b x+c x^2}} \, dx}{40 c^2}\\ &=\frac{(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{10 c^2}+\frac{2 e \left (a+b x+c x^2\right )^{7/4}}{7 c}-\frac{\left (3 \left (b^2-4 a c\right ) (2 c d-b e) \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 )}{10 c^2 (b+2 c x)}\\ &=\frac{(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{10 c^2}+\frac{2 e \left (a+b x+c x^2\right )^{7/4}}{7 c}-\frac{\left (3 \left (b^2-4 a c\right )^{3/2} (2 c d-b e) \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 )}{20 c^{5/2} (b+2 c x)}+\frac{\left (3 \left (b^2-4 a c\right )^{3/2} (2 c d-b e) \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 )}{20 c^{5/2} (b+2 c x)}\\ &=\frac{(2 c d-b e) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{10 c^2}+\frac{2 e \left (a+b x+c x^2\right )^{7/4}}{7 c}-\frac{3 \sqrt{b^2-4 a c} (2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{20 c^{5/2} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}+\frac{3 \left (b^2-4 a c\right )^{7/4} (2 c d-b e) \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 )}{20 \sqrt{2} c^{11/4} (b+2 c x)}-\frac{3 \left (b^2-4 a c\right )^{7/4} (2 c d-b e) \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 )}{40 \sqrt{2} c^{11/4} (b+2 c x)}\\ \end{align*}
Mathematica [C] time = 0.242966, size = 141, normalized size = 0.28 \[ \frac{(b+2 c x) (2 c d-b e) \left (8 c (a+x (b+c x))-3 \sqrt{2} \left (b^2-4 a c\right ) \sqrt [4]{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )\right )}{80 c^3 \sqrt [4]{a+x (b+c x)}}+\frac{2 e (a+x (b+c x))^{7/4}}{7 c} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.049, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) \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{\left (c x^{2} + b x + a\right )}^{\frac{3}{4}}{\left (e x + d\right )}\,{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 x^{2} + b x + a\right )}^{\frac{3}{4}}{\left (e x + d\right )}, 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 ) \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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