Optimal. Leaf size=82 \[ -\frac{2 \left (a+b x+c x^2\right )^{7/2} (b d+2 c d x)^{m+1} \, _2F_1\left (1,\frac{m+8}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right )} \]
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Rubi [A] time = 0.119357, antiderivative size = 114, normalized size of antiderivative = 1.39, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {694, 365, 364} \[ \frac{\left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} (d (b+2 c x))^{m+1} \, _2F_1\left (-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{32 c^3 d (m+1) \sqrt{1-\frac{(b+2 c x)^2}{b^2-4 a c}}} \]
Antiderivative was successfully verified.
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Rule 694
Rule 365
Rule 364
Rubi steps
\begin{align*} \int (b d+2 c d x)^m \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int x^m \left (a-\frac{b^2}{4 c}+\frac{x^2}{4 c d^2}\right )^{5/2} \, dx,x,b d+2 c d x\right )}{2 c d}\\ &=\frac{\left (\left (a-\frac{b^2}{4 c}\right )^2 \sqrt{a+b x+c x^2}\right ) \operatorname{Subst}\left (\int x^m \left (1+\frac{x^2}{4 \left (a-\frac{b^2}{4 c}\right ) c d^2}\right )^{5/2} \, dx,x,b d+2 c d x\right )}{c d \sqrt{4+\frac{(b d+2 c d x)^2}{\left (a-\frac{b^2}{4 c}\right ) c d^2}}}\\ &=\frac{\left (b^2-4 a c\right )^2 (d (b+2 c x))^{1+m} \sqrt{a+b x+c x^2} \, _2F_1\left (-\frac{5}{2},\frac{1+m}{2};\frac{3+m}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{32 c^3 d (1+m) \sqrt{1-\frac{(b+2 c x)^2}{b^2-4 a c}}}\\ \end{align*}
Mathematica [A] time = 0.0841635, size = 115, normalized size = 1.4 \[ \frac{\left (b^2-4 a c\right )^2 (b+2 c x) \sqrt{a+x (b+c x)} (d (b+2 c x))^m \, _2F_1\left (-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{64 c^3 (m+1) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.167, size = 0, normalized size = 0. \begin{align*} \int \left ( 2\,cdx+bd \right ) ^{m} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}\, 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}{2}}{\left (2 \, c d x + b d\right )}^{m}\,{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^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{c x^{2} + b x + a}{\left (2 \, c d x + b d\right )}^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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}{2}}{\left (2 \, c d x + b d\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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