Optimal. Leaf size=96 \[ -\frac{\sqrt{4 a-\frac{b^2}{c}+\frac{(b+2 c x)^2}{c}} (b d+2 c d x)^{m+1} \, _2F_1\left (1,\frac{m+2}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{d (m+1) \left (b^2-4 a c\right )} \]
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Rubi [A] time = 0.106837, antiderivative size = 104, normalized size of antiderivative = 1.08, 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} \[ \frac{\sqrt{1-\frac{(b+2 c x)^2}{b^2-4 a c}} (d (b+2 c x))^{m+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{2 c d (m+1) \sqrt{a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 694
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^m}{\sqrt{a+b x+c x^2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^m}{\sqrt{a-\frac{b^2}{4 c}+\frac{x^2}{4 c d^2}}} \, dx,x,b d+2 c d x\right )}{2 c d}\\ &=\frac{\sqrt{4+\frac{(b d+2 c d x)^2}{\left (a-\frac{b^2}{4 c}\right ) c d^2}} \operatorname{Subst}\left (\int \frac{x^m}{\sqrt{1+\frac{x^2}{4 \left (a-\frac{b^2}{4 c}\right ) c d^2}}} \, dx,x,b d+2 c d x\right )}{4 c d \sqrt{a+b x+c x^2}}\\ &=\frac{(d (b+2 c x))^{1+m} \sqrt{1-\frac{(b+2 c x)^2}{b^2-4 a c}} \, _2F_1\left (\frac{1}{2},\frac{1+m}{2};\frac{3+m}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{2 c d (1+m) \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.046378, size = 102, normalized size = 1.06 \[ \frac{(b+2 c x) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^m \, _2F_1\left (\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{c (m+1) \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.159, size = 0, normalized size = 0. \begin{align*} \int{ \left ( 2\,cdx+bd \right ) ^{m}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}\, 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 (2 \, c d x + b d\right )}^{m}}{\sqrt{c x^{2} + b x + a}}\,{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 (2 \, c d x + b d\right )}^{m}}{\sqrt{c x^{2} + b x + a}}, 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 \left (b + 2 c x\right )\right )^{m}}{\sqrt{a + b x + c x^{2}}}\, 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 (2 \, c d x + b d\right )}^{m}}{\sqrt{c x^{2} + b x + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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