Optimal. Leaf size=231 \[ \frac{5 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{231 c^2 d^{13/2} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}}+\frac{10 \sqrt{a+b x+c x^2}}{231 c d^5 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}+\frac{2 \sqrt{a+b x+c x^2}}{77 c d^3 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}-\frac{\sqrt{a+b x+c x^2}}{11 c d (b d+2 c d x)^{11/2}} \]
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Rubi [A] time = 0.186901, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {684, 693, 691, 689, 221} \[ \frac{5 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{231 c^2 d^{13/2} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}}+\frac{10 \sqrt{a+b x+c x^2}}{231 c d^5 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}+\frac{2 \sqrt{a+b x+c x^2}}{77 c d^3 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}-\frac{\sqrt{a+b x+c x^2}}{11 c d (b d+2 c d x)^{11/2}} \]
Antiderivative was successfully verified.
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Rule 684
Rule 693
Rule 691
Rule 689
Rule 221
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^{13/2}} \, dx &=-\frac{\sqrt{a+b x+c x^2}}{11 c d (b d+2 c d x)^{11/2}}+\frac{\int \frac{1}{(b d+2 c d x)^{9/2} \sqrt{a+b x+c x^2}} \, dx}{22 c d^2}\\ &=-\frac{\sqrt{a+b x+c x^2}}{11 c d (b d+2 c d x)^{11/2}}+\frac{2 \sqrt{a+b x+c x^2}}{77 c \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{7/2}}+\frac{5 \int \frac{1}{(b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}} \, dx}{154 c \left (b^2-4 a c\right ) d^4}\\ &=-\frac{\sqrt{a+b x+c x^2}}{11 c d (b d+2 c d x)^{11/2}}+\frac{2 \sqrt{a+b x+c x^2}}{77 c \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{7/2}}+\frac{10 \sqrt{a+b x+c x^2}}{231 c \left (b^2-4 a c\right )^2 d^5 (b d+2 c d x)^{3/2}}+\frac{5 \int \frac{1}{\sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}} \, dx}{462 c \left (b^2-4 a c\right )^2 d^6}\\ &=-\frac{\sqrt{a+b x+c x^2}}{11 c d (b d+2 c d x)^{11/2}}+\frac{2 \sqrt{a+b x+c x^2}}{77 c \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{7/2}}+\frac{10 \sqrt{a+b x+c x^2}}{231 c \left (b^2-4 a c\right )^2 d^5 (b d+2 c d x)^{3/2}}+\frac{\left (5 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \int \frac{1}{\sqrt{b d+2 c d x} \sqrt{-\frac{a c}{b^2-4 a c}-\frac{b c x}{b^2-4 a c}-\frac{c^2 x^2}{b^2-4 a c}}} \, dx}{462 c \left (b^2-4 a c\right )^2 d^6 \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{a+b x+c x^2}}{11 c d (b d+2 c d x)^{11/2}}+\frac{2 \sqrt{a+b x+c x^2}}{77 c \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{7/2}}+\frac{10 \sqrt{a+b x+c x^2}}{231 c \left (b^2-4 a c\right )^2 d^5 (b d+2 c d x)^{3/2}}+\frac{\left (5 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt{b d+2 c d x}\right )}{231 c^2 \left (b^2-4 a c\right )^2 d^7 \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{a+b x+c x^2}}{11 c d (b d+2 c d x)^{11/2}}+\frac{2 \sqrt{a+b x+c x^2}}{77 c \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{7/2}}+\frac{10 \sqrt{a+b x+c x^2}}{231 c \left (b^2-4 a c\right )^2 d^5 (b d+2 c d x)^{3/2}}+\frac{5 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{231 c^2 \left (b^2-4 a c\right )^{7/4} d^{13/2} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0775007, size = 99, normalized size = 0.43 \[ -\frac{\sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \, _2F_1\left (-\frac{11}{4},-\frac{1}{2};-\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{22 c d^7 (b+2 c x)^6 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.274, size = 1016, normalized size = 4.4 \begin{align*} \text{result too large to display} \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{\sqrt{c x^{2} + b x + a}}{{\left (2 \, c d x + b d\right )}^{\frac{13}{2}}}\,{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{\sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}}{128 \, c^{7} d^{7} x^{7} + 448 \, b c^{6} d^{7} x^{6} + 672 \, b^{2} c^{5} d^{7} x^{5} + 560 \, b^{3} c^{4} d^{7} x^{4} + 280 \, b^{4} c^{3} d^{7} x^{3} + 84 \, b^{5} c^{2} d^{7} x^{2} + 14 \, b^{6} c d^{7} x + b^{7} d^{7}}, 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 \frac{\sqrt{c x^{2} + b x + a}}{{\left (2 \, c d x + b d\right )}^{\frac{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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