Optimal. Leaf size=184 \[ \frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{21 c^2 d^{9/2} \left (b^2-4 a c\right )^{3/4} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{a+b x+c x^2}}{21 c d^3 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}-\frac{\sqrt{a+b x+c x^2}}{7 c d (b d+2 c d x)^{7/2}} \]
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Rubi [A] time = 0.155442, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ \frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{21 c^2 d^{9/2} \left (b^2-4 a c\right )^{3/4} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{a+b x+c x^2}}{21 c d^3 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}-\frac{\sqrt{a+b x+c x^2}}{7 c d (b d+2 c d x)^{7/2}} \]
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)^{9/2}} \, dx &=-\frac{\sqrt{a+b x+c x^2}}{7 c d (b d+2 c d x)^{7/2}}+\frac{\int \frac{1}{(b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}} \, dx}{14 c d^2}\\ &=-\frac{\sqrt{a+b x+c x^2}}{7 c d (b d+2 c d x)^{7/2}}+\frac{2 \sqrt{a+b x+c x^2}}{21 c \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}}+\frac{\int \frac{1}{\sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}} \, dx}{42 c \left (b^2-4 a c\right ) d^4}\\ &=-\frac{\sqrt{a+b x+c x^2}}{7 c d (b d+2 c d x)^{7/2}}+\frac{2 \sqrt{a+b x+c x^2}}{21 c \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}}+\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \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}{42 c \left (b^2-4 a c\right ) d^4 \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{a+b x+c x^2}}{7 c d (b d+2 c d x)^{7/2}}+\frac{2 \sqrt{a+b x+c x^2}}{21 c \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}}+\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \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 )}{21 c^2 \left (b^2-4 a c\right ) d^5 \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{a+b x+c x^2}}{7 c d (b d+2 c d x)^{7/2}}+\frac{2 \sqrt{a+b x+c x^2}}{21 c \left (b^2-4 a c\right ) d^3 (b d+2 c d x)^{3/2}}+\frac{\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 )}{21 c^2 \left (b^2-4 a c\right )^{3/4} d^{9/2} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0728355, size = 99, normalized size = 0.54 \[ -\frac{\sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \, _2F_1\left (-\frac{7}{4},-\frac{1}{2};-\frac{3}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{14 c d^5 (b+2 c x)^4 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.268, size = 659, normalized size = 3.6 \begin{align*} -{\frac{1}{42\,{d}^{5} \left ( 4\,ac-{b}^{2} \right ) \left ( 2\,cx+b \right ) ^{4}{c}^{2}} \left ( 8\,\sqrt{-4\,ac+{b}^{2}}\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){x}^{3}{c}^{3}+12\,\sqrt{-4\,ac+{b}^{2}}\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){x}^{2}b{c}^{2}+6\,\sqrt{-4\,ac+{b}^{2}}\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) x{b}^{2}c+\sqrt{-4\,ac+{b}^{2}}\sqrt{{ \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{-{(2\,cx+b){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}\sqrt{{ \left ( -b-2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}{\it EllipticF} \left ({\frac{\sqrt{2}}{2}\sqrt{{ \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}}}},\sqrt{2} \right ){b}^{3}+16\,{c}^{4}{x}^{4}+32\,b{c}^{3}{x}^{3}+40\,{x}^{2}a{c}^{3}+14\,{x}^{2}{b}^{2}{c}^{2}+40\,ba{c}^{2}x-2\,{b}^{3}cx+24\,{a}^{2}{c}^{2}-2\,ac{b}^{2} \right ) \sqrt{d \left ( 2\,cx+b \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}} \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{9}{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}}{32 \, c^{5} d^{5} x^{5} + 80 \, b c^{4} d^{5} x^{4} + 80 \, b^{2} c^{3} d^{5} x^{3} + 40 \, b^{3} c^{2} d^{5} x^{2} + 10 \, b^{4} c d^{5} x + b^{5} d^{5}}, 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{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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