Optimal. Leaf size=188 \[ \frac{10 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{21 c d^{9/2} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}}+\frac{20 \sqrt{a+b x+c x^2}}{21 d^3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}+\frac{4 \sqrt{a+b x+c x^2}}{7 d \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}} \]
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Rubi [A] time = 0.146826, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {693, 691, 689, 221} \[ \frac{20 \sqrt{a+b x+c x^2}}{21 d^3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}+\frac{10 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{21 c d^{9/2} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt{a+b x+c x^2}}{7 d \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 693
Rule 691
Rule 689
Rule 221
Rubi steps
\begin{align*} \int \frac{1}{(b d+2 c d x)^{9/2} \sqrt{a+b x+c x^2}} \, dx &=\frac{4 \sqrt{a+b x+c x^2}}{7 \left (b^2-4 a c\right ) d (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}{7 \left (b^2-4 a c\right ) d^2}\\ &=\frac{4 \sqrt{a+b x+c x^2}}{7 \left (b^2-4 a c\right ) d (b d+2 c d x)^{7/2}}+\frac{20 \sqrt{a+b x+c x^2}}{21 \left (b^2-4 a c\right )^2 d^3 (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}{21 \left (b^2-4 a c\right )^2 d^4}\\ &=\frac{4 \sqrt{a+b x+c x^2}}{7 \left (b^2-4 a c\right ) d (b d+2 c d x)^{7/2}}+\frac{20 \sqrt{a+b x+c x^2}}{21 \left (b^2-4 a c\right )^2 d^3 (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}{21 \left (b^2-4 a c\right )^2 d^4 \sqrt{a+b x+c x^2}}\\ &=\frac{4 \sqrt{a+b x+c x^2}}{7 \left (b^2-4 a c\right ) d (b d+2 c d x)^{7/2}}+\frac{20 \sqrt{a+b x+c x^2}}{21 \left (b^2-4 a c\right )^2 d^3 (b d+2 c d x)^{3/2}}+\frac{\left (10 \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 )}{21 c \left (b^2-4 a c\right )^2 d^5 \sqrt{a+b x+c x^2}}\\ &=\frac{4 \sqrt{a+b x+c x^2}}{7 \left (b^2-4 a c\right ) d (b d+2 c d x)^{7/2}}+\frac{20 \sqrt{a+b x+c x^2}}{21 \left (b^2-4 a c\right )^2 d^3 (b d+2 c d x)^{3/2}}+\frac{10 \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 \left (b^2-4 a c\right )^{7/4} d^{9/2} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0747777, size = 99, normalized size = 0.53 \[ -\frac{2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \sqrt{d (b+2 c x)} \, _2F_1\left (-\frac{7}{4},\frac{1}{2};-\frac{3}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{7 c d^5 (b+2 c x)^4 \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.224, size = 691, normalized size = 3.7 \begin{align*}{\frac{1}{21\,{d}^{5} \left ( 2\,{c}^{2}{x}^{3}+3\,bc{x}^{2}+2\,acx+{b}^{2}x+ab \right ) \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( 2\,cx+b \right ) ^{3}c}\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a} \left ( 40\,\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}+60\,\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}+30\,\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+5\,\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 ){b}^{3}+80\,{c}^{4}{x}^{4}+160\,b{c}^{3}{x}^{3}+32\,{x}^{2}a{c}^{3}+112\,{x}^{2}{b}^{2}{c}^{2}+32\,ba{c}^{2}x+32\,{b}^{3}cx-48\,{a}^{2}{c}^{2}+32\,ac{b}^{2} \right ) } \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{1}{{\left (2 \, c d x + b d\right )}^{\frac{9}{2}} \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{\sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}}{32 \, c^{6} d^{5} x^{7} + 112 \, b c^{5} d^{5} x^{6} + a b^{5} d^{5} + 32 \,{\left (5 \, b^{2} c^{4} + a c^{5}\right )} d^{5} x^{5} + 40 \,{\left (3 \, b^{3} c^{3} + 2 \, a b c^{4}\right )} d^{5} x^{4} + 10 \,{\left (5 \, b^{4} c^{2} + 8 \, a b^{2} c^{3}\right )} d^{5} x^{3} +{\left (11 \, b^{5} c + 40 \, a b^{3} c^{2}\right )} d^{5} x^{2} +{\left (b^{6} + 10 \, a b^{4} c\right )} d^{5} x}, 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{1}{{\left (2 \, c d x + b d\right )}^{\frac{9}{2}} \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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