Optimal. Leaf size=174 \[ \frac{10 d^{7/2} \left (b^2-4 a c\right )^{9/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{21 c \sqrt{a+b x+c x^2}}+\frac{20}{21} d^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}+\frac{4}{7} d \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2} \]
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Rubi [A] time = 0.144182, antiderivative size = 174, 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 = {692, 691, 689, 221} \[ \frac{20}{21} d^3 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}+\frac{10 d^{7/2} \left (b^2-4 a c\right )^{9/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{21 c \sqrt{a+b x+c x^2}}+\frac{4}{7} d \sqrt{a+b x+c x^2} (b d+2 c d x)^{5/2} \]
Antiderivative was successfully verified.
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Rule 692
Rule 691
Rule 689
Rule 221
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^{7/2}}{\sqrt{a+b x+c x^2}} \, dx &=\frac{4}{7} d (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}+\frac{1}{7} \left (5 \left (b^2-4 a c\right ) d^2\right ) \int \frac{(b d+2 c d x)^{3/2}}{\sqrt{a+b x+c x^2}} \, dx\\ &=\frac{20}{21} \left (b^2-4 a c\right ) d^3 \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}+\frac{4}{7} d (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}+\frac{1}{21} \left (5 \left (b^2-4 a c\right )^2 d^4\right ) \int \frac{1}{\sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}} \, dx\\ &=\frac{20}{21} \left (b^2-4 a c\right ) d^3 \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}+\frac{4}{7} d (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}+\frac{\left (5 \left (b^2-4 a c\right )^2 d^4 \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 \sqrt{a+b x+c x^2}}\\ &=\frac{20}{21} \left (b^2-4 a c\right ) d^3 \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}+\frac{4}{7} d (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}+\frac{\left (10 \left (b^2-4 a c\right )^2 d^3 \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 \sqrt{a+b x+c x^2}}\\ &=\frac{20}{21} \left (b^2-4 a c\right ) d^3 \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}+\frac{4}{7} d (b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}+\frac{10 \left (b^2-4 a c\right )^{9/4} d^{7/2} \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 \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.181379, size = 166, normalized size = 0.95 \[ \frac{2 d^3 \sqrt{d (b+2 c x)} \left (8 c \left (-5 a^2 c+2 a \left (b^2-b c x-c^2 x^2\right )+x \left (5 b^2 c x+2 b^3+6 b c^2 x^2+3 c^3 x^3\right )\right )+5 \left (b^2-4 a c\right )^2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )\right )}{21 c \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.232, size = 567, normalized size = 3.3 \begin{align*}{\frac{{d}^{3}}{21\,c \left ( 2\,{c}^{2}{x}^{3}+3\,bc{x}^{2}+2\,acx+{b}^{2}x+ab \right ) }\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a} \left ( 96\,{x}^{5}{c}^{5}+80\,\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 ) \sqrt{-4\,ac+{b}^{2}}{a}^{2}{c}^{2}-40\,\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 ) \sqrt{-4\,ac+{b}^{2}}a{b}^{2}c+5\,\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 ) \sqrt{-4\,ac+{b}^{2}}{b}^{4}+240\,{x}^{4}b{c}^{4}-64\,{x}^{3}a{c}^{4}+256\,{x}^{3}{b}^{2}{c}^{3}-96\,{x}^{2}ab{c}^{3}+144\,{x}^{2}{b}^{3}{c}^{2}-160\,x{a}^{2}{c}^{3}+32\,xa{b}^{2}{c}^{2}+32\,x{b}^{4}c-80\,{a}^{2}b{c}^{2}+32\,a{b}^{3}c \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{{\left (2 \, c d x + b d\right )}^{\frac{7}{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{{\left (8 \, c^{3} d^{3} x^{3} + 12 \, b c^{2} d^{3} x^{2} + 6 \, b^{2} c d^{3} x + b^{3} d^{3}\right )} \sqrt{2 \, c d x + b d}}{\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(-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{{\left (2 \, c d x + b d\right )}^{\frac{7}{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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