Optimal. Leaf size=182 \[ \frac{\left (b^2-4 a c\right )^{9/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{14 c^3 \sqrt{d} \sqrt{a+b x+c x^2}}-\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{14 c^2 d}+\frac{\left (a+b x+c x^2\right )^{3/2} \sqrt{b d+2 c d x}}{7 c d} \]
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Rubi [A] time = 0.149949, antiderivative size = 182, 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 = {685, 691, 689, 221} \[ -\frac{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{14 c^2 d}+\frac{\left (b^2-4 a c\right )^{9/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{14 c^3 \sqrt{d} \sqrt{a+b x+c x^2}}+\frac{\left (a+b x+c x^2\right )^{3/2} \sqrt{b d+2 c d x}}{7 c d} \]
Antiderivative was successfully verified.
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Rule 685
Rule 691
Rule 689
Rule 221
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{3/2}}{\sqrt{b d+2 c d x}} \, dx &=\frac{\sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{7 c d}-\frac{\left (3 \left (b^2-4 a c\right )\right ) \int \frac{\sqrt{a+b x+c x^2}}{\sqrt{b d+2 c d x}} \, dx}{14 c}\\ &=-\frac{\left (b^2-4 a c\right ) \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{14 c^2 d}+\frac{\sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{7 c d}+\frac{\left (b^2-4 a c\right )^2 \int \frac{1}{\sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}} \, dx}{28 c^2}\\ &=-\frac{\left (b^2-4 a c\right ) \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{14 c^2 d}+\frac{\sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{7 c d}+\frac{\left (\left (b^2-4 a c\right )^2 \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}{28 c^2 \sqrt{a+b x+c x^2}}\\ &=-\frac{\left (b^2-4 a c\right ) \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{14 c^2 d}+\frac{\sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{7 c d}+\frac{\left (\left (b^2-4 a c\right )^2 \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 )}{14 c^3 d \sqrt{a+b x+c x^2}}\\ &=-\frac{\left (b^2-4 a c\right ) \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{14 c^2 d}+\frac{\sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{7 c d}+\frac{\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 d x}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{14 c^3 \sqrt{d} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0705426, size = 99, normalized size = 0.54 \[ -\frac{\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{8 c^2 d \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.207, size = 566, normalized size = 3.1 \begin{align*}{\frac{1}{28\,d \left ( 2\,{c}^{2}{x}^{3}+3\,bc{x}^{2}+2\,acx+{b}^{2}x+ab \right ){c}^{3}}\sqrt{c{x}^{2}+bx+a}\sqrt{d \left ( 2\,cx+b \right ) } \left ( 8\,{x}^{5}{c}^{5}+16\,\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}-8\,\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+\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 ) \sqrt{-4\,ac+{b}^{2}}{b}^{4}+20\,{x}^{4}b{c}^{4}+32\,{x}^{3}a{c}^{4}+12\,{x}^{3}{b}^{2}{c}^{3}+48\,{x}^{2}ab{c}^{3}-2\,{x}^{2}{b}^{3}{c}^{2}+24\,x{a}^{2}{c}^{3}+12\,xa{b}^{2}{c}^{2}-2\,x{b}^{4}c+12\,{a}^{2}b{c}^{2}-2\,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 (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{\sqrt{2 \, c d x + b d}}\,{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 (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{\sqrt{2 \, c d x + b d}}, 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 (a + b x + c x^{2}\right )^{\frac{3}{2}}}{\sqrt{d \left (b + 2 c x\right )}}\, 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 (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{\sqrt{2 \, c d x + b d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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