Optimal. Leaf size=174 \[ -\frac{\left (b^2-4 a c\right )^{5/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{6 c^3 d^{5/2} \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{6 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{3/2}}{3 c d (b d+2 c d x)^{3/2}} \]
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Rubi [A] time = 0.143503, antiderivative size = 174, 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, 685, 691, 689, 221} \[ -\frac{\left (b^2-4 a c\right )^{5/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{6 c^3 d^{5/2} \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{6 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{3/2}}{3 c d (b d+2 c d x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 684
Rule 685
Rule 691
Rule 689
Rule 221
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{5/2}} \, dx &=-\frac{\left (a+b x+c x^2\right )^{3/2}}{3 c d (b d+2 c d x)^{3/2}}+\frac{\int \frac{\sqrt{a+b x+c x^2}}{\sqrt{b d+2 c d x}} \, dx}{2 c d^2}\\ &=\frac{\sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{6 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{3/2}}{3 c d (b d+2 c d x)^{3/2}}-\frac{\left (b^2-4 a c\right ) \int \frac{1}{\sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}} \, dx}{12 c^2 d^2}\\ &=\frac{\sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{6 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{3/2}}{3 c d (b d+2 c d x)^{3/2}}-\frac{\left (\left (b^2-4 a c\right ) \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}{12 c^2 d^2 \sqrt{a+b x+c x^2}}\\ &=\frac{\sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{6 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{3/2}}{3 c d (b d+2 c d x)^{3/2}}-\frac{\left (\left (b^2-4 a c\right ) \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 )}{6 c^3 d^3 \sqrt{a+b x+c x^2}}\\ &=\frac{\sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{6 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{3/2}}{3 c d (b d+2 c d x)^{3/2}}-\frac{\left (b^2-4 a c\right )^{5/4} \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 )}{6 c^3 d^{5/2} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0578706, size = 99, normalized size = 0.57 \[ \frac{\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \, _2F_1\left (-\frac{3}{2},-\frac{3}{4};\frac{1}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{24 c^2 d \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.216, size = 626, normalized size = 3.6 \begin{align*}{\frac{1}{12\,{d}^{3} \left ( 2\,cx+b \right ) ^{2}{c}^{3}} \left ( 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}}xa{c}^{2}-2\,\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+4\,\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}}abc-\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}+4\,{c}^{4}{x}^{4}+8\,b{c}^{3}{x}^{3}+6\,{x}^{2}{b}^{2}{c}^{2}+2\,{b}^{3}cx-4\,{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{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{5}{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}{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{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}}, 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}}}{\left (d \left (b + 2 c x\right )\right )^{\frac{5}{2}}}\, 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}}}{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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