Optimal. Leaf size=144 \[ \frac{2 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{3 c d^{5/2} \left (b^2-4 a c\right )^{3/4} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt{a+b x+c x^2}}{3 d \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}} \]
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Rubi [A] time = 0.116817, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {693, 691, 689, 221} \[ \frac{2 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{3 c d^{5/2} \left (b^2-4 a c\right )^{3/4} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt{a+b x+c x^2}}{3 d \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}} \]
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)^{5/2} \sqrt{a+b x+c x^2}} \, dx &=\frac{4 \sqrt{a+b x+c x^2}}{3 \left (b^2-4 a c\right ) d (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}{3 \left (b^2-4 a c\right ) d^2}\\ &=\frac{4 \sqrt{a+b x+c x^2}}{3 \left (b^2-4 a c\right ) d (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}{3 \left (b^2-4 a c\right ) d^2 \sqrt{a+b x+c x^2}}\\ &=\frac{4 \sqrt{a+b x+c x^2}}{3 \left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2}}+\frac{\left (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 )}{3 c \left (b^2-4 a c\right ) d^3 \sqrt{a+b x+c x^2}}\\ &=\frac{4 \sqrt{a+b x+c x^2}}{3 \left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2}}+\frac{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 )}{3 c \left (b^2-4 a c\right )^{3/4} d^{5/2} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0512377, size = 91, normalized size = 0.63 \[ -\frac{2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left (-\frac{3}{4},\frac{1}{2};\frac{1}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{3 c d \sqrt{a+x (b+c x)} (d (b+2 c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.217, size = 362, normalized size = 2.5 \begin{align*} -{\frac{1}{3\,{d}^{3} \left ( 2\,{c}^{2}{x}^{3}+3\,bc{x}^{2}+2\,acx+{b}^{2}x+ab \right ) \left ( 4\,ac-{b}^{2} \right ) c \left ( 2\,cx+b \right ) }\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a} \left ( 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 ) xc+\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+4\,{c}^{2}{x}^{2}+4\,bcx+4\,ac \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{5}{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}}{8 \, c^{4} d^{3} x^{5} + 20 \, b c^{3} d^{3} x^{4} + a b^{3} d^{3} + 2 \,{\left (9 \, b^{2} c^{2} + 4 \, a c^{3}\right )} d^{3} x^{3} +{\left (7 \, b^{3} c + 12 \, a b c^{2}\right )} d^{3} x^{2} +{\left (b^{4} + 6 \, a b^{2} c\right )} d^{3} x}, 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{1}{\left (d \left (b + 2 c x\right )\right )^{\frac{5}{2}} \sqrt{a + b x + c x^{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{1}{{\left (2 \, c d x + b d\right )}^{\frac{5}{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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