Optimal. Leaf size=125 \[ \frac{4 d^{3/2} \sqrt [4]{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{\sqrt{a+b x+c x^2}}-\frac{2 d \sqrt{b d+2 c d x}}{\sqrt{a+b x+c x^2}} \]
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Rubi [A] time = 0.113233, antiderivative size = 125, 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 = {686, 691, 689, 221} \[ \frac{4 d^{3/2} \sqrt [4]{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\sqrt{a+b x+c x^2}}-\frac{2 d \sqrt{b d+2 c d x}}{\sqrt{a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 686
Rule 691
Rule 689
Rule 221
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 d \sqrt{b d+2 c d x}}{\sqrt{a+b x+c x^2}}+\left (2 c d^2\right ) \int \frac{1}{\sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}} \, dx\\ &=-\frac{2 d \sqrt{b d+2 c d x}}{\sqrt{a+b x+c x^2}}+\frac{\left (2 c d^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}{\sqrt{a+b x+c x^2}}\\ &=-\frac{2 d \sqrt{b d+2 c d x}}{\sqrt{a+b x+c x^2}}+\frac{\left (4 d \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 )}{\sqrt{a+b x+c x^2}}\\ &=-\frac{2 d \sqrt{b d+2 c d x}}{\sqrt{a+b x+c x^2}}+\frac{4 \sqrt [4]{b^2-4 a c} d^{3/2} \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 )}{\sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0651446, size = 88, normalized size = 0.7 \[ \frac{2 d \sqrt{d (b+2 c x)} \left (2 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )-1\right )}{\sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.282, size = 194, normalized size = 1.6 \begin{align*} 2\,{\frac{\sqrt{d \left ( 2\,cx+b \right ) }\sqrt{c{x}^{2}+bx+a}d}{2\,{c}^{2}{x}^{3}+3\,bc{x}^{2}+2\,acx+{b}^{2}x+ab} \left ({\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}}\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}}}}}-2\,cx-b \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{3}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{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{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} \sqrt{c x^{2} + b x + a}}{c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}}, 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{3}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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