Optimal. Leaf size=268 \[ \frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{462 c^3 d^{17/2} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2}}{231 c^2 d^7 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}+\frac{\sqrt{a+b x+c x^2}}{385 c^2 d^5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}-\frac{\sqrt{a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}} \]
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Rubi [A] time = 0.218567, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {684, 693, 691, 689, 221} \[ \frac{\sqrt{a+b x+c x^2}}{231 c^2 d^7 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}+\frac{\sqrt{a+b x+c x^2}}{385 c^2 d^5 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}+\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{462 c^3 d^{17/2} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}}-\frac{\sqrt{a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}} \]
Antiderivative was successfully verified.
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Rule 684
Rule 693
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)^{17/2}} \, dx &=-\frac{\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}}+\frac{\int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^{13/2}} \, dx}{10 c d^2}\\ &=-\frac{\sqrt{a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}}+\frac{\int \frac{1}{(b d+2 c d x)^{9/2} \sqrt{a+b x+c x^2}} \, dx}{220 c^2 d^4}\\ &=-\frac{\sqrt{a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}+\frac{\sqrt{a+b x+c x^2}}{385 c^2 \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{7/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}}+\frac{\int \frac{1}{(b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}} \, dx}{308 c^2 \left (b^2-4 a c\right ) d^6}\\ &=-\frac{\sqrt{a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}+\frac{\sqrt{a+b x+c x^2}}{385 c^2 \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{7/2}}+\frac{\sqrt{a+b x+c x^2}}{231 c^2 \left (b^2-4 a c\right )^2 d^7 (b d+2 c d x)^{3/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}}+\frac{\int \frac{1}{\sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}} \, dx}{924 c^2 \left (b^2-4 a c\right )^2 d^8}\\ &=-\frac{\sqrt{a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}+\frac{\sqrt{a+b x+c x^2}}{385 c^2 \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{7/2}}+\frac{\sqrt{a+b x+c x^2}}{231 c^2 \left (b^2-4 a c\right )^2 d^7 (b d+2 c d x)^{3/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}}+\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}{924 c^2 \left (b^2-4 a c\right )^2 d^8 \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}+\frac{\sqrt{a+b x+c x^2}}{385 c^2 \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{7/2}}+\frac{\sqrt{a+b x+c x^2}}{231 c^2 \left (b^2-4 a c\right )^2 d^7 (b d+2 c d x)^{3/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}}+\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \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 )}{462 c^3 \left (b^2-4 a c\right )^2 d^9 \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{a+b x+c x^2}}{110 c^2 d^3 (b d+2 c d x)^{11/2}}+\frac{\sqrt{a+b x+c x^2}}{385 c^2 \left (b^2-4 a c\right ) d^5 (b d+2 c d x)^{7/2}}+\frac{\sqrt{a+b x+c x^2}}{231 c^2 \left (b^2-4 a c\right )^2 d^7 (b d+2 c d x)^{3/2}}-\frac{\left (a+b x+c x^2\right )^{3/2}}{15 c d (b d+2 c d x)^{15/2}}+\frac{\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 )}{462 c^3 \left (b^2-4 a c\right )^{7/4} d^{17/2} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0937255, size = 107, normalized size = 0.4 \[ \frac{\left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \, _2F_1\left (-\frac{15}{4},-\frac{3}{2};-\frac{11}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{120 c^2 d^9 (b+2 c x)^8 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.284, size = 1431, normalized size = 5.3 \begin{align*} \text{result too large to display} \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{17}{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}}}{512 \, c^{9} d^{9} x^{9} + 2304 \, b c^{8} d^{9} x^{8} + 4608 \, b^{2} c^{7} d^{9} x^{7} + 5376 \, b^{3} c^{6} d^{9} x^{6} + 4032 \, b^{4} c^{5} d^{9} x^{5} + 2016 \, b^{5} c^{4} d^{9} x^{4} + 672 \, b^{6} c^{3} d^{9} x^{3} + 144 \, b^{7} c^{2} d^{9} x^{2} + 18 \, b^{8} c d^{9} x + b^{9} d^{9}}, 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 (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{17}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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