Optimal. Leaf size=278 \[ \frac{8 c \sqrt{d} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{\left (b^2-4 a c\right )^{5/4} \sqrt{a+b x+c x^2}}+\frac{4 c (b d+2 c d x)^{3/2}}{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (b d+2 c d x)^{3/2}}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{8 c \sqrt{d} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\left (b^2-4 a c\right )^{5/4} \sqrt{a+b x+c x^2}} \]
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Rubi [A] time = 0.247288, antiderivative size = 278, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {687, 691, 690, 307, 221, 1199, 424} \[ \frac{4 c (b d+2 c d x)^{3/2}}{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (b d+2 c d x)^{3/2}}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{8 c \sqrt{d} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\left (b^2-4 a c\right )^{5/4} \sqrt{a+b x+c x^2}}-\frac{8 c \sqrt{d} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\left (b^2-4 a c\right )^{5/4} \sqrt{a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 687
Rule 691
Rule 690
Rule 307
Rule 221
Rule 1199
Rule 424
Rubi steps
\begin{align*} \int \frac{\sqrt{b d+2 c d x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 (b d+2 c d x)^{3/2}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}-\frac{(2 c) \int \frac{\sqrt{b d+2 c d x}}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{b^2-4 a c}\\ &=-\frac{2 (b d+2 c d x)^{3/2}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac{4 c (b d+2 c d x)^{3/2}}{\left (b^2-4 a c\right )^2 d \sqrt{a+b x+c x^2}}-\frac{\left (4 c^2\right ) \int \frac{\sqrt{b d+2 c d x}}{\sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\frac{2 (b d+2 c d x)^{3/2}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac{4 c (b d+2 c d x)^{3/2}}{\left (b^2-4 a c\right )^2 d \sqrt{a+b x+c x^2}}-\frac{\left (4 c^2 \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \int \frac{\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}{\left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}\\ &=-\frac{2 (b d+2 c d x)^{3/2}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac{4 c (b d+2 c d x)^{3/2}}{\left (b^2-4 a c\right )^2 d \sqrt{a+b x+c x^2}}-\frac{\left (8 c \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-\frac{x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt{b d+2 c d x}\right )}{\left (b^2-4 a c\right )^2 d \sqrt{a+b x+c x^2}}\\ &=-\frac{2 (b d+2 c d x)^{3/2}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac{4 c (b d+2 c d x)^{3/2}}{\left (b^2-4 a c\right )^2 d \sqrt{a+b x+c x^2}}+\frac{\left (8 c \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 )}{\left (b^2-4 a c\right )^{3/2} \sqrt{a+b x+c x^2}}-\frac{\left (8 c \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1+\frac{x^2}{\sqrt{b^2-4 a c} d}}{\sqrt{1-\frac{x^4}{\left (b^2-4 a c\right ) d^2}}} \, dx,x,\sqrt{b d+2 c d x}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{a+b x+c x^2}}\\ &=-\frac{2 (b d+2 c d x)^{3/2}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac{4 c (b d+2 c d x)^{3/2}}{\left (b^2-4 a c\right )^2 d \sqrt{a+b x+c x^2}}+\frac{8 c \sqrt{d} \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 )}{\left (b^2-4 a c\right )^{5/4} \sqrt{a+b x+c x^2}}-\frac{\left (8 c \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x^2}{\sqrt{b^2-4 a c} d}}}{\sqrt{1-\frac{x^2}{\sqrt{b^2-4 a c} d}}} \, dx,x,\sqrt{b d+2 c d x}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{a+b x+c x^2}}\\ &=-\frac{2 (b d+2 c d x)^{3/2}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac{4 c (b d+2 c d x)^{3/2}}{\left (b^2-4 a c\right )^2 d \sqrt{a+b x+c x^2}}-\frac{8 c \sqrt{d} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\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 )}{\left (b^2-4 a c\right )^{5/4} \sqrt{a+b x+c x^2}}+\frac{8 c \sqrt{d} \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 )}{\left (b^2-4 a c\right )^{5/4} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0589341, size = 99, normalized size = 0.36 \[ \frac{32 c \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^{3/2} \, _2F_1\left (\frac{3}{4},\frac{5}{2};\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{3 d \left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.217, size = 866, normalized size = 3.1 \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{\sqrt{2 \, c d x + b d}}{{\left (c x^{2} + b x + a\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} \sqrt{c x^{2} + b x + a}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \,{\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x +{\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} x^{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{\sqrt{2 \, c d x + b d}}{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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