Optimal. Leaf size=228 \[ \frac{40 c \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{d^{5/2} \left (b^2-4 a c\right )^{11/4} \sqrt{a+b x+c x^2}}+\frac{80 c^2 \sqrt{a+b x+c x^2}}{d \left (b^2-4 a c\right )^3 (b d+2 c d x)^{3/2}}+\frac{12 c}{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}-\frac{2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}} \]
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Rubi [A] time = 0.176612, antiderivative size = 228, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {687, 693, 691, 689, 221} \[ \frac{80 c^2 \sqrt{a+b x+c x^2}}{d \left (b^2-4 a c\right )^3 (b d+2 c d x)^{3/2}}+\frac{40 c \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{d^{5/2} \left (b^2-4 a c\right )^{11/4} \sqrt{a+b x+c x^2}}+\frac{12 c}{d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}}-\frac{2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} (b d+2 c d x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 687
Rule 693
Rule 691
Rule 689
Rule 221
Rubi steps
\begin{align*} \int \frac{1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2}{3 \left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}-\frac{(6 c) \int \frac{1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^{3/2}} \, dx}{b^2-4 a c}\\ &=-\frac{2}{3 \left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}+\frac{12 c}{\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}}+\frac{\left (60 c^2\right ) \int \frac{1}{(b d+2 c d x)^{5/2} \sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\frac{2}{3 \left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}+\frac{12 c}{\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}}+\frac{80 c^2 \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2}}+\frac{\left (20 c^2\right ) \int \frac{1}{\sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right )^3 d^2}\\ &=-\frac{2}{3 \left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}+\frac{12 c}{\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}}+\frac{80 c^2 \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2}}+\frac{\left (20 c^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}{\left (b^2-4 a c\right )^3 d^2 \sqrt{a+b x+c x^2}}\\ &=-\frac{2}{3 \left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}+\frac{12 c}{\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}}+\frac{80 c^2 \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2}}+\frac{\left (40 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 d^3 \sqrt{a+b x+c x^2}}\\ &=-\frac{2}{3 \left (b^2-4 a c\right ) d (b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}+\frac{12 c}{\left (b^2-4 a c\right )^2 d (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}}+\frac{80 c^2 \sqrt{a+b x+c x^2}}{\left (b^2-4 a c\right )^3 d (b d+2 c d x)^{3/2}}+\frac{40 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 d x}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{\left (b^2-4 a c\right )^{11/4} d^{5/2} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.07151, size = 99, normalized size = 0.43 \[ -\frac{32 c \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left (-\frac{3}{4},\frac{5}{2};\frac{1}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{3 d \left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)} (d (b+2 c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.24, size = 797, normalized size = 3.5 \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{1}{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}{\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}}{8 \, c^{6} d^{3} x^{9} + 36 \, b c^{5} d^{3} x^{8} + 6 \,{\left (11 \, b^{2} c^{4} + 4 \, a c^{5}\right )} d^{3} x^{7} + 21 \,{\left (3 \, b^{3} c^{3} + 4 \, a b c^{4}\right )} d^{3} x^{6} + a^{3} b^{3} d^{3} + 3 \,{\left (11 \, b^{4} c^{2} + 38 \, a b^{2} c^{3} + 8 \, a^{2} c^{4}\right )} d^{3} x^{5} + 3 \,{\left (3 \, b^{5} c + 25 \, a b^{3} c^{2} + 20 \, a^{2} b c^{3}\right )} d^{3} x^{4} +{\left (b^{6} + 24 \, a b^{4} c + 54 \, a^{2} b^{2} c^{2} + 8 \, a^{3} c^{3}\right )} d^{3} x^{3} + 3 \,{\left (a b^{5} + 7 \, a^{2} b^{3} c + 4 \, a^{3} b c^{2}\right )} d^{3} x^{2} + 3 \,{\left (a^{2} b^{4} + 2 \, a^{3} 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}} \left (a + b x + c x^{2}\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{1}{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}{\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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