Optimal. Leaf size=219 \[ \frac{5 \left (b^2-4 a c\right )^{9/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{84 c^4 d^{5/2} \sqrt{a+b x+c x^2}}-\frac{5 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{84 c^3 d^3}+\frac{5 \left (a+b x+c x^2\right )^{3/2} \sqrt{b d+2 c d x}}{42 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}} \]
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Rubi [A] time = 0.183386, antiderivative size = 219, 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 = {684, 685, 691, 689, 221} \[ -\frac{5 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \sqrt{b d+2 c d x}}{84 c^3 d^3}+\frac{5 \left (b^2-4 a c\right )^{9/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{84 c^4 d^{5/2} \sqrt{a+b x+c x^2}}+\frac{5 \left (a+b x+c x^2\right )^{3/2} \sqrt{b d+2 c d x}}{42 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 684
Rule 685
Rule 691
Rule 689
Rule 221
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{5/2}} \, dx &=-\frac{\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}}+\frac{5 \int \frac{\left (a+b x+c x^2\right )^{3/2}}{\sqrt{b d+2 c d x}} \, dx}{6 c d^2}\\ &=\frac{5 \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{42 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}}-\frac{\left (5 \left (b^2-4 a c\right )\right ) \int \frac{\sqrt{a+b x+c x^2}}{\sqrt{b d+2 c d x}} \, dx}{28 c^2 d^2}\\ &=-\frac{5 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{84 c^3 d^3}+\frac{5 \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{42 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}}+\frac{\left (5 \left (b^2-4 a c\right )^2\right ) \int \frac{1}{\sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}} \, dx}{168 c^3 d^2}\\ &=-\frac{5 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{84 c^3 d^3}+\frac{5 \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{42 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}}+\frac{\left (5 \left (b^2-4 a c\right )^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}{168 c^3 d^2 \sqrt{a+b x+c x^2}}\\ &=-\frac{5 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{84 c^3 d^3}+\frac{5 \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{42 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}}+\frac{\left (5 \left (b^2-4 a c\right )^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 )}{84 c^4 d^3 \sqrt{a+b x+c x^2}}\\ &=-\frac{5 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x} \sqrt{a+b x+c x^2}}{84 c^3 d^3}+\frac{5 \sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}}{42 c^2 d^3}-\frac{\left (a+b x+c x^2\right )^{5/2}}{3 c d (b d+2 c d x)^{3/2}}+\frac{5 \left (b^2-4 a c\right )^{9/4} \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 )}{84 c^4 d^{5/2} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0628769, size = 101, normalized size = 0.46 \[ -\frac{\left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)} \, _2F_1\left (-\frac{5}{2},-\frac{3}{4};\frac{1}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{96 c^3 d \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} (d (b+2 c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.245, size = 1000, normalized size = 4.6 \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{5}{2}}}{{\left (2 \, c d x + b d\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{{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}}{8 \, c^{3} d^{3} x^{3} + 12 \, b c^{2} d^{3} x^{2} + 6 \, b^{2} c d^{3} x + b^{3} d^{3}}, 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{\left (a + b x + c x^{2}\right )^{\frac{5}{2}}}{\left (d \left (b + 2 c x\right )\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{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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