Optimal. Leaf size=357 \[ \frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{156 c^4 d^{15/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}+\frac{\sqrt{a+b x+c x^2}}{78 c^3 d^7 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}}-\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{156 c^4 d^{15/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac{\sqrt{a+b x+c x^2}}{156 c^3 d^5 (b d+2 c d x)^{5/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}} \]
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Rubi [A] time = 0.337042, antiderivative size = 357, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {684, 693, 691, 690, 307, 221, 1199, 424} \[ \frac{\sqrt{a+b x+c x^2}}{78 c^3 d^7 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}}+\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{156 c^4 d^{15/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}-\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{156 c^4 d^{15/2} \sqrt [4]{b^2-4 a c} \sqrt{a+b x+c x^2}}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac{\sqrt{a+b x+c x^2}}{156 c^3 d^5 (b d+2 c d x)^{5/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}} \]
Antiderivative was successfully verified.
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Rule 684
Rule 693
Rule 691
Rule 690
Rule 307
Rule 221
Rule 1199
Rule 424
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{5/2}}{(b d+2 c d x)^{15/2}} \, dx &=-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}}+\frac{5 \int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{11/2}} \, dx}{26 c d^2}\\ &=-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}}+\frac{5 \int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^{7/2}} \, dx}{156 c^2 d^4}\\ &=-\frac{\sqrt{a+b x+c x^2}}{156 c^3 d^5 (b d+2 c d x)^{5/2}}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}}+\frac{\int \frac{1}{(b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}} \, dx}{312 c^3 d^6}\\ &=-\frac{\sqrt{a+b x+c x^2}}{156 c^3 d^5 (b d+2 c d x)^{5/2}}+\frac{\sqrt{a+b x+c x^2}}{78 c^3 \left (b^2-4 a c\right ) d^7 \sqrt{b d+2 c d x}}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}}-\frac{\int \frac{\sqrt{b d+2 c d x}}{\sqrt{a+b x+c x^2}} \, dx}{312 c^3 \left (b^2-4 a c\right ) d^8}\\ &=-\frac{\sqrt{a+b x+c x^2}}{156 c^3 d^5 (b d+2 c d x)^{5/2}}+\frac{\sqrt{a+b x+c x^2}}{78 c^3 \left (b^2-4 a c\right ) d^7 \sqrt{b d+2 c d x}}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}}-\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \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}{312 c^3 \left (b^2-4 a c\right ) d^8 \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{a+b x+c x^2}}{156 c^3 d^5 (b d+2 c d x)^{5/2}}+\frac{\sqrt{a+b x+c x^2}}{78 c^3 \left (b^2-4 a c\right ) d^7 \sqrt{b d+2 c d x}}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}}-\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \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 )}{156 c^4 \left (b^2-4 a c\right ) d^9 \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{a+b x+c x^2}}{156 c^3 d^5 (b d+2 c d x)^{5/2}}+\frac{\sqrt{a+b x+c x^2}}{78 c^3 \left (b^2-4 a c\right ) d^7 \sqrt{b d+2 c d x}}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}}+\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 )}{156 c^4 \sqrt{b^2-4 a c} d^8 \sqrt{a+b x+c x^2}}-\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \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 )}{156 c^4 \sqrt{b^2-4 a c} d^8 \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{a+b x+c x^2}}{156 c^3 d^5 (b d+2 c d x)^{5/2}}+\frac{\sqrt{a+b x+c x^2}}{78 c^3 \left (b^2-4 a c\right ) d^7 \sqrt{b d+2 c d x}}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}}+\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 )}{156 c^4 \sqrt [4]{b^2-4 a c} d^{15/2} \sqrt{a+b x+c x^2}}-\frac{\sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \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 )}{156 c^4 \sqrt{b^2-4 a c} d^8 \sqrt{a+b x+c x^2}}\\ &=-\frac{\sqrt{a+b x+c x^2}}{156 c^3 d^5 (b d+2 c d x)^{5/2}}+\frac{\sqrt{a+b x+c x^2}}{78 c^3 \left (b^2-4 a c\right ) d^7 \sqrt{b d+2 c d x}}-\frac{5 \left (a+b x+c x^2\right )^{3/2}}{234 c^2 d^3 (b d+2 c d x)^{9/2}}-\frac{\left (a+b x+c x^2\right )^{5/2}}{13 c d (b d+2 c d x)^{13/2}}-\frac{\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 )}{156 c^4 \sqrt [4]{b^2-4 a c} d^{15/2} \sqrt{a+b x+c x^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 )}{156 c^4 \sqrt [4]{b^2-4 a c} d^{15/2} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.0876682, size = 109, normalized size = 0.31 \[ -\frac{\left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)} \sqrt{d (b+2 c x)} \, _2F_1\left (-\frac{13}{4},-\frac{5}{2};-\frac{9}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{416 c^3 d^8 (b+2 c x)^7 \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.33, size = 2125, normalized size = 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{15}{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}}{256 \, c^{8} d^{8} x^{8} + 1024 \, b c^{7} d^{8} x^{7} + 1792 \, b^{2} c^{6} d^{8} x^{6} + 1792 \, b^{3} c^{5} d^{8} x^{5} + 1120 \, b^{4} c^{4} d^{8} x^{4} + 448 \, b^{5} c^{3} d^{8} x^{3} + 112 \, b^{6} c^{2} d^{8} x^{2} + 16 \, b^{7} c d^{8} x + b^{8} d^{8}}, 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{5}{2}}}{{\left (2 \, c d x + b d\right )}^{\frac{15}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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