Optimal. Leaf size=187 \[ \frac{40 c \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{3 \sqrt{d} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}}+\frac{20 c \sqrt{b d+2 c d x}}{3 d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{b d+2 c d x}}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.140025, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {687, 691, 689, 221} \[ \frac{20 c \sqrt{b d+2 c d x}}{3 d \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{b d+2 c d x}}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{40 c \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{3 \sqrt{d} \left (b^2-4 a c\right )^{7/4} \sqrt{a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 687
Rule 691
Rule 689
Rule 221
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 \sqrt{b d+2 c d x}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}-\frac{(10 c) \int \frac{1}{\sqrt{b d+2 c d x} \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac{2 \sqrt{b d+2 c d x}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac{20 c \sqrt{b d+2 c d x}}{3 \left (b^2-4 a c\right )^2 d \sqrt{a+b x+c x^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}{3 \left (b^2-4 a c\right )^2}\\ &=-\frac{2 \sqrt{b d+2 c d x}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac{20 c \sqrt{b d+2 c d x}}{3 \left (b^2-4 a c\right )^2 d \sqrt{a+b x+c x^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}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}\\ &=-\frac{2 \sqrt{b d+2 c d x}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac{20 c \sqrt{b d+2 c d x}}{3 \left (b^2-4 a c\right )^2 d \sqrt{a+b x+c x^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 )}{3 \left (b^2-4 a c\right )^2 d \sqrt{a+b x+c x^2}}\\ &=-\frac{2 \sqrt{b d+2 c d x}}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac{20 c \sqrt{b d+2 c d x}}{3 \left (b^2-4 a c\right )^2 d \sqrt{a+b x+c x^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 )}{3 \left (b^2-4 a c\right )^{7/4} \sqrt{d} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.167233, size = 132, normalized size = 0.71 \[ -\frac{2 \sqrt{d (b+2 c x)} \left (-20 c (a+x (b+c x)) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )-2 c \left (7 a+5 c x^2\right )+b^2-10 b c x\right )}{3 d \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.235, size = 491, normalized size = 2.6 \begin{align*}{\frac{2}{3\,d \left ( 2\,cx+b \right ) \left ( 4\,ac-{b}^{2} \right ) ^{2}}\sqrt{d \left ( 2\,cx+b \right ) } \left ( 10\,{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){x}^{2}{c}^{2}\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-4\,ac+{b}^{2}}+10\,{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) xbc\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-4\,ac+{b}^{2}}+10\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{-{\frac{2\,cx+b}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{{\frac{-b-2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}{\it EllipticF} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) \sqrt{-4\,ac+{b}^{2}}ac+20\,{c}^{3}{x}^{3}+30\,b{c}^{2}{x}^{2}+28\,xa{c}^{2}+8\,x{b}^{2}c+14\,abc-{b}^{3} \right ) \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \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}{\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}}{2 \, c^{4} d x^{7} + 7 \, b c^{3} d x^{6} + 3 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d x^{5} + 5 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d x^{4} + a^{3} b d +{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d x^{3} + 3 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d x^{2} +{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d 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}{\sqrt{d \left (b + 2 c x\right )} \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}{\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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