Optimal. Leaf size=258 \[ -\frac{84 d^{9/2} \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{b d+2 c d x}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right ),-1\right )}{5 \sqrt{a+b x+c x^2}}+\frac{84 d^{9/2} \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 \sqrt{a+b x+c x^2}}+\frac{56}{5} c d^3 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}-\frac{2 d (b d+2 c d x)^{7/2}}{\sqrt{a+b x+c x^2}} \]
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Rubi [A] time = 0.246077, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {686, 692, 691, 690, 307, 221, 1199, 424} \[ -\frac{84 d^{9/2} \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 \sqrt{a+b x+c x^2}}+\frac{84 d^{9/2} \left (b^2-4 a c\right )^{7/4} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt{d}}\right )\right |-1\right )}{5 \sqrt{a+b x+c x^2}}+\frac{56}{5} c d^3 \sqrt{a+b x+c x^2} (b d+2 c d x)^{3/2}-\frac{2 d (b d+2 c d x)^{7/2}}{\sqrt{a+b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 686
Rule 692
Rule 691
Rule 690
Rule 307
Rule 221
Rule 1199
Rule 424
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 d (b d+2 c d x)^{7/2}}{\sqrt{a+b x+c x^2}}+\left (14 c d^2\right ) \int \frac{(b d+2 c d x)^{5/2}}{\sqrt{a+b x+c x^2}} \, dx\\ &=-\frac{2 d (b d+2 c d x)^{7/2}}{\sqrt{a+b x+c x^2}}+\frac{56}{5} c d^3 (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}+\frac{1}{5} \left (42 c \left (b^2-4 a c\right ) d^4\right ) \int \frac{\sqrt{b d+2 c d x}}{\sqrt{a+b x+c x^2}} \, dx\\ &=-\frac{2 d (b d+2 c d x)^{7/2}}{\sqrt{a+b x+c x^2}}+\frac{56}{5} c d^3 (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}+\frac{\left (42 c \left (b^2-4 a c\right ) d^4 \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}{5 \sqrt{a+b x+c x^2}}\\ &=-\frac{2 d (b d+2 c d x)^{7/2}}{\sqrt{a+b x+c x^2}}+\frac{56}{5} c d^3 (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}+\frac{\left (84 \left (b^2-4 a c\right ) d^3 \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 )}{5 \sqrt{a+b x+c x^2}}\\ &=-\frac{2 d (b d+2 c d x)^{7/2}}{\sqrt{a+b x+c x^2}}+\frac{56}{5} c d^3 (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}-\frac{\left (84 \left (b^2-4 a c\right )^{3/2} d^4 \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 )}{5 \sqrt{a+b x+c x^2}}+\frac{\left (84 \left (b^2-4 a c\right )^{3/2} d^4 \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 )}{5 \sqrt{a+b x+c x^2}}\\ &=-\frac{2 d (b d+2 c d x)^{7/2}}{\sqrt{a+b x+c x^2}}+\frac{56}{5} c d^3 (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}-\frac{84 \left (b^2-4 a c\right )^{7/4} d^{9/2} \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 )}{5 \sqrt{a+b x+c x^2}}+\frac{\left (84 \left (b^2-4 a c\right )^{3/2} d^4 \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 )}{5 \sqrt{a+b x+c x^2}}\\ &=-\frac{2 d (b d+2 c d x)^{7/2}}{\sqrt{a+b x+c x^2}}+\frac{56}{5} c d^3 (b d+2 c d x)^{3/2} \sqrt{a+b x+c x^2}+\frac{84 \left (b^2-4 a c\right )^{7/4} d^{9/2} \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 )}{5 \sqrt{a+b x+c x^2}}-\frac{84 \left (b^2-4 a c\right )^{7/4} d^{9/2} \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 )}{5 \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 0.146354, size = 122, normalized size = 0.47 \[ -\frac{8 d^3 (d (b+2 c x))^{3/2} \left (7 \left (b^2-4 a c\right ) \sqrt{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left (\frac{3}{4},\frac{3}{2};\frac{7}{4};\frac{(b+2 c x)^2}{b^2-4 a c}\right )-2 \left (c \left (c x^2-7 a\right )+2 b^2+b c x\right )\right )}{5 \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.277, size = 498, normalized size = 1.9 \begin{align*} -{\frac{2\,{d}^{4}}{10\,{c}^{2}{x}^{3}+15\,bc{x}^{2}+10\,acx+5\,{b}^{2}x+5\,ab} \left ( 336\,\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 EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){a}^{2}{c}^{2}-168\,\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 EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ) a{b}^{2}c+21\,\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 EllipticE} \left ( 1/2\,\sqrt{{\frac{b+2\,cx+\sqrt{-4\,ac+{b}^{2}}}{\sqrt{-4\,ac+{b}^{2}}}}}\sqrt{2},\sqrt{2} \right ){b}^{4}-32\,{c}^{4}{x}^{4}-64\,b{c}^{3}{x}^{3}-112\,{x}^{2}a{c}^{3}-20\,{x}^{2}{b}^{2}{c}^{2}-112\,ba{c}^{2}x+12\,{b}^{3}cx-28\,ac{b}^{2}+5\,{b}^{4} \right ) \sqrt{c{x}^{2}+bx+a}\sqrt{d \left ( 2\,cx+b \right ) }} \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 (2 \, c d x + b d\right )}^{\frac{9}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{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 (16 \, c^{4} d^{4} x^{4} + 32 \, b c^{3} d^{4} x^{3} + 24 \, b^{2} c^{2} d^{4} x^{2} + 8 \, b^{3} c d^{4} x + b^{4} d^{4}\right )} \sqrt{2 \, c d x + b d} \sqrt{c x^{2} + b x + a}}{c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{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{{\left (2 \, c d x + b d\right )}^{\frac{9}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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