Optimal. Leaf size=207 \[ \frac{3 \sqrt{a+b x+c x^2}}{1024 c^2 d^9 \left (b^2-4 a c\right )^2 (b+2 c x)^2}+\frac{\sqrt{a+b x+c x^2}}{512 c^2 d^9 \left (b^2-4 a c\right ) (b+2 c x)^4}+\frac{3 \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{2048 c^{5/2} d^9 \left (b^2-4 a c\right )^{5/2}}-\frac{\sqrt{a+b x+c x^2}}{128 c^2 d^9 (b+2 c x)^6}-\frac{\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8} \]
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Rubi [A] time = 0.146302, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {684, 693, 688, 205} \[ \frac{3 \sqrt{a+b x+c x^2}}{1024 c^2 d^9 \left (b^2-4 a c\right )^2 (b+2 c x)^2}+\frac{\sqrt{a+b x+c x^2}}{512 c^2 d^9 \left (b^2-4 a c\right ) (b+2 c x)^4}+\frac{3 \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{2048 c^{5/2} d^9 \left (b^2-4 a c\right )^{5/2}}-\frac{\sqrt{a+b x+c x^2}}{128 c^2 d^9 (b+2 c x)^6}-\frac{\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8} \]
Antiderivative was successfully verified.
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Rule 684
Rule 693
Rule 688
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^9} \, dx &=-\frac{\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8}+\frac{3 \int \frac{\sqrt{a+b x+c x^2}}{(b d+2 c d x)^7} \, dx}{32 c d^2}\\ &=-\frac{\sqrt{a+b x+c x^2}}{128 c^2 d^9 (b+2 c x)^6}-\frac{\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8}+\frac{\int \frac{1}{(b d+2 c d x)^5 \sqrt{a+b x+c x^2}} \, dx}{256 c^2 d^4}\\ &=-\frac{\sqrt{a+b x+c x^2}}{128 c^2 d^9 (b+2 c x)^6}+\frac{\sqrt{a+b x+c x^2}}{512 c^2 \left (b^2-4 a c\right ) d^9 (b+2 c x)^4}-\frac{\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8}+\frac{3 \int \frac{1}{(b d+2 c d x)^3 \sqrt{a+b x+c x^2}} \, dx}{1024 c^2 \left (b^2-4 a c\right ) d^6}\\ &=-\frac{\sqrt{a+b x+c x^2}}{128 c^2 d^9 (b+2 c x)^6}+\frac{\sqrt{a+b x+c x^2}}{512 c^2 \left (b^2-4 a c\right ) d^9 (b+2 c x)^4}+\frac{3 \sqrt{a+b x+c x^2}}{1024 c^2 \left (b^2-4 a c\right )^2 d^9 (b+2 c x)^2}-\frac{\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8}+\frac{3 \int \frac{1}{(b d+2 c d x) \sqrt{a+b x+c x^2}} \, dx}{2048 c^2 \left (b^2-4 a c\right )^2 d^8}\\ &=-\frac{\sqrt{a+b x+c x^2}}{128 c^2 d^9 (b+2 c x)^6}+\frac{\sqrt{a+b x+c x^2}}{512 c^2 \left (b^2-4 a c\right ) d^9 (b+2 c x)^4}+\frac{3 \sqrt{a+b x+c x^2}}{1024 c^2 \left (b^2-4 a c\right )^2 d^9 (b+2 c x)^2}-\frac{\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{2 b^2 c d-8 a c^2 d+8 c^2 d x^2} \, dx,x,\sqrt{a+b x+c x^2}\right )}{512 c \left (b^2-4 a c\right )^2 d^8}\\ &=-\frac{\sqrt{a+b x+c x^2}}{128 c^2 d^9 (b+2 c x)^6}+\frac{\sqrt{a+b x+c x^2}}{512 c^2 \left (b^2-4 a c\right ) d^9 (b+2 c x)^4}+\frac{3 \sqrt{a+b x+c x^2}}{1024 c^2 \left (b^2-4 a c\right )^2 d^9 (b+2 c x)^2}-\frac{\left (a+b x+c x^2\right )^{3/2}}{16 c d^9 (b+2 c x)^8}+\frac{3 \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{2048 c^{5/2} \left (b^2-4 a c\right )^{5/2} d^9}\\ \end{align*}
Mathematica [C] time = 0.0297943, size = 62, normalized size = 0.3 \[ \frac{2 (a+x (b+c x))^{5/2} \, _2F_1\left (\frac{5}{2},5;\frac{7}{2};\frac{4 c (a+x (b+c x))}{4 a c-b^2}\right )}{5 d^9 \left (b^2-4 a c\right )^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.238, size = 742, normalized size = 3.6 \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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