Optimal. Leaf size=79 \[ \frac{4 \left (a+b x+c x^2\right )^{5/2}}{35 d^8 \left (b^2-4 a c\right )^2 (b+2 c x)^5}+\frac{2 \left (a+b x+c x^2\right )^{5/2}}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7} \]
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Rubi [A] time = 0.0348012, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {693, 682} \[ \frac{4 \left (a+b x+c x^2\right )^{5/2}}{35 d^8 \left (b^2-4 a c\right )^2 (b+2 c x)^5}+\frac{2 \left (a+b x+c x^2\right )^{5/2}}{7 d^8 \left (b^2-4 a c\right ) (b+2 c x)^7} \]
Antiderivative was successfully verified.
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Rule 693
Rule 682
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^8} \, dx &=\frac{2 \left (a+b x+c x^2\right )^{5/2}}{7 \left (b^2-4 a c\right ) d^8 (b+2 c x)^7}+\frac{2 \int \frac{\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^6} \, dx}{7 \left (b^2-4 a c\right ) d^2}\\ &=\frac{2 \left (a+b x+c x^2\right )^{5/2}}{7 \left (b^2-4 a c\right ) d^8 (b+2 c x)^7}+\frac{4 \left (a+b x+c x^2\right )^{5/2}}{35 \left (b^2-4 a c\right )^2 d^8 (b+2 c x)^5}\\ \end{align*}
Mathematica [A] time = 0.0367531, size = 62, normalized size = 0.78 \[ \frac{2 (a+x (b+c x))^{5/2} \left (4 c \left (2 c x^2-5 a\right )+7 b^2+8 b c x\right )}{35 d^8 \left (b^2-4 a c\right )^2 (b+2 c x)^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 70, normalized size = 0.9 \begin{align*} -{\frac{-16\,{c}^{2}{x}^{2}-16\,bcx+40\,ac-14\,{b}^{2}}{35\, \left ( 2\,cx+b \right ) ^{7}{d}^{8} \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) } \left ( c{x}^{2}+bx+a \right ) ^{{\frac{5}{2}}}} \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 [B] time = 119.981, size = 841, normalized size = 10.65 \begin{align*} \frac{2 \,{\left (8 \, c^{4} x^{6} + 24 \, b c^{3} x^{5} +{\left (31 \, b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + 7 \, a^{2} b^{2} - 20 \, a^{3} c + 2 \,{\left (11 \, b^{3} c - 4 \, a b c^{2}\right )} x^{3} +{\left (7 \, b^{4} + 10 \, a b^{2} c - 32 \, a^{2} c^{2}\right )} x^{2} + 2 \,{\left (7 \, a b^{3} - 16 \, a^{2} b c\right )} x\right )} \sqrt{c x^{2} + b x + a}}{35 \,{\left (128 \,{\left (b^{4} c^{7} - 8 \, a b^{2} c^{8} + 16 \, a^{2} c^{9}\right )} d^{8} x^{7} + 448 \,{\left (b^{5} c^{6} - 8 \, a b^{3} c^{7} + 16 \, a^{2} b c^{8}\right )} d^{8} x^{6} + 672 \,{\left (b^{6} c^{5} - 8 \, a b^{4} c^{6} + 16 \, a^{2} b^{2} c^{7}\right )} d^{8} x^{5} + 560 \,{\left (b^{7} c^{4} - 8 \, a b^{5} c^{5} + 16 \, a^{2} b^{3} c^{6}\right )} d^{8} x^{4} + 280 \,{\left (b^{8} c^{3} - 8 \, a b^{6} c^{4} + 16 \, a^{2} b^{4} c^{5}\right )} d^{8} x^{3} + 84 \,{\left (b^{9} c^{2} - 8 \, a b^{7} c^{3} + 16 \, a^{2} b^{5} c^{4}\right )} d^{8} x^{2} + 14 \,{\left (b^{10} c - 8 \, a b^{8} c^{2} + 16 \, a^{2} b^{6} c^{3}\right )} d^{8} x +{\left (b^{11} - 8 \, a b^{9} c + 16 \, a^{2} b^{7} c^{2}\right )} d^{8}\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 [B] time = 2.00361, size = 1354, normalized size = 17.14 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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