Optimal. Leaf size=59 \[ \frac{4}{63} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}+\frac{2}{9} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{7/2} \]
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Rubi [A] time = 0.0264323, antiderivative size = 59, 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 = {692, 629} \[ \frac{4}{63} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}+\frac{2}{9} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{7/2} \]
Antiderivative was successfully verified.
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Rule 692
Rule 629
Rubi steps
\begin{align*} \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx &=\frac{2}{9} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{7/2}+\frac{1}{9} \left (2 \left (b^2-4 a c\right ) d^2\right ) \int (b d+2 c d x) \left (a+b x+c x^2\right )^{5/2} \, dx\\ &=\frac{4}{63} \left (b^2-4 a c\right ) d^3 \left (a+b x+c x^2\right )^{7/2}+\frac{2}{9} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0614551, size = 44, normalized size = 0.75 \[ \frac{2}{63} d^3 (a+x (b+c x))^{7/2} \left (4 c \left (7 c x^2-2 a\right )+9 b^2+28 b c x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 41, normalized size = 0.7 \begin{align*} -{\frac{ \left ( -56\,{c}^{2}{x}^{2}-56\,bcx+16\,ac-18\,{b}^{2} \right ){d}^{3}}{63} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{7}{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 = 3.11147, size = 467, normalized size = 7.92 \begin{align*} \frac{2}{63} \,{\left (28 \, c^{5} d^{3} x^{8} + 112 \, b c^{4} d^{3} x^{7} +{\left (177 \, b^{2} c^{3} + 76 \, a c^{4}\right )} d^{3} x^{6} +{\left (139 \, b^{3} c^{2} + 228 \, a b c^{3}\right )} d^{3} x^{5} + 5 \,{\left (11 \, b^{4} c + 51 \, a b^{2} c^{2} + 12 \, a^{2} c^{3}\right )} d^{3} x^{4} +{\left (9 \, b^{5} + 130 \, a b^{3} c + 120 \, a^{2} b c^{2}\right )} d^{3} x^{3} +{\left (27 \, a b^{4} + 87 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} d^{3} x^{2} +{\left (27 \, a^{2} b^{3} + 4 \, a^{3} b c\right )} d^{3} x +{\left (9 \, a^{3} b^{2} - 8 \, a^{4} c\right )} d^{3}\right )} \sqrt{c x^{2} + b x + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 7.86453, size = 559, normalized size = 9.47 \begin{align*} - \frac{16 a^{4} c d^{3} \sqrt{a + b x + c x^{2}}}{63} + \frac{2 a^{3} b^{2} d^{3} \sqrt{a + b x + c x^{2}}}{7} + \frac{8 a^{3} b c d^{3} x \sqrt{a + b x + c x^{2}}}{63} + \frac{8 a^{3} c^{2} d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{63} + \frac{6 a^{2} b^{3} d^{3} x \sqrt{a + b x + c x^{2}}}{7} + \frac{58 a^{2} b^{2} c d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{21} + \frac{80 a^{2} b c^{2} d^{3} x^{3} \sqrt{a + b x + c x^{2}}}{21} + \frac{40 a^{2} c^{3} d^{3} x^{4} \sqrt{a + b x + c x^{2}}}{21} + \frac{6 a b^{4} d^{3} x^{2} \sqrt{a + b x + c x^{2}}}{7} + \frac{260 a b^{3} c d^{3} x^{3} \sqrt{a + b x + c x^{2}}}{63} + \frac{170 a b^{2} c^{2} d^{3} x^{4} \sqrt{a + b x + c x^{2}}}{21} + \frac{152 a b c^{3} d^{3} x^{5} \sqrt{a + b x + c x^{2}}}{21} + \frac{152 a c^{4} d^{3} x^{6} \sqrt{a + b x + c x^{2}}}{63} + \frac{2 b^{5} d^{3} x^{3} \sqrt{a + b x + c x^{2}}}{7} + \frac{110 b^{4} c d^{3} x^{4} \sqrt{a + b x + c x^{2}}}{63} + \frac{278 b^{3} c^{2} d^{3} x^{5} \sqrt{a + b x + c x^{2}}}{63} + \frac{118 b^{2} c^{3} d^{3} x^{6} \sqrt{a + b x + c x^{2}}}{21} + \frac{32 b c^{4} d^{3} x^{7} \sqrt{a + b x + c x^{2}}}{9} + \frac{8 c^{5} d^{3} x^{8} \sqrt{a + b x + c x^{2}}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17485, size = 389, normalized size = 6.59 \begin{align*} \frac{2}{63} \, \sqrt{c x^{2} + b x + a}{\left ({\left ({\left ({\left ({\left ({\left ({\left (28 \,{\left (c^{5} d^{3} x + 4 \, b c^{4} d^{3}\right )} x + \frac{177 \, b^{2} c^{11} d^{3} + 76 \, a c^{12} d^{3}}{c^{8}}\right )} x + \frac{139 \, b^{3} c^{10} d^{3} + 228 \, a b c^{11} d^{3}}{c^{8}}\right )} x + \frac{5 \,{\left (11 \, b^{4} c^{9} d^{3} + 51 \, a b^{2} c^{10} d^{3} + 12 \, a^{2} c^{11} d^{3}\right )}}{c^{8}}\right )} x + \frac{9 \, b^{5} c^{8} d^{3} + 130 \, a b^{3} c^{9} d^{3} + 120 \, a^{2} b c^{10} d^{3}}{c^{8}}\right )} x + \frac{27 \, a b^{4} c^{8} d^{3} + 87 \, a^{2} b^{2} c^{9} d^{3} + 4 \, a^{3} c^{10} d^{3}}{c^{8}}\right )} x + \frac{27 \, a^{2} b^{3} c^{8} d^{3} + 4 \, a^{3} b c^{9} d^{3}}{c^{8}}\right )} x + \frac{9 \, a^{3} b^{2} c^{8} d^{3} - 8 \, a^{4} c^{9} d^{3}}{c^{8}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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