Optimal. Leaf size=59 \[ \frac {4}{35} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}+\frac {2}{7} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{5/2} \]
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Rubi [A] time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, 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} \[ \frac {4}{35} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}+\frac {2}{7} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{5/2} \]
Antiderivative was successfully verified.
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Rule 629
Rule 692
Rubi steps
\begin {align*} \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac {2}{7} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{5/2}+\frac {1}{7} \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 )^{3/2} \, dx\\ &=\frac {4}{35} \left (b^2-4 a c\right ) d^3 \left (a+b x+c x^2\right )^{5/2}+\frac {2}{7} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 44, normalized size = 0.75 \[ \frac {2}{35} d^3 (a+x (b+c x))^{5/2} \left (4 c \left (5 c x^2-2 a\right )+7 b^2+20 b c x\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.02, size = 149, normalized size = 2.53 \[ \frac {2}{35} \, {\left (20 \, c^{4} d^{3} x^{6} + 60 \, b c^{3} d^{3} x^{5} + {\left (67 \, b^{2} c^{2} + 32 \, a c^{3}\right )} d^{3} x^{4} + 2 \, {\left (17 \, b^{3} c + 32 \, a b c^{2}\right )} d^{3} x^{3} + {\left (7 \, b^{4} + 46 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d^{3} x^{2} + 2 \, {\left (7 \, a b^{3} + 2 \, a^{2} b c\right )} d^{3} x + {\left (7 \, a^{2} b^{2} - 8 \, a^{3} c\right )} d^{3}\right )} \sqrt {c x^{2} + b x + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 58, normalized size = 0.98 \[ \frac {2}{5} \, {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} b^{2} d^{3} + \frac {8}{7} \, {\left (c x^{2} + b x + a\right )}^{\frac {7}{2}} c d^{3} - \frac {8}{5} \, {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} a c d^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 41, normalized size = 0.69 \[ -\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} \left (-20 c^{2} x^{2}-20 b c x +8 a c -7 b^{2}\right ) d^{3}}{35} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.75, size = 58, normalized size = 0.98 \[ \frac {8\,c\,d^3\,{\left (c\,x^2+b\,x+a\right )}^{7/2}}{7}+\frac {2\,b^2\,d^3\,{\left (c\,x^2+b\,x+a\right )}^{5/2}}{5}-\frac {8\,a\,c\,d^3\,{\left (c\,x^2+b\,x+a\right )}^{5/2}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.52, size = 371, normalized size = 6.29 \[ - \frac {16 a^{3} c d^{3} \sqrt {a + b x + c x^{2}}}{35} + \frac {2 a^{2} b^{2} d^{3} \sqrt {a + b x + c x^{2}}}{5} + \frac {8 a^{2} b c d^{3} x \sqrt {a + b x + c x^{2}}}{35} + \frac {8 a^{2} c^{2} d^{3} x^{2} \sqrt {a + b x + c x^{2}}}{35} + \frac {4 a b^{3} d^{3} x \sqrt {a + b x + c x^{2}}}{5} + \frac {92 a b^{2} c d^{3} x^{2} \sqrt {a + b x + c x^{2}}}{35} + \frac {128 a b c^{2} d^{3} x^{3} \sqrt {a + b x + c x^{2}}}{35} + \frac {64 a c^{3} d^{3} x^{4} \sqrt {a + b x + c x^{2}}}{35} + \frac {2 b^{4} d^{3} x^{2} \sqrt {a + b x + c x^{2}}}{5} + \frac {68 b^{3} c d^{3} x^{3} \sqrt {a + b x + c x^{2}}}{35} + \frac {134 b^{2} c^{2} d^{3} x^{4} \sqrt {a + b x + c x^{2}}}{35} + \frac {24 b c^{3} d^{3} x^{5} \sqrt {a + b x + c x^{2}}}{7} + \frac {8 c^{4} d^{3} x^{6} \sqrt {a + b x + c x^{2}}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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