Optimal. Leaf size=121 \[ -\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}{448 c^4 d^5}+\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}{64 c^4 d^3}+\frac {\left (b^2-4 a c\right )^3}{64 c^4 d \sqrt {b d+2 c d x}}+\frac {(b d+2 c d x)^{11/2}}{704 c^4 d^7} \]
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Rubi [A] time = 0.05, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {683} \[ -\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}{448 c^4 d^5}+\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}{64 c^4 d^3}+\frac {\left (b^2-4 a c\right )^3}{64 c^4 d \sqrt {b d+2 c d x}}+\frac {(b d+2 c d x)^{11/2}}{704 c^4 d^7} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{3/2}} \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^3}{64 c^3 (b d+2 c d x)^{3/2}}+\frac {3 \left (-b^2+4 a c\right )^2 \sqrt {b d+2 c d x}}{64 c^3 d^2}+\frac {3 \left (-b^2+4 a c\right ) (b d+2 c d x)^{5/2}}{64 c^3 d^4}+\frac {(b d+2 c d x)^{9/2}}{64 c^3 d^6}\right ) \, dx\\ &=\frac {\left (b^2-4 a c\right )^3}{64 c^4 d \sqrt {b d+2 c d x}}+\frac {\left (b^2-4 a c\right )^2 (b d+2 c d x)^{3/2}}{64 c^4 d^3}-\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{7/2}}{448 c^4 d^5}+\frac {(b d+2 c d x)^{11/2}}{704 c^4 d^7}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 83, normalized size = 0.69 \[ \frac {-33 \left (b^2-4 a c\right ) (b+2 c x)^4+77 \left (b^2-4 a c\right )^2 (b+2 c x)^2+77 \left (b^2-4 a c\right )^3+7 (b+2 c x)^6}{4928 c^4 d \sqrt {d (b+2 c x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 178, normalized size = 1.47 \[ \frac {{\left (7 \, c^{6} x^{6} + 21 \, b c^{5} x^{5} + 2 \, b^{6} - 22 \, a b^{4} c + 77 \, a^{2} b^{2} c^{2} - 77 \, a^{3} c^{3} + 3 \, {\left (6 \, b^{2} c^{4} + 11 \, a c^{5}\right )} x^{4} + {\left (b^{3} c^{3} + 66 \, a b c^{4}\right )} x^{3} - {\left (b^{4} c^{2} - 11 \, a b^{2} c^{3} - 77 \, a^{2} c^{4}\right )} x^{2} + {\left (2 \, b^{5} c - 22 \, a b^{3} c^{2} + 77 \, a^{2} b c^{3}\right )} x\right )} \sqrt {2 \, c d x + b d}}{77 \, {\left (2 \, c^{5} d^{2} x + b c^{4} d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 187, normalized size = 1.55 \[ \frac {b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}{64 \, \sqrt {2 \, c d x + b d} c^{4} d} + \frac {77 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{4} c^{40} d^{74} - 616 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} a b^{2} c^{41} d^{74} + 1232 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} a^{2} c^{42} d^{74} - 33 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} b^{2} c^{40} d^{72} + 132 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} a c^{41} d^{72} + 7 \, {\left (2 \, c d x + b d\right )}^{\frac {11}{2}} c^{40} d^{70}}{4928 \, c^{44} d^{77}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 173, normalized size = 1.43 \[ -\frac {\left (2 c x +b \right ) \left (-7 c^{6} x^{6}-21 b \,c^{5} x^{5}-33 a \,c^{5} x^{4}-18 b^{2} c^{4} x^{4}-66 a b \,c^{4} x^{3}-b^{3} c^{3} x^{3}-77 a^{2} c^{4} x^{2}-11 a \,b^{2} c^{3} x^{2}+b^{4} c^{2} x^{2}-77 a^{2} b \,c^{3} x +22 a \,b^{3} c^{2} x -2 b^{5} c x +77 a^{3} c^{3}-77 a^{2} b^{2} c^{2}+22 a \,b^{4} c -2 b^{6}\right )}{77 \left (2 c d x +b d \right )^{\frac {3}{2}} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 136, normalized size = 1.12 \[ \frac {\frac {77 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )}}{\sqrt {2 \, c d x + b d} c^{3}} - \frac {33 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} {\left (b^{2} - 4 \, a c\right )} d^{2} - 77 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} d^{4} - 7 \, {\left (2 \, c d x + b d\right )}^{\frac {11}{2}}}{c^{3} d^{6}}}{4928 \, c d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.50, size = 129, normalized size = 1.07 \[ \frac {{\left (b\,d+2\,c\,d\,x\right )}^{11/2}}{704\,c^4\,d^7}+\frac {3\,{\left (b\,d+2\,c\,d\,x\right )}^{7/2}\,\left (4\,a\,c-b^2\right )}{448\,c^4\,d^5}+\frac {{\left (b\,d+2\,c\,d\,x\right )}^{3/2}\,{\left (4\,a\,c-b^2\right )}^2}{64\,c^4\,d^3}+\frac {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}{64\,c^4\,d\,\sqrt {b\,d+2\,c\,d\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 59.83, size = 128, normalized size = 1.06 \[ - \frac {\left (4 a c - b^{2}\right )^{3}}{64 c^{4} d \sqrt {b d + 2 c d x}} + \frac {\left (b d + 2 c d x\right )^{\frac {3}{2}} \left (48 a^{2} c^{2} - 24 a b^{2} c + 3 b^{4}\right )}{192 c^{4} d^{3}} + \frac {\left (12 a c - 3 b^{2}\right ) \left (b d + 2 c d x\right )^{\frac {7}{2}}}{448 c^{4} d^{5}} + \frac {\left (b d + 2 c d x\right )^{\frac {11}{2}}}{704 c^{4} d^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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