Optimal. Leaf size=121 \[ -\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}{320 c^4 d^5}+\frac {3 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x}}{64 c^4 d^3}+\frac {\left (b^2-4 a c\right )^3}{192 c^4 d (b d+2 c d x)^{3/2}}+\frac {(b d+2 c d x)^{9/2}}{576 c^4 d^7} \]
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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} \begin {gather*} -\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}{320 c^4 d^5}+\frac {3 \left (b^2-4 a c\right )^2 \sqrt {b d+2 c d x}}{64 c^4 d^3}+\frac {\left (b^2-4 a c\right )^3}{192 c^4 d (b d+2 c d x)^{3/2}}+\frac {(b d+2 c d x)^{9/2}}{576 c^4 d^7} \end {gather*}
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)^{5/2}} \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^3}{64 c^3 (b d+2 c d x)^{5/2}}+\frac {3 \left (-b^2+4 a c\right )^2}{64 c^3 d^2 \sqrt {b d+2 c d x}}+\frac {3 \left (-b^2+4 a c\right ) (b d+2 c d x)^{3/2}}{64 c^3 d^4}+\frac {(b d+2 c d x)^{7/2}}{64 c^3 d^6}\right ) \, dx\\ &=\frac {\left (b^2-4 a c\right )^3}{192 c^4 d (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^4 d^3}-\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}}{320 c^4 d^5}+\frac {(b d+2 c d x)^{9/2}}{576 c^4 d^7}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 83, normalized size = 0.69 \begin {gather*} \frac {-27 \left (b^2-4 a c\right ) (b+2 c x)^4+135 \left (b^2-4 a c\right )^2 (b+2 c x)^2+15 \left (b^2-4 a c\right )^3+5 (b+2 c x)^6}{2880 c^4 d (d (b+2 c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 174, normalized size = 1.44 \begin {gather*} \frac {-15 a^3 c^3+45 a^2 b^2 c^2+135 a^2 b c^3 x+135 a^2 c^4 x^2-18 a b^4 c-54 a b^3 c^2 x-27 a b^2 c^3 x^2+54 a b c^4 x^3+27 a c^5 x^4+2 b^6+6 b^5 c x+3 b^4 c^2 x^2-b^3 c^3 x^3+12 b^2 c^4 x^4+15 b c^5 x^5+5 c^6 x^6}{45 c^4 d (b d+2 c d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 194, normalized size = 1.60 \begin {gather*} \frac {{\left (5 \, c^{6} x^{6} + 15 \, b c^{5} x^{5} + 2 \, b^{6} - 18 \, a b^{4} c + 45 \, a^{2} b^{2} c^{2} - 15 \, a^{3} c^{3} + 3 \, {\left (4 \, b^{2} c^{4} + 9 \, a c^{5}\right )} x^{4} - {\left (b^{3} c^{3} - 54 \, a b c^{4}\right )} x^{3} + 3 \, {\left (b^{4} c^{2} - 9 \, a b^{2} c^{3} + 45 \, a^{2} c^{4}\right )} x^{2} + 3 \, {\left (2 \, b^{5} c - 18 \, a b^{3} c^{2} + 45 \, a^{2} b c^{3}\right )} x\right )} \sqrt {2 \, c d x + b d}}{45 \, {\left (4 \, c^{6} d^{3} x^{2} + 4 \, b c^{5} d^{3} x + b^{2} c^{4} d^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 187, normalized size = 1.55 \begin {gather*} \frac {b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}{192 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} c^{4} d} + \frac {135 \, \sqrt {2 \, c d x + b d} b^{4} c^{32} d^{60} - 1080 \, \sqrt {2 \, c d x + b d} a b^{2} c^{33} d^{60} + 2160 \, \sqrt {2 \, c d x + b d} a^{2} c^{34} d^{60} - 27 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b^{2} c^{32} d^{58} + 108 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} a c^{33} d^{58} + 5 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}} c^{32} d^{56}}{2880 \, c^{36} d^{63}} \end {gather*}
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 \begin {gather*} -\frac {\left (2 c x +b \right ) \left (-5 c^{6} x^{6}-15 b \,c^{5} x^{5}-27 a \,c^{5} x^{4}-12 b^{2} c^{4} x^{4}-54 a b \,c^{4} x^{3}+b^{3} c^{3} x^{3}-135 a^{2} c^{4} x^{2}+27 a \,b^{2} c^{3} x^{2}-3 b^{4} c^{2} x^{2}-135 a^{2} b \,c^{3} x +54 a \,b^{3} c^{2} x -6 b^{5} c x +15 a^{3} c^{3}-45 a^{2} b^{2} c^{2}+18 a \,b^{4} c -2 b^{6}\right )}{45 \left (2 c d x +b d \right )^{\frac {5}{2}} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 136, normalized size = 1.12 \begin {gather*} \frac {\frac {15 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )}}{{\left (2 \, c d x + b d\right )}^{\frac {3}{2}} c^{3}} - \frac {27 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} {\left (b^{2} - 4 \, a c\right )} d^{2} - 135 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt {2 \, c d x + b d} d^{4} - 5 \, {\left (2 \, c d x + b d\right )}^{\frac {9}{2}}}{c^{3} d^{6}}}{2880 \, c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 131, normalized size = 1.08 \begin {gather*} \frac {{\left (b\,d+2\,c\,d\,x\right )}^{9/2}}{576\,c^4\,d^7}+\frac {3\,{\left (b\,d+2\,c\,d\,x\right )}^{5/2}\,\left (4\,a\,c-b^2\right )}{320\,c^4\,d^5}+\frac {-\frac {64\,a^3\,c^3}{3}+16\,a^2\,b^2\,c^2-4\,a\,b^4\,c+\frac {b^6}{3}}{64\,c^4\,d\,{\left (b\,d+2\,c\,d\,x\right )}^{3/2}}+\frac {3\,\sqrt {b\,d+2\,c\,d\,x}\,{\left (4\,a\,c-b^2\right )}^2}{64\,c^4\,d^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 86.40, size = 128, normalized size = 1.06 \begin {gather*} - \frac {\left (4 a c - b^{2}\right )^{3}}{192 c^{4} d \left (b d + 2 c d x\right )^{\frac {3}{2}}} + \frac {\sqrt {b d + 2 c d x} \left (48 a^{2} c^{2} - 24 a b^{2} c + 3 b^{4}\right )}{64 c^{4} d^{3}} + \frac {\left (12 a c - 3 b^{2}\right ) \left (b d + 2 c d x\right )^{\frac {5}{2}}}{320 c^{4} d^{5}} + \frac {\left (b d + 2 c d x\right )^{\frac {9}{2}}}{576 c^{4} d^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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