Optimal. Leaf size=88 \[ \frac {b^2-4 a c}{8 c^3 d^3 \sqrt {b d+2 c d x}}-\frac {\left (b^2-4 a c\right )^2}{80 c^3 d (b d+2 c d x)^{5/2}}+\frac {(b d+2 c d x)^{3/2}}{48 c^3 d^5} \]
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Rubi [A] time = 0.04, antiderivative size = 88, 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} \[ \frac {b^2-4 a c}{8 c^3 d^3 \sqrt {b d+2 c d x}}-\frac {\left (b^2-4 a c\right )^2}{80 c^3 d (b d+2 c d x)^{5/2}}+\frac {(b d+2 c d x)^{3/2}}{48 c^3 d^5} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{7/2}} \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^2}{16 c^2 (b d+2 c d x)^{7/2}}+\frac {-b^2+4 a c}{8 c^2 d^2 (b d+2 c d x)^{3/2}}+\frac {\sqrt {b d+2 c d x}}{16 c^2 d^4}\right ) \, dx\\ &=-\frac {\left (b^2-4 a c\right )^2}{80 c^3 d (b d+2 c d x)^{5/2}}+\frac {b^2-4 a c}{8 c^3 d^3 \sqrt {b d+2 c d x}}+\frac {(b d+2 c d x)^{3/2}}{48 c^3 d^5}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 92, normalized size = 1.05 \[ \frac {c^2 \left (-3 a^2-30 a c x^2+5 c^2 x^4\right )+3 b^2 c \left (5 c x^2-2 a\right )+10 b c^2 x \left (c x^2-3 a\right )+2 b^4+10 b^3 c x}{15 c^3 d (d (b+2 c x))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 134, normalized size = 1.52 \[ \frac {{\left (5 \, c^{4} x^{4} + 10 \, b c^{3} x^{3} + 2 \, b^{4} - 6 \, a b^{2} c - 3 \, a^{2} c^{2} + 15 \, {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} x^{2} + 10 \, {\left (b^{3} c - 3 \, a b c^{2}\right )} x\right )} \sqrt {2 \, c d x + b d}}{15 \, {\left (8 \, c^{6} d^{4} x^{3} + 12 \, b c^{5} d^{4} x^{2} + 6 \, b^{2} c^{4} d^{4} x + b^{3} c^{3} d^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 99, normalized size = 1.12 \[ \frac {{\left (2 \, c d x + b d\right )}^{\frac {3}{2}}}{48 \, c^{3} d^{5}} - \frac {b^{4} d^{2} - 8 \, a b^{2} c d^{2} + 16 \, a^{2} c^{2} d^{2} - 10 \, {\left (2 \, c d x + b d\right )}^{2} b^{2} + 40 \, {\left (2 \, c d x + b d\right )}^{2} a c}{80 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} c^{3} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 96, normalized size = 1.09 \[ -\frac {\left (2 c x +b \right ) \left (-5 c^{4} x^{4}-10 b \,c^{3} x^{3}+30 a \,c^{3} x^{2}-15 x^{2} b^{2} c^{2}+30 a b \,c^{2} x -10 x \,b^{3} c +3 a^{2} c^{2}+6 a \,b^{2} c -2 b^{4}\right )}{15 \left (2 c d x +b d \right )^{\frac {7}{2}} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 93, normalized size = 1.06 \[ \frac {\frac {5 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}}}{c^{2} d^{4}} + \frac {3 \, {\left (10 \, {\left (2 \, c d x + b d\right )}^{2} {\left (b^{2} - 4 \, a c\right )} - {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{2}\right )}}{{\left (2 \, c d x + b d\right )}^{\frac {5}{2}} c^{2} d^{2}}}{240 \, c d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.50, size = 92, normalized size = 1.05 \[ \frac {-3\,a^2\,c^2-6\,a\,b^2\,c-30\,a\,b\,c^2\,x-30\,a\,c^3\,x^2+2\,b^4+10\,b^3\,c\,x+15\,b^2\,c^2\,x^2+10\,b\,c^3\,x^3+5\,c^4\,x^4}{15\,c^3\,d\,{\left (b\,d+2\,c\,d\,x\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.68, size = 688, normalized size = 7.82 \[ \begin {cases} - \frac {3 a^{2} c^{2} \sqrt {b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} - \frac {6 a b^{2} c \sqrt {b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} - \frac {30 a b c^{2} x \sqrt {b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} - \frac {30 a c^{3} x^{2} \sqrt {b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} + \frac {2 b^{4} \sqrt {b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} + \frac {10 b^{3} c x \sqrt {b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} + \frac {15 b^{2} c^{2} x^{2} \sqrt {b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} + \frac {10 b c^{3} x^{3} \sqrt {b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} + \frac {5 c^{4} x^{4} \sqrt {b d + 2 c d x}}{15 b^{3} c^{3} d^{4} + 90 b^{2} c^{4} d^{4} x + 180 b c^{5} d^{4} x^{2} + 120 c^{6} d^{4} x^{3}} & \text {for}\: c \neq 0 \\\frac {a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3}}{\left (b d\right )^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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