Optimal. Leaf size=88 \[ -\frac {\left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}{8 c^3 d^3}-\frac {\left (b^2-4 a c\right )^2}{48 c^3 d (b d+2 c d x)^{3/2}}+\frac {(b d+2 c d x)^{5/2}}{80 c^3 d^5} \]
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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} \begin {gather*} -\frac {\left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}{8 c^3 d^3}-\frac {\left (b^2-4 a c\right )^2}{48 c^3 d (b d+2 c d x)^{3/2}}+\frac {(b d+2 c d x)^{5/2}}{80 c^3 d^5} \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 )^2}{(b d+2 c d x)^{5/2}} \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^2}{16 c^2 (b d+2 c d x)^{5/2}}+\frac {-b^2+4 a c}{8 c^2 d^2 \sqrt {b d+2 c d x}}+\frac {(b d+2 c d x)^{3/2}}{16 c^2 d^4}\right ) \, dx\\ &=-\frac {\left (b^2-4 a c\right )^2}{48 c^3 d (b d+2 c d x)^{3/2}}-\frac {\left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}{8 c^3 d^3}+\frac {(b d+2 c d x)^{5/2}}{80 c^3 d^5}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 91, normalized size = 1.03 \begin {gather*} \frac {c^2 \left (-5 a^2+30 a c x^2+3 c^2 x^4\right )+b^2 c \left (10 a-3 c x^2\right )+6 b c^2 x \left (5 a+c x^2\right )-2 b^4-6 b^3 c x}{15 c^3 d (d (b+2 c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 96, normalized size = 1.09 \begin {gather*} \frac {-5 a^2 c^2+10 a b^2 c+30 a b c^2 x+30 a c^3 x^2-2 b^4-6 b^3 c x-3 b^2 c^2 x^2+6 b c^3 x^3+3 c^4 x^4}{15 c^3 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 = 120, normalized size = 1.36 \begin {gather*} \frac {{\left (3 \, c^{4} x^{4} + 6 \, b c^{3} x^{3} - 2 \, b^{4} + 10 \, a b^{2} c - 5 \, a^{2} c^{2} - 3 \, {\left (b^{2} c^{2} - 10 \, a c^{3}\right )} x^{2} - 6 \, {\left (b^{3} c - 5 \, a b c^{2}\right )} x\right )} \sqrt {2 \, c d x + b d}}{15 \, {\left (4 \, c^{5} d^{3} x^{2} + 4 \, b c^{4} d^{3} x + b^{2} c^{3} d^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 109, normalized size = 1.24 \begin {gather*} -\frac {b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}{48 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} c^{3} d} - \frac {10 \, \sqrt {2 \, c d x + b d} b^{2} c^{12} d^{22} - 40 \, \sqrt {2 \, c d x + b d} a c^{13} d^{22} - {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} c^{12} d^{20}}{80 \, c^{15} d^{25}} \end {gather*}
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 \begin {gather*} -\frac {\left (2 c x +b \right ) \left (-3 c^{4} x^{4}-6 b \,c^{3} x^{3}-30 a \,c^{3} x^{2}+3 x^{2} b^{2} c^{2}-30 a b \,c^{2} x +6 x \,b^{3} c +5 a^{2} c^{2}-10 a \,b^{2} c +2 b^{4}\right )}{15 \left (2 c d x +b d \right )^{\frac {5}{2}} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 90, normalized size = 1.02 \begin {gather*} -\frac {\frac {5 \, {\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}} c^{2}} + \frac {3 \, {\left (10 \, \sqrt {2 \, c d x + b d} {\left (b^{2} - 4 \, a c\right )} d^{2} - {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}\right )}}{c^{2} d^{4}}}{240 \, c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.49, size = 99, normalized size = 1.12 \begin {gather*} \frac {3\,{\left (b\,d+2\,c\,d\,x\right )}^4-5\,b^4\,d^4-30\,b^2\,d^2\,{\left (b\,d+2\,c\,d\,x\right )}^2-80\,a^2\,c^2\,d^4+120\,a\,c\,d^2\,{\left (b\,d+2\,c\,d\,x\right )}^2+40\,a\,b^2\,c\,d^4}{240\,c^3\,d^5\,{\left (b\,d+2\,c\,d\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 54.22, size = 82, normalized size = 0.93 \begin {gather*} - \frac {\left (4 a c - b^{2}\right )^{2}}{48 c^{3} d \left (b d + 2 c d x\right )^{\frac {3}{2}}} + \frac {\left (4 a c - b^{2}\right ) \sqrt {b d + 2 c d x}}{8 c^{3} d^{3}} + \frac {\left (b d + 2 c d x\right )^{\frac {5}{2}}}{80 c^{3} d^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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