Optimal. Leaf size=122 \[ \frac {256 c^2 \sqrt {a+b x+c x^2}}{3 d^2 \left (b^2-4 a c\right )^3 (b+2 c x)}+\frac {32 c}{3 d^2 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2}}-\frac {2}{3 d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {687, 682} \[ \frac {256 c^2 \sqrt {a+b x+c x^2}}{3 d^2 \left (b^2-4 a c\right )^3 (b+2 c x)}+\frac {32 c}{3 d^2 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2}}-\frac {2}{3 d^2 \left (b^2-4 a c\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 682
Rule 687
Rubi steps
\begin {align*} \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2}{3 \left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}-\frac {(16 c) \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}+\frac {32 c}{3 \left (b^2-4 a c\right )^2 d^2 (b+2 c x) \sqrt {a+b x+c x^2}}+\frac {\left (128 c^2\right ) \int \frac {1}{(b d+2 c d x)^2 \sqrt {a+b x+c x^2}} \, dx}{3 \left (b^2-4 a c\right )^2}\\ &=-\frac {2}{3 \left (b^2-4 a c\right ) d^2 (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}+\frac {32 c}{3 \left (b^2-4 a c\right )^2 d^2 (b+2 c x) \sqrt {a+b x+c x^2}}+\frac {256 c^2 \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^3 d^2 (b+2 c x)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 108, normalized size = 0.89 \[ \frac {32 c^2 \left (3 a^2+12 a c x^2+8 c^2 x^4\right )+48 b^2 c \left (a+6 c x^2\right )+128 b c^2 x \left (3 a+4 c x^2\right )-2 b^4+32 b^3 c x}{3 d^2 \left (b^2-4 a c\right )^3 (b+2 c x) (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 4.54, size = 374, normalized size = 3.07 \[ \frac {2 \, {\left (128 \, c^{4} x^{4} + 256 \, b c^{3} x^{3} - b^{4} + 24 \, a b^{2} c + 48 \, a^{2} c^{2} + 48 \, {\left (3 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 16 \, {\left (b^{3} c + 12 \, a b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (2 \, {\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )} d^{2} x^{5} + 5 \, {\left (b^{7} c^{2} - 12 \, a b^{5} c^{3} + 48 \, a^{2} b^{3} c^{4} - 64 \, a^{3} b c^{5}\right )} d^{2} x^{4} + 4 \, {\left (b^{8} c - 11 \, a b^{6} c^{2} + 36 \, a^{2} b^{4} c^{3} - 16 \, a^{3} b^{2} c^{4} - 64 \, a^{4} c^{5}\right )} d^{2} x^{3} + {\left (b^{9} - 6 \, a b^{7} c - 24 \, a^{2} b^{5} c^{2} + 224 \, a^{3} b^{3} c^{3} - 384 \, a^{4} b c^{4}\right )} d^{2} x^{2} + 2 \, {\left (a b^{8} - 11 \, a^{2} b^{6} c + 36 \, a^{3} b^{4} c^{2} - 16 \, a^{4} b^{2} c^{3} - 64 \, a^{5} c^{4}\right )} d^{2} x + {\left (a^{2} b^{7} - 12 \, a^{3} b^{5} c + 48 \, a^{4} b^{3} c^{2} - 64 \, a^{5} b c^{3}\right )} d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.60, size = 374, normalized size = 3.07 \[ \frac {16 \, {\left (\frac {{\left (\frac {3 \, \sqrt {-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c} c^{2}}{b^{2} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\relax (c) \mathrm {sgn}\relax (d) - 4 \, a c \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\relax (c) \mathrm {sgn}\relax (d)} + \frac {6 \, {\left (\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - c\right )} c^{3} + c^{4}}{{\left (b^{2} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\relax (c) \mathrm {sgn}\relax (d) - 4 \, a c \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\relax (c) \mathrm {sgn}\relax (d)\right )} {\left (\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - c\right )} \sqrt {-\frac {b^{2} c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {4 \, a c^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + c}}\right )} c^{5} d^{8} {\left | c \right |}}{{\left (b^{2} c^{3} d^{4} - 4 \, a c^{4} d^{4}\right )}^{2}} - \frac {8 \, c^{\frac {3}{2}} {\left | c \right |} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\relax (c) \mathrm {sgn}\relax (d)}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )}}{3 \, d^{2} {\left | c \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 133, normalized size = 1.09 \[ -\frac {2 \left (128 c^{4} x^{4}+256 b \,c^{3} x^{3}+192 a \,c^{3} x^{2}+144 x^{2} b^{2} c^{2}+192 a b \,c^{2} x +16 x \,b^{3} c +48 a^{2} c^{2}+24 a \,b^{2} c -b^{4}\right )}{3 \left (2 c x +b \right ) \left (64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} d^{2}} \]
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 = 1.04, size = 110, normalized size = 0.90 \[ -\frac {2\,\left (48\,a^2\,c^2+24\,a\,b^2\,c+192\,a\,b\,c^2\,x+192\,a\,c^3\,x^2-b^4+16\,b^3\,c\,x+144\,b^2\,c^2\,x^2+256\,b\,c^3\,x^3+128\,c^4\,x^4\right )}{3\,d^2\,{\left (4\,a\,c-b^2\right )}^3\,\left (b+2\,c\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{a^{2} b^{2} \sqrt {a + b x + c x^{2}} + 4 a^{2} b c x \sqrt {a + b x + c x^{2}} + 4 a^{2} c^{2} x^{2} \sqrt {a + b x + c x^{2}} + 2 a b^{3} x \sqrt {a + b x + c x^{2}} + 10 a b^{2} c x^{2} \sqrt {a + b x + c x^{2}} + 16 a b c^{2} x^{3} \sqrt {a + b x + c x^{2}} + 8 a c^{3} x^{4} \sqrt {a + b x + c x^{2}} + b^{4} x^{2} \sqrt {a + b x + c x^{2}} + 6 b^{3} c x^{3} \sqrt {a + b x + c x^{2}} + 13 b^{2} c^{2} x^{4} \sqrt {a + b x + c x^{2}} + 12 b c^{3} x^{5} \sqrt {a + b x + c x^{2}} + 4 c^{4} x^{6} \sqrt {a + b x + c x^{2}}}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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