Optimal. Leaf size=132 \[ -\frac {2}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \sqrt {a+b x+c x^2}}-\frac {12 c \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2}-\frac {6 \sqrt {c} \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} d^3} \]
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Rubi [A]
time = 0.05, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {701, 707, 702,
211} \begin {gather*} -\frac {6 \sqrt {c} \text {ArcTan}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{d^3 \left (b^2-4 a c\right )^{5/2}}-\frac {12 c \sqrt {a+b x+c x^2}}{d^3 \left (b^2-4 a c\right )^2 (b+2 c x)^2}-\frac {2}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \sqrt {a+b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 701
Rule 702
Rule 707
Rubi steps
\begin {align*} \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac {2}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \sqrt {a+b x+c x^2}}-\frac {(12 c) \int \frac {1}{(b d+2 c d x)^3 \sqrt {a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=-\frac {2}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \sqrt {a+b x+c x^2}}-\frac {12 c \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2}-\frac {(6 c) \int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right )^2 d^2}\\ &=-\frac {2}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \sqrt {a+b x+c x^2}}-\frac {12 c \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2}-\frac {\left (24 c^2\right ) \text {Subst}\left (\int \frac {1}{2 b^2 c d-8 a c^2 d+8 c^2 d x^2} \, dx,x,\sqrt {a+b x+c x^2}\right )}{\left (b^2-4 a c\right )^2 d^2}\\ &=-\frac {2}{\left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \sqrt {a+b x+c x^2}}-\frac {12 c \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2}-\frac {6 \sqrt {c} \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} d^3}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 10.03, size = 60, normalized size = 0.45 \begin {gather*} -\frac {2 \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {4 c (a+x (b+c x))}{-b^2+4 a c}\right )}{\left (b^2-4 a c\right )^2 d^3 \sqrt {a+x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.71, size = 235, normalized size = 1.78
method | result | size |
default | \(\frac {-\frac {2 c}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{2} \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}-\frac {6 c^{2} \left (\frac {4 c}{\left (4 a c -b^{2}\right ) \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}-\frac {8 c \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{\left (4 a c -b^{2}\right ) \sqrt {\frac {4 a c -b^{2}}{c}}}\right )}{4 a c -b^{2}}}{8 d^{3} c^{3}}\) | \(235\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 336 vs.
\(2 (118) = 236\).
time = 2.55, size = 702, normalized size = 5.32 \begin {gather*} \left [\frac {3 \, {\left (4 \, c^{3} x^{4} + 8 \, b c^{2} x^{3} + a b^{2} + {\left (5 \, b^{2} c + 4 \, a c^{2}\right )} x^{2} + {\left (b^{3} + 4 \, a b c\right )} x\right )} \sqrt {-\frac {c}{b^{2} - 4 \, a c}} \log \left (-\frac {4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt {c x^{2} + b x + a} {\left (b^{2} - 4 \, a c\right )} \sqrt {-\frac {c}{b^{2} - 4 \, a c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) - 2 \, {\left (6 \, c^{2} x^{2} + 6 \, b c x + b^{2} + 2 \, a c\right )} \sqrt {c x^{2} + b x + a}}{4 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{3} x^{4} + 8 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{3} x^{3} + {\left (5 \, b^{6} c - 36 \, a b^{4} c^{2} + 48 \, a^{2} b^{2} c^{3} + 64 \, a^{3} c^{4}\right )} d^{3} x^{2} + {\left (b^{7} - 4 \, a b^{5} c - 16 \, a^{2} b^{3} c^{2} + 64 \, a^{3} b c^{3}\right )} d^{3} x + {\left (a b^{6} - 8 \, a^{2} b^{4} c + 16 \, a^{3} b^{2} c^{2}\right )} d^{3}}, -\frac {2 \, {\left (3 \, {\left (4 \, c^{3} x^{4} + 8 \, b c^{2} x^{3} + a b^{2} + {\left (5 \, b^{2} c + 4 \, a c^{2}\right )} x^{2} + {\left (b^{3} + 4 \, a b c\right )} x\right )} \sqrt {\frac {c}{b^{2} - 4 \, a c}} \arctan \left (-\frac {\sqrt {c x^{2} + b x + a} {\left (b^{2} - 4 \, a c\right )} \sqrt {\frac {c}{b^{2} - 4 \, a c}}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + {\left (6 \, c^{2} x^{2} + 6 \, b c x + b^{2} + 2 \, a c\right )} \sqrt {c x^{2} + b x + a}\right )}}{4 \, {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{3} x^{4} + 8 \, {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{3} x^{3} + {\left (5 \, b^{6} c - 36 \, a b^{4} c^{2} + 48 \, a^{2} b^{2} c^{3} + 64 \, a^{3} c^{4}\right )} d^{3} x^{2} + {\left (b^{7} - 4 \, a b^{5} c - 16 \, a^{2} b^{3} c^{2} + 64 \, a^{3} b c^{3}\right )} d^{3} x + {\left (a b^{6} - 8 \, a^{2} b^{4} c + 16 \, a^{3} b^{2} c^{2}\right )} d^{3}}\right ] \end {gather*}
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 \begin {gather*} \frac {\int \frac {1}{a b^{3} \sqrt {a + b x + c x^{2}} + 6 a b^{2} c x \sqrt {a + b x + c x^{2}} + 12 a b c^{2} x^{2} \sqrt {a + b x + c x^{2}} + 8 a c^{3} x^{3} \sqrt {a + b x + c x^{2}} + b^{4} x \sqrt {a + b x + c x^{2}} + 7 b^{3} c x^{2} \sqrt {a + b x + c x^{2}} + 18 b^{2} c^{2} x^{3} \sqrt {a + b x + c x^{2}} + 20 b c^{3} x^{4} \sqrt {a + b x + c x^{2}} + 8 c^{4} x^{5} \sqrt {a + b x + c x^{2}}}\, dx}{d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (b\,d+2\,c\,d\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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