Optimal. Leaf size=107 \[ -\frac {\sqrt {a+b x+c x^2}}{8 c^2 d^4 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d^4 (b+2 c x)^3}+\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{5/2} d^4} \]
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Rubi [A]
time = 0.03, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {698, 635, 212}
\begin {gather*} \frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{5/2} d^4}-\frac {\sqrt {a+b x+c x^2}}{8 c^2 d^4 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d^4 (b+2 c x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 698
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^4} \, dx &=-\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d^4 (b+2 c x)^3}+\frac {\int \frac {\sqrt {a+b x+c x^2}}{(b d+2 c d x)^2} \, dx}{4 c d^2}\\ &=-\frac {\sqrt {a+b x+c x^2}}{8 c^2 d^4 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d^4 (b+2 c x)^3}+\frac {\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{16 c^2 d^4}\\ &=-\frac {\sqrt {a+b x+c x^2}}{8 c^2 d^4 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d^4 (b+2 c x)^3}+\frac {\text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{8 c^2 d^4}\\ &=-\frac {\sqrt {a+b x+c x^2}}{8 c^2 d^4 (b+2 c x)}-\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d^4 (b+2 c x)^3}+\frac {\tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16 c^{5/2} d^4}\\ \end {align*}
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Mathematica [A]
time = 0.53, size = 95, normalized size = 0.89 \begin {gather*} \frac {-\frac {\sqrt {a+x (b+c x)} \left (3 b^2+16 b c x+4 c \left (a+4 c x^2\right )\right )}{24 c^2 (b+2 c x)^3}-\frac {\log \left (c^2 \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{16 c^{5/2}}}{d^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(310\) vs.
\(2(89)=178\).
time = 0.70, size = 311, normalized size = 2.91
method | result | size |
default | \(\frac {-\frac {4 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{3 \left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{3}}+\frac {8 c^{2} \left (-\frac {4 c \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )}+\frac {16 c^{2} \left (\frac {\left (x +\frac {b}{2 c}\right ) \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{4}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (x +\frac {b}{2 c}\right ) \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}{2}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\sqrt {c}\, \left (x +\frac {b}{2 c}\right )+\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{4 a c -b^{2}}\right )}{3 \left (4 a c -b^{2}\right )}}{16 d^{4} c^{4}}\) | \(311\) |
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 [A]
time = 5.21, size = 345, normalized size = 3.22 \begin {gather*} \left [\frac {3 \, {\left (8 \, c^{3} x^{3} + 12 \, b c^{2} x^{2} + 6 \, b^{2} c x + b^{3}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, {\left (16 \, c^{3} x^{2} + 16 \, b c^{2} x + 3 \, b^{2} c + 4 \, a c^{2}\right )} \sqrt {c x^{2} + b x + a}}{96 \, {\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 )}}, -\frac {3 \, {\left (8 \, c^{3} x^{3} + 12 \, b c^{2} x^{2} + 6 \, b^{2} c x + b^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (16 \, c^{3} x^{2} + 16 \, b c^{2} x + 3 \, b^{2} c + 4 \, a c^{2}\right )} \sqrt {c x^{2} + b x + a}}{48 \, {\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 )}}\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 {a \sqrt {a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac {b x \sqrt {a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx + \int \frac {c x^{2} \sqrt {a + b x + c x^{2}}}{b^{4} + 8 b^{3} c x + 24 b^{2} c^{2} x^{2} + 32 b c^{3} x^{3} + 16 c^{4} x^{4}}\, dx}{d^{4}} \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 {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (b\,d+2\,c\,d\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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