Optimal. Leaf size=115 \[ \frac {\left (b^2-4 a c\right )^{3/2} \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{16 c^{5/2} d}-\frac {\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{8 c^2 d}+\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d} \]
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Rubi [A] time = 0.08, antiderivative size = 115, 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 = {685, 688, 205} \[ -\frac {\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{8 c^2 d}+\frac {\left (b^2-4 a c\right )^{3/2} \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{16 c^{5/2} d}+\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d} \]
Antiderivative was successfully verified.
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Rule 205
Rule 685
Rule 688
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{b d+2 c d x} \, dx &=\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d}-\frac {\left (b^2-4 a c\right ) \int \frac {\sqrt {a+b x+c x^2}}{b d+2 c d x} \, dx}{4 c}\\ &=-\frac {\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{8 c^2 d}+\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d}+\frac {\left (b^2-4 a c\right )^2 \int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx}{16 c^2}\\ &=-\frac {\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{8 c^2 d}+\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d}+\frac {\left (b^2-4 a c\right )^2 \operatorname {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 )}{4 c}\\ &=-\frac {\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}{8 c^2 d}+\frac {\left (a+b x+c x^2\right )^{3/2}}{6 c d}+\frac {\left (b^2-4 a c\right )^{3/2} \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{16 c^{5/2} d}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 103, normalized size = 0.90 \[ \frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (4 c \left (4 a+c x^2\right )-3 b^2+4 b c x\right )+3 \left (b^2-4 a c\right )^{3/2} \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+x (b+c x)}}{\sqrt {b^2-4 a c}}\right )}{48 c^{5/2} d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.14, size = 246, normalized size = 2.14 \[ \left [-\frac {3 \, {\left (b^{2} - 4 \, a c\right )} \sqrt {-\frac {b^{2} - 4 \, a c}{c}} \log \left (-\frac {4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt {c x^{2} + b x + a} c \sqrt {-\frac {b^{2} - 4 \, a c}{c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) - 4 \, {\left (4 \, c^{2} x^{2} + 4 \, b c x - 3 \, b^{2} + 16 \, a c\right )} \sqrt {c x^{2} + b x + a}}{96 \, c^{2} d}, -\frac {3 \, {\left (b^{2} - 4 \, a c\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c}} \arctan \left (\frac {\sqrt {\frac {b^{2} - 4 \, a c}{c}}}{2 \, \sqrt {c x^{2} + b x + a}}\right ) - 2 \, {\left (4 \, c^{2} x^{2} + 4 \, b c x - 3 \, b^{2} + 16 \, a c\right )} \sqrt {c x^{2} + b x + a}}{48 \, c^{2} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 149, normalized size = 1.30 \[ \frac {1}{24} \, \sqrt {c x^{2} + b x + a} {\left (4 \, x {\left (\frac {x}{d} + \frac {b}{c d}\right )} - \frac {3 \, b^{2} c^{3} d^{3} - 16 \, a c^{4} d^{3}}{c^{5} d^{4}}\right )} + \frac {{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \arctan \left (-\frac {2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} c + b \sqrt {c}}{\sqrt {b^{2} c - 4 \, a c^{2}}}\right )}{8 \, \sqrt {b^{2} c - 4 \, a c^{2}} c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 430, normalized size = 3.74 \[ -\frac {a^{2} \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{\sqrt {\frac {4 a c -b^{2}}{c}}\, c d}+\frac {a \,b^{2} \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{2 \sqrt {\frac {4 a c -b^{2}}{c}}\, c^{2} d}-\frac {b^{4} \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{16 \sqrt {\frac {4 a c -b^{2}}{c}}\, c^{3} d}+\frac {\sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}\, a}{4 c d}-\frac {\sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}\, b^{2}}{16 c^{2} d}+\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{6 c d} \]
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 [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{b\,d+2\,c\,d\,x} \,d x \]
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 \[ \frac {\int \frac {a \sqrt {a + b x + c x^{2}}}{b + 2 c x}\, dx + \int \frac {b x \sqrt {a + b x + c x^{2}}}{b + 2 c x}\, dx + \int \frac {c x^{2} \sqrt {a + b x + c x^{2}}}{b + 2 c x}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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