Optimal. Leaf size=113 \[ -\frac {3 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32 c^{5/2} d^2}+\frac {3 (b+2 c x) \sqrt {a+b x+c x^2}}{16 c^2 d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 c d^2 (b+2 c x)} \]
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Rubi [A] time = 0.05, antiderivative size = 113, 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 = {684, 612, 621, 206} \begin {gather*} -\frac {3 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32 c^{5/2} d^2}+\frac {3 (b+2 c x) \sqrt {a+b x+c x^2}}{16 c^2 d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 c d^2 (b+2 c x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 684
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^2} \, dx &=-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 c d^2 (b+2 c x)}+\frac {3 \int \sqrt {a+b x+c x^2} \, dx}{4 c d^2}\\ &=\frac {3 (b+2 c x) \sqrt {a+b x+c x^2}}{16 c^2 d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 c d^2 (b+2 c x)}-\frac {\left (3 \left (b^2-4 a c\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{32 c^2 d^2}\\ &=\frac {3 (b+2 c x) \sqrt {a+b x+c x^2}}{16 c^2 d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 c d^2 (b+2 c x)}-\frac {\left (3 \left (b^2-4 a c\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{16 c^2 d^2}\\ &=\frac {3 (b+2 c x) \sqrt {a+b x+c x^2}}{16 c^2 d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{2 c d^2 (b+2 c x)}-\frac {3 \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32 c^{5/2} d^2}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 95, normalized size = 0.84 \begin {gather*} \frac {\left (b^2-4 a c\right ) \sqrt {a+x (b+c x)} \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{16 c^2 d^2 (b+2 c x) \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.54, size = 104, normalized size = 0.92 \begin {gather*} \frac {3 \left (b^2-4 a c\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{32 c^{5/2} d^2}+\frac {\sqrt {a+b x+c x^2} \left (-8 a c+3 b^2+4 b c x+4 c^2 x^2\right )}{16 c^2 d^2 (b+2 c x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 281, normalized size = 2.49 \begin {gather*} \left [-\frac {3 \, {\left (b^{3} - 4 \, a b c + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} x\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 (4 \, c^{3} x^{2} + 4 \, b c^{2} x + 3 \, b^{2} c - 8 \, a c^{2}\right )} \sqrt {c x^{2} + b x + a}}{64 \, {\left (2 \, c^{4} d^{2} x + b c^{3} d^{2}\right )}}, \frac {3 \, {\left (b^{3} - 4 \, a b c + 2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} x\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 (4 \, c^{3} x^{2} + 4 \, b c^{2} x + 3 \, b^{2} c - 8 \, a c^{2}\right )} \sqrt {c x^{2} + b x + a}}{32 \, {\left (2 \, c^{4} d^{2} x + b c^{3} d^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.51, size = 440, normalized size = 3.89 \begin {gather*} \frac {1}{32} \, d^{2} {\left (\frac {3 \, {\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 )} \arctan \left (\frac {\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}}{\sqrt {-c}}\right )}{\sqrt {-c} c^{2} d^{4} {\left | c \right |}} + \frac {2 \, {\left (\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} b^{2} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\relax (c) \mathrm {sgn}\relax (d) - 4 \, \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} a c \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\relax (c) \mathrm {sgn}\relax (d)\right )}}{c^{3} d^{4} {\left | c \right |}} + \frac {\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} b^{2} \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\relax (c) \mathrm {sgn}\relax (d) - 4 \, \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} a c \mathrm {sgn}\left (\frac {1}{2 \, c d x + b d}\right ) \mathrm {sgn}\relax (c) \mathrm {sgn}\relax (d)}{{\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}}\right )} c^{2} d^{4} {\left | c \right |}}\right )} {\left | c \right |} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 570, normalized size = 5.04 \begin {gather*} \frac {3 a^{2} \ln \left (\left (x +\frac {b}{2 c}\right ) \sqrt {c}+\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\right )}{2 \left (4 a c -b^{2}\right ) \sqrt {c}\, d^{2}}-\frac {3 a \,b^{2} \ln \left (\left (x +\frac {b}{2 c}\right ) \sqrt {c}+\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\right )}{4 \left (4 a c -b^{2}\right ) c^{\frac {3}{2}} d^{2}}+\frac {3 b^{4} \ln \left (\left (x +\frac {b}{2 c}\right ) \sqrt {c}+\sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\right )}{32 \left (4 a c -b^{2}\right ) c^{\frac {5}{2}} d^{2}}+\frac {3 \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, a x}{2 \left (4 a c -b^{2}\right ) d^{2}}-\frac {3 \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, b^{2} x}{8 \left (4 a c -b^{2}\right ) c \,d^{2}}+\frac {3 \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, a b}{4 \left (4 a c -b^{2}\right ) c \,d^{2}}-\frac {3 \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}\, b^{3}}{16 \left (4 a c -b^{2}\right ) c^{2} d^{2}}+\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}} x}{\left (4 a c -b^{2}\right ) d^{2}}+\frac {\left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}} b}{2 \left (4 a c -b^{2}\right ) c \,d^{2}}-\frac {\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 ) c \,d^{2}} \end {gather*}
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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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 )}^2} \,d x \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^{2} + 4 b c x + 4 c^{2} x^{2}}\, dx + \int \frac {b x \sqrt {a + b x + c x^{2}}}{b^{2} + 4 b c x + 4 c^{2} x^{2}}\, dx + \int \frac {c x^{2} \sqrt {a + b x + c x^{2}}}{b^{2} + 4 b c x + 4 c^{2} x^{2}}\, dx}{d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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