Optimal. Leaf size=165 \[ \frac {d^2 \left (b^2-4 a c\right )^3 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{5/2}}+\frac {d^2 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^2}-\frac {d^2 \left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt {a+b x+c x^2}}{64 c^2}+\frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{12 c} \]
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Rubi [A] time = 0.09, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {685, 692, 621, 206} \begin {gather*} \frac {d^2 \left (b^2-4 a c\right )^2 (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^2}-\frac {d^2 \left (b^2-4 a c\right ) (b+2 c x)^3 \sqrt {a+b x+c x^2}}{64 c^2}+\frac {d^2 \left (b^2-4 a c\right )^3 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{5/2}}+\frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{12 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 685
Rule 692
Rubi steps
\begin {align*} \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{12 c}-\frac {\left (b^2-4 a c\right ) \int (b d+2 c d x)^2 \sqrt {a+b x+c x^2} \, dx}{8 c}\\ &=-\frac {\left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{64 c^2}+\frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{12 c}+\frac {\left (b^2-4 a c\right )^2 \int \frac {(b d+2 c d x)^2}{\sqrt {a+b x+c x^2}} \, dx}{128 c^2}\\ &=\frac {\left (b^2-4 a c\right )^2 d^2 (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^2}-\frac {\left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{64 c^2}+\frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{12 c}+\frac {\left (\left (b^2-4 a c\right )^3 d^2\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{256 c^2}\\ &=\frac {\left (b^2-4 a c\right )^2 d^2 (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^2}-\frac {\left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{64 c^2}+\frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{12 c}+\frac {\left (\left (b^2-4 a c\right )^3 d^2\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 )}{128 c^2}\\ &=\frac {\left (b^2-4 a c\right )^2 d^2 (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^2}-\frac {\left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{64 c^2}+\frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{12 c}+\frac {\left (b^2-4 a c\right )^3 d^2 \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 211, normalized size = 1.28 \begin {gather*} \frac {1}{3} d^2 (b+2 c x) \sqrt {a+x (b+c x)} \left ((a+x (b+c x))^2-\frac {(a+x (b+c x)) \left (2 (b+2 c x) \sqrt {\frac {c (a+x (b+c x))}{4 a c-b^2}} \left (4 c \left (5 a+2 c x^2\right )-3 b^2+8 b c x\right )+3 \sqrt {c} \sqrt {4 a-\frac {b^2}{c}} \left (4 a c-b^2\right ) \sinh ^{-1}\left (\frac {b+2 c x}{\sqrt {c} \sqrt {4 a-\frac {b^2}{c}}}\right )\right )}{256 c (b+2 c x) \left (\frac {c (a+x (b+c x))}{4 a c-b^2}\right )^{3/2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.76, size = 245, normalized size = 1.48 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (48 a^2 b c^2 d^2+96 a^2 c^3 d^2 x+32 a b^3 c d^2+288 a b^2 c^2 d^2 x+672 a b c^3 d^2 x^2+448 a c^4 d^2 x^3-3 b^5 d^2+2 b^4 c d^2 x+152 b^3 c^2 d^2 x^2+528 b^2 c^3 d^2 x^3+640 b c^4 d^2 x^4+256 c^5 d^2 x^5\right )}{384 c^2}+\frac {\left (64 a^3 c^3 d^2-48 a^2 b^2 c^2 d^2+12 a b^4 c d^2+b^6 \left (-d^2\right )\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{256 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 473, normalized size = 2.87 \begin {gather*} \left [-\frac {3 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {c} d^{2} \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 (256 \, c^{6} d^{2} x^{5} + 640 \, b c^{5} d^{2} x^{4} + 16 \, {\left (33 \, b^{2} c^{4} + 28 \, a c^{5}\right )} d^{2} x^{3} + 8 \, {\left (19 \, b^{3} c^{3} + 84 \, a b c^{4}\right )} d^{2} x^{2} + 2 \, {\left (b^{4} c^{2} + 144 \, a b^{2} c^{3} + 48 \, a^{2} c^{4}\right )} d^{2} x - {\left (3 \, b^{5} c - 32 \, a b^{3} c^{2} - 48 \, a^{2} b c^{3}\right )} d^{2}\right )} \sqrt {c x^{2} + b x + a}}{1536 \, c^{3}}, -\frac {3 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {-c} d^{2} \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 (256 \, c^{6} d^{2} x^{5} + 640 \, b c^{5} d^{2} x^{4} + 16 \, {\left (33 \, b^{2} c^{4} + 28 \, a c^{5}\right )} d^{2} x^{3} + 8 \, {\left (19 \, b^{3} c^{3} + 84 \, a b c^{4}\right )} d^{2} x^{2} + 2 \, {\left (b^{4} c^{2} + 144 \, a b^{2} c^{3} + 48 \, a^{2} c^{4}\right )} d^{2} x - {\left (3 \, b^{5} c - 32 \, a b^{3} c^{2} - 48 \, a^{2} b c^{3}\right )} d^{2}\right )} \sqrt {c x^{2} + b x + a}}{768 \, c^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 259, normalized size = 1.57 \begin {gather*} \frac {1}{384} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, c^{3} d^{2} x + 5 \, b c^{2} d^{2}\right )} x + \frac {33 \, b^{2} c^{6} d^{2} + 28 \, a c^{7} d^{2}}{c^{5}}\right )} x + \frac {19 \, b^{3} c^{5} d^{2} + 84 \, a b c^{6} d^{2}}{c^{5}}\right )} x + \frac {b^{4} c^{4} d^{2} + 144 \, a b^{2} c^{5} d^{2} + 48 \, a^{2} c^{6} d^{2}}{c^{5}}\right )} x - \frac {3 \, b^{5} c^{3} d^{2} - 32 \, a b^{3} c^{4} d^{2} - 48 \, a^{2} b c^{5} d^{2}}{c^{5}}\right )} - \frac {{\left (b^{6} d^{2} - 12 \, a b^{4} c d^{2} + 48 \, a^{2} b^{2} c^{2} d^{2} - 64 \, a^{3} c^{3} d^{2}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{256 \, c^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 406, normalized size = 2.46 \begin {gather*} -\frac {a^{3} \sqrt {c}\, d^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{4}+\frac {3 a^{2} b^{2} d^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{16 \sqrt {c}}-\frac {3 a \,b^{4} d^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{64 c^{\frac {3}{2}}}+\frac {b^{6} d^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {5}{2}}}-\frac {\sqrt {c \,x^{2}+b x +a}\, a^{2} c \,d^{2} x}{4}+\frac {\sqrt {c \,x^{2}+b x +a}\, a \,b^{2} d^{2} x}{8}-\frac {\sqrt {c \,x^{2}+b x +a}\, b^{4} d^{2} x}{64 c}-\frac {\sqrt {c \,x^{2}+b x +a}\, a^{2} b \,d^{2}}{8}+\frac {\sqrt {c \,x^{2}+b x +a}\, a \,b^{3} d^{2}}{16 c}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a c \,d^{2} x}{6}-\frac {\sqrt {c \,x^{2}+b x +a}\, b^{5} d^{2}}{128 c^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{2} d^{2} x}{24}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a b \,d^{2}}{12}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{3} d^{2}}{48 c}+\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} c \,d^{2} x}{3}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} b \,d^{2}}{3} \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 {\left (b\,d+2\,c\,d\,x\right )}^2\,{\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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} d^{2} \left (\int a b^{2} \sqrt {a + b x + c x^{2}}\, dx + \int b^{3} x \sqrt {a + b x + c x^{2}}\, dx + \int 4 c^{3} x^{4} \sqrt {a + b x + c x^{2}}\, dx + \int 4 a c^{2} x^{2} \sqrt {a + b x + c x^{2}}\, dx + \int 8 b c^{2} x^{3} \sqrt {a + b x + c x^{2}}\, dx + \int 5 b^{2} c x^{2} \sqrt {a + b x + c x^{2}}\, dx + \int 4 a b c x \sqrt {a + b x + c x^{2}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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