Optimal. Leaf size=153 \[ \frac {\sqrt {a+b x+c x^2} \left (-8 a B c-2 A b c+3 b^2 B\right )}{c^2 \left (b^2-4 a c\right )}-\frac {2 x \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {(3 b B-2 A c) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{5/2}} \]
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Rubi [A] time = 0.14, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {818, 640, 621, 206} \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-8 a B c-2 A b c+3 b^2 B\right )}{c^2 \left (b^2-4 a c\right )}-\frac {2 x \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {(3 b B-2 A c) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 640
Rule 818
Rubi steps
\begin {align*} \int \frac {x^2 (A+B x)}{\left (a+b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 x \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 \int \frac {a (b B-2 A c)+\frac {1}{2} \left (3 b^2 B-2 A b c-8 a B c\right ) x}{\sqrt {a+b x+c x^2}} \, dx}{c \left (b^2-4 a c\right )}\\ &=-\frac {2 x \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {\left (3 b^2 B-2 A b c-8 a B c\right ) \sqrt {a+b x+c x^2}}{c^2 \left (b^2-4 a c\right )}-\frac {(3 b B-2 A c) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{2 c^2}\\ &=-\frac {2 x \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {\left (3 b^2 B-2 A b c-8 a B c\right ) \sqrt {a+b x+c x^2}}{c^2 \left (b^2-4 a c\right )}-\frac {(3 b B-2 A c) \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 )}{c^2}\\ &=-\frac {2 x \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {\left (3 b^2 B-2 A b c-8 a B c\right ) \sqrt {a+b x+c x^2}}{c^2 \left (b^2-4 a c\right )}-\frac {(3 b B-2 A c) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 148, normalized size = 0.97 \begin {gather*} \frac {\frac {2 \sqrt {c} \left (8 a^2 B c+a \left (2 b c (A+5 B x)+4 c^2 x (B x-A)-3 b^2 B\right )-b^2 x (-2 A c+3 b B+B c x)\right )}{\sqrt {a+x (b+c x)}}+\left (b^2-4 a c\right ) (3 b B-2 A c) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{2 c^{5/2} \left (4 a c-b^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.71, size = 148, normalized size = 0.97 \begin {gather*} \frac {8 a^2 B c+2 a A b c-4 a A c^2 x-3 a b^2 B+10 a b B c x+4 a B c^2 x^2+2 A b^2 c x-3 b^3 B x-b^2 B c x^2}{c^2 \left (4 a c-b^2\right ) \sqrt {a+b x+c x^2}}+\frac {(3 b B-2 A c) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{2 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 603, normalized size = 3.94 \begin {gather*} \left [-\frac {{\left (3 \, B a b^{3} + 8 \, A a^{2} c^{2} + {\left (3 \, B b^{3} c + 8 \, A a c^{3} - 2 \, {\left (6 \, B a b + A b^{2}\right )} c^{2}\right )} x^{2} - 2 \, {\left (6 \, B a^{2} b + A a b^{2}\right )} c + {\left (3 \, B b^{4} + 8 \, A a b c^{2} - 2 \, {\left (6 \, B a b^{2} + A b^{3}\right )} c\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 (3 \, B a b^{2} c - 2 \, {\left (4 \, B a^{2} + A a b\right )} c^{2} + {\left (B b^{2} c^{2} - 4 \, B a c^{3}\right )} x^{2} + {\left (3 \, B b^{3} c + 4 \, A a c^{3} - 2 \, {\left (5 \, B a b + A b^{2}\right )} c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{4 \, {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} + {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{2} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x\right )}}, \frac {{\left (3 \, B a b^{3} + 8 \, A a^{2} c^{2} + {\left (3 \, B b^{3} c + 8 \, A a c^{3} - 2 \, {\left (6 \, B a b + A b^{2}\right )} c^{2}\right )} x^{2} - 2 \, {\left (6 \, B a^{2} b + A a b^{2}\right )} c + {\left (3 \, B b^{4} + 8 \, A a b c^{2} - 2 \, {\left (6 \, B a b^{2} + A b^{3}\right )} c\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 (3 \, B a b^{2} c - 2 \, {\left (4 \, B a^{2} + A a b\right )} c^{2} + {\left (B b^{2} c^{2} - 4 \, B a c^{3}\right )} x^{2} + {\left (3 \, B b^{3} c + 4 \, A a c^{3} - 2 \, {\left (5 \, B a b + A b^{2}\right )} c^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} + {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{2} + {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 177, normalized size = 1.16 \begin {gather*} \frac {{\left (\frac {{\left (B b^{2} c - 4 \, B a c^{2}\right )} x}{b^{2} c^{2} - 4 \, a c^{3}} + \frac {3 \, B b^{3} - 10 \, B a b c - 2 \, A b^{2} c + 4 \, A a c^{2}}{b^{2} c^{2} - 4 \, a c^{3}}\right )} x + \frac {3 \, B a b^{2} - 8 \, B a^{2} c - 2 \, A a b c}{b^{2} c^{2} - 4 \, a c^{3}}}{\sqrt {c x^{2} + b x + a}} + \frac {{\left (3 \, B b - 2 \, A c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{2 \, 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 = 382, normalized size = 2.50 \begin {gather*} \frac {A \,b^{2} x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c}+\frac {4 B a b x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c}-\frac {3 B \,b^{3} x}{2 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{2}}+\frac {A \,b^{3}}{2 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{2}}+\frac {2 B a \,b^{2}}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{2}}-\frac {3 B \,b^{4}}{4 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{3}}+\frac {B \,x^{2}}{\sqrt {c \,x^{2}+b x +a}\, c}-\frac {A x}{\sqrt {c \,x^{2}+b x +a}\, c}+\frac {3 B b x}{2 \sqrt {c \,x^{2}+b x +a}\, c^{2}}+\frac {A \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}-\frac {3 B b \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {5}{2}}}+\frac {A b}{2 \sqrt {c \,x^{2}+b x +a}\, c^{2}}+\frac {2 B a}{\sqrt {c \,x^{2}+b x +a}\, c^{2}}-\frac {3 B \,b^{2}}{4 \sqrt {c \,x^{2}+b x +a}\, c^{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 \frac {x^2\,\left (A+B\,x\right )}{{\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*} \int \frac {x^{2} \left (A + B x\right )}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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