Optimal. Leaf size=189 \[ -\frac {2 \left (x \left (24 a^2 B c^2+8 a A b c^2-22 a b^2 B c+3 b^4 B\right )+a \left (16 a A c^2-20 a b B c+3 b^3 B\right )\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 x^2 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {B \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{5/2}} \]
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Rubi [A] time = 0.13, antiderivative size = 189, 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, 777, 621, 206} \begin {gather*} -\frac {2 \left (x \left (24 a^2 B c^2+8 a A b c^2-22 a b^2 B c+3 b^4 B\right )+a \left (16 a A c^2-20 a b B c+3 b^3 B\right )\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 x^2 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {B \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 777
Rule 818
Rubi steps
\begin {align*} \int \frac {x^3 (A+B x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 x^2 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \int \frac {x \left (2 a (b B-2 A c)+\frac {3}{2} B \left (b^2-4 a c\right ) x\right )}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 c \left (b^2-4 a c\right )}\\ &=-\frac {2 x^2 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (a \left (3 b^3 B-20 a b B c+16 a A c^2\right )+\left (3 b^4 B-22 a b^2 B c+8 a A b c^2+24 a^2 B c^2\right ) x\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {B \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{c^2}\\ &=-\frac {2 x^2 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (a \left (3 b^3 B-20 a b B c+16 a A c^2\right )+\left (3 b^4 B-22 a b^2 B c+8 a A b c^2+24 a^2 B c^2\right ) x\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {(2 B) \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^2 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (a \left (3 b^3 B-20 a b B c+16 a A c^2\right )+\left (3 b^4 B-22 a b^2 B c+8 a A b c^2+24 a^2 B c^2\right ) x\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {B \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 201, normalized size = 1.06 \begin {gather*} \frac {B \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{c^{5/2}}-\frac {2 \left (4 a^3 c (4 A c-5 b B+6 B c x)+a^2 \left (24 A b c^2 x+8 c^3 x^2 (3 A+4 B x)+3 b^3 B-42 b^2 B c x\right )+2 a b x \left (b c^2 x (3 A-14 B x)+6 A c^3 x^2+3 b^3 B-9 b^2 B c x\right )+b^3 x^2 \left (-A c^2 x+3 b^2 B+4 b B c x\right )\right )}{3 c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.73, size = 246, normalized size = 1.30 \begin {gather*} -\frac {2 \left (16 a^3 A c^2-20 a^3 b B c+24 a^3 B c^2 x+24 a^2 A b c^2 x+24 a^2 A c^3 x^2+3 a^2 b^3 B-42 a^2 b^2 B c x+32 a^2 B c^3 x^3+6 a A b^2 c^2 x^2+12 a A b c^3 x^3+6 a b^4 B x-18 a b^3 B c x^2-28 a b^2 B c^2 x^3-A b^3 c^2 x^3+3 b^5 B x^2+4 b^4 B c x^3\right )}{3 c^2 \left (4 a c-b^2\right )^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {B \log \left (-2 c^{5/2} \sqrt {a+b x+c x^2}+b c^2+2 c^3 x\right )}{c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.49, size = 1061, normalized size = 5.61 \begin {gather*} \left [\frac {3 \, {\left (B a^{2} b^{4} - 8 \, B a^{3} b^{2} c + 16 \, B a^{4} c^{2} + {\left (B b^{4} c^{2} - 8 \, B a b^{2} c^{3} + 16 \, B a^{2} c^{4}\right )} x^{4} + 2 \, {\left (B b^{5} c - 8 \, B a b^{3} c^{2} + 16 \, B a^{2} b c^{3}\right )} x^{3} + {\left (B b^{6} - 6 \, B a b^{4} c + 32 \, B a^{3} c^{3}\right )} x^{2} + 2 \, {\left (B a b^{5} - 8 \, B a^{2} b^{3} c + 16 \, B a^{3} b 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 (3 \, B a^{2} b^{3} c - 20 \, B a^{3} b c^{2} + 16 \, A a^{3} c^{3} + {\left (4 \, B b^{4} c^{2} + 4 \, {\left (8 \, B a^{2} + 3 \, A a b\right )} c^{4} - {\left (28 \, B a b^{2} + A b^{3}\right )} c^{3}\right )} x^{3} + 3 \, {\left (B b^{5} c - 6 \, B a b^{3} c^{2} + 2 \, A a b^{2} c^{3} + 8 \, A a^{2} c^{4}\right )} x^{2} + 6 \, {\left (B a b^{4} c - 7 \, B a^{2} b^{2} c^{2} + 4 \, {\left (B a^{3} + A a^{2} b\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{6 \, {\left (a^{2} b^{4} c^{3} - 8 \, a^{3} b^{2} c^{4} + 16 \, a^{4} c^{5} + {\left (b^{4} c^{5} - 8 \, a b^{2} c^{6} + 16 \, a^{2} c^{7}\right )} x^{4} + 2 \, {\left (b^{5} c^{4} - 8 \, a b^{3} c^{5} + 16 \, a^{2} b c^{6}\right )} x^{3} + {\left (b^{6} c^{3} - 6 \, a b^{4} c^{4} + 32 \, a^{3} c^{6}\right )} x^{2} + 2 \, {\left (a b^{5} c^{3} - 8 \, a^{2} b^{3} c^{4} + 16 \, a^{3} b c^{5}\right )} x\right )}}, -\frac {3 \, {\left (B a^{2} b^{4} - 8 \, B a^{3} b^{2} c + 16 \, B a^{4} c^{2} + {\left (B b^{4} c^{2} - 8 \, B a b^{2} c^{3} + 16 \, B a^{2} c^{4}\right )} x^{4} + 2 \, {\left (B b^{5} c - 8 \, B a b^{3} c^{2} + 16 \, B a^{2} b c^{3}\right )} x^{3} + {\left (B b^{6} - 6 \, B a b^{4} c + 32 \, B a^{3} c^{3}\right )} x^{2} + 2 \, {\left (B a b^{5} - 8 \, B a^{2} b^{3} c + 16 \, B a^{3} b 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 (3 \, B a^{2} b^{3} c - 20 \, B a^{3} b c^{2} + 16 \, A a^{3} c^{3} + {\left (4 \, B b^{4} c^{2} + 4 \, {\left (8 \, B a^{2} + 3 \, A a b\right )} c^{4} - {\left (28 \, B a b^{2} + A b^{3}\right )} c^{3}\right )} x^{3} + 3 \, {\left (B b^{5} c - 6 \, B a b^{3} c^{2} + 2 \, A a b^{2} c^{3} + 8 \, A a^{2} c^{4}\right )} x^{2} + 6 \, {\left (B a b^{4} c - 7 \, B a^{2} b^{2} c^{2} + 4 \, {\left (B a^{3} + A a^{2} b\right )} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (a^{2} b^{4} c^{3} - 8 \, a^{3} b^{2} c^{4} + 16 \, a^{4} c^{5} + {\left (b^{4} c^{5} - 8 \, a b^{2} c^{6} + 16 \, a^{2} c^{7}\right )} x^{4} + 2 \, {\left (b^{5} c^{4} - 8 \, a b^{3} c^{5} + 16 \, a^{2} b c^{6}\right )} x^{3} + {\left (b^{6} c^{3} - 6 \, a b^{4} c^{4} + 32 \, a^{3} c^{6}\right )} x^{2} + 2 \, {\left (a b^{5} c^{3} - 8 \, a^{2} b^{3} c^{4} + 16 \, a^{3} b c^{5}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 314, normalized size = 1.66 \begin {gather*} -\frac {2 \, {\left ({\left ({\left (\frac {{\left (4 \, B b^{4} c - 28 \, B a b^{2} c^{2} - A b^{3} c^{2} + 32 \, B a^{2} c^{3} + 12 \, A a b c^{3}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac {3 \, {\left (B b^{5} - 6 \, B a b^{3} c + 2 \, A a b^{2} c^{2} + 8 \, A a^{2} c^{3}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac {6 \, {\left (B a b^{4} - 7 \, B a^{2} b^{2} c + 4 \, B a^{3} c^{2} + 4 \, A a^{2} b c^{2}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac {3 \, B a^{2} b^{3} - 20 \, B a^{3} b c + 16 \, A a^{3} c^{2}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} - \frac {B \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{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 = 860, normalized size = 4.55 \begin {gather*} -\frac {8 A a b x}{\left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}+\frac {2 A \,b^{3} x}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}\, c}+\frac {4 B a \,b^{2} x}{\left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}\, c}-\frac {B \,b^{4} x}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}\, c^{2}}-\frac {4 A a \,b^{2}}{\left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}\, c}-\frac {A a b x}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c}+\frac {A \,b^{4}}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}\, c^{2}}+\frac {A \,b^{3} x}{12 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{2}}+\frac {2 B a \,b^{3}}{\left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}\, c^{2}}+\frac {B a \,b^{2} x}{2 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{2}}-\frac {B \,b^{5}}{6 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}\, c^{3}}-\frac {B \,b^{4} x}{24 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{3}}-\frac {B \,x^{3}}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c}-\frac {A a \,b^{2}}{2 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{2}}+\frac {A \,b^{4}}{24 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{3}}-\frac {A \,x^{2}}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c}+\frac {B a \,b^{3}}{4 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{3}}-\frac {B \,b^{5}}{48 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{4}}+\frac {B \,b^{2} x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{2}}+\frac {B b \,x^{2}}{2 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{2}}-\frac {A b x}{4 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{2}}+\frac {B \,b^{3}}{2 \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, c^{3}}+\frac {B \,b^{2} x}{8 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{3}}-\frac {2 A a}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{2}}+\frac {A \,b^{2}}{24 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{3}}+\frac {B a b}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{3}}-\frac {B \,b^{3}}{48 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{4}}-\frac {B x}{\sqrt {c \,x^{2}+b x +a}\, c^{2}}+\frac {B \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {5}{2}}}+\frac {B b}{2 \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^3\,\left (A+B\,x\right )}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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