Optimal. Leaf size=184 \[ -\frac {A \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{a^{5/2}}+\frac {2 \left (c x \left (16 a^2 B c-20 a A b c+3 A b^3\right )+A \left (24 a^2 c^2-22 a b^2 c+3 b^4\right )+8 a^2 b B c\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.13, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {822, 12, 724, 206} \begin {gather*} \frac {2 \left (c x \left (16 a^2 B c-20 a A b c+3 A b^3\right )+A \left (24 a^2 c^2-22 a b^2 c+3 b^4\right )+8 a^2 b B c\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {A \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{a^{5/2}}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 724
Rule 822
Rubi steps
\begin {align*} \int \frac {A+B x}{x \left (a+b x+c x^2\right )^{5/2}} \, dx &=\frac {2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \int \frac {-\frac {3}{2} A \left (b^2-4 a c\right )-2 (A b-2 a B) c x}{x \left (a+b x+c x^2\right )^{3/2}} \, dx}{3 a \left (b^2-4 a c\right )}\\ &=\frac {2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (8 a^2 b B c+A \left (3 b^4-22 a b^2 c+24 a^2 c^2\right )+c \left (3 A b^3-20 a A b c+16 a^2 B c\right ) x\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {4 \int \frac {3 A \left (b^2-4 a c\right )^2}{4 x \sqrt {a+b x+c x^2}} \, dx}{3 a^2 \left (b^2-4 a c\right )^2}\\ &=\frac {2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (8 a^2 b B c+A \left (3 b^4-22 a b^2 c+24 a^2 c^2\right )+c \left (3 A b^3-20 a A b c+16 a^2 B c\right ) x\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {A \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{a^2}\\ &=\frac {2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (8 a^2 b B c+A \left (3 b^4-22 a b^2 c+24 a^2 c^2\right )+c \left (3 A b^3-20 a A b c+16 a^2 B c\right ) x\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {(2 A) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x}{\sqrt {a+b x+c x^2}}\right )}{a^2}\\ &=\frac {2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (8 a^2 b B c+A \left (3 b^4-22 a b^2 c+24 a^2 c^2\right )+c \left (3 A b^3-20 a A b c+16 a^2 B c\right ) x\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {A \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{a^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 179, normalized size = 0.97 \begin {gather*} -\frac {A \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )}{a^{5/2}}+\frac {A \left (48 a^2 c^2-44 a b^2 c-40 a b c^2 x+6 b^4+6 b^3 c x\right )+16 a^2 B c (b+2 c x)}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+x (b+c x)}}+\frac {2 a B (b+2 c x)-2 A \left (-2 a c+b^2+b c x\right )}{3 a \left (4 a c-b^2\right ) (a+x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.47, size = 241, normalized size = 1.31 \begin {gather*} \frac {2 A \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}-\frac {\sqrt {a+b x+c x^2}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {2 \left (32 a^3 A c^2+12 a^3 b B c+24 a^3 B c^2 x-28 a^2 A b^2 c+24 a^2 A c^3 x^2-a^2 b^3 B+6 a^2 b^2 B c x+24 a^2 b B c^2 x^2+16 a^2 B c^3 x^3+4 a A b^4-18 a A b^3 c x-42 a A b^2 c^2 x^2-20 a A b c^3 x^3+3 A b^5 x+6 A b^4 c x^2+3 A b^3 c^2 x^3\right )}{3 a^2 \left (4 a c-b^2\right )^2 \left (a+b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 2.37, size = 1077, normalized size = 5.85 \begin {gather*} \left [\frac {3 \, {\left (A a^{2} b^{4} - 8 \, A a^{3} b^{2} c + 16 \, A a^{4} c^{2} + {\left (A b^{4} c^{2} - 8 \, A a b^{2} c^{3} + 16 \, A a^{2} c^{4}\right )} x^{4} + 2 \, {\left (A b^{5} c - 8 \, A a b^{3} c^{2} + 16 \, A a^{2} b c^{3}\right )} x^{3} + {\left (A b^{6} - 6 \, A a b^{4} c + 32 \, A a^{3} c^{3}\right )} x^{2} + 2 \, {\left (A a b^{5} - 8 \, A a^{2} b^{3} c + 16 \, A a^{3} b c^{2}\right )} x\right )} \sqrt {a} \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) - 4 \, {\left (B a^{3} b^{3} - 4 \, A a^{2} b^{4} - 32 \, A a^{4} c^{2} - {\left (3 \, A a b^{3} c^{2} + 4 \, {\left (4 \, B a^{3} - 5 \, A a^{2} b\right )} c^{3}\right )} x^{3} - 6 \, {\left (A a b^{4} c + 4 \, A a^{3} c^{3} + {\left (4 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} c^{2}\right )} x^{2} - 4 \, {\left (3 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} c - 3 \, {\left (A a b^{5} + 8 \, B a^{4} c^{2} + 2 \, {\left (B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} c\right )} x\right )} \sqrt {c x^{2} + b x + a}}{6 \, {\left (a^{5} b^{4} - 8 \, a^{6} b^{2} c + 16 \, a^{7} c^{2} + {\left (a^{3} b^{4} c^{2} - 8 \, a^{4} b^{2} c^{3} + 16 \, a^{5} c^{4}\right )} x^{4} + 2 \, {\left (a^{3} b^{5} c - 8 \, a^{4} b^{3} c^{2} + 16 \, a^{5} b c^{3}\right )} x^{3} + {\left (a^{3} b^{6} - 6 \, a^{4} b^{4} c + 32 \, a^{6} c^{3}\right )} x^{2} + 2 \, {\left (a^{4} b^{5} - 8 \, a^{5} b^{3} c + 16 \, a^{6} b c^{2}\right )} x\right )}}, \frac {3 \, {\left (A a^{2} b^{4} - 8 \, A a^{3} b^{2} c + 16 \, A a^{4} c^{2} + {\left (A b^{4} c^{2} - 8 \, A a b^{2} c^{3} + 16 \, A a^{2} c^{4}\right )} x^{4} + 2 \, {\left (A b^{5} c - 8 \, A a b^{3} c^{2} + 16 \, A a^{2} b c^{3}\right )} x^{3} + {\left (A b^{6} - 6 \, A a b^{4} c + 32 \, A a^{3} c^{3}\right )} x^{2} + 2 \, {\left (A a b^{5} - 8 \, A a^{2} b^{3} c + 16 \, A a^{3} b c^{2}\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) - 2 \, {\left (B a^{3} b^{3} - 4 \, A a^{2} b^{4} - 32 \, A a^{4} c^{2} - {\left (3 \, A a b^{3} c^{2} + 4 \, {\left (4 \, B a^{3} - 5 \, A a^{2} b\right )} c^{3}\right )} x^{3} - 6 \, {\left (A a b^{4} c + 4 \, A a^{3} c^{3} + {\left (4 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} c^{2}\right )} x^{2} - 4 \, {\left (3 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} c - 3 \, {\left (A a b^{5} + 8 \, B a^{4} c^{2} + 2 \, {\left (B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} c\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (a^{5} b^{4} - 8 \, a^{6} b^{2} c + 16 \, a^{7} c^{2} + {\left (a^{3} b^{4} c^{2} - 8 \, a^{4} b^{2} c^{3} + 16 \, a^{5} c^{4}\right )} x^{4} + 2 \, {\left (a^{3} b^{5} c - 8 \, a^{4} b^{3} c^{2} + 16 \, a^{5} b c^{3}\right )} x^{3} + {\left (a^{3} b^{6} - 6 \, a^{4} b^{4} c + 32 \, a^{6} c^{3}\right )} x^{2} + 2 \, {\left (a^{4} b^{5} - 8 \, a^{5} b^{3} c + 16 \, a^{6} b c^{2}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 333, normalized size = 1.81 \begin {gather*} \frac {2 \, {\left ({\left ({\left (\frac {{\left (3 \, A a^{5} b^{3} c^{2} + 16 \, B a^{7} c^{3} - 20 \, A a^{6} b c^{3}\right )} x}{a^{7} b^{4} - 8 \, a^{8} b^{2} c + 16 \, a^{9} c^{2}} + \frac {6 \, {\left (A a^{5} b^{4} c + 4 \, B a^{7} b c^{2} - 7 \, A a^{6} b^{2} c^{2} + 4 \, A a^{7} c^{3}\right )}}{a^{7} b^{4} - 8 \, a^{8} b^{2} c + 16 \, a^{9} c^{2}}\right )} x + \frac {3 \, {\left (A a^{5} b^{5} + 2 \, B a^{7} b^{2} c - 6 \, A a^{6} b^{3} c + 8 \, B a^{8} c^{2}\right )}}{a^{7} b^{4} - 8 \, a^{8} b^{2} c + 16 \, a^{9} c^{2}}\right )} x - \frac {B a^{7} b^{3} - 4 \, A a^{6} b^{4} - 12 \, B a^{8} b c + 28 \, A a^{7} b^{2} c - 32 \, A a^{8} c^{2}}{a^{7} b^{4} - 8 \, a^{8} b^{2} c + 16 \, a^{9} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} + \frac {2 \, A \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 390, normalized size = 2.12 \begin {gather*} -\frac {16 A b \,c^{2} x}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}\, a}+\frac {32 B \,c^{2} x}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}-\frac {8 A \,b^{2} c}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}\, a}-\frac {2 A b c x}{3 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a}+\frac {16 B b c}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}+\frac {4 B c x}{3 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {A \,b^{2}}{3 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a}-\frac {2 A b c x}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{2}}+\frac {2 B b}{3 \left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {A \,b^{2}}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{2}}+\frac {A}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a}-\frac {A \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{a^{\frac {5}{2}}}+\frac {A}{\sqrt {c \,x^{2}+b x +a}\, a^{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 {A+B\,x}{x\,{\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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