Optimal. Leaf size=206 \[ -\frac {\left (6 a B \left (4 a c+b^2\right )-A \left (b^3-12 a b c\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{16 a^{3/2}}+\frac {\sqrt {a+b x+c x^2} \left (2 c x (6 a B+A b)-8 a A c-6 a b B+A b^2\right )}{8 a x}+\frac {1}{2} \sqrt {c} (2 A c+3 b B) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )-\frac {\left (a+b x+c x^2\right )^{3/2} (3 x (2 a B+A b)+4 a A)}{12 a x^3} \]
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Rubi [A] time = 0.22, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {810, 812, 843, 621, 206, 724} \begin {gather*} -\frac {\left (6 a B \left (4 a c+b^2\right )-A \left (b^3-12 a b c\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{16 a^{3/2}}+\frac {\sqrt {a+b x+c x^2} \left (2 c x (6 a B+A b)-8 a A c-6 a b B+A b^2\right )}{8 a x}-\frac {\left (a+b x+c x^2\right )^{3/2} (3 x (2 a B+A b)+4 a A)}{12 a x^3}+\frac {1}{2} \sqrt {c} (2 A c+3 b B) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 810
Rule 812
Rule 843
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^4} \, dx &=-\frac {(4 a A+3 (A b+2 a B) x) \left (a+b x+c x^2\right )^{3/2}}{12 a x^3}-\frac {\int \frac {\left (\frac {1}{2} \left (-6 a b B+A \left (b^2-8 a c\right )\right )-(A b+6 a B) c x\right ) \sqrt {a+b x+c x^2}}{x^2} \, dx}{4 a}\\ &=\frac {\left (A b^2-6 a b B-8 a A c+2 (A b+6 a B) c x\right ) \sqrt {a+b x+c x^2}}{8 a x}-\frac {(4 a A+3 (A b+2 a B) x) \left (a+b x+c x^2\right )^{3/2}}{12 a x^3}+\frac {\int \frac {\frac {1}{2} \left (6 a B \left (b^2+4 a c\right )-2 A \left (\frac {b^3}{2}-6 a b c\right )\right )+4 a c (3 b B+2 A c) x}{x \sqrt {a+b x+c x^2}} \, dx}{8 a}\\ &=\frac {\left (A b^2-6 a b B-8 a A c+2 (A b+6 a B) c x\right ) \sqrt {a+b x+c x^2}}{8 a x}-\frac {(4 a A+3 (A b+2 a B) x) \left (a+b x+c x^2\right )^{3/2}}{12 a x^3}+\frac {1}{2} (c (3 b B+2 A c)) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx+\frac {\left (6 a B \left (b^2+4 a c\right )-A \left (b^3-12 a b c\right )\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx}{16 a}\\ &=\frac {\left (A b^2-6 a b B-8 a A c+2 (A b+6 a B) c x\right ) \sqrt {a+b x+c x^2}}{8 a x}-\frac {(4 a A+3 (A b+2 a B) x) \left (a+b x+c x^2\right )^{3/2}}{12 a x^3}+(c (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 )-\frac {\left (6 a B \left (b^2+4 a c\right )-A \left (b^3-12 a b c\right )\right ) \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 )}{8 a}\\ &=\frac {\left (A b^2-6 a b B-8 a A c+2 (A b+6 a B) c x\right ) \sqrt {a+b x+c x^2}}{8 a x}-\frac {(4 a A+3 (A b+2 a B) x) \left (a+b x+c x^2\right )^{3/2}}{12 a x^3}-\frac {\left (6 a B \left (b^2+4 a c\right )-A \left (b^3-12 a b c\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{16 a^{3/2}}+\frac {1}{2} \sqrt {c} (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 )\\ \end {align*}
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Mathematica [A] time = 0.50, size = 182, normalized size = 0.88 \begin {gather*} \frac {1}{48} \left (\frac {3 \left (A \left (b^3-12 a b c\right )-6 a B \left (4 a c+b^2\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )}{a^{3/2}}-\frac {2 \sqrt {a+x (b+c x)} \left (4 a^2 (2 A+3 B x)+2 a x (A (7 b+16 c x)+3 B x (5 b-4 c x))+3 A b^2 x^2\right )}{a x^3}+24 \sqrt {c} (2 A c+3 b B) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.51, size = 193, normalized size = 0.94 \begin {gather*} \frac {\sqrt {a+b x+c x^2} \left (-8 a^2 A-12 a^2 B x-14 a A b x-32 a A c x^2-30 a b B x^2+24 a B c x^3-3 A b^2 x^2\right )}{24 a x^3}+\frac {\left (-24 a^2 B c-12 a A b c-6 a b^2 B+A b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x+c x^2}-\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{3/2}}+\frac {1}{2} \left (-2 A c^{3/2}-3 b B \sqrt {c}\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.01, size = 953, normalized size = 4.63 \begin {gather*} \left [\frac {24 \, {\left (3 \, B a^{2} b + 2 \, A a^{2} c\right )} \sqrt {c} x^{3} \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 ) + 3 \, {\left (6 \, B a b^{2} - A b^{3} + 12 \, {\left (2 \, B a^{2} + A a b\right )} c\right )} \sqrt {a} x^{3} \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 (24 \, B a^{2} c x^{3} - 8 \, A a^{3} - {\left (30 \, B a^{2} b + 3 \, A a b^{2} + 32 \, A a^{2} c\right )} x^{2} - 2 \, {\left (6 \, B a^{3} + 7 \, A a^{2} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{96 \, a^{2} x^{3}}, -\frac {48 \, {\left (3 \, B a^{2} b + 2 \, A a^{2} c\right )} \sqrt {-c} x^{3} \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 ) - 3 \, {\left (6 \, B a b^{2} - A b^{3} + 12 \, {\left (2 \, B a^{2} + A a b\right )} c\right )} \sqrt {a} x^{3} \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 (24 \, B a^{2} c x^{3} - 8 \, A a^{3} - {\left (30 \, B a^{2} b + 3 \, A a b^{2} + 32 \, A a^{2} c\right )} x^{2} - 2 \, {\left (6 \, B a^{3} + 7 \, A a^{2} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{96 \, a^{2} x^{3}}, \frac {3 \, {\left (6 \, B a b^{2} - A b^{3} + 12 \, {\left (2 \, B a^{2} + A a b\right )} c\right )} \sqrt {-a} x^{3} \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 ) + 12 \, {\left (3 \, B a^{2} b + 2 \, A a^{2} c\right )} \sqrt {c} x^{3} \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 ) + 2 \, {\left (24 \, B a^{2} c x^{3} - 8 \, A a^{3} - {\left (30 \, B a^{2} b + 3 \, A a b^{2} + 32 \, A a^{2} c\right )} x^{2} - 2 \, {\left (6 \, B a^{3} + 7 \, A a^{2} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{48 \, a^{2} x^{3}}, \frac {3 \, {\left (6 \, B a b^{2} - A b^{3} + 12 \, {\left (2 \, B a^{2} + A a b\right )} c\right )} \sqrt {-a} x^{3} \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 ) - 24 \, {\left (3 \, B a^{2} b + 2 \, A a^{2} c\right )} \sqrt {-c} x^{3} \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 (24 \, B a^{2} c x^{3} - 8 \, A a^{3} - {\left (30 \, B a^{2} b + 3 \, A a b^{2} + 32 \, A a^{2} c\right )} x^{2} - 2 \, {\left (6 \, B a^{3} + 7 \, A a^{2} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{48 \, a^{2} x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.44, size = 630, normalized size = 3.06 \begin {gather*} \sqrt {c x^{2} + b x + a} B c - \frac {{\left (3 \, B b c^{\frac {3}{2}} + 2 \, A c^{\frac {5}{2}}\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} c - b \sqrt {c} \right |}\right )}{2 \, c} + \frac {{\left (6 \, B a b^{2} - A b^{3} + 24 \, B a^{2} c + 12 \, A a b c\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a} + \frac {30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a b^{2} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A b^{3} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} B a^{2} c + 60 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{5} A a b c + 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} B a^{2} b \sqrt {c} + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} A a b^{2} \sqrt {c} + 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{4} A a^{2} c^{\frac {3}{2}} - 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a^{2} b^{2} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a b^{3} - 144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{3} b \sqrt {c} - 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a^{3} c^{\frac {3}{2}} + 18 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{3} b^{2} - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{2} b^{3} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{4} c + 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{3} b c + 48 \, B a^{4} b \sqrt {c} + 64 \, A a^{4} c^{\frac {3}{2}}}{24 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{3} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 635, normalized size = 3.08 \begin {gather*} -\frac {3 A b c \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{4 \sqrt {a}}+\frac {A \,b^{3} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{16 a^{\frac {3}{2}}}+A \,c^{\frac {3}{2}} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )-\frac {3 B \sqrt {a}\, c \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{2}-\frac {3 B \,b^{2} \ln \left (\frac {b x +2 a +2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a}}{x}\right )}{8 \sqrt {a}}+\frac {3 B b \sqrt {c}\, \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2}+\frac {\sqrt {c \,x^{2}+b x +a}\, A \,c^{2} x}{a}-\frac {\sqrt {c \,x^{2}+b x +a}\, A \,b^{2} c x}{8 a^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, B b c x}{4 a}+\frac {5 \sqrt {c \,x^{2}+b x +a}\, A b c}{4 a}-\frac {\sqrt {c \,x^{2}+b x +a}\, A \,b^{3}}{8 a^{2}}+\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,c^{2} x}{3 a^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{2} c x}{24 a^{3}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, B \,b^{2}}{4 a}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B b c x}{4 a^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, B c}{2}+\frac {7 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A b c}{12 a^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{3}}{24 a^{3}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B c}{2 a}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B \,b^{2}}{4 a^{2}}-\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A c}{3 a^{2} x}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A \,b^{2}}{24 a^{3} x}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B b}{4 a^{2} x}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A b}{12 a^{2} x^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B}{2 a \,x^{2}}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A}{3 a \,x^{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 = -0.00 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{x^4} \,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 {\left (A + B x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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