Optimal. Leaf size=179 \[ -\frac {3 \left (A \left (4 a c+b^2\right )+4 a b B\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {a}}+\frac {3 \left (4 a B c+4 A b c+b^2 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c}}-\frac {(A-B x) \left (a+b x+c x^2\right )^{3/2}}{2 x^2}-\frac {3 \sqrt {a+b x+c x^2} (2 a B-x (2 A c+b B)+A b)}{4 x} \]
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Rubi [A] time = 0.17, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {812, 843, 621, 206, 724} \begin {gather*} -\frac {3 \left (A \left (4 a c+b^2\right )+4 a b B\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {a}}+\frac {3 \left (4 a B c+4 A b c+b^2 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c}}-\frac {(A-B x) \left (a+b x+c x^2\right )^{3/2}}{2 x^2}-\frac {3 \sqrt {a+b x+c x^2} (2 a B-x (2 A c+b B)+A b)}{4 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 812
Rule 843
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^{3/2}}{x^3} \, dx &=-\frac {(A-B x) \left (a+b x+c x^2\right )^{3/2}}{2 x^2}-\frac {3}{8} \int \frac {(-2 (A b+2 a B)-2 (b B+2 A c) x) \sqrt {a+b x+c x^2}}{x^2} \, dx\\ &=-\frac {3 (A b+2 a B-(b B+2 A c) x) \sqrt {a+b x+c x^2}}{4 x}-\frac {(A-B x) \left (a+b x+c x^2\right )^{3/2}}{2 x^2}+\frac {3}{16} \int \frac {2 \left (4 a b B+A \left (b^2+4 a c\right )\right )+2 \left (b^2 B+4 A b c+4 a B c\right ) x}{x \sqrt {a+b x+c x^2}} \, dx\\ &=-\frac {3 (A b+2 a B-(b B+2 A c) x) \sqrt {a+b x+c x^2}}{4 x}-\frac {(A-B x) \left (a+b x+c x^2\right )^{3/2}}{2 x^2}+\frac {1}{8} \left (3 \left (b^2 B+4 A b c+4 a B c\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx+\frac {1}{8} \left (3 \left (4 a b B+A \left (b^2+4 a c\right )\right )\right ) \int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx\\ &=-\frac {3 (A b+2 a B-(b B+2 A c) x) \sqrt {a+b x+c x^2}}{4 x}-\frac {(A-B x) \left (a+b x+c x^2\right )^{3/2}}{2 x^2}+\frac {1}{4} \left (3 \left (b^2 B+4 A b c+4 a B c\right )\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 )-\frac {1}{4} \left (3 \left (4 a b B+A \left (b^2+4 a c\right )\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 )\\ &=-\frac {3 (A b+2 a B-(b B+2 A c) x) \sqrt {a+b x+c x^2}}{4 x}-\frac {(A-B x) \left (a+b x+c x^2\right )^{3/2}}{2 x^2}-\frac {3 \left (4 a b B+A \left (b^2+4 a c\right )\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {a}}+\frac {3 \left (b^2 B+4 A b c+4 a B c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 162, normalized size = 0.91 \begin {gather*} \frac {1}{8} \left (-\frac {3 \left (A \left (4 a c+b^2\right )+4 a b B\right ) \tanh ^{-1}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+x (b+c x)}}\right )}{\sqrt {a}}+\frac {3 \left (4 a B c+4 A b c+b^2 B\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{\sqrt {c}}+\frac {2 \sqrt {a+x (b+c x)} (x (A (4 c x-5 b)+B x (5 b+2 c x))-2 a (A+2 B x))}{x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.28, size = 166, normalized size = 0.93 \begin {gather*} -\frac {3 \left (4 a B c+4 A b c+b^2 B\right ) \log \left (-2 \sqrt {c} \sqrt {a+b x+c x^2}+b+2 c x\right )}{8 \sqrt {c}}-\frac {3 \left (4 a A c+4 a b B+A b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x+c x^2}-\sqrt {c} x}{\sqrt {a}}\right )}{4 \sqrt {a}}+\frac {\sqrt {a+b x+c x^2} \left (-2 a A-4 a B x-5 A b x+4 A c x^2+5 b B x^2+2 B c x^3\right )}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.90, size = 921, normalized size = 5.15 \begin {gather*} \left [\frac {3 \, {\left (B a b^{2} + 4 \, {\left (B a^{2} + A a b\right )} c\right )} \sqrt {c} x^{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 ) + 3 \, {\left (4 \, A a c^{2} + {\left (4 \, B a b + A b^{2}\right )} c\right )} \sqrt {a} x^{2} \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 (2 \, B a c^{2} x^{3} - 2 \, A a^{2} c - {\left (4 \, B a^{2} + 5 \, A a b\right )} c x + {\left (5 \, B a b c + 4 \, A a c^{2}\right )} x^{2}\right )} \sqrt {c x^{2} + b x + a}}{16 \, a c x^{2}}, -\frac {6 \, {\left (B a b^{2} + 4 \, {\left (B a^{2} + A a b\right )} c\right )} \sqrt {-c} x^{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 ) - 3 \, {\left (4 \, A a c^{2} + {\left (4 \, B a b + A b^{2}\right )} c\right )} \sqrt {a} x^{2} \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 (2 \, B a c^{2} x^{3} - 2 \, A a^{2} c - {\left (4 \, B a^{2} + 5 \, A a b\right )} c x + {\left (5 \, B a b c + 4 \, A a c^{2}\right )} x^{2}\right )} \sqrt {c x^{2} + b x + a}}{16 \, a c x^{2}}, \frac {6 \, {\left (4 \, A a c^{2} + {\left (4 \, B a b + A b^{2}\right )} c\right )} \sqrt {-a} x^{2} \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 ) + 3 \, {\left (B a b^{2} + 4 \, {\left (B a^{2} + A a b\right )} c\right )} \sqrt {c} x^{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 (2 \, B a c^{2} x^{3} - 2 \, A a^{2} c - {\left (4 \, B a^{2} + 5 \, A a b\right )} c x + {\left (5 \, B a b c + 4 \, A a c^{2}\right )} x^{2}\right )} \sqrt {c x^{2} + b x + a}}{16 \, a c x^{2}}, \frac {3 \, {\left (4 \, A a c^{2} + {\left (4 \, B a b + A b^{2}\right )} c\right )} \sqrt {-a} x^{2} \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 ) - 3 \, {\left (B a b^{2} + 4 \, {\left (B a^{2} + A a b\right )} c\right )} \sqrt {-c} x^{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 (2 \, B a c^{2} x^{3} - 2 \, A a^{2} c - {\left (4 \, B a^{2} + 5 \, A a b\right )} c x + {\left (5 \, B a b c + 4 \, A a c^{2}\right )} x^{2}\right )} \sqrt {c x^{2} + b x + a}}{8 \, a c x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.38, size = 412, normalized size = 2.30 \begin {gather*} \frac {1}{4} \, {\left (2 \, B c x + \frac {5 \, B b c + 4 \, A c^{2}}{c}\right )} \sqrt {c x^{2} + b x + a} + \frac {3 \, {\left (4 \, B a b + A b^{2} + 4 \, A a c\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a}} - \frac {3 \, {\left (B b^{2} + 4 \, B a c + 4 \, A b c\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{8 \, \sqrt {c}} + \frac {4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} B a b + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A b^{2} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} A a c + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} B a^{2} \sqrt {c} + 16 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} A a b \sqrt {c} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} B a^{2} b - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a b^{2} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} A a^{2} c - 8 \, B a^{3} \sqrt {c} - 8 \, A a^{2} b \sqrt {c}}{4 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 463, normalized size = 2.59 \begin {gather*} -\frac {3 A \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 A \,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 A b \sqrt {c}\, \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2}+\frac {3 B a \sqrt {c}\, \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2}-\frac {3 B \sqrt {a}\, b \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 {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 \sqrt {c}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, A b c x}{4 a}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, B c x}{2}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, A \,b^{2}}{4 a}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A b c x}{4 a^{2}}+\frac {3 \sqrt {c \,x^{2}+b x +a}\, A c}{2}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B c x}{a}+\frac {9 \sqrt {c \,x^{2}+b x +a}\, B b}{4}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A c}{2 a}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} A \,b^{2}}{4 a^{2}}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} B b}{a}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A b}{4 a^{2} x}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} B}{a x}-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} A}{2 a \,x^{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 {\left (A+B\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{x^3} \,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^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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