Optimal. Leaf size=142 \[ -\frac {c^2 (6 b B-5 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{8 b^{7/2}}+\frac {c \sqrt {b x+c x^2} (6 b B-5 A c)}{8 b^3 x^{3/2}}-\frac {\sqrt {b x+c x^2} (6 b B-5 A c)}{12 b^2 x^{5/2}}-\frac {A \sqrt {b x+c x^2}}{3 b x^{7/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {792, 672, 660, 207} \begin {gather*} -\frac {c^2 (6 b B-5 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{8 b^{7/2}}+\frac {c \sqrt {b x+c x^2} (6 b B-5 A c)}{8 b^3 x^{3/2}}-\frac {\sqrt {b x+c x^2} (6 b B-5 A c)}{12 b^2 x^{5/2}}-\frac {A \sqrt {b x+c x^2}}{3 b x^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 207
Rule 660
Rule 672
Rule 792
Rubi steps
\begin {align*} \int \frac {A+B x}{x^{7/2} \sqrt {b x+c x^2}} \, dx &=-\frac {A \sqrt {b x+c x^2}}{3 b x^{7/2}}+\frac {\left (-\frac {7}{2} (-b B+A c)+\frac {1}{2} (-b B+2 A c)\right ) \int \frac {1}{x^{5/2} \sqrt {b x+c x^2}} \, dx}{3 b}\\ &=-\frac {A \sqrt {b x+c x^2}}{3 b x^{7/2}}-\frac {(6 b B-5 A c) \sqrt {b x+c x^2}}{12 b^2 x^{5/2}}-\frac {(c (6 b B-5 A c)) \int \frac {1}{x^{3/2} \sqrt {b x+c x^2}} \, dx}{8 b^2}\\ &=-\frac {A \sqrt {b x+c x^2}}{3 b x^{7/2}}-\frac {(6 b B-5 A c) \sqrt {b x+c x^2}}{12 b^2 x^{5/2}}+\frac {c (6 b B-5 A c) \sqrt {b x+c x^2}}{8 b^3 x^{3/2}}+\frac {\left (c^2 (6 b B-5 A c)\right ) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{16 b^3}\\ &=-\frac {A \sqrt {b x+c x^2}}{3 b x^{7/2}}-\frac {(6 b B-5 A c) \sqrt {b x+c x^2}}{12 b^2 x^{5/2}}+\frac {c (6 b B-5 A c) \sqrt {b x+c x^2}}{8 b^3 x^{3/2}}+\frac {\left (c^2 (6 b B-5 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{8 b^3}\\ &=-\frac {A \sqrt {b x+c x^2}}{3 b x^{7/2}}-\frac {(6 b B-5 A c) \sqrt {b x+c x^2}}{12 b^2 x^{5/2}}+\frac {c (6 b B-5 A c) \sqrt {b x+c x^2}}{8 b^3 x^{3/2}}-\frac {c^2 (6 b B-5 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{8 b^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 61, normalized size = 0.43 \begin {gather*} -\frac {\sqrt {x (b+c x)} \left (A b^3+c^2 x^3 (6 b B-5 A c) \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};\frac {c x}{b}+1\right )\right )}{3 b^4 x^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.35, size = 111, normalized size = 0.78 \begin {gather*} \frac {\left (5 A c^3-6 b B c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x+c x^2}}\right )}{8 b^{7/2}}+\frac {\sqrt {b x+c x^2} \left (-8 A b^2+10 A b c x-15 A c^2 x^2-12 b^2 B x+18 b B c x^2\right )}{24 b^3 x^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 241, normalized size = 1.70 \begin {gather*} \left [-\frac {3 \, {\left (6 \, B b c^{2} - 5 \, A c^{3}\right )} \sqrt {b} x^{4} \log \left (-\frac {c x^{2} + 2 \, b x + 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) + 2 \, {\left (8 \, A b^{3} - 3 \, {\left (6 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{2} + 2 \, {\left (6 \, B b^{3} - 5 \, A b^{2} c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{48 \, b^{4} x^{4}}, \frac {3 \, {\left (6 \, B b c^{2} - 5 \, A c^{3}\right )} \sqrt {-b} x^{4} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) - {\left (8 \, A b^{3} - 3 \, {\left (6 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{2} + 2 \, {\left (6 \, B b^{3} - 5 \, A b^{2} c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{24 \, b^{4} x^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 144, normalized size = 1.01 \begin {gather*} \frac {\frac {3 \, {\left (6 \, B b c^{3} - 5 \, A c^{4}\right )} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{3}} + \frac {18 \, {\left (c x + b\right )}^{\frac {5}{2}} B b c^{3} - 48 \, {\left (c x + b\right )}^{\frac {3}{2}} B b^{2} c^{3} + 30 \, \sqrt {c x + b} B b^{3} c^{3} - 15 \, {\left (c x + b\right )}^{\frac {5}{2}} A c^{4} + 40 \, {\left (c x + b\right )}^{\frac {3}{2}} A b c^{4} - 33 \, \sqrt {c x + b} A b^{2} c^{4}}{b^{3} c^{3} x^{3}}}{24 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 147, normalized size = 1.04 \begin {gather*} \frac {\sqrt {\left (c x +b \right ) x}\, \left (15 A \,c^{3} x^{3} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-18 B b \,c^{2} x^{3} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-15 \sqrt {c x +b}\, A \sqrt {b}\, c^{2} x^{2}+18 \sqrt {c x +b}\, B \,b^{\frac {3}{2}} c \,x^{2}+10 \sqrt {c x +b}\, A \,b^{\frac {3}{2}} c x -12 \sqrt {c x +b}\, B \,b^{\frac {5}{2}} x -8 \sqrt {c x +b}\, A \,b^{\frac {5}{2}}\right )}{24 \sqrt {c x +b}\, b^{\frac {7}{2}} x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {B x + A}{\sqrt {c x^{2} + b x} x^{\frac {7}{2}}}\,{d x} \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^{7/2}\,\sqrt {c\,x^2+b\,x}} \,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 {A + B x}{x^{\frac {7}{2}} \sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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