Optimal. Leaf size=179 \[ -\frac {5 c^2 (6 b B-7 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{8 b^{9/2}}+\frac {5 c^2 \sqrt {x} (6 b B-7 A c)}{8 b^4 \sqrt {b x+c x^2}}+\frac {5 c (6 b B-7 A c)}{24 b^3 \sqrt {x} \sqrt {b x+c x^2}}-\frac {6 b B-7 A c}{12 b^2 x^{3/2} \sqrt {b x+c x^2}}-\frac {A}{3 b x^{5/2} \sqrt {b x+c x^2}} \]
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Rubi [A] time = 0.15, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {792, 672, 666, 660, 207} \begin {gather*} \frac {5 c^2 \sqrt {x} (6 b B-7 A c)}{8 b^4 \sqrt {b x+c x^2}}-\frac {5 c^2 (6 b B-7 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{8 b^{9/2}}+\frac {5 c (6 b B-7 A c)}{24 b^3 \sqrt {x} \sqrt {b x+c x^2}}-\frac {6 b B-7 A c}{12 b^2 x^{3/2} \sqrt {b x+c x^2}}-\frac {A}{3 b x^{5/2} \sqrt {b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 207
Rule 660
Rule 666
Rule 672
Rule 792
Rubi steps
\begin {align*} \int \frac {A+B x}{x^{5/2} \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {A}{3 b x^{5/2} \sqrt {b x+c x^2}}+\frac {\left (\frac {1}{2} (b B-2 A c)-\frac {5}{2} (-b B+A c)\right ) \int \frac {1}{x^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx}{3 b}\\ &=-\frac {A}{3 b x^{5/2} \sqrt {b x+c x^2}}-\frac {6 b B-7 A c}{12 b^2 x^{3/2} \sqrt {b x+c x^2}}-\frac {(5 c (6 b B-7 A c)) \int \frac {1}{\sqrt {x} \left (b x+c x^2\right )^{3/2}} \, dx}{24 b^2}\\ &=-\frac {A}{3 b x^{5/2} \sqrt {b x+c x^2}}-\frac {6 b B-7 A c}{12 b^2 x^{3/2} \sqrt {b x+c x^2}}+\frac {5 c (6 b B-7 A c)}{24 b^3 \sqrt {x} \sqrt {b x+c x^2}}+\frac {\left (5 c^2 (6 b B-7 A c)\right ) \int \frac {\sqrt {x}}{\left (b x+c x^2\right )^{3/2}} \, dx}{16 b^3}\\ &=-\frac {A}{3 b x^{5/2} \sqrt {b x+c x^2}}-\frac {6 b B-7 A c}{12 b^2 x^{3/2} \sqrt {b x+c x^2}}+\frac {5 c (6 b B-7 A c)}{24 b^3 \sqrt {x} \sqrt {b x+c x^2}}+\frac {5 c^2 (6 b B-7 A c) \sqrt {x}}{8 b^4 \sqrt {b x+c x^2}}+\frac {\left (5 c^2 (6 b B-7 A c)\right ) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{16 b^4}\\ &=-\frac {A}{3 b x^{5/2} \sqrt {b x+c x^2}}-\frac {6 b B-7 A c}{12 b^2 x^{3/2} \sqrt {b x+c x^2}}+\frac {5 c (6 b B-7 A c)}{24 b^3 \sqrt {x} \sqrt {b x+c x^2}}+\frac {5 c^2 (6 b B-7 A c) \sqrt {x}}{8 b^4 \sqrt {b x+c x^2}}+\frac {\left (5 c^2 (6 b B-7 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^4}\\ &=-\frac {A}{3 b x^{5/2} \sqrt {b x+c x^2}}-\frac {6 b B-7 A c}{12 b^2 x^{3/2} \sqrt {b x+c x^2}}+\frac {5 c (6 b B-7 A c)}{24 b^3 \sqrt {x} \sqrt {b x+c x^2}}+\frac {5 c^2 (6 b B-7 A c) \sqrt {x}}{8 b^4 \sqrt {b x+c x^2}}-\frac {5 c^2 (6 b B-7 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{8 b^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 62, normalized size = 0.35 \begin {gather*} \frac {c^2 x^3 (6 b B-7 A c) \, _2F_1\left (-\frac {1}{2},3;\frac {1}{2};\frac {c x}{b}+1\right )-A b^3}{3 b^4 x^{5/2} \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.93, size = 142, normalized size = 0.79 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (-8 A b^3+14 A b^2 c x-35 A b c^2 x^2-105 A c^3 x^3-12 b^3 B x+30 b^2 B c x^2+90 b B c^2 x^3\right )}{24 b^4 x^{7/2} (b+c x)}-\frac {5 \left (6 b B c^2-7 A c^3\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x+c x^2}}\right )}{8 b^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 359, normalized size = 2.01 \begin {gather*} \left [-\frac {15 \, {\left ({\left (6 \, B b c^{3} - 7 \, A c^{4}\right )} x^{5} + {\left (6 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{4}\right )} \sqrt {b} \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^{4} - 15 \, {\left (6 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{3} - 5 \, {\left (6 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{2} + 2 \, {\left (6 \, B b^{4} - 7 \, A b^{3} c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{48 \, {\left (b^{5} c x^{5} + b^{6} x^{4}\right )}}, \frac {15 \, {\left ({\left (6 \, B b c^{3} - 7 \, A c^{4}\right )} x^{5} + {\left (6 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{4}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) - {\left (8 \, A b^{4} - 15 \, {\left (6 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{3} - 5 \, {\left (6 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{2} + 2 \, {\left (6 \, B b^{4} - 7 \, A b^{3} c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{24 \, {\left (b^{5} c x^{5} + b^{6} x^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 165, normalized size = 0.92 \begin {gather*} \frac {5 \, {\left (6 \, B b c^{2} - 7 \, A c^{3}\right )} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{8 \, \sqrt {-b} b^{4}} + \frac {2 \, {\left (B b c^{2} - A c^{3}\right )}}{\sqrt {c x + b} b^{4}} + \frac {42 \, {\left (c x + b\right )}^{\frac {5}{2}} B b c^{2} - 96 \, {\left (c x + b\right )}^{\frac {3}{2}} B b^{2} c^{2} + 54 \, \sqrt {c x + b} B b^{3} c^{2} - 57 \, {\left (c x + b\right )}^{\frac {5}{2}} A c^{3} + 136 \, {\left (c x + b\right )}^{\frac {3}{2}} A b c^{3} - 87 \, \sqrt {c x + b} A b^{2} c^{3}}{24 \, b^{4} c^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 150, normalized size = 0.84 \begin {gather*} \frac {\sqrt {\left (c x +b \right ) x}\, \left (105 \sqrt {c x +b}\, A \,c^{3} x^{3} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-90 \sqrt {c x +b}\, B b \,c^{2} x^{3} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-105 A \sqrt {b}\, c^{3} x^{3}+90 B \,b^{\frac {3}{2}} c^{2} x^{3}-35 A \,b^{\frac {3}{2}} c^{2} x^{2}+30 B \,b^{\frac {5}{2}} c \,x^{2}+14 A \,b^{\frac {5}{2}} c x -12 B \,b^{\frac {7}{2}} x -8 A \,b^{\frac {7}{2}}\right )}{24 \left (c x +b \right ) b^{\frac {9}{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}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} x^{\frac {5}{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^{5/2}\,{\left (c\,x^2+b\,x\right )}^{3/2}} \,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 {5}{2}} \left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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