Optimal. Leaf size=140 \[ \frac {3 c (4 b B-5 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{7/2}}-\frac {3 c \sqrt {x} (4 b B-5 A c)}{4 b^3 \sqrt {b x+c x^2}}-\frac {4 b B-5 A c}{4 b^2 \sqrt {x} \sqrt {b x+c x^2}}-\frac {A}{2 b x^{3/2} \sqrt {b x+c x^2}} \]
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Rubi [A] time = 0.11, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {3 c \sqrt {x} (4 b B-5 A c)}{4 b^3 \sqrt {b x+c x^2}}-\frac {4 b B-5 A c}{4 b^2 \sqrt {x} \sqrt {b x+c x^2}}+\frac {3 c (4 b B-5 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{7/2}}-\frac {A}{2 b x^{3/2} \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^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {A}{2 b x^{3/2} \sqrt {b x+c x^2}}+\frac {\left (\frac {1}{2} (b B-2 A c)-\frac {3}{2} (-b B+A c)\right ) \int \frac {1}{\sqrt {x} \left (b x+c x^2\right )^{3/2}} \, dx}{2 b}\\ &=-\frac {A}{2 b x^{3/2} \sqrt {b x+c x^2}}-\frac {4 b B-5 A c}{4 b^2 \sqrt {x} \sqrt {b x+c x^2}}-\frac {(3 c (4 b B-5 A c)) \int \frac {\sqrt {x}}{\left (b x+c x^2\right )^{3/2}} \, dx}{8 b^2}\\ &=-\frac {A}{2 b x^{3/2} \sqrt {b x+c x^2}}-\frac {4 b B-5 A c}{4 b^2 \sqrt {x} \sqrt {b x+c x^2}}-\frac {3 c (4 b B-5 A c) \sqrt {x}}{4 b^3 \sqrt {b x+c x^2}}-\frac {(3 c (4 b B-5 A c)) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{8 b^3}\\ &=-\frac {A}{2 b x^{3/2} \sqrt {b x+c x^2}}-\frac {4 b B-5 A c}{4 b^2 \sqrt {x} \sqrt {b x+c x^2}}-\frac {3 c (4 b B-5 A c) \sqrt {x}}{4 b^3 \sqrt {b x+c x^2}}-\frac {(3 c (4 b B-5 A c)) \operatorname {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{4 b^3}\\ &=-\frac {A}{2 b x^{3/2} \sqrt {b x+c x^2}}-\frac {4 b B-5 A c}{4 b^2 \sqrt {x} \sqrt {b x+c x^2}}-\frac {3 c (4 b B-5 A c) \sqrt {x}}{4 b^3 \sqrt {b x+c x^2}}+\frac {3 c (4 b B-5 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 60, normalized size = 0.43 \begin {gather*} \frac {c x^2 (5 A c-4 b B) \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {c x}{b}+1\right )-A b^2}{2 b^3 x^{3/2} \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.35, size = 116, normalized size = 0.83 \begin {gather*} \frac {3 \left (4 b B c-5 A c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x+c x^2}}\right )}{4 b^{7/2}}+\frac {\sqrt {b x+c x^2} \left (-2 A b^2+5 A b c x+15 A c^2 x^2-4 b^2 B x-12 b B c x^2\right )}{4 b^3 x^{5/2} (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 304, normalized size = 2.17 \begin {gather*} \left [-\frac {3 \, {\left ({\left (4 \, B b c^{2} - 5 \, A c^{3}\right )} x^{4} + {\left (4 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{3}\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 (2 \, A b^{3} + 3 \, {\left (4 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{2} + {\left (4 \, B b^{3} - 5 \, A b^{2} c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{8 \, {\left (b^{4} c x^{4} + b^{5} x^{3}\right )}}, -\frac {3 \, {\left ({\left (4 \, B b c^{2} - 5 \, A c^{3}\right )} x^{4} + {\left (4 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + {\left (2 \, A b^{3} + 3 \, {\left (4 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{2} + {\left (4 \, B b^{3} - 5 \, A b^{2} c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{4 \, {\left (b^{4} c x^{4} + b^{5} x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 125, normalized size = 0.89 \begin {gather*} -\frac {3 \, {\left (4 \, B b c - 5 \, A c^{2}\right )} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{4 \, \sqrt {-b} b^{3}} - \frac {2 \, {\left (B b c - A c^{2}\right )}}{\sqrt {c x + b} b^{3}} - \frac {4 \, {\left (c x + b\right )}^{\frac {3}{2}} B b c - 4 \, \sqrt {c x + b} B b^{2} c - 7 \, {\left (c x + b\right )}^{\frac {3}{2}} A c^{2} + 9 \, \sqrt {c x + b} A b c^{2}}{4 \, b^{3} c^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 124, normalized size = 0.89 \begin {gather*} -\frac {\sqrt {\left (c x +b \right ) x}\, \left (15 \sqrt {c x +b}\, A \,c^{2} x^{2} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-12 \sqrt {c x +b}\, B b c \,x^{2} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-15 A \sqrt {b}\, c^{2} x^{2}+12 B \,b^{\frac {3}{2}} c \,x^{2}-5 A \,b^{\frac {3}{2}} c x +4 B \,b^{\frac {5}{2}} x +2 A \,b^{\frac {5}{2}}\right )}{4 \left (c x +b \right ) b^{\frac {7}{2}} x^{\frac {5}{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 {3}{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^{3/2}\,{\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 {3}{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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