Optimal. Leaf size=137 \[ \frac {3 c \sqrt {b x+c x^2} (A c+4 b B)}{4 b \sqrt {x}}-\frac {3 c (A c+4 b B) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 \sqrt {b}}-\frac {\left (b x+c x^2\right )^{3/2} (A c+4 b B)}{4 b x^{5/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{2 b x^{9/2}} \]
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Rubi [A] time = 0.13, antiderivative size = 137, 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, 662, 664, 660, 207} \begin {gather*} -\frac {\left (b x+c x^2\right )^{3/2} (A c+4 b B)}{4 b x^{5/2}}+\frac {3 c \sqrt {b x+c x^2} (A c+4 b B)}{4 b \sqrt {x}}-\frac {3 c (A c+4 b B) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 \sqrt {b}}-\frac {A \left (b x+c x^2\right )^{5/2}}{2 b x^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 207
Rule 660
Rule 662
Rule 664
Rule 792
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^{9/2}} \, dx &=-\frac {A \left (b x+c x^2\right )^{5/2}}{2 b x^{9/2}}+\frac {\left (-\frac {9}{2} (-b B+A c)+\frac {5}{2} (-b B+2 A c)\right ) \int \frac {\left (b x+c x^2\right )^{3/2}}{x^{7/2}} \, dx}{2 b}\\ &=-\frac {(4 b B+A c) \left (b x+c x^2\right )^{3/2}}{4 b x^{5/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{2 b x^{9/2}}+\frac {(3 c (4 b B+A c)) \int \frac {\sqrt {b x+c x^2}}{x^{3/2}} \, dx}{8 b}\\ &=\frac {3 c (4 b B+A c) \sqrt {b x+c x^2}}{4 b \sqrt {x}}-\frac {(4 b B+A c) \left (b x+c x^2\right )^{3/2}}{4 b x^{5/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{2 b x^{9/2}}+\frac {1}{8} (3 c (4 b B+A c)) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx\\ &=\frac {3 c (4 b B+A c) \sqrt {b x+c x^2}}{4 b \sqrt {x}}-\frac {(4 b B+A c) \left (b x+c x^2\right )^{3/2}}{4 b x^{5/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{2 b x^{9/2}}+\frac {1}{4} (3 c (4 b B+A c)) \operatorname {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )\\ &=\frac {3 c (4 b B+A c) \sqrt {b x+c x^2}}{4 b \sqrt {x}}-\frac {(4 b B+A c) \left (b x+c x^2\right )^{3/2}}{4 b x^{5/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{2 b x^{9/2}}-\frac {3 c (4 b B+A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 \sqrt {b}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 59, normalized size = 0.43 \begin {gather*} \frac {(x (b+c x))^{5/2} \left (c x^2 (A c+4 b B) \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {c x}{b}+1\right )-5 A b^2\right )}{10 b^3 x^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.80, size = 90, normalized size = 0.66 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (-2 A b-5 A c x-4 b B x+8 B c x^2\right )}{4 x^{5/2}}-\frac {3 \left (A c^2+4 b B c\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x+c x^2}}\right )}{4 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 206, normalized size = 1.50 \begin {gather*} \left [\frac {3 \, {\left (4 \, B b c + A c^{2}\right )} \sqrt {b} x^{3} \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 \, B b c x^{2} - 2 \, A b^{2} - {\left (4 \, B b^{2} + 5 \, A b c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{8 \, b x^{3}}, \frac {3 \, {\left (4 \, B b c + A c^{2}\right )} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + {\left (8 \, B b c x^{2} - 2 \, A b^{2} - {\left (4 \, B b^{2} + 5 \, A b c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{4 \, b x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 119, normalized size = 0.87 \begin {gather*} \frac {8 \, \sqrt {c x + b} B c^{2} + \frac {3 \, {\left (4 \, B b c^{2} + A c^{3}\right )} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {4 \, {\left (c x + b\right )}^{\frac {3}{2}} B b c^{2} - 4 \, \sqrt {c x + b} B b^{2} c^{2} + 5 \, {\left (c x + b\right )}^{\frac {3}{2}} A c^{3} - 3 \, \sqrt {c x + b} A b c^{3}}{c^{2} x^{2}}}{4 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 126, normalized size = 0.92 \begin {gather*} -\frac {\sqrt {\left (c x +b \right ) x}\, \left (3 A \,c^{2} x^{2} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )+12 B b c \,x^{2} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-8 \sqrt {c x +b}\, B \sqrt {b}\, c \,x^{2}+5 \sqrt {c x +b}\, A \sqrt {b}\, c x +4 \sqrt {c x +b}\, B \,b^{\frac {3}{2}} x +2 \sqrt {c x +b}\, A \,b^{\frac {3}{2}}\right )}{4 \sqrt {c x +b}\, \sqrt {b}\, 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 {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{x^{\frac {9}{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 {{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right )}{x^{9/2}} \,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 {\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )}{x^{\frac {9}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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