Optimal. Leaf size=128 \[ \frac {\sqrt {b x+c x^2} (3 A c+2 b B)}{\sqrt {x}}-\sqrt {b} (3 A c+2 b B) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )+\frac {\left (b x+c x^2\right )^{3/2} (3 A c+2 b B)}{3 b x^{3/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{b x^{7/2}} \]
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Rubi [A] time = 0.13, antiderivative size = 128, 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, 664, 660, 207} \begin {gather*} \frac {\left (b x+c x^2\right )^{3/2} (3 A c+2 b B)}{3 b x^{3/2}}+\frac {\sqrt {b x+c x^2} (3 A c+2 b B)}{\sqrt {x}}-\sqrt {b} (3 A c+2 b B) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )-\frac {A \left (b x+c x^2\right )^{5/2}}{b x^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 207
Rule 660
Rule 664
Rule 792
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^{7/2}} \, dx &=-\frac {A \left (b x+c x^2\right )^{5/2}}{b x^{7/2}}+\frac {\left (-\frac {7}{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^{5/2}} \, dx}{b}\\ &=\frac {(2 b B+3 A c) \left (b x+c x^2\right )^{3/2}}{3 b x^{3/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{b x^{7/2}}+\frac {1}{2} (2 b B+3 A c) \int \frac {\sqrt {b x+c x^2}}{x^{3/2}} \, dx\\ &=\frac {(2 b B+3 A c) \sqrt {b x+c x^2}}{\sqrt {x}}+\frac {(2 b B+3 A c) \left (b x+c x^2\right )^{3/2}}{3 b x^{3/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{b x^{7/2}}+\frac {1}{2} (b (2 b B+3 A c)) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx\\ &=\frac {(2 b B+3 A c) \sqrt {b x+c x^2}}{\sqrt {x}}+\frac {(2 b B+3 A c) \left (b x+c x^2\right )^{3/2}}{3 b x^{3/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{b x^{7/2}}+(b (2 b B+3 A c)) \operatorname {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )\\ &=\frac {(2 b B+3 A c) \sqrt {b x+c x^2}}{\sqrt {x}}+\frac {(2 b B+3 A c) \left (b x+c x^2\right )^{3/2}}{3 b x^{3/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{b x^{7/2}}-\sqrt {b} (2 b B+3 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 94, normalized size = 0.73 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\sqrt {b+c x} (2 B x (4 b+c x)-3 A (b-2 c x))-3 \sqrt {b} x (3 A c+2 b B) \tanh ^{-1}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )\right )}{3 x^{3/2} \sqrt {b+c x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.75, size = 89, normalized size = 0.70 \begin {gather*} \left (-3 A \sqrt {b} c-2 b^{3/2} B\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x+c x^2}}\right )+\frac {\sqrt {b x+c x^2} \left (-3 A b+6 A c x+8 b B x+2 B c x^2\right )}{3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 184, normalized size = 1.44 \begin {gather*} \left [\frac {3 \, {\left (2 \, B b + 3 \, A c\right )} \sqrt {b} x^{2} \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 \, B c x^{2} - 3 \, A b + 2 \, {\left (4 \, B b + 3 \, A c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{6 \, x^{2}}, \frac {3 \, {\left (2 \, B b + 3 \, A c\right )} \sqrt {-b} x^{2} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + {\left (2 \, B c x^{2} - 3 \, A b + 2 \, {\left (4 \, B b + 3 \, A c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{3 \, x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 93, normalized size = 0.73 \begin {gather*} \frac {2 \, {\left (c x + b\right )}^{\frac {3}{2}} B c + 6 \, \sqrt {c x + b} B b c + 6 \, \sqrt {c x + b} A c^{2} - \frac {3 \, \sqrt {c x + b} A b c}{x} + \frac {3 \, {\left (2 \, B b^{2} c + 3 \, A b c^{2}\right )} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}}}{3 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 122, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {\left (c x +b \right ) x}\, \left (9 A b c x \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )+6 B \,b^{2} x \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-2 \sqrt {c x +b}\, B \sqrt {b}\, c \,x^{2}-6 \sqrt {c x +b}\, A \sqrt {b}\, c x -8 \sqrt {c x +b}\, B \,b^{\frac {3}{2}} x +3 \sqrt {c x +b}\, A \,b^{\frac {3}{2}}\right )}{3 \sqrt {c x +b}\, \sqrt {b}\, x^{\frac {3}{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*} \frac {2}{3} \, {\left (B c x + B b\right )} \sqrt {c x + b} + \int \frac {{\left (A b + {\left (B b + A c\right )} x\right )} \sqrt {c x + b}}{x^{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^{7/2}} \,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 {7}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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