Optimal. Leaf size=216 \[ -\frac {3 c^4 (2 b B-A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{128 b^{7/2}}+\frac {3 c^3 \sqrt {b x+c x^2} (2 b B-A c)}{128 b^3 x^{3/2}}-\frac {c^2 \sqrt {b x+c x^2} (2 b B-A c)}{64 b^2 x^{5/2}}-\frac {c \sqrt {b x+c x^2} (2 b B-A c)}{16 b x^{7/2}}-\frac {\left (b x+c x^2\right )^{3/2} (2 b B-A c)}{8 b x^{11/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{5 b x^{15/2}} \]
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Rubi [A] time = 0.19, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {792, 662, 672, 660, 207} \[ \frac {3 c^3 \sqrt {b x+c x^2} (2 b B-A c)}{128 b^3 x^{3/2}}-\frac {c^2 \sqrt {b x+c x^2} (2 b B-A c)}{64 b^2 x^{5/2}}-\frac {3 c^4 (2 b B-A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{128 b^{7/2}}-\frac {c \sqrt {b x+c x^2} (2 b B-A c)}{16 b x^{7/2}}-\frac {\left (b x+c x^2\right )^{3/2} (2 b B-A c)}{8 b x^{11/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{5 b x^{15/2}} \]
Antiderivative was successfully verified.
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Rule 207
Rule 660
Rule 662
Rule 672
Rule 792
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{x^{15/2}} \, dx &=-\frac {A \left (b x+c x^2\right )^{5/2}}{5 b x^{15/2}}+\frac {\left (-\frac {15}{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^{13/2}} \, dx}{5 b}\\ &=-\frac {(2 b B-A c) \left (b x+c x^2\right )^{3/2}}{8 b x^{11/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{5 b x^{15/2}}+\frac {(3 c (2 b B-A c)) \int \frac {\sqrt {b x+c x^2}}{x^{9/2}} \, dx}{16 b}\\ &=-\frac {c (2 b B-A c) \sqrt {b x+c x^2}}{16 b x^{7/2}}-\frac {(2 b B-A c) \left (b x+c x^2\right )^{3/2}}{8 b x^{11/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{5 b x^{15/2}}+\frac {\left (c^2 (2 b B-A c)\right ) \int \frac {1}{x^{5/2} \sqrt {b x+c x^2}} \, dx}{32 b}\\ &=-\frac {c (2 b B-A c) \sqrt {b x+c x^2}}{16 b x^{7/2}}-\frac {c^2 (2 b B-A c) \sqrt {b x+c x^2}}{64 b^2 x^{5/2}}-\frac {(2 b B-A c) \left (b x+c x^2\right )^{3/2}}{8 b x^{11/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{5 b x^{15/2}}-\frac {\left (3 c^3 (2 b B-A c)\right ) \int \frac {1}{x^{3/2} \sqrt {b x+c x^2}} \, dx}{128 b^2}\\ &=-\frac {c (2 b B-A c) \sqrt {b x+c x^2}}{16 b x^{7/2}}-\frac {c^2 (2 b B-A c) \sqrt {b x+c x^2}}{64 b^2 x^{5/2}}+\frac {3 c^3 (2 b B-A c) \sqrt {b x+c x^2}}{128 b^3 x^{3/2}}-\frac {(2 b B-A c) \left (b x+c x^2\right )^{3/2}}{8 b x^{11/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{5 b x^{15/2}}+\frac {\left (3 c^4 (2 b B-A c)\right ) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{256 b^3}\\ &=-\frac {c (2 b B-A c) \sqrt {b x+c x^2}}{16 b x^{7/2}}-\frac {c^2 (2 b B-A c) \sqrt {b x+c x^2}}{64 b^2 x^{5/2}}+\frac {3 c^3 (2 b B-A c) \sqrt {b x+c x^2}}{128 b^3 x^{3/2}}-\frac {(2 b B-A c) \left (b x+c x^2\right )^{3/2}}{8 b x^{11/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{5 b x^{15/2}}+\frac {\left (3 c^4 (2 b B-A c)\right ) \operatorname {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{128 b^3}\\ &=-\frac {c (2 b B-A c) \sqrt {b x+c x^2}}{16 b x^{7/2}}-\frac {c^2 (2 b B-A c) \sqrt {b x+c x^2}}{64 b^2 x^{5/2}}+\frac {3 c^3 (2 b B-A c) \sqrt {b x+c x^2}}{128 b^3 x^{3/2}}-\frac {(2 b B-A c) \left (b x+c x^2\right )^{3/2}}{8 b x^{11/2}}-\frac {A \left (b x+c x^2\right )^{5/2}}{5 b x^{15/2}}-\frac {3 c^4 (2 b B-A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{128 b^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 61, normalized size = 0.28 \[ -\frac {(x (b+c x))^{5/2} \left (A b^5+c^4 x^5 (2 b B-A c) \, _2F_1\left (\frac {5}{2},5;\frac {7}{2};\frac {c x}{b}+1\right )\right )}{5 b^6 x^{15/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 335, normalized size = 1.55 \[ \left [-\frac {15 \, {\left (2 \, B b c^{4} - A c^{5}\right )} \sqrt {b} x^{6} \log \left (-\frac {c x^{2} + 2 \, b x + 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) + 2 \, {\left (128 \, A b^{5} - 15 \, {\left (2 \, B b^{2} c^{3} - A b c^{4}\right )} x^{4} + 10 \, {\left (2 \, B b^{3} c^{2} - A b^{2} c^{3}\right )} x^{3} + 8 \, {\left (30 \, B b^{4} c + A b^{3} c^{2}\right )} x^{2} + 16 \, {\left (10 \, B b^{5} + 11 \, A b^{4} c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{1280 \, b^{4} x^{6}}, \frac {15 \, {\left (2 \, B b c^{4} - A c^{5}\right )} \sqrt {-b} x^{6} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) - {\left (128 \, A b^{5} - 15 \, {\left (2 \, B b^{2} c^{3} - A b c^{4}\right )} x^{4} + 10 \, {\left (2 \, B b^{3} c^{2} - A b^{2} c^{3}\right )} x^{3} + 8 \, {\left (30 \, B b^{4} c + A b^{3} c^{2}\right )} x^{2} + 16 \, {\left (10 \, B b^{5} + 11 \, A b^{4} c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{640 \, b^{4} x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 192, normalized size = 0.89 \[ \frac {\frac {15 \, {\left (2 \, B b c^{5} - A c^{6}\right )} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{3}} + \frac {30 \, {\left (c x + b\right )}^{\frac {9}{2}} B b c^{5} - 140 \, {\left (c x + b\right )}^{\frac {7}{2}} B b^{2} c^{5} + 140 \, {\left (c x + b\right )}^{\frac {3}{2}} B b^{4} c^{5} - 30 \, \sqrt {c x + b} B b^{5} c^{5} - 15 \, {\left (c x + b\right )}^{\frac {9}{2}} A c^{6} + 70 \, {\left (c x + b\right )}^{\frac {7}{2}} A b c^{6} - 128 \, {\left (c x + b\right )}^{\frac {5}{2}} A b^{2} c^{6} - 70 \, {\left (c x + b\right )}^{\frac {3}{2}} A b^{3} c^{6} + 15 \, \sqrt {c x + b} A b^{4} c^{6}}{b^{3} c^{5} x^{5}}}{640 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 223, normalized size = 1.03 \[ \frac {\sqrt {\left (c x +b \right ) x}\, \left (15 A \,c^{5} x^{5} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-30 B b \,c^{4} x^{5} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-15 \sqrt {c x +b}\, A \sqrt {b}\, c^{4} x^{4}+30 \sqrt {c x +b}\, B \,b^{\frac {3}{2}} c^{3} x^{4}+10 \sqrt {c x +b}\, A \,b^{\frac {3}{2}} c^{3} x^{3}-20 \sqrt {c x +b}\, B \,b^{\frac {5}{2}} c^{2} x^{3}-8 \sqrt {c x +b}\, A \,b^{\frac {5}{2}} c^{2} x^{2}-240 \sqrt {c x +b}\, B \,b^{\frac {7}{2}} c \,x^{2}-176 \sqrt {c x +b}\, A \,b^{\frac {7}{2}} c x -160 \sqrt {c x +b}\, B \,b^{\frac {9}{2}} x -128 \sqrt {c x +b}\, A \,b^{\frac {9}{2}}\right )}{640 \sqrt {c x +b}\, b^{\frac {7}{2}} x^{\frac {11}{2}}} \]
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 \[ \int \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} {\left (B x + A\right )}}{x^{\frac {15}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right )}{x^{15/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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