Optimal. Leaf size=169 \[ -\frac {7 b^{3/2} (9 b B-5 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 c^{11/2}}+\frac {7 b \sqrt {x} (9 b B-5 A c)}{4 c^5}-\frac {7 x^{3/2} (9 b B-5 A c)}{12 c^4}+\frac {7 x^{5/2} (9 b B-5 A c)}{20 b c^3}-\frac {x^{7/2} (9 b B-5 A c)}{4 b c^2 (b+c x)}-\frac {x^{9/2} (b B-A c)}{2 b c (b+c x)^2} \]
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Rubi [A] time = 0.09, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {781, 78, 47, 50, 63, 205} \[ -\frac {7 b^{3/2} (9 b B-5 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 c^{11/2}}-\frac {x^{7/2} (9 b B-5 A c)}{4 b c^2 (b+c x)}+\frac {7 x^{5/2} (9 b B-5 A c)}{20 b c^3}-\frac {7 x^{3/2} (9 b B-5 A c)}{12 c^4}+\frac {7 b \sqrt {x} (9 b B-5 A c)}{4 c^5}-\frac {x^{9/2} (b B-A c)}{2 b c (b+c x)^2} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 78
Rule 205
Rule 781
Rubi steps
\begin {align*} \int \frac {x^{13/2} (A+B x)}{\left (b x+c x^2\right )^3} \, dx &=\int \frac {x^{7/2} (A+B x)}{(b+c x)^3} \, dx\\ &=-\frac {(b B-A c) x^{9/2}}{2 b c (b+c x)^2}-\frac {\left (-\frac {9 b B}{2}+\frac {5 A c}{2}\right ) \int \frac {x^{7/2}}{(b+c x)^2} \, dx}{2 b c}\\ &=-\frac {(b B-A c) x^{9/2}}{2 b c (b+c x)^2}-\frac {(9 b B-5 A c) x^{7/2}}{4 b c^2 (b+c x)}+\frac {(7 (9 b B-5 A c)) \int \frac {x^{5/2}}{b+c x} \, dx}{8 b c^2}\\ &=\frac {7 (9 b B-5 A c) x^{5/2}}{20 b c^3}-\frac {(b B-A c) x^{9/2}}{2 b c (b+c x)^2}-\frac {(9 b B-5 A c) x^{7/2}}{4 b c^2 (b+c x)}-\frac {(7 (9 b B-5 A c)) \int \frac {x^{3/2}}{b+c x} \, dx}{8 c^3}\\ &=-\frac {7 (9 b B-5 A c) x^{3/2}}{12 c^4}+\frac {7 (9 b B-5 A c) x^{5/2}}{20 b c^3}-\frac {(b B-A c) x^{9/2}}{2 b c (b+c x)^2}-\frac {(9 b B-5 A c) x^{7/2}}{4 b c^2 (b+c x)}+\frac {(7 b (9 b B-5 A c)) \int \frac {\sqrt {x}}{b+c x} \, dx}{8 c^4}\\ &=\frac {7 b (9 b B-5 A c) \sqrt {x}}{4 c^5}-\frac {7 (9 b B-5 A c) x^{3/2}}{12 c^4}+\frac {7 (9 b B-5 A c) x^{5/2}}{20 b c^3}-\frac {(b B-A c) x^{9/2}}{2 b c (b+c x)^2}-\frac {(9 b B-5 A c) x^{7/2}}{4 b c^2 (b+c x)}-\frac {\left (7 b^2 (9 b B-5 A c)\right ) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{8 c^5}\\ &=\frac {7 b (9 b B-5 A c) \sqrt {x}}{4 c^5}-\frac {7 (9 b B-5 A c) x^{3/2}}{12 c^4}+\frac {7 (9 b B-5 A c) x^{5/2}}{20 b c^3}-\frac {(b B-A c) x^{9/2}}{2 b c (b+c x)^2}-\frac {(9 b B-5 A c) x^{7/2}}{4 b c^2 (b+c x)}-\frac {\left (7 b^2 (9 b B-5 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{b+c x^2} \, dx,x,\sqrt {x}\right )}{4 c^5}\\ &=\frac {7 b (9 b B-5 A c) \sqrt {x}}{4 c^5}-\frac {7 (9 b B-5 A c) x^{3/2}}{12 c^4}+\frac {7 (9 b B-5 A c) x^{5/2}}{20 b c^3}-\frac {(b B-A c) x^{9/2}}{2 b c (b+c x)^2}-\frac {(9 b B-5 A c) x^{7/2}}{4 b c^2 (b+c x)}-\frac {7 b^{3/2} (9 b B-5 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{4 c^{11/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 61, normalized size = 0.36 \[ \frac {x^{9/2} \left (\frac {9 b^2 (A c-b B)}{(b+c x)^2}+(9 b B-5 A c) \, _2F_1\left (2,\frac {9}{2};\frac {11}{2};-\frac {c x}{b}\right )\right )}{18 b^3 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.18, size = 408, normalized size = 2.41 \[ \left [-\frac {105 \, {\left (9 \, B b^{4} - 5 \, A b^{3} c + {\left (9 \, B b^{2} c^{2} - 5 \, A b c^{3}\right )} x^{2} + 2 \, {\left (9 \, B b^{3} c - 5 \, A b^{2} c^{2}\right )} x\right )} \sqrt {-\frac {b}{c}} \log \left (\frac {c x + 2 \, c \sqrt {x} \sqrt {-\frac {b}{c}} - b}{c x + b}\right ) - 2 \, {\left (24 \, B c^{4} x^{4} + 945 \, B b^{4} - 525 \, A b^{3} c - 8 \, {\left (9 \, B b c^{3} - 5 \, A c^{4}\right )} x^{3} + 56 \, {\left (9 \, B b^{2} c^{2} - 5 \, A b c^{3}\right )} x^{2} + 175 \, {\left (9 \, B b^{3} c - 5 \, A b^{2} c^{2}\right )} x\right )} \sqrt {x}}{120 \, {\left (c^{7} x^{2} + 2 \, b c^{6} x + b^{2} c^{5}\right )}}, -\frac {105 \, {\left (9 \, B b^{4} - 5 \, A b^{3} c + {\left (9 \, B b^{2} c^{2} - 5 \, A b c^{3}\right )} x^{2} + 2 \, {\left (9 \, B b^{3} c - 5 \, A b^{2} c^{2}\right )} x\right )} \sqrt {\frac {b}{c}} \arctan \left (\frac {c \sqrt {x} \sqrt {\frac {b}{c}}}{b}\right ) - {\left (24 \, B c^{4} x^{4} + 945 \, B b^{4} - 525 \, A b^{3} c - 8 \, {\left (9 \, B b c^{3} - 5 \, A c^{4}\right )} x^{3} + 56 \, {\left (9 \, B b^{2} c^{2} - 5 \, A b c^{3}\right )} x^{2} + 175 \, {\left (9 \, B b^{3} c - 5 \, A b^{2} c^{2}\right )} x\right )} \sqrt {x}}{60 \, {\left (c^{7} x^{2} + 2 \, b c^{6} x + b^{2} c^{5}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 146, normalized size = 0.86 \[ -\frac {7 \, {\left (9 \, B b^{3} - 5 \, A b^{2} c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \, \sqrt {b c} c^{5}} + \frac {17 \, B b^{3} c x^{\frac {3}{2}} - 13 \, A b^{2} c^{2} x^{\frac {3}{2}} + 15 \, B b^{4} \sqrt {x} - 11 \, A b^{3} c \sqrt {x}}{4 \, {\left (c x + b\right )}^{2} c^{5}} + \frac {2 \, {\left (3 \, B c^{12} x^{\frac {5}{2}} - 15 \, B b c^{11} x^{\frac {3}{2}} + 5 \, A c^{12} x^{\frac {3}{2}} + 90 \, B b^{2} c^{10} \sqrt {x} - 45 \, A b c^{11} \sqrt {x}\right )}}{15 \, c^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 178, normalized size = 1.05 \[ -\frac {13 A \,b^{2} x^{\frac {3}{2}}}{4 \left (c x +b \right )^{2} c^{3}}+\frac {17 B \,b^{3} x^{\frac {3}{2}}}{4 \left (c x +b \right )^{2} c^{4}}-\frac {11 A \,b^{3} \sqrt {x}}{4 \left (c x +b \right )^{2} c^{4}}+\frac {15 B \,b^{4} \sqrt {x}}{4 \left (c x +b \right )^{2} c^{5}}+\frac {2 B \,x^{\frac {5}{2}}}{5 c^{3}}+\frac {35 A \,b^{2} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \sqrt {b c}\, c^{4}}-\frac {63 B \,b^{3} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \sqrt {b c}\, c^{5}}+\frac {2 A \,x^{\frac {3}{2}}}{3 c^{3}}-\frac {2 B b \,x^{\frac {3}{2}}}{c^{4}}-\frac {6 A b \sqrt {x}}{c^{4}}+\frac {12 B \,b^{2} \sqrt {x}}{c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.13, size = 151, normalized size = 0.89 \[ \frac {{\left (17 \, B b^{3} c - 13 \, A b^{2} c^{2}\right )} x^{\frac {3}{2}} + {\left (15 \, B b^{4} - 11 \, A b^{3} c\right )} \sqrt {x}}{4 \, {\left (c^{7} x^{2} + 2 \, b c^{6} x + b^{2} c^{5}\right )}} - \frac {7 \, {\left (9 \, B b^{3} - 5 \, A b^{2} c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{4 \, \sqrt {b c} c^{5}} + \frac {2 \, {\left (3 \, B c^{2} x^{\frac {5}{2}} - 5 \, {\left (3 \, B b c - A c^{2}\right )} x^{\frac {3}{2}} + 45 \, {\left (2 \, B b^{2} - A b c\right )} \sqrt {x}\right )}}{15 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 183, normalized size = 1.08 \[ x^{3/2}\,\left (\frac {2\,A}{3\,c^3}-\frac {2\,B\,b}{c^4}\right )-\frac {x^{3/2}\,\left (\frac {13\,A\,b^2\,c^2}{4}-\frac {17\,B\,b^3\,c}{4}\right )-\sqrt {x}\,\left (\frac {15\,B\,b^4}{4}-\frac {11\,A\,b^3\,c}{4}\right )}{b^2\,c^5+2\,b\,c^6\,x+c^7\,x^2}-\sqrt {x}\,\left (\frac {3\,b\,\left (\frac {2\,A}{c^3}-\frac {6\,B\,b}{c^4}\right )}{c}+\frac {6\,B\,b^2}{c^5}\right )+\frac {2\,B\,x^{5/2}}{5\,c^3}-\frac {7\,b^{3/2}\,\mathrm {atan}\left (\frac {b^{3/2}\,\sqrt {c}\,\sqrt {x}\,\left (5\,A\,c-9\,B\,b\right )}{9\,B\,b^3-5\,A\,b^2\,c}\right )\,\left (5\,A\,c-9\,B\,b\right )}{4\,c^{11/2}} \]
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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