Optimal. Leaf size=156 \[ -\frac {c^{5/2} (7 b B-9 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{11/2}}-\frac {c^2 (7 b B-9 A c)}{b^5 \sqrt {x}}+\frac {c (7 b B-9 A c)}{3 b^4 x^{3/2}}-\frac {7 b B-9 A c}{5 b^3 x^{5/2}}+\frac {7 b B-9 A c}{7 b^2 c x^{7/2}}-\frac {b B-A c}{b c x^{7/2} (b+c x)} \]
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Rubi [A] time = 0.08, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {781, 78, 51, 63, 205} \begin {gather*} -\frac {c^2 (7 b B-9 A c)}{b^5 \sqrt {x}}-\frac {c^{5/2} (7 b B-9 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{11/2}}+\frac {c (7 b B-9 A c)}{3 b^4 x^{3/2}}-\frac {7 b B-9 A c}{5 b^3 x^{5/2}}+\frac {7 b B-9 A c}{7 b^2 c x^{7/2}}-\frac {b B-A c}{b c x^{7/2} (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 205
Rule 781
Rubi steps
\begin {align*} \int \frac {A+B x}{x^{5/2} \left (b x+c x^2\right )^2} \, dx &=\int \frac {A+B x}{x^{9/2} (b+c x)^2} \, dx\\ &=-\frac {b B-A c}{b c x^{7/2} (b+c x)}-\frac {\left (\frac {7 b B}{2}-\frac {9 A c}{2}\right ) \int \frac {1}{x^{9/2} (b+c x)} \, dx}{b c}\\ &=\frac {7 b B-9 A c}{7 b^2 c x^{7/2}}-\frac {b B-A c}{b c x^{7/2} (b+c x)}+\frac {(7 b B-9 A c) \int \frac {1}{x^{7/2} (b+c x)} \, dx}{2 b^2}\\ &=\frac {7 b B-9 A c}{7 b^2 c x^{7/2}}-\frac {7 b B-9 A c}{5 b^3 x^{5/2}}-\frac {b B-A c}{b c x^{7/2} (b+c x)}-\frac {(c (7 b B-9 A c)) \int \frac {1}{x^{5/2} (b+c x)} \, dx}{2 b^3}\\ &=\frac {7 b B-9 A c}{7 b^2 c x^{7/2}}-\frac {7 b B-9 A c}{5 b^3 x^{5/2}}+\frac {c (7 b B-9 A c)}{3 b^4 x^{3/2}}-\frac {b B-A c}{b c x^{7/2} (b+c x)}+\frac {\left (c^2 (7 b B-9 A c)\right ) \int \frac {1}{x^{3/2} (b+c x)} \, dx}{2 b^4}\\ &=\frac {7 b B-9 A c}{7 b^2 c x^{7/2}}-\frac {7 b B-9 A c}{5 b^3 x^{5/2}}+\frac {c (7 b B-9 A c)}{3 b^4 x^{3/2}}-\frac {c^2 (7 b B-9 A c)}{b^5 \sqrt {x}}-\frac {b B-A c}{b c x^{7/2} (b+c x)}-\frac {\left (c^3 (7 b B-9 A c)\right ) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{2 b^5}\\ &=\frac {7 b B-9 A c}{7 b^2 c x^{7/2}}-\frac {7 b B-9 A c}{5 b^3 x^{5/2}}+\frac {c (7 b B-9 A c)}{3 b^4 x^{3/2}}-\frac {c^2 (7 b B-9 A c)}{b^5 \sqrt {x}}-\frac {b B-A c}{b c x^{7/2} (b+c x)}-\frac {\left (c^3 (7 b B-9 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{b+c x^2} \, dx,x,\sqrt {x}\right )}{b^5}\\ &=\frac {7 b B-9 A c}{7 b^2 c x^{7/2}}-\frac {7 b B-9 A c}{5 b^3 x^{5/2}}+\frac {c (7 b B-9 A c)}{3 b^4 x^{3/2}}-\frac {c^2 (7 b B-9 A c)}{b^5 \sqrt {x}}-\frac {b B-A c}{b c x^{7/2} (b+c x)}-\frac {c^{5/2} (7 b B-9 A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{11/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 64, normalized size = 0.41 \begin {gather*} \frac {(b+c x) (7 b B-9 A c) \, _2F_1\left (-\frac {7}{2},1;-\frac {5}{2};-\frac {c x}{b}\right )+7 b (A c-b B)}{7 b^2 c x^{7/2} (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 146, normalized size = 0.94 \begin {gather*} \frac {\left (9 A c^{7/2}-7 b B c^{5/2}\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{11/2}}+\frac {-30 A b^4+54 A b^3 c x-126 A b^2 c^2 x^2+630 A b c^3 x^3+945 A c^4 x^4-42 b^4 B x+98 b^3 B c x^2-490 b^2 B c^2 x^3-735 b B c^3 x^4}{105 b^5 x^{7/2} (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 372, normalized size = 2.38 \begin {gather*} \left [-\frac {105 \, {\left ({\left (7 \, B b c^{3} - 9 \, A c^{4}\right )} x^{5} + {\left (7 \, B b^{2} c^{2} - 9 \, A b c^{3}\right )} x^{4}\right )} \sqrt {-\frac {c}{b}} \log \left (\frac {c x + 2 \, b \sqrt {x} \sqrt {-\frac {c}{b}} - b}{c x + b}\right ) + 2 \, {\left (30 \, A b^{4} + 105 \, {\left (7 \, B b c^{3} - 9 \, A c^{4}\right )} x^{4} + 70 \, {\left (7 \, B b^{2} c^{2} - 9 \, A b c^{3}\right )} x^{3} - 14 \, {\left (7 \, B b^{3} c - 9 \, A b^{2} c^{2}\right )} x^{2} + 6 \, {\left (7 \, B b^{4} - 9 \, A b^{3} c\right )} x\right )} \sqrt {x}}{210 \, {\left (b^{5} c x^{5} + b^{6} x^{4}\right )}}, \frac {105 \, {\left ({\left (7 \, B b c^{3} - 9 \, A c^{4}\right )} x^{5} + {\left (7 \, B b^{2} c^{2} - 9 \, A b c^{3}\right )} x^{4}\right )} \sqrt {\frac {c}{b}} \arctan \left (\frac {b \sqrt {\frac {c}{b}}}{c \sqrt {x}}\right ) - {\left (30 \, A b^{4} + 105 \, {\left (7 \, B b c^{3} - 9 \, A c^{4}\right )} x^{4} + 70 \, {\left (7 \, B b^{2} c^{2} - 9 \, A b c^{3}\right )} x^{3} - 14 \, {\left (7 \, B b^{3} c - 9 \, A b^{2} c^{2}\right )} x^{2} + 6 \, {\left (7 \, B b^{4} - 9 \, A b^{3} c\right )} x\right )} \sqrt {x}}{105 \, {\left (b^{5} c x^{5} + b^{6} x^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 136, normalized size = 0.87 \begin {gather*} -\frac {{\left (7 \, B b c^{3} - 9 \, A c^{4}\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b^{5}} - \frac {B b c^{3} \sqrt {x} - A c^{4} \sqrt {x}}{{\left (c x + b\right )} b^{5}} - \frac {2 \, {\left (315 \, B b c^{2} x^{3} - 420 \, A c^{3} x^{3} - 70 \, B b^{2} c x^{2} + 105 \, A b c^{2} x^{2} + 21 \, B b^{3} x - 42 \, A b^{2} c x + 15 \, A b^{3}\right )}}{105 \, b^{5} x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 163, normalized size = 1.04 \begin {gather*} \frac {9 A \,c^{4} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, b^{5}}-\frac {7 B \,c^{3} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, b^{4}}+\frac {A \,c^{4} \sqrt {x}}{\left (c x +b \right ) b^{5}}-\frac {B \,c^{3} \sqrt {x}}{\left (c x +b \right ) b^{4}}+\frac {8 A \,c^{3}}{b^{5} \sqrt {x}}-\frac {6 B \,c^{2}}{b^{4} \sqrt {x}}-\frac {2 A \,c^{2}}{b^{4} x^{\frac {3}{2}}}+\frac {4 B c}{3 b^{3} x^{\frac {3}{2}}}+\frac {4 A c}{5 b^{3} x^{\frac {5}{2}}}-\frac {2 B}{5 b^{2} x^{\frac {5}{2}}}-\frac {2 A}{7 b^{2} x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.19, size = 143, normalized size = 0.92 \begin {gather*} -\frac {30 \, A b^{4} + 105 \, {\left (7 \, B b c^{3} - 9 \, A c^{4}\right )} x^{4} + 70 \, {\left (7 \, B b^{2} c^{2} - 9 \, A b c^{3}\right )} x^{3} - 14 \, {\left (7 \, B b^{3} c - 9 \, A b^{2} c^{2}\right )} x^{2} + 6 \, {\left (7 \, B b^{4} - 9 \, A b^{3} c\right )} x}{105 \, {\left (b^{5} c x^{\frac {9}{2}} + b^{6} x^{\frac {7}{2}}\right )}} - \frac {{\left (7 \, B b c^{3} - 9 \, A c^{4}\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 121, normalized size = 0.78 \begin {gather*} \frac {\frac {2\,x\,\left (9\,A\,c-7\,B\,b\right )}{35\,b^2}-\frac {2\,A}{7\,b}+\frac {2\,c^2\,x^3\,\left (9\,A\,c-7\,B\,b\right )}{3\,b^4}+\frac {c^3\,x^4\,\left (9\,A\,c-7\,B\,b\right )}{b^5}-\frac {2\,c\,x^2\,\left (9\,A\,c-7\,B\,b\right )}{15\,b^3}}{b\,x^{7/2}+c\,x^{9/2}}+\frac {c^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {x}}{\sqrt {b}}\right )\,\left (9\,A\,c-7\,B\,b\right )}{b^{11/2}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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