Optimal. Leaf size=136 \[ \frac {2 c^{7/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{11/2}}+\frac {2 c^3 (b B-A c)}{b^5 \sqrt {x}}-\frac {2 c^2 (b B-A c)}{3 b^4 x^{3/2}}+\frac {2 c (b B-A c)}{5 b^3 x^{5/2}}-\frac {2 (b B-A c)}{7 b^2 x^{7/2}}-\frac {2 A}{9 b x^{9/2}} \]
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Rubi [A] time = 0.08, antiderivative size = 136, 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 {2 c^2 (b B-A c)}{3 b^4 x^{3/2}}+\frac {2 c^3 (b B-A c)}{b^5 \sqrt {x}}+\frac {2 c^{7/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{11/2}}+\frac {2 c (b B-A c)}{5 b^3 x^{5/2}}-\frac {2 (b B-A c)}{7 b^2 x^{7/2}}-\frac {2 A}{9 b x^{9/2}} \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^{9/2} \left (b x+c x^2\right )} \, dx &=\int \frac {A+B x}{x^{11/2} (b+c x)} \, dx\\ &=-\frac {2 A}{9 b x^{9/2}}+\frac {\left (2 \left (\frac {9 b B}{2}-\frac {9 A c}{2}\right )\right ) \int \frac {1}{x^{9/2} (b+c x)} \, dx}{9 b}\\ &=-\frac {2 A}{9 b x^{9/2}}-\frac {2 (b B-A c)}{7 b^2 x^{7/2}}-\frac {(c (b B-A c)) \int \frac {1}{x^{7/2} (b+c x)} \, dx}{b^2}\\ &=-\frac {2 A}{9 b x^{9/2}}-\frac {2 (b B-A c)}{7 b^2 x^{7/2}}+\frac {2 c (b B-A c)}{5 b^3 x^{5/2}}+\frac {\left (c^2 (b B-A c)\right ) \int \frac {1}{x^{5/2} (b+c x)} \, dx}{b^3}\\ &=-\frac {2 A}{9 b x^{9/2}}-\frac {2 (b B-A c)}{7 b^2 x^{7/2}}+\frac {2 c (b B-A c)}{5 b^3 x^{5/2}}-\frac {2 c^2 (b B-A c)}{3 b^4 x^{3/2}}-\frac {\left (c^3 (b B-A c)\right ) \int \frac {1}{x^{3/2} (b+c x)} \, dx}{b^4}\\ &=-\frac {2 A}{9 b x^{9/2}}-\frac {2 (b B-A c)}{7 b^2 x^{7/2}}+\frac {2 c (b B-A c)}{5 b^3 x^{5/2}}-\frac {2 c^2 (b B-A c)}{3 b^4 x^{3/2}}+\frac {2 c^3 (b B-A c)}{b^5 \sqrt {x}}+\frac {\left (c^4 (b B-A c)\right ) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{b^5}\\ &=-\frac {2 A}{9 b x^{9/2}}-\frac {2 (b B-A c)}{7 b^2 x^{7/2}}+\frac {2 c (b B-A c)}{5 b^3 x^{5/2}}-\frac {2 c^2 (b B-A c)}{3 b^4 x^{3/2}}+\frac {2 c^3 (b B-A c)}{b^5 \sqrt {x}}+\frac {\left (2 c^4 (b B-A c)\right ) \operatorname {Subst}\left (\int \frac {1}{b+c x^2} \, dx,x,\sqrt {x}\right )}{b^5}\\ &=-\frac {2 A}{9 b x^{9/2}}-\frac {2 (b B-A c)}{7 b^2 x^{7/2}}+\frac {2 c (b B-A c)}{5 b^3 x^{5/2}}-\frac {2 c^2 (b B-A c)}{3 b^4 x^{3/2}}+\frac {2 c^3 (b B-A c)}{b^5 \sqrt {x}}+\frac {2 c^{7/2} (b B-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 = 0.01, size = 44, normalized size = 0.32 \begin {gather*} \frac {2 \left (\, _2F_1\left (-\frac {7}{2},1;-\frac {5}{2};-\frac {c x}{b}\right ) (9 A c x-9 b B x)-7 A b\right )}{63 b^2 x^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 139, normalized size = 1.02 \begin {gather*} \frac {2 \left (b B c^{7/2}-A c^{9/2}\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{11/2}}-\frac {2 \left (35 A b^4-45 A b^3 c x+63 A b^2 c^2 x^2-105 A b c^3 x^3+315 A c^4 x^4+45 b^4 B x-63 b^3 B c x^2+105 b^2 B c^2 x^3-315 b B c^3 x^4\right )}{315 b^5 x^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 291, normalized size = 2.14 \begin {gather*} \left [-\frac {315 \, {\left (B b c^{3} - A c^{4}\right )} x^{5} \sqrt {-\frac {c}{b}} \log \left (\frac {c x - 2 \, b \sqrt {x} \sqrt {-\frac {c}{b}} - b}{c x + b}\right ) + 2 \, {\left (35 \, A b^{4} - 315 \, {\left (B b c^{3} - A c^{4}\right )} x^{4} + 105 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} x^{3} - 63 \, {\left (B b^{3} c - A b^{2} c^{2}\right )} x^{2} + 45 \, {\left (B b^{4} - A b^{3} c\right )} x\right )} \sqrt {x}}{315 \, b^{5} x^{5}}, -\frac {2 \, {\left (315 \, {\left (B b c^{3} - A c^{4}\right )} x^{5} \sqrt {\frac {c}{b}} \arctan \left (\frac {b \sqrt {\frac {c}{b}}}{c \sqrt {x}}\right ) + {\left (35 \, A b^{4} - 315 \, {\left (B b c^{3} - A c^{4}\right )} x^{4} + 105 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} x^{3} - 63 \, {\left (B b^{3} c - A b^{2} c^{2}\right )} x^{2} + 45 \, {\left (B b^{4} - A b^{3} c\right )} x\right )} \sqrt {x}\right )}}{315 \, b^{5} x^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 128, normalized size = 0.94 \begin {gather*} \frac {2 \, {\left (B b c^{4} - A c^{5}\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b^{5}} + \frac {2 \, {\left (315 \, B b c^{3} x^{4} - 315 \, A c^{4} x^{4} - 105 \, B b^{2} c^{2} x^{3} + 105 \, A b c^{3} x^{3} + 63 \, B b^{3} c x^{2} - 63 \, A b^{2} c^{2} x^{2} - 45 \, B b^{4} x + 45 \, A b^{3} c x - 35 \, A b^{4}\right )}}{315 \, b^{5} x^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 150, normalized size = 1.10 \begin {gather*} -\frac {2 A \,c^{5} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, b^{5}}+\frac {2 B \,c^{4} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, b^{4}}-\frac {2 A \,c^{4}}{b^{5} \sqrt {x}}+\frac {2 B \,c^{3}}{b^{4} \sqrt {x}}+\frac {2 A \,c^{3}}{3 b^{4} x^{\frac {3}{2}}}-\frac {2 B \,c^{2}}{3 b^{3} x^{\frac {3}{2}}}-\frac {2 A \,c^{2}}{5 b^{3} x^{\frac {5}{2}}}+\frac {2 B c}{5 b^{2} x^{\frac {5}{2}}}+\frac {2 A c}{7 b^{2} x^{\frac {7}{2}}}-\frac {2 B}{7 b \,x^{\frac {7}{2}}}-\frac {2 A}{9 b \,x^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 126, normalized size = 0.93 \begin {gather*} \frac {2 \, {\left (B b c^{4} - A c^{5}\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b^{5}} - \frac {2 \, {\left (35 \, A b^{4} - 315 \, {\left (B b c^{3} - A c^{4}\right )} x^{4} + 105 \, {\left (B b^{2} c^{2} - A b c^{3}\right )} x^{3} - 63 \, {\left (B b^{3} c - A b^{2} c^{2}\right )} x^{2} + 45 \, {\left (B b^{4} - A b^{3} c\right )} x\right )}}{315 \, b^{5} x^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 109, normalized size = 0.80 \begin {gather*} -\frac {\frac {2\,A}{9\,b}-\frac {2\,x\,\left (A\,c-B\,b\right )}{7\,b^2}-\frac {2\,c^2\,x^3\,\left (A\,c-B\,b\right )}{3\,b^4}+\frac {2\,c^3\,x^4\,\left (A\,c-B\,b\right )}{b^5}+\frac {2\,c\,x^2\,\left (A\,c-B\,b\right )}{5\,b^3}}{x^{9/2}}-\frac {2\,c^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {x}}{\sqrt {b}}\right )\,\left (A\,c-B\,b\right )}{b^{11/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 126.31, size = 360, normalized size = 2.65 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{11 x^{\frac {11}{2}}} - \frac {2 B}{9 x^{\frac {9}{2}}}\right ) & \text {for}\: b = 0 \wedge c = 0 \\\frac {- \frac {2 A}{11 x^{\frac {11}{2}}} - \frac {2 B}{9 x^{\frac {9}{2}}}}{c} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{9 x^{\frac {9}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}}{b} & \text {for}\: c = 0 \\- \frac {2 A}{9 b x^{\frac {9}{2}}} + \frac {2 A c}{7 b^{2} x^{\frac {7}{2}}} - \frac {2 A c^{2}}{5 b^{3} x^{\frac {5}{2}}} + \frac {2 A c^{3}}{3 b^{4} x^{\frac {3}{2}}} - \frac {2 A c^{4}}{b^{5} \sqrt {x}} + \frac {i A c^{4} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{b^{\frac {11}{2}} \sqrt {\frac {1}{c}}} - \frac {i A c^{4} \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{b^{\frac {11}{2}} \sqrt {\frac {1}{c}}} - \frac {2 B}{7 b x^{\frac {7}{2}}} + \frac {2 B c}{5 b^{2} x^{\frac {5}{2}}} - \frac {2 B c^{2}}{3 b^{3} x^{\frac {3}{2}}} + \frac {2 B c^{3}}{b^{4} \sqrt {x}} - \frac {i B c^{3} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{b^{\frac {9}{2}} \sqrt {\frac {1}{c}}} + \frac {i B c^{3} \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{b^{\frac {9}{2}} \sqrt {\frac {1}{c}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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