Optimal. Leaf size=113 \[ -\frac {2 c^{5/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{9/2}}-\frac {2 c^2 (b B-A c)}{b^4 \sqrt {x}}+\frac {2 c (b B-A c)}{3 b^3 x^{3/2}}-\frac {2 (b B-A c)}{5 b^2 x^{5/2}}-\frac {2 A}{7 b x^{7/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 7, 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)}{b^4 \sqrt {x}}-\frac {2 c^{5/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{9/2}}+\frac {2 c (b B-A c)}{3 b^3 x^{3/2}}-\frac {2 (b B-A c)}{5 b^2 x^{5/2}}-\frac {2 A}{7 b x^{7/2}} \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^{7/2} \left (b x+c x^2\right )} \, dx &=\int \frac {A+B x}{x^{9/2} (b+c x)} \, dx\\ &=-\frac {2 A}{7 b x^{7/2}}+\frac {\left (2 \left (\frac {7 b B}{2}-\frac {7 A c}{2}\right )\right ) \int \frac {1}{x^{7/2} (b+c x)} \, dx}{7 b}\\ &=-\frac {2 A}{7 b x^{7/2}}-\frac {2 (b B-A c)}{5 b^2 x^{5/2}}-\frac {(c (b B-A c)) \int \frac {1}{x^{5/2} (b+c x)} \, dx}{b^2}\\ &=-\frac {2 A}{7 b x^{7/2}}-\frac {2 (b B-A c)}{5 b^2 x^{5/2}}+\frac {2 c (b B-A c)}{3 b^3 x^{3/2}}+\frac {\left (c^2 (b B-A c)\right ) \int \frac {1}{x^{3/2} (b+c x)} \, dx}{b^3}\\ &=-\frac {2 A}{7 b x^{7/2}}-\frac {2 (b B-A c)}{5 b^2 x^{5/2}}+\frac {2 c (b B-A c)}{3 b^3 x^{3/2}}-\frac {2 c^2 (b B-A c)}{b^4 \sqrt {x}}-\frac {\left (c^3 (b B-A c)\right ) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{b^4}\\ &=-\frac {2 A}{7 b x^{7/2}}-\frac {2 (b B-A c)}{5 b^2 x^{5/2}}+\frac {2 c (b B-A c)}{3 b^3 x^{3/2}}-\frac {2 c^2 (b B-A c)}{b^4 \sqrt {x}}-\frac {\left (2 c^3 (b B-A c)\right ) \operatorname {Subst}\left (\int \frac {1}{b+c x^2} \, dx,x,\sqrt {x}\right )}{b^4}\\ &=-\frac {2 A}{7 b x^{7/2}}-\frac {2 (b B-A c)}{5 b^2 x^{5/2}}+\frac {2 c (b B-A c)}{3 b^3 x^{3/2}}-\frac {2 c^2 (b B-A c)}{b^4 \sqrt {x}}-\frac {2 c^{5/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 44, normalized size = 0.39 \begin {gather*} \frac {2 \left (\, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};-\frac {c x}{b}\right ) (7 A c x-7 b B x)-5 A b\right )}{35 b^2 x^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 115, normalized size = 1.02 \begin {gather*} -\frac {2 \left (b B c^{5/2}-A c^{7/2}\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{9/2}}-\frac {2 \left (15 A b^3-21 A b^2 c x+35 A b c^2 x^2-105 A c^3 x^3+21 b^3 B x-35 b^2 B c x^2+105 b B c^2 x^3\right )}{105 b^4 x^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 246, normalized size = 2.18 \begin {gather*} \left [-\frac {105 \, {\left (B b c^{2} - A c^{3}\right )} x^{4} \sqrt {-\frac {c}{b}} \log \left (\frac {c x + 2 \, b \sqrt {x} \sqrt {-\frac {c}{b}} - b}{c x + b}\right ) + 2 \, {\left (15 \, A b^{3} + 105 \, {\left (B b c^{2} - A c^{3}\right )} x^{3} - 35 \, {\left (B b^{2} c - A b c^{2}\right )} x^{2} + 21 \, {\left (B b^{3} - A b^{2} c\right )} x\right )} \sqrt {x}}{105 \, b^{4} x^{4}}, \frac {2 \, {\left (105 \, {\left (B b c^{2} - A c^{3}\right )} x^{4} \sqrt {\frac {c}{b}} \arctan \left (\frac {b \sqrt {\frac {c}{b}}}{c \sqrt {x}}\right ) - {\left (15 \, A b^{3} + 105 \, {\left (B b c^{2} - A c^{3}\right )} x^{3} - 35 \, {\left (B b^{2} c - A b c^{2}\right )} x^{2} + 21 \, {\left (B b^{3} - A b^{2} c\right )} x\right )} \sqrt {x}\right )}}{105 \, b^{4} x^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 104, normalized size = 0.92 \begin {gather*} -\frac {2 \, {\left (B b c^{3} - A c^{4}\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b^{4}} - \frac {2 \, {\left (105 \, B b c^{2} x^{3} - 105 \, A c^{3} x^{3} - 35 \, B b^{2} c x^{2} + 35 \, A b c^{2} x^{2} + 21 \, B b^{3} x - 21 \, A b^{2} c x + 15 \, A b^{3}\right )}}{105 \, b^{4} x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 126, normalized size = 1.12 \begin {gather*} \frac {2 A \,c^{4} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, b^{4}}-\frac {2 B \,c^{3} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, b^{3}}+\frac {2 A \,c^{3}}{b^{4} \sqrt {x}}-\frac {2 B \,c^{2}}{b^{3} \sqrt {x}}-\frac {2 A \,c^{2}}{3 b^{3} x^{\frac {3}{2}}}+\frac {2 B c}{3 b^{2} x^{\frac {3}{2}}}+\frac {2 A c}{5 b^{2} x^{\frac {5}{2}}}-\frac {2 B}{5 b \,x^{\frac {5}{2}}}-\frac {2 A}{7 b \,x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 103, normalized size = 0.91 \begin {gather*} -\frac {2 \, {\left (B b c^{3} - A c^{4}\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b^{4}} - \frac {2 \, {\left (15 \, A b^{3} + 105 \, {\left (B b c^{2} - A c^{3}\right )} x^{3} - 35 \, {\left (B b^{2} c - A b c^{2}\right )} x^{2} + 21 \, {\left (B b^{3} - A b^{2} c\right )} x\right )}}{105 \, b^{4} x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 90, normalized size = 0.80 \begin {gather*} \frac {2\,c^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {x}}{\sqrt {b}}\right )\,\left (A\,c-B\,b\right )}{b^{9/2}}-\frac {\frac {2\,A}{7\,b}-\frac {2\,x\,\left (A\,c-B\,b\right )}{5\,b^2}-\frac {2\,c^2\,x^3\,\left (A\,c-B\,b\right )}{b^4}+\frac {2\,c\,x^2\,\left (A\,c-B\,b\right )}{3\,b^3}}{x^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 52.05, size = 326, normalized size = 2.88 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{9 x^{\frac {9}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}\right ) & \text {for}\: b = 0 \wedge c = 0 \\\frac {- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{5 x^{\frac {5}{2}}}}{b} & \text {for}\: c = 0 \\\frac {- \frac {2 A}{9 x^{\frac {9}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}}{c} & \text {for}\: b = 0 \\- \frac {2 A}{7 b x^{\frac {7}{2}}} + \frac {2 A c}{5 b^{2} x^{\frac {5}{2}}} - \frac {2 A c^{2}}{3 b^{3} x^{\frac {3}{2}}} + \frac {2 A c^{3}}{b^{4} \sqrt {x}} - \frac {i A c^{3} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{b^{\frac {9}{2}} \sqrt {\frac {1}{c}}} + \frac {i A c^{3} \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{b^{\frac {9}{2}} \sqrt {\frac {1}{c}}} - \frac {2 B}{5 b x^{\frac {5}{2}}} + \frac {2 B c}{3 b^{2} x^{\frac {3}{2}}} - \frac {2 B c^{2}}{b^{3} \sqrt {x}} + \frac {i B c^{2} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{b^{\frac {7}{2}} \sqrt {\frac {1}{c}}} - \frac {i B c^{2} \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{b^{\frac {7}{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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