Optimal. Leaf size=113 \[ \frac {2 b^{5/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{9/2}}-\frac {2 b^2 \sqrt {x} (b B-A c)}{c^4}+\frac {2 b x^{3/2} (b B-A c)}{3 c^3}-\frac {2 x^{5/2} (b B-A c)}{5 c^2}+\frac {2 B x^{7/2}}{7 c} \]
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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, 80, 50, 63, 205} \begin {gather*} -\frac {2 b^2 \sqrt {x} (b B-A c)}{c^4}+\frac {2 b^{5/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{9/2}}-\frac {2 x^{5/2} (b B-A c)}{5 c^2}+\frac {2 b x^{3/2} (b B-A c)}{3 c^3}+\frac {2 B x^{7/2}}{7 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 205
Rule 781
Rubi steps
\begin {align*} \int \frac {x^{7/2} (A+B x)}{b x+c x^2} \, dx &=\int \frac {x^{5/2} (A+B x)}{b+c x} \, dx\\ &=\frac {2 B x^{7/2}}{7 c}+\frac {\left (2 \left (-\frac {7 b B}{2}+\frac {7 A c}{2}\right )\right ) \int \frac {x^{5/2}}{b+c x} \, dx}{7 c}\\ &=-\frac {2 (b B-A c) x^{5/2}}{5 c^2}+\frac {2 B x^{7/2}}{7 c}+\frac {(b (b B-A c)) \int \frac {x^{3/2}}{b+c x} \, dx}{c^2}\\ &=\frac {2 b (b B-A c) x^{3/2}}{3 c^3}-\frac {2 (b B-A c) x^{5/2}}{5 c^2}+\frac {2 B x^{7/2}}{7 c}-\frac {\left (b^2 (b B-A c)\right ) \int \frac {\sqrt {x}}{b+c x} \, dx}{c^3}\\ &=-\frac {2 b^2 (b B-A c) \sqrt {x}}{c^4}+\frac {2 b (b B-A c) x^{3/2}}{3 c^3}-\frac {2 (b B-A c) x^{5/2}}{5 c^2}+\frac {2 B x^{7/2}}{7 c}+\frac {\left (b^3 (b B-A c)\right ) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{c^4}\\ &=-\frac {2 b^2 (b B-A c) \sqrt {x}}{c^4}+\frac {2 b (b B-A c) x^{3/2}}{3 c^3}-\frac {2 (b B-A c) x^{5/2}}{5 c^2}+\frac {2 B x^{7/2}}{7 c}+\frac {\left (2 b^3 (b B-A c)\right ) \operatorname {Subst}\left (\int \frac {1}{b+c x^2} \, dx,x,\sqrt {x}\right )}{c^4}\\ &=-\frac {2 b^2 (b B-A c) \sqrt {x}}{c^4}+\frac {2 b (b B-A c) x^{3/2}}{3 c^3}-\frac {2 (b B-A c) x^{5/2}}{5 c^2}+\frac {2 B x^{7/2}}{7 c}+\frac {2 b^{5/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 101, normalized size = 0.89 \begin {gather*} \frac {2 b^{5/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{9/2}}+\frac {2 \sqrt {x} \left (35 b^2 c (3 A+B x)-7 b c^2 x (5 A+3 B x)+3 c^3 x^2 (7 A+5 B x)-105 b^3 B\right )}{105 c^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 131, normalized size = 1.16 \begin {gather*} \frac {2 \left (b^{7/2} B-A b^{5/2} c\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{c^{9/2}}+\frac {2 \left (105 A b^2 c \sqrt {x}-35 A b c^2 x^{3/2}+21 A c^3 x^{5/2}-105 b^3 B \sqrt {x}+35 b^2 B c x^{3/2}-21 b B c^2 x^{5/2}+15 B c^3 x^{7/2}\right )}{105 c^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 229, normalized size = 2.03 \begin {gather*} \left [-\frac {105 \, {\left (B b^{3} - A b^{2} c\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 (15 \, B c^{3} x^{3} - 105 \, B b^{3} + 105 \, A b^{2} c - 21 \, {\left (B b c^{2} - A c^{3}\right )} x^{2} + 35 \, {\left (B b^{2} c - A b c^{2}\right )} x\right )} \sqrt {x}}{105 \, c^{4}}, \frac {2 \, {\left (105 \, {\left (B b^{3} - A b^{2} c\right )} \sqrt {\frac {b}{c}} \arctan \left (\frac {c \sqrt {x} \sqrt {\frac {b}{c}}}{b}\right ) + {\left (15 \, B c^{3} x^{3} - 105 \, B b^{3} + 105 \, A b^{2} c - 21 \, {\left (B b c^{2} - A c^{3}\right )} x^{2} + 35 \, {\left (B b^{2} c - A b c^{2}\right )} x\right )} \sqrt {x}\right )}}{105 \, c^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 115, normalized size = 1.02 \begin {gather*} \frac {2 \, {\left (B b^{4} - A b^{3} c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} c^{4}} + \frac {2 \, {\left (15 \, B c^{6} x^{\frac {7}{2}} - 21 \, B b c^{5} x^{\frac {5}{2}} + 21 \, A c^{6} x^{\frac {5}{2}} + 35 \, B b^{2} c^{4} x^{\frac {3}{2}} - 35 \, A b c^{5} x^{\frac {3}{2}} - 105 \, B b^{3} c^{3} \sqrt {x} + 105 \, A b^{2} c^{4} \sqrt {x}\right )}}{105 \, c^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 126, normalized size = 1.12 \begin {gather*} \frac {2 B \,x^{\frac {7}{2}}}{7 c}+\frac {2 A \,x^{\frac {5}{2}}}{5 c}-\frac {2 B b \,x^{\frac {5}{2}}}{5 c^{2}}-\frac {2 A \,b^{3} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, c^{3}}+\frac {2 B \,b^{4} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, c^{4}}-\frac {2 A b \,x^{\frac {3}{2}}}{3 c^{2}}+\frac {2 B \,b^{2} x^{\frac {3}{2}}}{3 c^{3}}+\frac {2 A \,b^{2} \sqrt {x}}{c^{3}}-\frac {2 B \,b^{3} \sqrt {x}}{c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 105, normalized size = 0.93 \begin {gather*} \frac {2 \, {\left (B b^{4} - A b^{3} c\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} c^{4}} + \frac {2 \, {\left (15 \, B c^{3} x^{\frac {7}{2}} - 21 \, {\left (B b c^{2} - A c^{3}\right )} x^{\frac {5}{2}} + 35 \, {\left (B b^{2} c - A b c^{2}\right )} x^{\frac {3}{2}} - 105 \, {\left (B b^{3} - A b^{2} c\right )} \sqrt {x}\right )}}{105 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 125, normalized size = 1.11 \begin {gather*} x^{5/2}\,\left (\frac {2\,A}{5\,c}-\frac {2\,B\,b}{5\,c^2}\right )+\frac {2\,B\,x^{7/2}}{7\,c}+\frac {b^2\,\sqrt {x}\,\left (\frac {2\,A}{c}-\frac {2\,B\,b}{c^2}\right )}{c^2}+\frac {2\,b^{5/2}\,\mathrm {atan}\left (\frac {b^{5/2}\,\sqrt {c}\,\sqrt {x}\,\left (A\,c-B\,b\right )}{B\,b^4-A\,b^3\,c}\right )\,\left (A\,c-B\,b\right )}{c^{9/2}}-\frac {b\,x^{3/2}\,\left (\frac {2\,A}{c}-\frac {2\,B\,b}{c^2}\right )}{3\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 45.39, size = 279, normalized size = 2.47 \begin {gather*} \begin {cases} \frac {i A b^{\frac {5}{2}} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{c^{4} \sqrt {\frac {1}{c}}} - \frac {i A b^{\frac {5}{2}} \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{c^{4} \sqrt {\frac {1}{c}}} + \frac {2 A b^{2} \sqrt {x}}{c^{3}} - \frac {2 A b x^{\frac {3}{2}}}{3 c^{2}} + \frac {2 A x^{\frac {5}{2}}}{5 c} - \frac {i B b^{\frac {7}{2}} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{c^{5} \sqrt {\frac {1}{c}}} + \frac {i B b^{\frac {7}{2}} \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{c^{5} \sqrt {\frac {1}{c}}} - \frac {2 B b^{3} \sqrt {x}}{c^{4}} + \frac {2 B b^{2} x^{\frac {3}{2}}}{3 c^{3}} - \frac {2 B b x^{\frac {5}{2}}}{5 c^{2}} + \frac {2 B x^{\frac {7}{2}}}{7 c} & \text {for}\: c \neq 0 \\\frac {\frac {2 A x^{\frac {7}{2}}}{7} + \frac {2 B x^{\frac {9}{2}}}{9}}{b} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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