Optimal. Leaf size=90 \[ \frac {2 c^{3/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}}+\frac {2 c (b B-A c)}{b^3 \sqrt {x}}-\frac {2 (b B-A c)}{3 b^2 x^{3/2}}-\frac {2 A}{5 b x^{5/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 6, 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^{3/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}}-\frac {2 (b B-A c)}{3 b^2 x^{3/2}}+\frac {2 c (b B-A c)}{b^3 \sqrt {x}}-\frac {2 A}{5 b x^{5/2}} \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^{5/2} \left (b x+c x^2\right )} \, dx &=\int \frac {A+B x}{x^{7/2} (b+c x)} \, dx\\ &=-\frac {2 A}{5 b x^{5/2}}+\frac {\left (2 \left (\frac {5 b B}{2}-\frac {5 A c}{2}\right )\right ) \int \frac {1}{x^{5/2} (b+c x)} \, dx}{5 b}\\ &=-\frac {2 A}{5 b x^{5/2}}-\frac {2 (b B-A c)}{3 b^2 x^{3/2}}-\frac {(c (b B-A c)) \int \frac {1}{x^{3/2} (b+c x)} \, dx}{b^2}\\ &=-\frac {2 A}{5 b x^{5/2}}-\frac {2 (b B-A c)}{3 b^2 x^{3/2}}+\frac {2 c (b B-A c)}{b^3 \sqrt {x}}+\frac {\left (c^2 (b B-A c)\right ) \int \frac {1}{\sqrt {x} (b+c x)} \, dx}{b^3}\\ &=-\frac {2 A}{5 b x^{5/2}}-\frac {2 (b B-A c)}{3 b^2 x^{3/2}}+\frac {2 c (b B-A c)}{b^3 \sqrt {x}}+\frac {\left (2 c^2 (b B-A c)\right ) \operatorname {Subst}\left (\int \frac {1}{b+c x^2} \, dx,x,\sqrt {x}\right )}{b^3}\\ &=-\frac {2 A}{5 b x^{5/2}}-\frac {2 (b B-A c)}{3 b^2 x^{3/2}}+\frac {2 c (b B-A c)}{b^3 \sqrt {x}}+\frac {2 c^{3/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 43, normalized size = 0.48 \begin {gather*} \frac {-10 x (b B-A c) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\frac {c x}{b}\right )-6 A b}{15 b^2 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 91, normalized size = 1.01 \begin {gather*} \frac {2 \left (b B c^{3/2}-A c^{5/2}\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}}-\frac {2 \left (3 A b^2-5 A b c x+15 A c^2 x^2+5 b^2 B x-15 b B c x^2\right )}{15 b^3 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 195, normalized size = 2.17 \begin {gather*} \left [-\frac {15 \, {\left (B b c - A c^{2}\right )} x^{3} \sqrt {-\frac {c}{b}} \log \left (\frac {c x - 2 \, b \sqrt {x} \sqrt {-\frac {c}{b}} - b}{c x + b}\right ) + 2 \, {\left (3 \, A b^{2} - 15 \, {\left (B b c - A c^{2}\right )} x^{2} + 5 \, {\left (B b^{2} - A b c\right )} x\right )} \sqrt {x}}{15 \, b^{3} x^{3}}, -\frac {2 \, {\left (15 \, {\left (B b c - A c^{2}\right )} x^{3} \sqrt {\frac {c}{b}} \arctan \left (\frac {b \sqrt {\frac {c}{b}}}{c \sqrt {x}}\right ) + {\left (3 \, A b^{2} - 15 \, {\left (B b c - A c^{2}\right )} x^{2} + 5 \, {\left (B b^{2} - A b c\right )} x\right )} \sqrt {x}\right )}}{15 \, b^{3} x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 80, normalized size = 0.89 \begin {gather*} \frac {2 \, {\left (B b c^{2} - A c^{3}\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b^{3}} + \frac {2 \, {\left (15 \, B b c x^{2} - 15 \, A c^{2} x^{2} - 5 \, B b^{2} x + 5 \, A b c x - 3 \, A b^{2}\right )}}{15 \, b^{3} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 102, normalized size = 1.13 \begin {gather*} -\frac {2 A \,c^{3} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, b^{3}}+\frac {2 B \,c^{2} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c}\, b^{2}}-\frac {2 A \,c^{2}}{b^{3} \sqrt {x}}+\frac {2 B c}{b^{2} \sqrt {x}}+\frac {2 A c}{3 b^{2} x^{\frac {3}{2}}}-\frac {2 B}{3 b \,x^{\frac {3}{2}}}-\frac {2 A}{5 b \,x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 80, normalized size = 0.89 \begin {gather*} \frac {2 \, {\left (B b c^{2} - A c^{3}\right )} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{\sqrt {b c} b^{3}} - \frac {2 \, {\left (3 \, A b^{2} - 15 \, {\left (B b c - A c^{2}\right )} x^{2} + 5 \, {\left (B b^{2} - A b c\right )} x\right )}}{15 \, b^{3} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 71, normalized size = 0.79 \begin {gather*} -\frac {\frac {2\,A}{5\,b}-\frac {2\,x\,\left (A\,c-B\,b\right )}{3\,b^2}+\frac {2\,c\,x^2\,\left (A\,c-B\,b\right )}{b^3}}{x^{5/2}}-\frac {2\,c^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {x}}{\sqrt {b}}\right )\,\left (A\,c-B\,b\right )}{b^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 18.37, size = 289, normalized size = 3.21 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{5 x^{\frac {5}{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}}}}{c} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{b} & \text {for}\: c = 0 \\- \frac {2 A}{5 b x^{\frac {5}{2}}} + \frac {2 A c}{3 b^{2} x^{\frac {3}{2}}} - \frac {2 A c^{2}}{b^{3} \sqrt {x}} + \frac {i A c^{2} \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{b^{\frac {7}{2}} \sqrt {\frac {1}{c}}} - \frac {i A c^{2} \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{b^{\frac {7}{2}} \sqrt {\frac {1}{c}}} - \frac {2 B}{3 b x^{\frac {3}{2}}} + \frac {2 B c}{b^{2} \sqrt {x}} - \frac {i B c \log {\left (- i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{b^{\frac {5}{2}} \sqrt {\frac {1}{c}}} + \frac {i B c \log {\left (i \sqrt {b} \sqrt {\frac {1}{c}} + \sqrt {x} \right )}}{b^{\frac {5}{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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