Optimal. Leaf size=90 \[ -\frac {2 A}{5 b x^{5/2}}-\frac {2 (b B-A c)}{3 b^2 x^{3/2}}+\frac {2 c (b B-A c)}{b^3 \sqrt {x}}+\frac {2 c^{3/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}} \]
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Rubi [A]
time = 0.04, 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 = {795, 79, 53, 65,
211} \begin {gather*} \frac {2 c^{3/2} (b B-A c) \text {ArcTan}\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 79
Rule 211
Rule 795
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 ) \text {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 [A]
time = 0.11, size = 83, normalized size = 0.92 \begin {gather*} -\frac {2 \left (5 b B x (b-3 c x)+A \left (3 b^2-5 b c x+15 c^2 x^2\right )\right )}{15 b^3 x^{5/2}}+\frac {2 c^{3/2} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.59, size = 76, normalized size = 0.84
method | result | size |
derivativedivides | \(-\frac {2 A}{5 b \,x^{\frac {5}{2}}}-\frac {2 \left (-A c +B b \right )}{3 b^{2} x^{\frac {3}{2}}}-\frac {2 \left (A c -B b \right ) c}{b^{3} \sqrt {x}}-\frac {2 c^{2} \left (A c -B b \right ) \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{b^{3} \sqrt {b c}}\) | \(76\) |
default | \(-\frac {2 A}{5 b \,x^{\frac {5}{2}}}-\frac {2 \left (-A c +B b \right )}{3 b^{2} x^{\frac {3}{2}}}-\frac {2 \left (A c -B b \right ) c}{b^{3} \sqrt {x}}-\frac {2 c^{2} \left (A c -B b \right ) \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right )}{b^{3} \sqrt {b c}}\) | \(76\) |
risch | \(-\frac {2 \left (15 A \,c^{2} x^{2}-15 b B \,x^{2} c -5 A b c x +5 b^{2} B x +3 b^{2} A \right )}{15 b^{3} x^{\frac {5}{2}}}-\frac {2 c^{3} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right ) A}{b^{3} \sqrt {b c}}+\frac {2 c^{2} \arctan \left (\frac {c \sqrt {x}}{\sqrt {b c}}\right ) B}{b^{2} \sqrt {b c}}\) | \(97\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, 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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Fricas [A]
time = 2.04, 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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 262 vs.
\(2 (87) = 174\).
time = 10.95, size = 262, normalized size = 2.91 \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 {A c^{2} \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{b^{3} \sqrt {- \frac {b}{c}}} + \frac {A c^{2} \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{b^{3} \sqrt {- \frac {b}{c}}} - \frac {2 A c^{2}}{b^{3} \sqrt {x}} - \frac {2 B}{3 b x^{\frac {3}{2}}} + \frac {B c \log {\left (\sqrt {x} - \sqrt {- \frac {b}{c}} \right )}}{b^{2} \sqrt {- \frac {b}{c}}} - \frac {B c \log {\left (\sqrt {x} + \sqrt {- \frac {b}{c}} \right )}}{b^{2} \sqrt {- \frac {b}{c}}} + \frac {2 B c}{b^{2} \sqrt {x}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.62, 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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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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