Optimal. Leaf size=88 \[ -\frac {3 A b-a B}{a^2 b \sqrt {x}}+\frac {A b-a B}{a b \sqrt {x} (a+b x)}-\frac {(3 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b}} \]
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Rubi [A]
time = 0.02, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {79, 53, 65, 211}
\begin {gather*} -\frac {(3 A b-a B) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b}}-\frac {3 A b-a B}{a^2 b \sqrt {x}}+\frac {A b-a B}{a b \sqrt {x} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 79
Rule 211
Rubi steps
\begin {align*} \int \frac {A+B x}{x^{3/2} (a+b x)^2} \, dx &=\frac {A b-a B}{a b \sqrt {x} (a+b x)}-\frac {\left (-\frac {3 A b}{2}+\frac {a B}{2}\right ) \int \frac {1}{x^{3/2} (a+b x)} \, dx}{a b}\\ &=-\frac {3 A b-a B}{a^2 b \sqrt {x}}+\frac {A b-a B}{a b \sqrt {x} (a+b x)}-\frac {(3 A b-a B) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{2 a^2}\\ &=-\frac {3 A b-a B}{a^2 b \sqrt {x}}+\frac {A b-a B}{a b \sqrt {x} (a+b x)}-\frac {(3 A b-a B) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{a^2}\\ &=-\frac {3 A b-a B}{a^2 b \sqrt {x}}+\frac {A b-a B}{a b \sqrt {x} (a+b x)}-\frac {(3 A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 67, normalized size = 0.76 \begin {gather*} \frac {-2 a A-3 A b x+a B x}{a^2 \sqrt {x} (a+b x)}+\frac {(-3 A b+a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 64, normalized size = 0.73
method | result | size |
derivativedivides | \(-\frac {2 A}{a^{2} \sqrt {x}}-\frac {2 \left (\frac {\left (\frac {A b}{2}-\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (3 A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}\) | \(64\) |
default | \(-\frac {2 A}{a^{2} \sqrt {x}}-\frac {2 \left (\frac {\left (\frac {A b}{2}-\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (3 A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}\) | \(64\) |
risch | \(-\frac {2 A}{a^{2} \sqrt {x}}-\frac {\sqrt {x}\, A b}{a^{2} \left (b x +a \right )}+\frac {\sqrt {x}\, B}{a \left (b x +a \right )}-\frac {3 \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) A b}{a^{2} \sqrt {a b}}+\frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) B}{a \sqrt {a b}}\) | \(87\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 65, normalized size = 0.74 \begin {gather*} -\frac {2 \, A a - {\left (B a - 3 \, A b\right )} x}{a^{2} b x^{\frac {3}{2}} + a^{3} \sqrt {x}} + \frac {{\left (B a - 3 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.40, size = 215, normalized size = 2.44 \begin {gather*} \left [\frac {{\left ({\left (B a b - 3 \, A b^{2}\right )} x^{2} + {\left (B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a + 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) - 2 \, {\left (2 \, A a^{2} b - {\left (B a^{2} b - 3 \, A a b^{2}\right )} x\right )} \sqrt {x}}{2 \, {\left (a^{3} b^{2} x^{2} + a^{4} b x\right )}}, -\frac {{\left ({\left (B a b - 3 \, A b^{2}\right )} x^{2} + {\left (B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (2 \, A a^{2} b - {\left (B a^{2} b - 3 \, A a b^{2}\right )} x\right )} \sqrt {x}}{a^{3} b^{2} x^{2} + a^{4} b x}\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. 794 vs.
\(2 (73) = 146\).
time = 8.10, size = 794, normalized size = 9.02 \begin {gather*} \begin {cases} \tilde {\infty } \left (- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{b^{2}} & \text {for}\: a = 0 \\\frac {- \frac {2 A}{\sqrt {x}} + 2 B \sqrt {x}}{a^{2}} & \text {for}\: b = 0 \\- \frac {3 A a b \sqrt {x} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} + \frac {3 A a b \sqrt {x} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} - \frac {4 A a b \sqrt {- \frac {a}{b}}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} - \frac {3 A b^{2} x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} + \frac {3 A b^{2} x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} - \frac {6 A b^{2} x \sqrt {- \frac {a}{b}}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} + \frac {B a^{2} \sqrt {x} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} - \frac {B a^{2} \sqrt {x} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} + \frac {B a b x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} - \frac {B a b x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} + \frac {2 B a b x \sqrt {- \frac {a}{b}}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.03, size = 60, normalized size = 0.68 \begin {gather*} \frac {{\left (B a - 3 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {B a x - 3 \, A b x - 2 \, A a}{{\left (b x^{\frac {3}{2}} + a \sqrt {x}\right )} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.43, size = 65, normalized size = 0.74 \begin {gather*} -\frac {\frac {2\,A}{a}+\frac {x\,\left (3\,A\,b-B\,a\right )}{a^2}}{a\,\sqrt {x}+b\,x^{3/2}}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (3\,A\,b-B\,a\right )}{a^{5/2}\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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