Optimal. Leaf size=76 \[ -\frac {3 \sqrt {a^2-b^2 x^2}}{b}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{b (a+b x)^2}-\frac {3 a \tan ^{-1}\left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{b} \]
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Rubi [A]
time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {677, 679, 223,
209} \begin {gather*} -\frac {3 a \text {ArcTan}\left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{b}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{b (a+b x)^2}-\frac {3 \sqrt {a^2-b^2 x^2}}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 677
Rule 679
Rubi steps
\begin {align*} \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^3} \, dx &=-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{b (a+b x)^2}-3 \int \frac {\sqrt {a^2-b^2 x^2}}{a+b x} \, dx\\ &=-\frac {3 \sqrt {a^2-b^2 x^2}}{b}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{b (a+b x)^2}-(3 a) \int \frac {1}{\sqrt {a^2-b^2 x^2}} \, dx\\ &=-\frac {3 \sqrt {a^2-b^2 x^2}}{b}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{b (a+b x)^2}-(3 a) \text {Subst}\left (\int \frac {1}{1+b^2 x^2} \, dx,x,\frac {x}{\sqrt {a^2-b^2 x^2}}\right )\\ &=-\frac {3 \sqrt {a^2-b^2 x^2}}{b}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{b (a+b x)^2}-\frac {3 a \tan ^{-1}\left (\frac {b x}{\sqrt {a^2-b^2 x^2}}\right )}{b}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 78, normalized size = 1.03 \begin {gather*} \frac {(-5 a-b x) \sqrt {a^2-b^2 x^2}}{b (a+b x)}+\frac {3 a \log \left (-\sqrt {-b^2} x+\sqrt {a^2-b^2 x^2}\right )}{\sqrt {-b^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(238\) vs.
\(2(70)=140\).
time = 0.51, size = 239, normalized size = 3.14
method | result | size |
risch | \(-\frac {\sqrt {-b^{2} x^{2}+a^{2}}}{b}-\frac {3 a \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} x^{2}+a^{2}}}\right )}{\sqrt {b^{2}}}-\frac {4 a \sqrt {-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )}}{b^{2} \left (x +\frac {a}{b}\right )}\) | \(94\) |
default | \(\frac {-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{a b \left (x +\frac {a}{b}\right )^{3}}-\frac {2 b \left (\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{a b \left (x +\frac {a}{b}\right )^{2}}+\frac {3 b \left (\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {3}{2}}}{3}+a b \left (-\frac {\left (-2 b^{2} \left (x +\frac {a}{b}\right )+2 a b \right ) \sqrt {-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )}}{4 b^{2}}+\frac {a^{2} \arctan \left (\frac {\sqrt {b^{2}}\, x}{\sqrt {-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )}}\right )}{2 \sqrt {b^{2}}}\right )\right )}{a}\right )}{a}}{b^{3}}\) | \(239\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 79, normalized size = 1.04 \begin {gather*} -\frac {3 \, a \arcsin \left (\frac {b x}{a}\right )}{b} + \frac {{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {3}{2}}}{b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b} - \frac {6 \, \sqrt {-b^{2} x^{2} + a^{2}} a}{b^{2} x + a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.35, size = 83, normalized size = 1.09 \begin {gather*} -\frac {5 \, a b x + 5 \, a^{2} - 6 \, {\left (a b x + a^{2}\right )} \arctan \left (-\frac {a - \sqrt {-b^{2} x^{2} + a^{2}}}{b x}\right ) + \sqrt {-b^{2} x^{2} + a^{2}} {\left (b x + 5 \, a\right )}}{b^{2} x + a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- \left (- a + b x\right ) \left (a + b x\right )\right )^{\frac {3}{2}}}{\left (a + b x\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.44, size = 77, normalized size = 1.01 \begin {gather*} -\frac {3 \, a \arcsin \left (\frac {b x}{a}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (b\right )}{{\left | b \right |}} - \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{b} + \frac {8 \, a}{{\left (\frac {a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}}{b^{2} x} + 1\right )} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a^2-b^2\,x^2\right )}^{3/2}}{{\left (a+b\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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