Optimal. Leaf size=84 \[ \frac {15 \sqrt {x}}{4 b^3}-\frac {x^{5/2}}{2 b (a-b x)^2}+\frac {5 x^{3/2}}{4 b^2 (a-b x)}-\frac {15 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{7/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {43, 52, 65, 214}
\begin {gather*} -\frac {15 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{7/2}}+\frac {5 x^{3/2}}{4 b^2 (a-b x)}-\frac {x^{5/2}}{2 b (a-b x)^2}+\frac {15 \sqrt {x}}{4 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 52
Rule 65
Rule 214
Rubi steps
\begin {align*} \int \frac {x^{5/2}}{(-a+b x)^3} \, dx &=-\frac {x^{5/2}}{2 b (a-b x)^2}+\frac {5 \int \frac {x^{3/2}}{(-a+b x)^2} \, dx}{4 b}\\ &=-\frac {x^{5/2}}{2 b (a-b x)^2}+\frac {5 x^{3/2}}{4 b^2 (a-b x)}+\frac {15 \int \frac {\sqrt {x}}{-a+b x} \, dx}{8 b^2}\\ &=\frac {15 \sqrt {x}}{4 b^3}-\frac {x^{5/2}}{2 b (a-b x)^2}+\frac {5 x^{3/2}}{4 b^2 (a-b x)}+\frac {(15 a) \int \frac {1}{\sqrt {x} (-a+b x)} \, dx}{8 b^3}\\ &=\frac {15 \sqrt {x}}{4 b^3}-\frac {x^{5/2}}{2 b (a-b x)^2}+\frac {5 x^{3/2}}{4 b^2 (a-b x)}+\frac {(15 a) \text {Subst}\left (\int \frac {1}{-a+b x^2} \, dx,x,\sqrt {x}\right )}{4 b^3}\\ &=\frac {15 \sqrt {x}}{4 b^3}-\frac {x^{5/2}}{2 b (a-b x)^2}+\frac {5 x^{3/2}}{4 b^2 (a-b x)}-\frac {15 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 71, normalized size = 0.85 \begin {gather*} \frac {\sqrt {x} \left (15 a^2-25 a b x+8 b^2 x^2\right )}{4 b^3 (a-b x)^2}-\frac {15 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{4 b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 35.49, size = 629, normalized size = 7.49 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\text {DirectedInfinity}\left [\sqrt {x}\right ],a\text {==}0\text {\&\&}b\text {==}0\right \},\left \{\frac {-2 x^{\frac {7}{2}}}{7 a^3},b\text {==}0\right \},\left \{\frac {2 \sqrt {x}}{b^3},a\text {==}0\right \}\right \},\frac {-15 a^3 \text {Log}\left [\sqrt {x}+\sqrt {\frac {a}{b}}\right ]}{8 a^2 b^4 \sqrt {\frac {a}{b}}-16 a b^5 x \sqrt {\frac {a}{b}}+8 b^6 x^2 \sqrt {\frac {a}{b}}}+\frac {15 a^3 \text {Log}\left [\sqrt {x}-\sqrt {\frac {a}{b}}\right ]}{8 a^2 b^4 \sqrt {\frac {a}{b}}-16 a b^5 x \sqrt {\frac {a}{b}}+8 b^6 x^2 \sqrt {\frac {a}{b}}}+\frac {30 a^2 b \sqrt {x} \sqrt {\frac {a}{b}}}{8 a^2 b^4 \sqrt {\frac {a}{b}}-16 a b^5 x \sqrt {\frac {a}{b}}+8 b^6 x^2 \sqrt {\frac {a}{b}}}-\frac {30 a^2 b x \text {Log}\left [\sqrt {x}-\sqrt {\frac {a}{b}}\right ]}{8 a^2 b^4 \sqrt {\frac {a}{b}}-16 a b^5 x \sqrt {\frac {a}{b}}+8 b^6 x^2 \sqrt {\frac {a}{b}}}+\frac {30 a^2 b x \text {Log}\left [\sqrt {x}+\sqrt {\frac {a}{b}}\right ]}{8 a^2 b^4 \sqrt {\frac {a}{b}}-16 a b^5 x \sqrt {\frac {a}{b}}+8 b^6 x^2 \sqrt {\frac {a}{b}}}-\frac {50 a b^2 x^{\frac {3}{2}} \sqrt {\frac {a}{b}}}{8 a^2 b^4 \sqrt {\frac {a}{b}}-16 a b^5 x \sqrt {\frac {a}{b}}+8 b^6 x^2 \sqrt {\frac {a}{b}}}-\frac {15 a b^2 x^2 \text {Log}\left [\sqrt {x}+\sqrt {\frac {a}{b}}\right ]}{8 a^2 b^4 \sqrt {\frac {a}{b}}-16 a b^5 x \sqrt {\frac {a}{b}}+8 b^6 x^2 \sqrt {\frac {a}{b}}}+\frac {15 a b^2 x^2 \text {Log}\left [\sqrt {x}-\sqrt {\frac {a}{b}}\right ]}{8 a^2 b^4 \sqrt {\frac {a}{b}}-16 a b^5 x \sqrt {\frac {a}{b}}+8 b^6 x^2 \sqrt {\frac {a}{b}}}+\frac {16 b^3 x^{\frac {5}{2}} \sqrt {\frac {a}{b}}}{8 a^2 b^4 \sqrt {\frac {a}{b}}-16 a b^5 x \sqrt {\frac {a}{b}}+8 b^6 x^2 \sqrt {\frac {a}{b}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.14, size = 57, normalized size = 0.68
method | result | size |
derivativedivides | \(\frac {2 \sqrt {x}}{b^{3}}-\frac {2 a \left (\frac {\frac {9 b \,x^{\frac {3}{2}}}{8}-\frac {7 a \sqrt {x}}{8}}{\left (-b x +a \right )^{2}}+\frac {15 \arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{3}}\) | \(57\) |
default | \(\frac {2 \sqrt {x}}{b^{3}}-\frac {2 a \left (\frac {\frac {9 b \,x^{\frac {3}{2}}}{8}-\frac {7 a \sqrt {x}}{8}}{\left (-b x +a \right )^{2}}+\frac {15 \arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{3}}\) | \(57\) |
risch | \(\frac {2 \sqrt {x}}{b^{3}}+\frac {a \left (\frac {-\frac {9 b \,x^{\frac {3}{2}}}{4}+\frac {7 a \sqrt {x}}{4}}{\left (b x -a \right )^{2}}-\frac {15 \arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{4 \sqrt {a b}}\right )}{b^{3}}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 90, normalized size = 1.07 \begin {gather*} -\frac {9 \, a b x^{\frac {3}{2}} - 7 \, a^{2} \sqrt {x}}{4 \, {\left (b^{5} x^{2} - 2 \, a b^{4} x + a^{2} b^{3}\right )}} + \frac {15 \, a \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{3}} + \frac {2 \, \sqrt {x}}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 199, normalized size = 2.37 \begin {gather*} \left [\frac {15 \, {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt {\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {\frac {a}{b}} + a}{b x - a}\right ) + 2 \, {\left (8 \, b^{2} x^{2} - 25 \, a b x + 15 \, a^{2}\right )} \sqrt {x}}{8 \, {\left (b^{5} x^{2} - 2 \, a b^{4} x + a^{2} b^{3}\right )}}, \frac {15 \, {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt {-\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {-\frac {a}{b}}}{a}\right ) + {\left (8 \, b^{2} x^{2} - 25 \, a b x + 15 \, a^{2}\right )} \sqrt {x}}{4 \, {\left (b^{5} x^{2} - 2 \, a b^{4} x + a^{2} b^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 34.69, size = 624, normalized size = 7.43 \begin {gather*} \begin {cases} \tilde {\infty } \sqrt {x} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2 x^{\frac {7}{2}}}{7 a^{3}} & \text {for}\: b = 0 \\\frac {2 \sqrt {x}}{b^{3}} & \text {for}\: a = 0 \\\frac {15 a^{3} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {\frac {a}{b}} - 16 a b^{5} x \sqrt {\frac {a}{b}} + 8 b^{6} x^{2} \sqrt {\frac {a}{b}}} - \frac {15 a^{3} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {\frac {a}{b}} - 16 a b^{5} x \sqrt {\frac {a}{b}} + 8 b^{6} x^{2} \sqrt {\frac {a}{b}}} + \frac {30 a^{2} b \sqrt {x} \sqrt {\frac {a}{b}}}{8 a^{2} b^{4} \sqrt {\frac {a}{b}} - 16 a b^{5} x \sqrt {\frac {a}{b}} + 8 b^{6} x^{2} \sqrt {\frac {a}{b}}} - \frac {30 a^{2} b x \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {\frac {a}{b}} - 16 a b^{5} x \sqrt {\frac {a}{b}} + 8 b^{6} x^{2} \sqrt {\frac {a}{b}}} + \frac {30 a^{2} b x \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {\frac {a}{b}} - 16 a b^{5} x \sqrt {\frac {a}{b}} + 8 b^{6} x^{2} \sqrt {\frac {a}{b}}} - \frac {50 a b^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b}}}{8 a^{2} b^{4} \sqrt {\frac {a}{b}} - 16 a b^{5} x \sqrt {\frac {a}{b}} + 8 b^{6} x^{2} \sqrt {\frac {a}{b}}} + \frac {15 a b^{2} x^{2} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {\frac {a}{b}} - 16 a b^{5} x \sqrt {\frac {a}{b}} + 8 b^{6} x^{2} \sqrt {\frac {a}{b}}} - \frac {15 a b^{2} x^{2} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{8 a^{2} b^{4} \sqrt {\frac {a}{b}} - 16 a b^{5} x \sqrt {\frac {a}{b}} + 8 b^{6} x^{2} \sqrt {\frac {a}{b}}} + \frac {16 b^{3} x^{\frac {5}{2}} \sqrt {\frac {a}{b}}}{8 a^{2} b^{4} \sqrt {\frac {a}{b}} - 16 a b^{5} x \sqrt {\frac {a}{b}} + 8 b^{6} x^{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 = 0.00, size = 92, normalized size = 1.10 \begin {gather*} -2 \left (-\frac {\sqrt {x}}{b^{3}}-\frac {-9 \sqrt {x} x b a+7 \sqrt {x} a^{2}}{8 b^{3} \left (x b-a\right )^{2}}-\frac {15 a \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{4 b^{3}\cdot 2 \sqrt {-a b}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 69, normalized size = 0.82 \begin {gather*} \frac {\frac {7\,a^2\,\sqrt {x}}{4}-\frac {9\,a\,b\,x^{3/2}}{4}}{a^2\,b^3-2\,a\,b^4\,x+b^5\,x^2}+\frac {2\,\sqrt {x}}{b^3}-\frac {15\,\sqrt {a}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{4\,b^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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