Optimal. Leaf size=95 \[ -\frac {15 b^2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{4 a^{7/2}}-\frac {15 b^2}{4 a^3 \sqrt {b x-a}}+\frac {5 b}{4 a^2 x \sqrt {b x-a}}+\frac {1}{2 a x^2 \sqrt {b x-a}} \]
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Rubi [A] time = 0.02, antiderivative size = 93, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {51, 63, 205} \begin {gather*} -\frac {15 b^2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{4 a^{7/2}}-\frac {5 \sqrt {b x-a}}{2 a^2 x^2}-\frac {15 b \sqrt {b x-a}}{4 a^3 x}-\frac {2}{a x^2 \sqrt {b x-a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 205
Rubi steps
\begin {align*} \int \frac {1}{x^3 (-a+b x)^{3/2}} \, dx &=-\frac {2}{a x^2 \sqrt {-a+b x}}-\frac {5 \int \frac {1}{x^3 \sqrt {-a+b x}} \, dx}{a}\\ &=-\frac {2}{a x^2 \sqrt {-a+b x}}-\frac {5 \sqrt {-a+b x}}{2 a^2 x^2}-\frac {(15 b) \int \frac {1}{x^2 \sqrt {-a+b x}} \, dx}{4 a^2}\\ &=-\frac {2}{a x^2 \sqrt {-a+b x}}-\frac {5 \sqrt {-a+b x}}{2 a^2 x^2}-\frac {15 b \sqrt {-a+b x}}{4 a^3 x}-\frac {\left (15 b^2\right ) \int \frac {1}{x \sqrt {-a+b x}} \, dx}{8 a^3}\\ &=-\frac {2}{a x^2 \sqrt {-a+b x}}-\frac {5 \sqrt {-a+b x}}{2 a^2 x^2}-\frac {15 b \sqrt {-a+b x}}{4 a^3 x}-\frac {(15 b) \operatorname {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b x}\right )}{4 a^3}\\ &=-\frac {2}{a x^2 \sqrt {-a+b x}}-\frac {5 \sqrt {-a+b x}}{2 a^2 x^2}-\frac {15 b \sqrt {-a+b x}}{4 a^3 x}-\frac {15 b^2 \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{4 a^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 36, normalized size = 0.38 \begin {gather*} -\frac {2 b^2 \, _2F_1\left (-\frac {1}{2},3;\frac {1}{2};1-\frac {b x}{a}\right )}{a^3 \sqrt {b x-a}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 79, normalized size = 0.83 \begin {gather*} -\frac {15 b^2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{4 a^{7/2}}-\frac {8 a^2+25 a (b x-a)+15 (b x-a)^2}{4 a^3 x^2 \sqrt {b x-a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 198, normalized size = 2.08 \begin {gather*} \left [-\frac {15 \, {\left (b^{3} x^{3} - a b^{2} x^{2}\right )} \sqrt {-a} \log \left (\frac {b x + 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) + 2 \, {\left (15 \, a b^{2} x^{2} - 5 \, a^{2} b x - 2 \, a^{3}\right )} \sqrt {b x - a}}{8 \, {\left (a^{4} b x^{3} - a^{5} x^{2}\right )}}, -\frac {15 \, {\left (b^{3} x^{3} - a b^{2} x^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + {\left (15 \, a b^{2} x^{2} - 5 \, a^{2} b x - 2 \, a^{3}\right )} \sqrt {b x - a}}{4 \, {\left (a^{4} b x^{3} - a^{5} x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 81, normalized size = 0.85 \begin {gather*} -\frac {15 \, b^{2} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{4 \, a^{\frac {7}{2}}} - \frac {2 \, b^{2}}{\sqrt {b x - a} a^{3}} - \frac {7 \, {\left (b x - a\right )}^{\frac {3}{2}} b^{2} + 9 \, \sqrt {b x - a} a b^{2}}{4 \, a^{3} b^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 75, normalized size = 0.79 \begin {gather*} -\frac {15 b^{2} \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{4 a^{\frac {7}{2}}}-\frac {2 b^{2}}{\sqrt {b x -a}\, a^{3}}-\frac {9 \sqrt {b x -a}}{4 a^{2} x^{2}}-\frac {7 \left (b x -a \right )^{\frac {3}{2}}}{4 a^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.93, size = 104, normalized size = 1.09 \begin {gather*} -\frac {15 \, {\left (b x - a\right )}^{2} b^{2} + 25 \, {\left (b x - a\right )} a b^{2} + 8 \, a^{2} b^{2}}{4 \, {\left ({\left (b x - a\right )}^{\frac {5}{2}} a^{3} + 2 \, {\left (b x - a\right )}^{\frac {3}{2}} a^{4} + \sqrt {b x - a} a^{5}\right )}} - \frac {15 \, b^{2} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{4 \, a^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 101, normalized size = 1.06 \begin {gather*} -\frac {\frac {2\,b^2}{a}+\frac {15\,b^2\,{\left (a-b\,x\right )}^2}{4\,a^3}-\frac {25\,b^2\,\left (a-b\,x\right )}{4\,a^2}}{2\,a\,{\left (b\,x-a\right )}^{3/2}+{\left (b\,x-a\right )}^{5/2}+a^2\,\sqrt {b\,x-a}}-\frac {15\,b^2\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )}{4\,a^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.57, size = 226, normalized size = 2.38 \begin {gather*} \begin {cases} - \frac {i}{2 a \sqrt {b} x^{\frac {5}{2}} \sqrt {\frac {a}{b x} - 1}} - \frac {5 i \sqrt {b}}{4 a^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} - 1}} + \frac {15 i b^{\frac {3}{2}}}{4 a^{3} \sqrt {x} \sqrt {\frac {a}{b x} - 1}} - \frac {15 i b^{2} \operatorname {acosh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{4 a^{\frac {7}{2}}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\\frac {1}{2 a \sqrt {b} x^{\frac {5}{2}} \sqrt {- \frac {a}{b x} + 1}} + \frac {5 \sqrt {b}}{4 a^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}} - \frac {15 b^{\frac {3}{2}}}{4 a^{3} \sqrt {x} \sqrt {- \frac {a}{b x} + 1}} + \frac {15 b^{2} \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{4 a^{\frac {7}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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