Optimal. Leaf size=68 \[ -\frac {2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}}+\frac {2 b^2}{a^3 \sqrt {x}}+\frac {2 b}{3 a^2 x^{3/2}}+\frac {2}{5 a x^{5/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, 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, 208} \begin {gather*} \frac {2 b^2}{a^3 \sqrt {x}}-\frac {2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}}+\frac {2 b}{3 a^2 x^{3/2}}+\frac {2}{5 a x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{x^{7/2} (-a+b x)} \, dx &=\frac {2}{5 a x^{5/2}}+\frac {b \int \frac {1}{x^{5/2} (-a+b x)} \, dx}{a}\\ &=\frac {2}{5 a x^{5/2}}+\frac {2 b}{3 a^2 x^{3/2}}+\frac {b^2 \int \frac {1}{x^{3/2} (-a+b x)} \, dx}{a^2}\\ &=\frac {2}{5 a x^{5/2}}+\frac {2 b}{3 a^2 x^{3/2}}+\frac {2 b^2}{a^3 \sqrt {x}}+\frac {b^3 \int \frac {1}{\sqrt {x} (-a+b x)} \, dx}{a^3}\\ &=\frac {2}{5 a x^{5/2}}+\frac {2 b}{3 a^2 x^{3/2}}+\frac {2 b^2}{a^3 \sqrt {x}}+\frac {\left (2 b^3\right ) \operatorname {Subst}\left (\int \frac {1}{-a+b x^2} \, dx,x,\sqrt {x}\right )}{a^3}\\ &=\frac {2}{5 a x^{5/2}}+\frac {2 b}{3 a^2 x^{3/2}}+\frac {2 b^2}{a^3 \sqrt {x}}-\frac {2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 26, normalized size = 0.38 \begin {gather*} \frac {2 \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\frac {b x}{a}\right )}{5 a x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 61, normalized size = 0.90 \begin {gather*} \frac {2 \left (3 a^2+5 a b x+15 b^2 x^2\right )}{15 a^3 x^{5/2}}-\frac {2 b^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 143, normalized size = 2.10 \begin {gather*} \left [\frac {15 \, b^{2} x^{3} \sqrt {\frac {b}{a}} \log \left (\frac {b x - 2 \, a \sqrt {x} \sqrt {\frac {b}{a}} + a}{b x - a}\right ) + 2 \, {\left (15 \, b^{2} x^{2} + 5 \, a b x + 3 \, a^{2}\right )} \sqrt {x}}{15 \, a^{3} x^{3}}, \frac {2 \, {\left (15 \, b^{2} x^{3} \sqrt {-\frac {b}{a}} \arctan \left (\frac {a \sqrt {-\frac {b}{a}}}{b \sqrt {x}}\right ) + {\left (15 \, b^{2} x^{2} + 5 \, a b x + 3 \, a^{2}\right )} \sqrt {x}\right )}}{15 \, a^{3} x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.97, size = 54, normalized size = 0.79 \begin {gather*} \frac {2 \, b^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{\sqrt {-a b} a^{3}} + \frac {2 \, {\left (15 \, b^{2} x^{2} + 5 \, a b x + 3 \, a^{2}\right )}}{15 \, a^{3} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 54, normalized size = 0.79 \begin {gather*} -\frac {2 b^{3} \arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{3}}+\frac {2 b^{2}}{a^{3} \sqrt {x}}+\frac {2 b}{3 a^{2} x^{\frac {3}{2}}}+\frac {2}{5 a \,x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.98, size = 68, normalized size = 1.00 \begin {gather*} \frac {b^{3} \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{\sqrt {a b} a^{3}} + \frac {2 \, {\left (15 \, b^{2} x^{2} + 5 \, a b x + 3 \, a^{2}\right )}}{15 \, a^{3} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 48, normalized size = 0.71 \begin {gather*} \frac {\frac {2}{5\,a}+\frac {2\,b^2\,x^2}{a^3}+\frac {2\,b\,x}{3\,a^2}}{x^{5/2}}-\frac {2\,b^{5/2}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 24.44, size = 131, normalized size = 1.93 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {7}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{7 b x^{\frac {7}{2}}} & \text {for}\: a = 0 \\\frac {2}{5 a x^{\frac {5}{2}}} & \text {for}\: b = 0 \\\frac {2}{5 a x^{\frac {5}{2}}} + \frac {2 b}{3 a^{2} x^{\frac {3}{2}}} + \frac {2 b^{2}}{a^{3} \sqrt {x}} + \frac {b^{2} \log {\left (- \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{a^{\frac {7}{2}} \sqrt {\frac {1}{b}}} - \frac {b^{2} \log {\left (\sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{a^{\frac {7}{2}} \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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