Optimal. Leaf size=68 \[ -\frac {2 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}}-\frac {2 b^2}{a^3 \sqrt {x}}+\frac {2 b}{3 a^2 x^{3/2}}-\frac {2}{5 a x^{5/2}} \]
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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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {51, 63, 205} \[ -\frac {2 b^2}{a^3 \sqrt {x}}-\frac {2 b^{5/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}}+\frac {2 b}{3 a^2 x^{3/2}}-\frac {2}{5 a x^{5/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 205
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} \tan ^{-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 = 27, normalized size = 0.40 \[ -\frac {2 \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};-\frac {b x}{a}\right )}{5 a x^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 144, normalized size = 2.12 \[ \left [\frac {15 \, b^{2} x^{3} \sqrt {-\frac {b}{a}} \log \left (\frac {b x - 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) - 2 \, {\left (15 \, b^{2} x^{2} - 5 \, a b x + 3 \, a^{2}\right )} \sqrt {x}}{15 \, a^{3} x^{3}}, \frac {2 \, {\left (15 \, b^{2} x^{3} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (15 \, b^{2} x^{2} - 5 \, a b x + 3 \, a^{2}\right )} \sqrt {x}\right )}}{15 \, a^{3} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 52, normalized size = 0.76 \[ -\frac {2 \, b^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} - \frac {2 \, {\left (15 \, b^{2} x^{2} - 5 \, a b x + 3 \, a^{2}\right )}}{15 \, a^{3} x^{\frac {5}{2}}} \]
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 \[ -\frac {2 b^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{3}}-\frac {2 b^{2}}{a^{3} \sqrt {x}}+\frac {2 b}{3 a^{2} x^{\frac {3}{2}}}-\frac {2}{5 a \,x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.91, size = 52, normalized size = 0.76 \[ -\frac {2 \, b^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} - \frac {2 \, {\left (15 \, b^{2} x^{2} - 5 \, a b x + 3 \, a^{2}\right )}}{15 \, a^{3} x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 49, normalized size = 0.72 \[ -\frac {\frac {2}{5\,a}+\frac {2\,b^2\,x^2}{a^3}-\frac {2\,b\,x}{3\,a^2}}{x^{5/2}}-\frac {2\,b^{5/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 24.82, size = 139, normalized size = 2.04 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {7}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{7 b x^{\frac {7}{2}}} & \text {for}\: a = 0 \\- \frac {2}{5 a x^{\frac {5}{2}}} & \text {for}\: b = 0 \\- \frac {2}{5 a x^{\frac {5}{2}}} + \frac {2 b}{3 a^{2} x^{\frac {3}{2}}} - \frac {2 b^{2}}{a^{3} \sqrt {x}} + \frac {i b^{2} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{a^{\frac {7}{2}} \sqrt {\frac {1}{b}}} - \frac {i b^{2} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{a^{\frac {7}{2}} \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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