Optimal. Leaf size=69 \[ \frac {5 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}}+\frac {5 b}{a^3 \sqrt {x}}-\frac {5}{3 a^2 x^{3/2}}+\frac {1}{a x^{3/2} (a+b x)} \]
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Rubi [A] time = 0.02, antiderivative size = 69, 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} \[ \frac {5 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}}+\frac {5 b}{a^3 \sqrt {x}}-\frac {5}{3 a^2 x^{3/2}}+\frac {1}{a x^{3/2} (a+b x)} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 205
Rubi steps
\begin {align*} \int \frac {1}{x^{5/2} (a+b x)^2} \, dx &=\frac {1}{a x^{3/2} (a+b x)}+\frac {5 \int \frac {1}{x^{5/2} (a+b x)} \, dx}{2 a}\\ &=-\frac {5}{3 a^2 x^{3/2}}+\frac {1}{a x^{3/2} (a+b x)}-\frac {(5 b) \int \frac {1}{x^{3/2} (a+b x)} \, dx}{2 a^2}\\ &=-\frac {5}{3 a^2 x^{3/2}}+\frac {5 b}{a^3 \sqrt {x}}+\frac {1}{a x^{3/2} (a+b x)}+\frac {\left (5 b^2\right ) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{2 a^3}\\ &=-\frac {5}{3 a^2 x^{3/2}}+\frac {5 b}{a^3 \sqrt {x}}+\frac {1}{a x^{3/2} (a+b x)}+\frac {\left (5 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{a^3}\\ &=-\frac {5}{3 a^2 x^{3/2}}+\frac {5 b}{a^3 \sqrt {x}}+\frac {1}{a x^{3/2} (a+b x)}+\frac {5 b^{3/2} \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 = 0.39 \[ -\frac {2 \, _2F_1\left (-\frac {3}{2},2;-\frac {1}{2};-\frac {b x}{a}\right )}{3 a^2 x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 184, normalized size = 2.67 \[ \left [\frac {15 \, {\left (b^{2} x^{3} + a b x^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (15 \, b^{2} x^{2} + 10 \, a b x - 2 \, a^{2}\right )} \sqrt {x}}{6 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}}, -\frac {15 \, {\left (b^{2} x^{3} + a b x^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (15 \, b^{2} x^{2} + 10 \, a b x - 2 \, a^{2}\right )} \sqrt {x}}{3 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 58, normalized size = 0.84 \[ \frac {5 \, b^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} + \frac {b^{2} \sqrt {x}}{{\left (b x + a\right )} a^{3}} + \frac {2 \, {\left (6 \, b x - a\right )}}{3 \, a^{3} x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 60, normalized size = 0.87 \[ \frac {5 b^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{3}}+\frac {b^{2} \sqrt {x}}{\left (b x +a \right ) a^{3}}+\frac {4 b}{a^{3} \sqrt {x}}-\frac {2}{3 a^{2} x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.88, size = 64, normalized size = 0.93 \[ \frac {15 \, b^{2} x^{2} + 10 \, a b x - 2 \, a^{2}}{3 \, {\left (a^{3} b x^{\frac {5}{2}} + a^{4} x^{\frac {3}{2}}\right )}} + \frac {5 \, b^{2} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 58, normalized size = 0.84 \[ \frac {\frac {5\,b^2\,x^2}{a^3}-\frac {2}{3\,a}+\frac {10\,b\,x}{3\,a^2}}{a\,x^{3/2}+b\,x^{5/2}}+\frac {5\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 50.52, size = 507, normalized size = 7.35 \[ \begin {cases} \frac {\tilde {\infty }}{x^{\frac {7}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{3 a^{2} x^{\frac {3}{2}}} & \text {for}\: b = 0 \\- \frac {2}{7 b^{2} x^{\frac {7}{2}}} & \text {for}\: a = 0 \\- \frac {4 i a^{\frac {5}{2}} \sqrt {\frac {1}{b}}}{6 i a^{\frac {9}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 6 i a^{\frac {7}{2}} b x^{\frac {5}{2}} \sqrt {\frac {1}{b}}} + \frac {20 i a^{\frac {3}{2}} b x \sqrt {\frac {1}{b}}}{6 i a^{\frac {9}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 6 i a^{\frac {7}{2}} b x^{\frac {5}{2}} \sqrt {\frac {1}{b}}} + \frac {30 i \sqrt {a} b^{2} x^{2} \sqrt {\frac {1}{b}}}{6 i a^{\frac {9}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 6 i a^{\frac {7}{2}} b x^{\frac {5}{2}} \sqrt {\frac {1}{b}}} + \frac {15 a b x^{\frac {3}{2}} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{6 i a^{\frac {9}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 6 i a^{\frac {7}{2}} b x^{\frac {5}{2}} \sqrt {\frac {1}{b}}} - \frac {15 a b x^{\frac {3}{2}} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{6 i a^{\frac {9}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 6 i a^{\frac {7}{2}} b x^{\frac {5}{2}} \sqrt {\frac {1}{b}}} + \frac {15 b^{2} x^{\frac {5}{2}} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{6 i a^{\frac {9}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 6 i a^{\frac {7}{2}} b x^{\frac {5}{2}} \sqrt {\frac {1}{b}}} - \frac {15 b^{2} x^{\frac {5}{2}} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{6 i a^{\frac {9}{2}} x^{\frac {3}{2}} \sqrt {\frac {1}{b}} + 6 i a^{\frac {7}{2}} b x^{\frac {5}{2}} \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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