Optimal. Leaf size=56 \[ -\frac {3 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a+b x)} \]
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Rubi [A] time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {51, 63, 205} \begin {gather*} -\frac {3 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 205
Rubi steps
\begin {align*} \int \frac {1}{x^{3/2} (a+b x)^2} \, dx &=\frac {1}{a \sqrt {x} (a+b x)}+\frac {3 \int \frac {1}{x^{3/2} (a+b x)} \, dx}{2 a}\\ &=-\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a+b x)}-\frac {(3 b) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{2 a^2}\\ &=-\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a+b x)}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{a^2}\\ &=-\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a+b x)}-\frac {3 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 25, normalized size = 0.45 \begin {gather*} -\frac {2 \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};-\frac {b x}{a}\right )}{a^2 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 54, normalized size = 0.96 \begin {gather*} \frac {-2 a-3 b x}{a^2 \sqrt {x} (a+b x)}-\frac {3 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 147, normalized size = 2.62 \begin {gather*} \left [\frac {3 \, {\left (b x^{2} + a x\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 (3 \, b x + 2 \, a\right )} \sqrt {x}}{2 \, {\left (a^{2} b x^{2} + a^{3} x\right )}}, \frac {3 \, {\left (b x^{2} + a x\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (3 \, b x + 2 \, a\right )} \sqrt {x}}{a^{2} b x^{2} + a^{3} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 49, normalized size = 0.88 \begin {gather*} -\frac {3 \, b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} - \frac {3 \, b x + 2 \, a}{{\left (b x^{\frac {3}{2}} + a \sqrt {x}\right )} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 48, normalized size = 0.86 \begin {gather*} -\frac {3 b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}\, a^{2}}-\frac {b \sqrt {x}}{\left (b x +a \right ) a^{2}}-\frac {2}{a^{2} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.94, size = 51, normalized size = 0.91 \begin {gather*} -\frac {3 \, b x + 2 \, a}{a^{2} b x^{\frac {3}{2}} + a^{3} \sqrt {x}} - \frac {3 \, b \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 48, normalized size = 0.86 \begin {gather*} -\frac {\frac {2}{a}+\frac {3\,b\,x}{a^2}}{a\,\sqrt {x}+b\,x^{3/2}}-\frac {3\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.73, size = 434, normalized size = 7.75 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{5 b^{2} x^{\frac {5}{2}}} & \text {for}\: a = 0 \\- \frac {2}{a^{2} \sqrt {x}} & \text {for}\: b = 0 \\- \frac {4 i a^{\frac {3}{2}} \sqrt {\frac {1}{b}}}{2 i a^{\frac {7}{2}} \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} - \frac {6 i \sqrt {a} b x \sqrt {\frac {1}{b}}}{2 i a^{\frac {7}{2}} \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} - \frac {3 a \sqrt {x} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {7}{2}} \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} + \frac {3 a \sqrt {x} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {7}{2}} \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} - \frac {3 b x^{\frac {3}{2}} \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {7}{2}} \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b x^{\frac {3}{2}} \sqrt {\frac {1}{b}}} + \frac {3 b x^{\frac {3}{2}} \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{2 i a^{\frac {7}{2}} \sqrt {x} \sqrt {\frac {1}{b}} + 2 i a^{\frac {5}{2}} b x^{\frac {3}{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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