Optimal. Leaf size=63 \[ \frac {16 b \sqrt {a+b x}}{3 a^3 \sqrt {x}}-\frac {8 \sqrt {a+b x}}{3 a^2 x^{3/2}}+\frac {2}{a x^{3/2} \sqrt {a+b x}} \]
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Rubi [A] time = 0.01, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {45, 37} \[ -\frac {8 \sqrt {a+b x}}{3 a^2 x^{3/2}}+\frac {16 b \sqrt {a+b x}}{3 a^3 \sqrt {x}}+\frac {2}{a x^{3/2} \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{x^{5/2} (a+b x)^{3/2}} \, dx &=\frac {2}{a x^{3/2} \sqrt {a+b x}}+\frac {4 \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{a}\\ &=\frac {2}{a x^{3/2} \sqrt {a+b x}}-\frac {8 \sqrt {a+b x}}{3 a^2 x^{3/2}}-\frac {(8 b) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{3 a^2}\\ &=\frac {2}{a x^{3/2} \sqrt {a+b x}}-\frac {8 \sqrt {a+b x}}{3 a^2 x^{3/2}}+\frac {16 b \sqrt {a+b x}}{3 a^3 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 38, normalized size = 0.60 \[ -\frac {2 \left (a^2-4 a b x-8 b^2 x^2\right )}{3 a^3 x^{3/2} \sqrt {a+b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 49, normalized size = 0.78 \[ \frac {2 \, {\left (8 \, b^{2} x^{2} + 4 \, a b x - a^{2}\right )} \sqrt {b x + a} \sqrt {x}}{3 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.20, size = 98, normalized size = 1.56 \[ \frac {2 \, \sqrt {b x + a} {\left (\frac {5 \, {\left (b x + a\right )} b^{2} {\left | b \right |}}{a^{3}} - \frac {6 \, b^{2} {\left | b \right |}}{a^{2}}\right )}}{3 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {3}{2}}} + \frac {4 \, b^{\frac {7}{2}}}{{\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} a^{2} {\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 33, normalized size = 0.52 \[ -\frac {2 \left (-8 b^{2} x^{2}-4 a b x +a^{2}\right )}{3 \sqrt {b x +a}\, a^{3} x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 50, normalized size = 0.79 \[ \frac {2 \, b^{2} \sqrt {x}}{\sqrt {b x + a} a^{3}} + \frac {2 \, {\left (\frac {6 \, \sqrt {b x + a} b}{\sqrt {x}} - \frac {{\left (b x + a\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}}\right )}}{3 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 46, normalized size = 0.73 \[ \frac {\sqrt {a+b\,x}\,\left (\frac {8\,x}{3\,a^2}-\frac {2}{3\,a\,b}+\frac {16\,b\,x^2}{3\,a^3}\right )}{x^{5/2}+\frac {a\,x^{3/2}}{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.98, size = 219, normalized size = 3.48 \[ - \frac {2 a^{3} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac {6 a^{2} b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac {24 a b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac {16 b^{\frac {15}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{3 a^{5} b^{4} x + 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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