Optimal. Leaf size=64 \[ \frac {2}{3 a \sqrt {x} (a+b x)^{3/2}}+\frac {8}{3 a^2 \sqrt {x} \sqrt {a+b x}}-\frac {16 \sqrt {a+b x}}{3 a^3 \sqrt {x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 64, 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 = {47, 37}
\begin {gather*} -\frac {16 \sqrt {a+b x}}{3 a^3 \sqrt {x}}+\frac {8}{3 a^2 \sqrt {x} \sqrt {a+b x}}+\frac {2}{3 a \sqrt {x} (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{x^{3/2} (a+b x)^{5/2}} \, dx &=\frac {2}{3 a \sqrt {x} (a+b x)^{3/2}}+\frac {4 \int \frac {1}{x^{3/2} (a+b x)^{3/2}} \, dx}{3 a}\\ &=\frac {2}{3 a \sqrt {x} (a+b x)^{3/2}}+\frac {8}{3 a^2 \sqrt {x} \sqrt {a+b x}}+\frac {8 \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{3 a^2}\\ &=\frac {2}{3 a \sqrt {x} (a+b x)^{3/2}}+\frac {8}{3 a^2 \sqrt {x} \sqrt {a+b x}}-\frac {16 \sqrt {a+b x}}{3 a^3 \sqrt {x}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 40, normalized size = 0.62 \begin {gather*} -\frac {2 \left (3 a^2+12 a b x+8 b^2 x^2\right )}{3 a^3 \sqrt {x} (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 54, normalized size = 0.84
method | result | size |
gosper | \(-\frac {2 \left (8 x^{2} b^{2}+12 a b x +3 a^{2}\right )}{3 \sqrt {x}\, \left (b x +a \right )^{\frac {3}{2}} a^{3}}\) | \(35\) |
risch | \(-\frac {2 \sqrt {b x +a}}{a^{3} \sqrt {x}}-\frac {2 b \left (5 b x +6 a \right ) \sqrt {x}}{3 \left (b x +a \right )^{\frac {3}{2}} a^{3}}\) | \(41\) |
default | \(-\frac {2}{a \left (b x +a \right )^{\frac {3}{2}} \sqrt {x}}-\frac {4 b \left (\frac {2 \sqrt {x}}{3 a \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 \sqrt {x}}{3 a^{2} \sqrt {b x +a}}\right )}{a}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 46, normalized size = 0.72 \begin {gather*} \frac {2 \, {\left (b^{2} - \frac {6 \, {\left (b x + a\right )} b}{x}\right )} x^{\frac {3}{2}}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3}} - \frac {2 \, \sqrt {b x + a}}{a^{3} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.91, size = 58, normalized size = 0.91 \begin {gather*} -\frac {2 \, {\left (8 \, b^{2} x^{2} + 12 \, a b x + 3 \, a^{2}\right )} \sqrt {b x + a} \sqrt {x}}{3 \, {\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 153 vs.
\(2 (58) = 116\).
time = 2.12, size = 153, normalized size = 2.39 \begin {gather*} - \frac {6 a^{2} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} - \frac {24 a b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} - \frac {16 b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{3 a^{5} b^{4} + 6 a^{4} b^{5} x + 3 a^{3} b^{6} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 159 vs.
\(2 (46) = 92\).
time = 1.81, size = 159, normalized size = 2.48 \begin {gather*} -\frac {2 \, \sqrt {b x + a} b^{2}}{\sqrt {{\left (b x + a\right )} b - a b} a^{3} {\left | b \right |}} - \frac {4 \, {\left (3 \, {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac {5}{2}} + 12 \, a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {7}{2}} + 5 \, a^{2} b^{\frac {9}{2}}\right )}}{3 \, {\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} a^{2} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.42, size = 71, normalized size = 1.11 \begin {gather*} -\frac {6\,a^2\,\sqrt {a+b\,x}+16\,b^2\,x^2\,\sqrt {a+b\,x}+24\,a\,b\,x\,\sqrt {a+b\,x}}{\sqrt {x}\,\left (x\,\left (6\,a^4\,b+3\,x\,a^3\,b^2\right )+3\,a^5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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