Optimal. Leaf size=84 \[ \frac {32 b \sqrt {a+b x}}{3 a^4 \sqrt {x}}-\frac {16 \sqrt {a+b x}}{3 a^3 x^{3/2}}+\frac {4}{a^2 x^{3/2} \sqrt {a+b x}}+\frac {2}{3 a x^{3/2} (a+b x)^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {45, 37} \begin {gather*} -\frac {16 \sqrt {a+b x}}{3 a^3 x^{3/2}}+\frac {4}{a^2 x^{3/2} \sqrt {a+b x}}+\frac {32 b \sqrt {a+b x}}{3 a^4 \sqrt {x}}+\frac {2}{3 a x^{3/2} (a+b x)^{3/2}} \end {gather*}
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)^{5/2}} \, dx &=\frac {2}{3 a x^{3/2} (a+b x)^{3/2}}+\frac {2 \int \frac {1}{x^{5/2} (a+b x)^{3/2}} \, dx}{a}\\ &=\frac {2}{3 a x^{3/2} (a+b x)^{3/2}}+\frac {4}{a^2 x^{3/2} \sqrt {a+b x}}+\frac {8 \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx}{a^2}\\ &=\frac {2}{3 a x^{3/2} (a+b x)^{3/2}}+\frac {4}{a^2 x^{3/2} \sqrt {a+b x}}-\frac {16 \sqrt {a+b x}}{3 a^3 x^{3/2}}-\frac {(16 b) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{3 a^3}\\ &=\frac {2}{3 a x^{3/2} (a+b x)^{3/2}}+\frac {4}{a^2 x^{3/2} \sqrt {a+b x}}-\frac {16 \sqrt {a+b x}}{3 a^3 x^{3/2}}+\frac {32 b \sqrt {a+b x}}{3 a^4 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 49, normalized size = 0.58 \begin {gather*} -\frac {2 \left (a^3-6 a^2 b x-24 a b^2 x^2-16 b^3 x^3\right )}{3 a^4 x^{3/2} (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 51, normalized size = 0.61 \begin {gather*} \frac {2 \left (-a^3+6 a^2 b x+24 a b^2 x^2+16 b^3 x^3\right )}{3 a^4 x^{3/2} (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 71, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (16 \, b^{3} x^{3} + 24 \, a b^{2} x^{2} + 6 \, a^{2} b x - a^{3}\right )} \sqrt {b x + a} \sqrt {x}}{3 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.24, size = 175, normalized size = 2.08 \begin {gather*} \frac {2 \, \sqrt {b x + a} {\left (\frac {8 \, {\left (b x + a\right )} b^{2} {\left | b \right |}}{a^{4}} - \frac {9 \, b^{2} {\left | b \right |}}{a^{3}}\right )}}{3 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {3}{2}}} + \frac {8 \, {\left (3 \, {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac {7}{2}} + 9 \, a {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {9}{2}} + 4 \, a^{2} b^{\frac {11}{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^{3} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 44, normalized size = 0.52 \begin {gather*} -\frac {2 \left (-16 b^{3} x^{3}-24 a \,b^{2} x^{2}-6 a^{2} b x +a^{3}\right )}{3 \left (b x +a \right )^{\frac {3}{2}} a^{4} x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 64, normalized size = 0.76 \begin {gather*} \frac {2 \, {\left (\frac {9 \, \sqrt {b x + a} b}{\sqrt {x}} - \frac {{\left (b x + a\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}}\right )}}{3 \, a^{4}} - \frac {2 \, {\left (b^{3} - \frac {9 \, {\left (b x + a\right )} b^{2}}{x}\right )} x^{\frac {3}{2}}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.47, size = 88, normalized size = 1.05 \begin {gather*} \frac {32\,b^3\,x^3\,\sqrt {a+b\,x}-2\,a^3\,\sqrt {a+b\,x}+12\,a^2\,b\,x\,\sqrt {a+b\,x}+48\,a\,b^2\,x^2\,\sqrt {a+b\,x}}{x^{3/2}\,\left (x\,\left (6\,a^5\,b+3\,x\,a^4\,b^2\right )+3\,a^6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 7.06, size = 337, normalized size = 4.01 \begin {gather*} - \frac {2 a^{4} b^{\frac {19}{2}} \sqrt {\frac {a}{b x} + 1}}{3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {10 a^{3} b^{\frac {21}{2}} x \sqrt {\frac {a}{b x} + 1}}{3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {60 a^{2} b^{\frac {23}{2}} x^{2} \sqrt {\frac {a}{b x} + 1}}{3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {80 a b^{\frac {25}{2}} x^{3} \sqrt {\frac {a}{b x} + 1}}{3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} + \frac {32 b^{\frac {27}{2}} x^{4} \sqrt {\frac {a}{b x} + 1}}{3 a^{7} b^{9} x + 9 a^{6} b^{10} x^{2} + 9 a^{5} b^{11} x^{3} + 3 a^{4} b^{12} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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