Optimal. Leaf size=74 \[ \frac {5 b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{7/2}}-\frac {5 b}{a^3 \sqrt {a+b x}}-\frac {5 b}{3 a^2 (a+b x)^{3/2}}-\frac {1}{a x (a+b x)^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 80, normalized size of antiderivative = 1.08, 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, 208} \begin {gather*} -\frac {5 \sqrt {a+b x}}{a^3 x}+\frac {10}{3 a^2 x \sqrt {a+b x}}+\frac {5 b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{7/2}}+\frac {2}{3 a x (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rubi steps
\begin {align*} \int \frac {1}{x^2 (a+b x)^{5/2}} \, dx &=\frac {2}{3 a x (a+b x)^{3/2}}+\frac {5 \int \frac {1}{x^2 (a+b x)^{3/2}} \, dx}{3 a}\\ &=\frac {2}{3 a x (a+b x)^{3/2}}+\frac {10}{3 a^2 x \sqrt {a+b x}}+\frac {5 \int \frac {1}{x^2 \sqrt {a+b x}} \, dx}{a^2}\\ &=\frac {2}{3 a x (a+b x)^{3/2}}+\frac {10}{3 a^2 x \sqrt {a+b x}}-\frac {5 \sqrt {a+b x}}{a^3 x}-\frac {(5 b) \int \frac {1}{x \sqrt {a+b x}} \, dx}{2 a^3}\\ &=\frac {2}{3 a x (a+b x)^{3/2}}+\frac {10}{3 a^2 x \sqrt {a+b x}}-\frac {5 \sqrt {a+b x}}{a^3 x}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{a^3}\\ &=\frac {2}{3 a x (a+b x)^{3/2}}+\frac {10}{3 a^2 x \sqrt {a+b x}}-\frac {5 \sqrt {a+b x}}{a^3 x}+\frac {5 b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 33, normalized size = 0.45 \begin {gather*} -\frac {2 b \, _2F_1\left (-\frac {3}{2},2;-\frac {1}{2};\frac {b x}{a}+1\right )}{3 a^2 (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 67, normalized size = 0.91 \begin {gather*} \frac {5 b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{7/2}}+\frac {2 a^2+10 a (a+b x)-15 (a+b x)^2}{3 a^3 x (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 221, normalized size = 2.99 \begin {gather*} \left [\frac {15 \, {\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + a^{2} b x\right )} \sqrt {a} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, {\left (15 \, a b^{2} x^{2} + 20 \, a^{2} b x + 3 \, a^{3}\right )} \sqrt {b x + a}}{6 \, {\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )}}, -\frac {15 \, {\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + a^{2} b x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (15 \, a b^{2} x^{2} + 20 \, a^{2} b x + 3 \, a^{3}\right )} \sqrt {b x + a}}{3 \, {\left (a^{4} b^{2} x^{3} + 2 \, a^{5} b x^{2} + a^{6} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.03, size = 65, normalized size = 0.88 \begin {gather*} -\frac {5 \, b \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3}} - \frac {2 \, {\left (6 \, {\left (b x + a\right )} b + a b\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3}} - \frac {\sqrt {b x + a}}{a^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 67, normalized size = 0.91 \begin {gather*} 2 \left (-\frac {1}{3 \left (b x +a \right )^{\frac {3}{2}} a^{2}}-\frac {2}{\sqrt {b x +a}\, a^{3}}-\frac {-\frac {5 \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2 \sqrt {a}}+\frac {\sqrt {b x +a}}{2 b x}}{a^{3}}\right ) b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.03, size = 89, normalized size = 1.20 \begin {gather*} -\frac {15 \, {\left (b x + a\right )}^{2} b - 10 \, {\left (b x + a\right )} a b - 2 \, a^{2} b}{3 \, {\left ({\left (b x + a\right )}^{\frac {5}{2}} a^{3} - {\left (b x + a\right )}^{\frac {3}{2}} a^{4}\right )}} - \frac {5 \, b \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{2 \, a^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 73, normalized size = 0.99 \begin {gather*} \frac {5\,b\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )}{a^{7/2}}-\frac {\frac {2\,b}{3\,a}+\frac {10\,b\,\left (a+b\,x\right )}{3\,a^2}-\frac {5\,b\,{\left (a+b\,x\right )}^2}{a^3}}{a\,{\left (a+b\,x\right )}^{3/2}-{\left (a+b\,x\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.60, size = 818, normalized size = 11.05 \begin {gather*} - \frac {6 a^{17} \sqrt {1 + \frac {b x}{a}}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} - \frac {46 a^{16} b x \sqrt {1 + \frac {b x}{a}}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} - \frac {15 a^{16} b x \log {\left (\frac {b x}{a} \right )}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} + \frac {30 a^{16} b x \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} - \frac {70 a^{15} b^{2} x^{2} \sqrt {1 + \frac {b x}{a}}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} - \frac {45 a^{15} b^{2} x^{2} \log {\left (\frac {b x}{a} \right )}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} + \frac {90 a^{15} b^{2} x^{2} \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} - \frac {30 a^{14} b^{3} x^{3} \sqrt {1 + \frac {b x}{a}}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} - \frac {45 a^{14} b^{3} x^{3} \log {\left (\frac {b x}{a} \right )}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} + \frac {90 a^{14} b^{3} x^{3} \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} - \frac {15 a^{13} b^{4} x^{4} \log {\left (\frac {b x}{a} \right )}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} + \frac {30 a^{13} b^{4} x^{4} \log {\left (\sqrt {1 + \frac {b x}{a}} + 1 \right )}}{6 a^{\frac {39}{2}} x + 18 a^{\frac {37}{2}} b x^{2} + 18 a^{\frac {35}{2}} b^{2} x^{3} + 6 a^{\frac {33}{2}} b^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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