Optimal. Leaf size=87 \[ -\frac{32 b^2 \sqrt{a+b x}}{5 a^4 \sqrt{x}}+\frac{16 b \sqrt{a+b x}}{5 a^3 x^{3/2}}-\frac{12 \sqrt{a+b x}}{5 a^2 x^{5/2}}+\frac{2}{a x^{5/2} \sqrt{a+b x}} \]
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Rubi [A] time = 0.016875, antiderivative size = 87, normalized size of antiderivative = 1., 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} \[ -\frac{32 b^2 \sqrt{a+b x}}{5 a^4 \sqrt{x}}+\frac{16 b \sqrt{a+b x}}{5 a^3 x^{3/2}}-\frac{12 \sqrt{a+b x}}{5 a^2 x^{5/2}}+\frac{2}{a x^{5/2} \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{1}{x^{7/2} (a+b x)^{3/2}} \, dx &=\frac{2}{a x^{5/2} \sqrt{a+b x}}+\frac{6 \int \frac{1}{x^{7/2} \sqrt{a+b x}} \, dx}{a}\\ &=\frac{2}{a x^{5/2} \sqrt{a+b x}}-\frac{12 \sqrt{a+b x}}{5 a^2 x^{5/2}}-\frac{(24 b) \int \frac{1}{x^{5/2} \sqrt{a+b x}} \, dx}{5 a^2}\\ &=\frac{2}{a x^{5/2} \sqrt{a+b x}}-\frac{12 \sqrt{a+b x}}{5 a^2 x^{5/2}}+\frac{16 b \sqrt{a+b x}}{5 a^3 x^{3/2}}+\frac{\left (16 b^2\right ) \int \frac{1}{x^{3/2} \sqrt{a+b x}} \, dx}{5 a^3}\\ &=\frac{2}{a x^{5/2} \sqrt{a+b x}}-\frac{12 \sqrt{a+b x}}{5 a^2 x^{5/2}}+\frac{16 b \sqrt{a+b x}}{5 a^3 x^{3/2}}-\frac{32 b^2 \sqrt{a+b x}}{5 a^4 \sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.010603, size = 49, normalized size = 0.56 \[ -\frac{2 \left (-2 a^2 b x+a^3+8 a b^2 x^2+16 b^3 x^3\right )}{5 a^4 x^{5/2} \sqrt{a+b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 44, normalized size = 0.5 \begin{align*} -{\frac{32\,{b}^{3}{x}^{3}+16\,a{b}^{2}{x}^{2}-4\,{a}^{2}bx+2\,{a}^{3}}{5\,{a}^{4}}{x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{bx+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03385, size = 86, normalized size = 0.99 \begin{align*} -\frac{2 \, b^{3} \sqrt{x}}{\sqrt{b x + a} a^{4}} - \frac{2 \,{\left (\frac{15 \, \sqrt{b x + a} b^{2}}{\sqrt{x}} - \frac{5 \,{\left (b x + a\right )}^{\frac{3}{2}} b}{x^{\frac{3}{2}}} + \frac{{\left (b x + a\right )}^{\frac{5}{2}}}{x^{\frac{5}{2}}}\right )}}{5 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04874, size = 128, normalized size = 1.47 \begin{align*} -\frac{2 \,{\left (16 \, b^{3} x^{3} + 8 \, a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}\right )} \sqrt{b x + a} \sqrt{x}}{5 \,{\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 106.382, size = 348, normalized size = 4. \begin{align*} - \frac{2 a^{5} b^{\frac{19}{2}} \sqrt{\frac{a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac{10 a^{3} b^{\frac{23}{2}} x^{2} \sqrt{\frac{a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac{60 a^{2} b^{\frac{25}{2}} x^{3} \sqrt{\frac{a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac{80 a b^{\frac{27}{2}} x^{4} \sqrt{\frac{a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} - \frac{32 b^{\frac{29}{2}} x^{5} \sqrt{\frac{a}{b x} + 1}}{5 a^{7} b^{9} x^{2} + 15 a^{6} b^{10} x^{3} + 15 a^{5} b^{11} x^{4} + 5 a^{4} b^{12} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10179, size = 147, normalized size = 1.69 \begin{align*} -\frac{4 \, b^{\frac{9}{2}}}{{\left ({\left (\sqrt{b x + a} \sqrt{b} - \sqrt{{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} a^{3}{\left | b \right |}} + \frac{{\left (\frac{15 \, a^{4}}{b} +{\left (\frac{11 \,{\left (b x + a\right )} a^{2}}{b} - \frac{25 \, a^{3}}{b}\right )}{\left (b x + a\right )}\right )} \sqrt{b x + a}}{40 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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