Optimal. Leaf size=121 \[ -\frac{544 a^4 \sqrt{1-a x}}{315 \sqrt{a x}}-\frac{272 a^4 \sqrt{1-a x}}{315 (a x)^{3/2}}-\frac{68 a^4 \sqrt{1-a x}}{105 (a x)^{5/2}}-\frac{34 a^4 \sqrt{1-a x}}{63 (a x)^{7/2}}-\frac{2 a^4 \sqrt{1-a x}}{9 (a x)^{9/2}} \]
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Rubi [A] time = 0.0384211, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {16, 78, 45, 37} \[ -\frac{544 a^4 \sqrt{1-a x}}{315 \sqrt{a x}}-\frac{272 a^4 \sqrt{1-a x}}{315 (a x)^{3/2}}-\frac{68 a^4 \sqrt{1-a x}}{105 (a x)^{5/2}}-\frac{34 a^4 \sqrt{1-a x}}{63 (a x)^{7/2}}-\frac{2 a^4 \sqrt{1-a x}}{9 (a x)^{9/2}} \]
Antiderivative was successfully verified.
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Rule 16
Rule 78
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{1+a x}{x^5 \sqrt{a x} \sqrt{1-a x}} \, dx &=a^5 \int \frac{1+a x}{(a x)^{11/2} \sqrt{1-a x}} \, dx\\ &=-\frac{2 a^4 \sqrt{1-a x}}{9 (a x)^{9/2}}+\frac{1}{9} \left (17 a^5\right ) \int \frac{1}{(a x)^{9/2} \sqrt{1-a x}} \, dx\\ &=-\frac{2 a^4 \sqrt{1-a x}}{9 (a x)^{9/2}}-\frac{34 a^4 \sqrt{1-a x}}{63 (a x)^{7/2}}+\frac{1}{21} \left (34 a^5\right ) \int \frac{1}{(a x)^{7/2} \sqrt{1-a x}} \, dx\\ &=-\frac{2 a^4 \sqrt{1-a x}}{9 (a x)^{9/2}}-\frac{34 a^4 \sqrt{1-a x}}{63 (a x)^{7/2}}-\frac{68 a^4 \sqrt{1-a x}}{105 (a x)^{5/2}}+\frac{1}{105} \left (136 a^5\right ) \int \frac{1}{(a x)^{5/2} \sqrt{1-a x}} \, dx\\ &=-\frac{2 a^4 \sqrt{1-a x}}{9 (a x)^{9/2}}-\frac{34 a^4 \sqrt{1-a x}}{63 (a x)^{7/2}}-\frac{68 a^4 \sqrt{1-a x}}{105 (a x)^{5/2}}-\frac{272 a^4 \sqrt{1-a x}}{315 (a x)^{3/2}}+\frac{1}{315} \left (272 a^5\right ) \int \frac{1}{(a x)^{3/2} \sqrt{1-a x}} \, dx\\ &=-\frac{2 a^4 \sqrt{1-a x}}{9 (a x)^{9/2}}-\frac{34 a^4 \sqrt{1-a x}}{63 (a x)^{7/2}}-\frac{68 a^4 \sqrt{1-a x}}{105 (a x)^{5/2}}-\frac{272 a^4 \sqrt{1-a x}}{315 (a x)^{3/2}}-\frac{544 a^4 \sqrt{1-a x}}{315 \sqrt{a x}}\\ \end{align*}
Mathematica [A] time = 0.0185621, size = 53, normalized size = 0.44 \[ -\frac{2 \sqrt{-a x (a x-1)} \left (272 a^4 x^4+136 a^3 x^3+102 a^2 x^2+85 a x+35\right )}{315 a x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 49, normalized size = 0.4 \begin{align*} -{\frac{544\,{a}^{4}{x}^{4}+272\,{a}^{3}{x}^{3}+204\,{a}^{2}{x}^{2}+170\,ax+70}{315\,{x}^{4}}\sqrt{-ax+1}{\frac{1}{\sqrt{ax}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40138, size = 131, normalized size = 1.08 \begin{align*} -\frac{2 \,{\left (272 \, a^{4} x^{4} + 136 \, a^{3} x^{3} + 102 \, a^{2} x^{2} + 85 \, a x + 35\right )} \sqrt{a x} \sqrt{-a x + 1}}{315 \, a x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 29.9426, size = 359, normalized size = 2.97 \begin{align*} a \left (\begin{cases} - \frac{32 a^{3} \sqrt{-1 + \frac{1}{a x}}}{35} - \frac{16 a^{2} \sqrt{-1 + \frac{1}{a x}}}{35 x} - \frac{12 a \sqrt{-1 + \frac{1}{a x}}}{35 x^{2}} - \frac{2 \sqrt{-1 + \frac{1}{a x}}}{7 x^{3}} & \text{for}\: \frac{1}{\left |{a x}\right |} > 1 \\- \frac{32 i a^{3} \sqrt{1 - \frac{1}{a x}}}{35} - \frac{16 i a^{2} \sqrt{1 - \frac{1}{a x}}}{35 x} - \frac{12 i a \sqrt{1 - \frac{1}{a x}}}{35 x^{2}} - \frac{2 i \sqrt{1 - \frac{1}{a x}}}{7 x^{3}} & \text{otherwise} \end{cases}\right ) + \begin{cases} - \frac{256 a^{4} \sqrt{-1 + \frac{1}{a x}}}{315} - \frac{128 a^{3} \sqrt{-1 + \frac{1}{a x}}}{315 x} - \frac{32 a^{2} \sqrt{-1 + \frac{1}{a x}}}{105 x^{2}} - \frac{16 a \sqrt{-1 + \frac{1}{a x}}}{63 x^{3}} - \frac{2 \sqrt{-1 + \frac{1}{a x}}}{9 x^{4}} & \text{for}\: \frac{1}{\left |{a x}\right |} > 1 \\- \frac{256 i a^{4} \sqrt{1 - \frac{1}{a x}}}{315} - \frac{128 i a^{3} \sqrt{1 - \frac{1}{a x}}}{315 x} - \frac{32 i a^{2} \sqrt{1 - \frac{1}{a x}}}{105 x^{2}} - \frac{16 i a \sqrt{1 - \frac{1}{a x}}}{63 x^{3}} - \frac{2 i \sqrt{1 - \frac{1}{a x}}}{9 x^{4}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.92981, size = 293, normalized size = 2.42 \begin{align*} -\frac{\frac{35 \, a^{5}{\left (\sqrt{-a x + 1} - 1\right )}^{9}}{\left (a x\right )^{\frac{9}{2}}} + \frac{585 \, a^{5}{\left (\sqrt{-a x + 1} - 1\right )}^{7}}{\left (a x\right )^{\frac{7}{2}}} + \frac{4032 \, a^{5}{\left (\sqrt{-a x + 1} - 1\right )}^{5}}{\left (a x\right )^{\frac{5}{2}}} + \frac{17640 \, a^{5}{\left (\sqrt{-a x + 1} - 1\right )}^{3}}{\left (a x\right )^{\frac{3}{2}}} + \frac{83790 \, a^{5}{\left (\sqrt{-a x + 1} - 1\right )}}{\sqrt{a x}} - \frac{{\left (35 \, a^{5} + \frac{585 \, a^{4}{\left (\sqrt{-a x + 1} - 1\right )}^{2}}{x} + \frac{4032 \, a^{3}{\left (\sqrt{-a x + 1} - 1\right )}^{4}}{x^{2}} + \frac{17640 \, a^{2}{\left (\sqrt{-a x + 1} - 1\right )}^{6}}{x^{3}} + \frac{83790 \, a{\left (\sqrt{-a x + 1} - 1\right )}^{8}}{x^{4}}\right )} \left (a x\right )^{\frac{9}{2}}}{{\left (\sqrt{-a x + 1} - 1\right )}^{9}}}{80640 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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