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 = 0.04, antiderivative size = 121, normalized size of antiderivative = 1.00, 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 37
Rule 45
Rule 78
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.93, size = 51, normalized size = 0.42 \[ -\frac {2 \, {\left (272 \, a^{4} x^{4} + 136 \, a^{3} x^{3} + 102 \, a^{2} x^{2} + 85 \, a x + 35\right )} \sqrt {a x} \sqrt {-a x + 1}}{315 \, a x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.32, size = 217, normalized size = 1.79 \[ -\frac {\frac {35 \, a^{5} {\left (\sqrt {-a x + 1} - 1\right )}^{9}}{\left (a x\right )^{\frac {9}{2}}} + \frac {585 \, a^{5} {\left (\sqrt {-a x + 1} - 1\right )}^{7}}{\left (a x\right )^{\frac {7}{2}}} + \frac {4032 \, a^{5} {\left (\sqrt {-a x + 1} - 1\right )}^{5}}{\left (a x\right )^{\frac {5}{2}}} + \frac {17640 \, a^{5} {\left (\sqrt {-a x + 1} - 1\right )}^{3}}{\left (a x\right )^{\frac {3}{2}}} + \frac {83790 \, a^{5} {\left (\sqrt {-a x + 1} - 1\right )}}{\sqrt {a x}} - \frac {{\left (35 \, a^{5} + \frac {585 \, a^{4} {\left (\sqrt {-a x + 1} - 1\right )}^{2}}{x} + \frac {4032 \, a^{3} {\left (\sqrt {-a x + 1} - 1\right )}^{4}}{x^{2}} + \frac {17640 \, a^{2} {\left (\sqrt {-a x + 1} - 1\right )}^{6}}{x^{3}} + \frac {83790 \, a {\left (\sqrt {-a x + 1} - 1\right )}^{8}}{x^{4}}\right )} \left (a x\right )^{\frac {9}{2}}}{{\left (\sqrt {-a x + 1} - 1\right )}^{9}}}{80640 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 49, normalized size = 0.40 \[ -\frac {2 \left (272 a^{4} x^{4}+136 a^{3} x^{3}+102 a^{2} x^{2}+85 a x +35\right ) \sqrt {-a x +1}}{315 \sqrt {a x}\, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 106, normalized size = 0.88 \[ -\frac {544 \, \sqrt {-a^{2} x^{2} + a x} a^{3}}{315 \, x} - \frac {272 \, \sqrt {-a^{2} x^{2} + a x} a^{2}}{315 \, x^{2}} - \frac {68 \, \sqrt {-a^{2} x^{2} + a x} a}{105 \, x^{3}} - \frac {34 \, \sqrt {-a^{2} x^{2} + a x}}{63 \, x^{4}} - \frac {2 \, \sqrt {-a^{2} x^{2} + a x}}{9 \, a x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.83, size = 48, normalized size = 0.40 \[ -\frac {\sqrt {1-a\,x}\,\left (\frac {544\,a^4\,x^4}{315}+\frac {272\,a^3\,x^3}{315}+\frac {68\,a^2\,x^2}{105}+\frac {34\,a\,x}{63}+\frac {2}{9}\right )}{x^4\,\sqrt {a\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 33.86, size = 359, normalized size = 2.97 \[ a \left (\begin {cases} - \frac {32 a^{3} \sqrt {-1 + \frac {1}{a x}}}{35} - \frac {16 a^{2} \sqrt {-1 + \frac {1}{a x}}}{35 x} - \frac {12 a \sqrt {-1 + \frac {1}{a x}}}{35 x^{2}} - \frac {2 \sqrt {-1 + \frac {1}{a x}}}{7 x^{3}} & \text {for}\: \frac {1}{\left |{a x}\right |} > 1 \\- \frac {32 i a^{3} \sqrt {1 - \frac {1}{a x}}}{35} - \frac {16 i a^{2} \sqrt {1 - \frac {1}{a x}}}{35 x} - \frac {12 i a \sqrt {1 - \frac {1}{a x}}}{35 x^{2}} - \frac {2 i \sqrt {1 - \frac {1}{a x}}}{7 x^{3}} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {256 a^{4} \sqrt {-1 + \frac {1}{a x}}}{315} - \frac {128 a^{3} \sqrt {-1 + \frac {1}{a x}}}{315 x} - \frac {32 a^{2} \sqrt {-1 + \frac {1}{a x}}}{105 x^{2}} - \frac {16 a \sqrt {-1 + \frac {1}{a x}}}{63 x^{3}} - \frac {2 \sqrt {-1 + \frac {1}{a x}}}{9 x^{4}} & \text {for}\: \frac {1}{\left |{a x}\right |} > 1 \\- \frac {256 i a^{4} \sqrt {1 - \frac {1}{a x}}}{315} - \frac {128 i a^{3} \sqrt {1 - \frac {1}{a x}}}{315 x} - \frac {32 i a^{2} \sqrt {1 - \frac {1}{a x}}}{105 x^{2}} - \frac {16 i a \sqrt {1 - \frac {1}{a x}}}{63 x^{3}} - \frac {2 i \sqrt {1 - \frac {1}{a x}}}{9 x^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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