Optimal. Leaf size=97 \[ -\frac {208 a^3 \sqrt {1-a x}}{105 \sqrt {a x}}-\frac {104 a^3 \sqrt {1-a x}}{105 (a x)^{3/2}}-\frac {26 a^3 \sqrt {1-a x}}{35 (a x)^{5/2}}-\frac {2 a^3 \sqrt {1-a x}}{7 (a x)^{7/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {16, 78, 45, 37} \[ -\frac {208 a^3 \sqrt {1-a x}}{105 \sqrt {a x}}-\frac {104 a^3 \sqrt {1-a x}}{105 (a x)^{3/2}}-\frac {26 a^3 \sqrt {1-a x}}{35 (a x)^{5/2}}-\frac {2 a^3 \sqrt {1-a x}}{7 (a x)^{7/2}} \]
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^4 \sqrt {a x} \sqrt {1-a x}} \, dx &=a^4 \int \frac {1+a x}{(a x)^{9/2} \sqrt {1-a x}} \, dx\\ &=-\frac {2 a^3 \sqrt {1-a x}}{7 (a x)^{7/2}}+\frac {1}{7} \left (13 a^4\right ) \int \frac {1}{(a x)^{7/2} \sqrt {1-a x}} \, dx\\ &=-\frac {2 a^3 \sqrt {1-a x}}{7 (a x)^{7/2}}-\frac {26 a^3 \sqrt {1-a x}}{35 (a x)^{5/2}}+\frac {1}{35} \left (52 a^4\right ) \int \frac {1}{(a x)^{5/2} \sqrt {1-a x}} \, dx\\ &=-\frac {2 a^3 \sqrt {1-a x}}{7 (a x)^{7/2}}-\frac {26 a^3 \sqrt {1-a x}}{35 (a x)^{5/2}}-\frac {104 a^3 \sqrt {1-a x}}{105 (a x)^{3/2}}+\frac {1}{105} \left (104 a^4\right ) \int \frac {1}{(a x)^{3/2} \sqrt {1-a x}} \, dx\\ &=-\frac {2 a^3 \sqrt {1-a x}}{7 (a x)^{7/2}}-\frac {26 a^3 \sqrt {1-a x}}{35 (a x)^{5/2}}-\frac {104 a^3 \sqrt {1-a x}}{105 (a x)^{3/2}}-\frac {208 a^3 \sqrt {1-a x}}{105 \sqrt {a x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 0.46 \[ -\frac {2 \sqrt {-a x (a x-1)} \left (104 a^3 x^3+52 a^2 x^2+39 a x+15\right )}{105 a x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 43, normalized size = 0.44 \[ -\frac {2 \, {\left (104 \, a^{3} x^{3} + 52 \, a^{2} x^{2} + 39 \, a x + 15\right )} \sqrt {a x} \sqrt {-a x + 1}}{105 \, a x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.41, size = 175, normalized size = 1.80 \[ -\frac {\frac {15 \, a^{4} {\left (\sqrt {-a x + 1} - 1\right )}^{7}}{\left (a x\right )^{\frac {7}{2}}} + \frac {231 \, a^{4} {\left (\sqrt {-a x + 1} - 1\right )}^{5}}{\left (a x\right )^{\frac {5}{2}}} + \frac {1435 \, a^{4} {\left (\sqrt {-a x + 1} - 1\right )}^{3}}{\left (a x\right )^{\frac {3}{2}}} + \frac {7875 \, a^{4} {\left (\sqrt {-a x + 1} - 1\right )}}{\sqrt {a x}} - \frac {{\left (15 \, a^{4} + \frac {231 \, a^{3} {\left (\sqrt {-a x + 1} - 1\right )}^{2}}{x} + \frac {1435 \, a^{2} {\left (\sqrt {-a x + 1} - 1\right )}^{4}}{x^{2}} + \frac {7875 \, a {\left (\sqrt {-a x + 1} - 1\right )}^{6}}{x^{3}}\right )} \left (a x\right )^{\frac {7}{2}}}{{\left (\sqrt {-a x + 1} - 1\right )}^{7}}}{6720 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 41, normalized size = 0.42 \[ -\frac {2 \left (104 a^{3} x^{3}+52 a^{2} x^{2}+39 a x +15\right ) \sqrt {-a x +1}}{105 \sqrt {a x}\, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 84, normalized size = 0.87 \[ -\frac {208 \, \sqrt {-a^{2} x^{2} + a x} a^{2}}{105 \, x} - \frac {104 \, \sqrt {-a^{2} x^{2} + a x} a}{105 \, x^{2}} - \frac {26 \, \sqrt {-a^{2} x^{2} + a x}}{35 \, x^{3}} - \frac {2 \, \sqrt {-a^{2} x^{2} + a x}}{7 \, a x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.77, size = 40, normalized size = 0.41 \[ -\frac {\sqrt {1-a\,x}\,\left (\frac {208\,a^3\,x^3}{105}+\frac {104\,a^2\,x^2}{105}+\frac {26\,a\,x}{35}+\frac {2}{7}\right )}{x^3\,\sqrt {a\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 23.15, size = 274, normalized size = 2.82 \[ a \left (\begin {cases} - \frac {16 a^{2} \sqrt {-1 + \frac {1}{a x}}}{15} - \frac {8 a \sqrt {-1 + \frac {1}{a x}}}{15 x} - \frac {2 \sqrt {-1 + \frac {1}{a x}}}{5 x^{2}} & \text {for}\: \frac {1}{\left |{a x}\right |} > 1 \\- \frac {16 i a^{2} \sqrt {1 - \frac {1}{a x}}}{15} - \frac {8 i a \sqrt {1 - \frac {1}{a x}}}{15 x} - \frac {2 i \sqrt {1 - \frac {1}{a x}}}{5 x^{2}} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {32 a^{3} \sqrt {-1 + \frac {1}{a x}}}{35} - \frac {16 a^{2} \sqrt {-1 + \frac {1}{a x}}}{35 x} - \frac {12 a \sqrt {-1 + \frac {1}{a x}}}{35 x^{2}} - \frac {2 \sqrt {-1 + \frac {1}{a x}}}{7 x^{3}} & \text {for}\: \frac {1}{\left |{a x}\right |} > 1 \\- \frac {32 i a^{3} \sqrt {1 - \frac {1}{a x}}}{35} - \frac {16 i a^{2} \sqrt {1 - \frac {1}{a x}}}{35 x} - \frac {12 i a \sqrt {1 - \frac {1}{a x}}}{35 x^{2}} - \frac {2 i \sqrt {1 - \frac {1}{a x}}}{7 x^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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