Optimal. Leaf size=63 \[ -\frac {\sqrt {1-a x} (a x)^{3/2}}{2 a^2}-\frac {7 \sqrt {1-a x} \sqrt {a x}}{4 a^2}-\frac {7 \sin ^{-1}(1-2 a x)}{8 a^2} \]
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Rubi [A] time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {16, 80, 50, 53, 619, 216} \[ -\frac {\sqrt {1-a x} (a x)^{3/2}}{2 a^2}-\frac {7 \sqrt {1-a x} \sqrt {a x}}{4 a^2}-\frac {7 \sin ^{-1}(1-2 a x)}{8 a^2} \]
Antiderivative was successfully verified.
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Rule 16
Rule 50
Rule 53
Rule 80
Rule 216
Rule 619
Rubi steps
\begin {align*} \int \frac {x (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx &=\frac {\int \frac {\sqrt {a x} (1+a x)}{\sqrt {1-a x}} \, dx}{a}\\ &=-\frac {(a x)^{3/2} \sqrt {1-a x}}{2 a^2}+\frac {7 \int \frac {\sqrt {a x}}{\sqrt {1-a x}} \, dx}{4 a}\\ &=-\frac {7 \sqrt {a x} \sqrt {1-a x}}{4 a^2}-\frac {(a x)^{3/2} \sqrt {1-a x}}{2 a^2}+\frac {7 \int \frac {1}{\sqrt {a x} \sqrt {1-a x}} \, dx}{8 a}\\ &=-\frac {7 \sqrt {a x} \sqrt {1-a x}}{4 a^2}-\frac {(a x)^{3/2} \sqrt {1-a x}}{2 a^2}+\frac {7 \int \frac {1}{\sqrt {a x-a^2 x^2}} \, dx}{8 a}\\ &=-\frac {7 \sqrt {a x} \sqrt {1-a x}}{4 a^2}-\frac {(a x)^{3/2} \sqrt {1-a x}}{2 a^2}-\frac {7 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,a-2 a^2 x\right )}{8 a^3}\\ &=-\frac {7 \sqrt {a x} \sqrt {1-a x}}{4 a^2}-\frac {(a x)^{3/2} \sqrt {1-a x}}{2 a^2}-\frac {7 \sin ^{-1}(1-2 a x)}{8 a^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 73, normalized size = 1.16 \[ \frac {\sqrt {a} x \left (2 a^2 x^2+5 a x-7\right )+7 \sqrt {x} \sqrt {1-a x} \sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{4 a^{3/2} \sqrt {-a x (a x-1)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 49, normalized size = 0.78 \[ -\frac {{\left (2 \, a x + 7\right )} \sqrt {a x} \sqrt {-a x + 1} + 7 \, \arctan \left (\frac {\sqrt {a x} \sqrt {-a x + 1}}{a x}\right )}{4 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.28, size = 40, normalized size = 0.63 \[ -\frac {\sqrt {a x} \sqrt {-a x + 1} {\left (2 \, x + \frac {7}{a}\right )} - \frac {7 \, \arcsin \left (\sqrt {a x}\right )}{a}}{4 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 90, normalized size = 1.43 \[ -\frac {\sqrt {-a x +1}\, \left (4 \sqrt {-\left (a x -1\right ) a x}\, a x \,\mathrm {csgn}\relax (a )-7 \arctan \left (\frac {\left (2 a x -1\right ) \mathrm {csgn}\relax (a )}{2 \sqrt {-\left (a x -1\right ) a x}}\right )+14 \sqrt {-\left (a x -1\right ) a x}\, \mathrm {csgn}\relax (a )\right ) x \,\mathrm {csgn}\relax (a )}{8 \sqrt {a x}\, \sqrt {-\left (a x -1\right ) a x}\, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 61, normalized size = 0.97 \[ -\frac {\sqrt {-a^{2} x^{2} + a x} x}{2 \, a} - \frac {7 \, \arcsin \left (-\frac {2 \, a^{2} x - a}{a}\right )}{8 \, a^{2}} - \frac {7 \, \sqrt {-a^{2} x^{2} + a x}}{4 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.53, size = 191, normalized size = 3.03 \[ \frac {7\,\mathrm {atan}\left (\frac {\sqrt {a\,x}}{\sqrt {1-a\,x}-1}\right )}{2\,a^2}-\frac {\frac {2\,\sqrt {a\,x}}{\sqrt {1-a\,x}-1}-\frac {2\,{\left (a\,x\right )}^{3/2}}{{\left (\sqrt {1-a\,x}-1\right )}^3}}{a^2\,{\left (\frac {a\,x}{{\left (\sqrt {1-a\,x}-1\right )}^2}+1\right )}^2}-\frac {\frac {3\,\sqrt {a\,x}}{2\,\left (\sqrt {1-a\,x}-1\right )}+\frac {11\,{\left (a\,x\right )}^{3/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^3}-\frac {11\,{\left (a\,x\right )}^{5/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^5}-\frac {3\,{\left (a\,x\right )}^{7/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^7}}{a^2\,{\left (\frac {a\,x}{{\left (\sqrt {1-a\,x}-1\right )}^2}+1\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 20.77, size = 269, normalized size = 4.27 \[ a \left (\begin {cases} - \frac {3 i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{4 a^{3}} - \frac {i x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {a x - 1}} - \frac {i x^{\frac {3}{2}}}{4 a^{\frac {3}{2}} \sqrt {a x - 1}} + \frac {3 i \sqrt {x}}{4 a^{\frac {5}{2}} \sqrt {a x - 1}} & \text {for}\: \left |{a x}\right | > 1 \\\frac {3 \operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{4 a^{3}} + \frac {x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {- a x + 1}} + \frac {x^{\frac {3}{2}}}{4 a^{\frac {3}{2}} \sqrt {- a x + 1}} - \frac {3 \sqrt {x}}{4 a^{\frac {5}{2}} \sqrt {- a x + 1}} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{a^{2}} - \frac {i \sqrt {x} \sqrt {a x - 1}}{a^{\frac {3}{2}}} & \text {for}\: \left |{a x}\right | > 1 \\\frac {\operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{a^{2}} + \frac {x^{\frac {3}{2}}}{\sqrt {a} \sqrt {- a x + 1}} - \frac {\sqrt {x}}{a^{\frac {3}{2}} \sqrt {- a x + 1}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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