Optimal. Leaf size=87 \[ -\frac{\sqrt{1-a x} (a x)^{5/2}}{3 a^3}-\frac{11 \sqrt{1-a x} (a x)^{3/2}}{12 a^3}-\frac{11 \sqrt{1-a x} \sqrt{a x}}{8 a^3}-\frac{11 \sin ^{-1}(1-2 a x)}{16 a^3} \]
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Rubi [A] time = 0.0335764, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {16, 80, 50, 53, 619, 216} \[ -\frac{\sqrt{1-a x} (a x)^{5/2}}{3 a^3}-\frac{11 \sqrt{1-a x} (a x)^{3/2}}{12 a^3}-\frac{11 \sqrt{1-a x} \sqrt{a x}}{8 a^3}-\frac{11 \sin ^{-1}(1-2 a x)}{16 a^3} \]
Antiderivative was successfully verified.
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Rule 16
Rule 80
Rule 50
Rule 53
Rule 619
Rule 216
Rubi steps
\begin{align*} \int \frac{x^2 (1+a x)}{\sqrt{a x} \sqrt{1-a x}} \, dx &=\frac{\int \frac{(a x)^{3/2} (1+a x)}{\sqrt{1-a x}} \, dx}{a^2}\\ &=-\frac{(a x)^{5/2} \sqrt{1-a x}}{3 a^3}+\frac{11 \int \frac{(a x)^{3/2}}{\sqrt{1-a x}} \, dx}{6 a^2}\\ &=-\frac{11 (a x)^{3/2} \sqrt{1-a x}}{12 a^3}-\frac{(a x)^{5/2} \sqrt{1-a x}}{3 a^3}+\frac{11 \int \frac{\sqrt{a x}}{\sqrt{1-a x}} \, dx}{8 a^2}\\ &=-\frac{11 \sqrt{a x} \sqrt{1-a x}}{8 a^3}-\frac{11 (a x)^{3/2} \sqrt{1-a x}}{12 a^3}-\frac{(a x)^{5/2} \sqrt{1-a x}}{3 a^3}+\frac{11 \int \frac{1}{\sqrt{a x} \sqrt{1-a x}} \, dx}{16 a^2}\\ &=-\frac{11 \sqrt{a x} \sqrt{1-a x}}{8 a^3}-\frac{11 (a x)^{3/2} \sqrt{1-a x}}{12 a^3}-\frac{(a x)^{5/2} \sqrt{1-a x}}{3 a^3}+\frac{11 \int \frac{1}{\sqrt{a x-a^2 x^2}} \, dx}{16 a^2}\\ &=-\frac{11 \sqrt{a x} \sqrt{1-a x}}{8 a^3}-\frac{11 (a x)^{3/2} \sqrt{1-a x}}{12 a^3}-\frac{(a x)^{5/2} \sqrt{1-a x}}{3 a^3}-\frac{11 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,a-2 a^2 x\right )}{16 a^4}\\ &=-\frac{11 \sqrt{a x} \sqrt{1-a x}}{8 a^3}-\frac{11 (a x)^{3/2} \sqrt{1-a x}}{12 a^3}-\frac{(a x)^{5/2} \sqrt{1-a x}}{3 a^3}-\frac{11 \sin ^{-1}(1-2 a x)}{16 a^3}\\ \end{align*}
Mathematica [A] time = 0.0311233, size = 81, normalized size = 0.93 \[ \frac{\sqrt{a} x \left (8 a^3 x^3+14 a^2 x^2+11 a x-33\right )+33 \sqrt{x} \sqrt{1-a x} \sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{24 a^{5/2} \sqrt{-a x (a x-1)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.013, size = 111, normalized size = 1.3 \begin{align*}{\frac{x{\it csgn} \left ( a \right ) }{48\,{a}^{2}}\sqrt{-ax+1} \left ( -16\,{a}^{2}{x}^{2}\sqrt{-x \left ( ax-1 \right ) a}{\it csgn} \left ( a \right ) -44\,\sqrt{-x \left ( ax-1 \right ) a}{\it csgn} \left ( a \right ) xa-66\,\sqrt{-x \left ( ax-1 \right ) a}{\it csgn} \left ( a \right ) +33\,\arctan \left ( 1/2\,{\frac{{\it csgn} \left ( a \right ) \left ( 2\,ax-1 \right ) }{\sqrt{-x \left ( ax-1 \right ) a}}} \right ) \right ){\frac{1}{\sqrt{ax}}}{\frac{1}{\sqrt{-x \left ( ax-1 \right ) a}}}} \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 = 1.37876, size = 146, normalized size = 1.68 \begin{align*} -\frac{{\left (8 \, a^{2} x^{2} + 22 \, a x + 33\right )} \sqrt{a x} \sqrt{-a x + 1} + 33 \, \arctan \left (\frac{\sqrt{a x} \sqrt{-a x + 1}}{a x}\right )}{24 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 18.821, size = 393, normalized size = 4.52 \begin{align*} a \left (\begin{cases} - \frac{5 i \operatorname{acosh}{\left (\sqrt{a} \sqrt{x} \right )}}{8 a^{4}} - \frac{i x^{\frac{7}{2}}}{3 \sqrt{a} \sqrt{a x - 1}} - \frac{i x^{\frac{5}{2}}}{12 a^{\frac{3}{2}} \sqrt{a x - 1}} - \frac{5 i x^{\frac{3}{2}}}{24 a^{\frac{5}{2}} \sqrt{a x - 1}} + \frac{5 i \sqrt{x}}{8 a^{\frac{7}{2}} \sqrt{a x - 1}} & \text{for}\: \left |{a x}\right | > 1 \\\frac{5 \operatorname{asin}{\left (\sqrt{a} \sqrt{x} \right )}}{8 a^{4}} + \frac{x^{\frac{7}{2}}}{3 \sqrt{a} \sqrt{- a x + 1}} + \frac{x^{\frac{5}{2}}}{12 a^{\frac{3}{2}} \sqrt{- a x + 1}} + \frac{5 x^{\frac{3}{2}}}{24 a^{\frac{5}{2}} \sqrt{- a x + 1}} - \frac{5 \sqrt{x}}{8 a^{\frac{7}{2}} \sqrt{- a x + 1}} & \text{otherwise} \end{cases}\right ) + \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.58971, size = 72, normalized size = 0.83 \begin{align*} -\frac{{\left (2 \, a x{\left (\frac{4 \, x}{a} + \frac{11}{a^{2}}\right )} + \frac{33}{a^{2}}\right )} \sqrt{a x} \sqrt{-a x + 1} - \frac{33 \, \arcsin \left (\sqrt{a x}\right )}{a^{2}}}{24 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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