Optimal. Leaf size=87 \[ -\frac {\sqrt {1-a x} (a x)^{5/2}}{3 a^3}-\frac {11 \sqrt {1-a x} (a x)^{3/2}}{12 a^3}-\frac {11 \sqrt {1-a x} \sqrt {a x}}{8 a^3}-\frac {11 \sin ^{-1}(1-2 a x)}{16 a^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 87, normalized size of antiderivative = 1.00, 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.
[In]
[Out]
Rule 16
Rule 50
Rule 53
Rule 80
Rule 216
Rule 619
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 = 0.03, 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.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.22, size = 57, normalized size = 0.66 \[ -\frac {{\left (8 \, a^{2} x^{2} + 22 \, a x + 33\right )} \sqrt {a x} \sqrt {-a x + 1} + 33 \, \arctan \left (\frac {\sqrt {a x} \sqrt {-a x + 1}}{a x}\right )}{24 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.34, size = 53, normalized size = 0.61 \[ -\frac {{\left (2 \, a x {\left (\frac {4 \, x}{a} + \frac {11}{a^{2}}\right )} + \frac {33}{a^{2}}\right )} \sqrt {a x} \sqrt {-a x + 1} - \frac {33 \, \arcsin \left (\sqrt {a x}\right )}{a^{2}}}{24 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.02, size = 111, normalized size = 1.28 \[ -\frac {\sqrt {-a x +1}\, \left (16 \sqrt {-\left (a x -1\right ) a x}\, a^{2} x^{2} \mathrm {csgn}\relax (a )+44 \sqrt {-\left (a x -1\right ) a x}\, a x \,\mathrm {csgn}\relax (a )-33 \arctan \left (\frac {\left (2 a x -1\right ) \mathrm {csgn}\relax (a )}{2 \sqrt {-\left (a x -1\right ) a x}}\right )+66 \sqrt {-\left (a x -1\right ) a x}\, \mathrm {csgn}\relax (a )\right ) x \,\mathrm {csgn}\relax (a )}{48 \sqrt {a x}\, \sqrt {-\left (a x -1\right ) a x}\, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.99, size = 83, normalized size = 0.95 \[ -\frac {\sqrt {-a^{2} x^{2} + a x} x^{2}}{3 \, a} - \frac {11 \, \sqrt {-a^{2} x^{2} + a x} x}{12 \, a^{2}} - \frac {11 \, \arcsin \left (-\frac {2 \, a^{2} x - a}{a}\right )}{16 \, a^{3}} - \frac {11 \, \sqrt {-a^{2} x^{2} + a x}}{8 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 5.92, size = 269, normalized size = 3.09 \[ \frac {11\,\mathrm {atan}\left (\frac {\sqrt {a\,x}}{\sqrt {1-a\,x}-1}\right )}{4\,a^3}-\frac {\frac {5\,\sqrt {a\,x}}{4\,\left (\sqrt {1-a\,x}-1\right )}+\frac {85\,{\left (a\,x\right )}^{3/2}}{12\,{\left (\sqrt {1-a\,x}-1\right )}^3}+\frac {33\,{\left (a\,x\right )}^{5/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^5}-\frac {33\,{\left (a\,x\right )}^{7/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^7}-\frac {85\,{\left (a\,x\right )}^{9/2}}{12\,{\left (\sqrt {1-a\,x}-1\right )}^9}-\frac {5\,{\left (a\,x\right )}^{11/2}}{4\,{\left (\sqrt {1-a\,x}-1\right )}^{11}}}{a^3\,{\left (\frac {a\,x}{{\left (\sqrt {1-a\,x}-1\right )}^2}+1\right )}^6}-\frac {\frac {3\,\sqrt {a\,x}}{2\,\left (\sqrt {1-a\,x}-1\right )}+\frac {11\,{\left (a\,x\right )}^{3/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^3}-\frac {11\,{\left (a\,x\right )}^{5/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^5}-\frac {3\,{\left (a\,x\right )}^{7/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^7}}{a^3\,{\left (\frac {a\,x}{{\left (\sqrt {1-a\,x}-1\right )}^2}+1\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 25.60, size = 393, normalized size = 4.52 \[ a \left (\begin {cases} - \frac {5 i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{8 a^{4}} - \frac {i x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {a x - 1}} - \frac {i x^{\frac {5}{2}}}{12 a^{\frac {3}{2}} \sqrt {a x - 1}} - \frac {5 i x^{\frac {3}{2}}}{24 a^{\frac {5}{2}} \sqrt {a x - 1}} + \frac {5 i \sqrt {x}}{8 a^{\frac {7}{2}} \sqrt {a x - 1}} & \text {for}\: \left |{a x}\right | > 1 \\\frac {5 \operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{8 a^{4}} + \frac {x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {- a x + 1}} + \frac {x^{\frac {5}{2}}}{12 a^{\frac {3}{2}} \sqrt {- a x + 1}} + \frac {5 x^{\frac {3}{2}}}{24 a^{\frac {5}{2}} \sqrt {- a x + 1}} - \frac {5 \sqrt {x}}{8 a^{\frac {7}{2}} \sqrt {- a x + 1}} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {3 i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{4 a^{3}} - \frac {i x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {a x - 1}} - \frac {i x^{\frac {3}{2}}}{4 a^{\frac {3}{2}} \sqrt {a x - 1}} + \frac {3 i \sqrt {x}}{4 a^{\frac {5}{2}} \sqrt {a x - 1}} & \text {for}\: \left |{a x}\right | > 1 \\\frac {3 \operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{4 a^{3}} + \frac {x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {- a x + 1}} + \frac {x^{\frac {3}{2}}}{4 a^{\frac {3}{2}} \sqrt {- a x + 1}} - \frac {3 \sqrt {x}}{4 a^{\frac {5}{2}} \sqrt {- a x + 1}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________