Optimal. Leaf size=111 \[ -\frac {\sqrt {1-a x} (a x)^{7/2}}{4 a^4}-\frac {5 \sqrt {1-a x} (a x)^{5/2}}{8 a^4}-\frac {25 \sqrt {1-a x} (a x)^{3/2}}{32 a^4}-\frac {75 \sqrt {1-a x} \sqrt {a x}}{64 a^4}-\frac {75 \sin ^{-1}(1-2 a x)}{128 a^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ -\frac {\sqrt {1-a x} (a x)^{7/2}}{4 a^4}-\frac {5 \sqrt {1-a x} (a x)^{5/2}}{8 a^4}-\frac {25 \sqrt {1-a x} (a x)^{3/2}}{32 a^4}-\frac {75 \sqrt {1-a x} \sqrt {a x}}{64 a^4}-\frac {75 \sin ^{-1}(1-2 a x)}{128 a^4} \]
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^3 (1+a x)}{\sqrt {a x} \sqrt {1-a x}} \, dx &=\frac {\int \frac {(a x)^{5/2} (1+a x)}{\sqrt {1-a x}} \, dx}{a^3}\\ &=-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a^4}+\frac {15 \int \frac {(a x)^{5/2}}{\sqrt {1-a x}} \, dx}{8 a^3}\\ &=-\frac {5 (a x)^{5/2} \sqrt {1-a x}}{8 a^4}-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a^4}+\frac {25 \int \frac {(a x)^{3/2}}{\sqrt {1-a x}} \, dx}{16 a^3}\\ &=-\frac {25 (a x)^{3/2} \sqrt {1-a x}}{32 a^4}-\frac {5 (a x)^{5/2} \sqrt {1-a x}}{8 a^4}-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a^4}+\frac {75 \int \frac {\sqrt {a x}}{\sqrt {1-a x}} \, dx}{64 a^3}\\ &=-\frac {75 \sqrt {a x} \sqrt {1-a x}}{64 a^4}-\frac {25 (a x)^{3/2} \sqrt {1-a x}}{32 a^4}-\frac {5 (a x)^{5/2} \sqrt {1-a x}}{8 a^4}-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a^4}+\frac {75 \int \frac {1}{\sqrt {a x} \sqrt {1-a x}} \, dx}{128 a^3}\\ &=-\frac {75 \sqrt {a x} \sqrt {1-a x}}{64 a^4}-\frac {25 (a x)^{3/2} \sqrt {1-a x}}{32 a^4}-\frac {5 (a x)^{5/2} \sqrt {1-a x}}{8 a^4}-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a^4}+\frac {75 \int \frac {1}{\sqrt {a x-a^2 x^2}} \, dx}{128 a^3}\\ &=-\frac {75 \sqrt {a x} \sqrt {1-a x}}{64 a^4}-\frac {25 (a x)^{3/2} \sqrt {1-a x}}{32 a^4}-\frac {5 (a x)^{5/2} \sqrt {1-a x}}{8 a^4}-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a^4}-\frac {75 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,a-2 a^2 x\right )}{128 a^5}\\ &=-\frac {75 \sqrt {a x} \sqrt {1-a x}}{64 a^4}-\frac {25 (a x)^{3/2} \sqrt {1-a x}}{32 a^4}-\frac {5 (a x)^{5/2} \sqrt {1-a x}}{8 a^4}-\frac {(a x)^{7/2} \sqrt {1-a x}}{4 a^4}-\frac {75 \sin ^{-1}(1-2 a x)}{128 a^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 89, normalized size = 0.80 \[ \frac {\sqrt {a} x \left (16 a^4 x^4+24 a^3 x^3+10 a^2 x^2+25 a x-75\right )+75 \sqrt {x} \sqrt {1-a x} \sin ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{64 a^{7/2} \sqrt {-a x (a x-1)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.88, size = 65, normalized size = 0.59 \[ -\frac {{\left (16 \, a^{3} x^{3} + 40 \, a^{2} x^{2} + 50 \, a x + 75\right )} \sqrt {a x} \sqrt {-a x + 1} + 75 \, \arctan \left (\frac {\sqrt {a x} \sqrt {-a x + 1}}{a x}\right )}{64 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.25, size = 63, normalized size = 0.57 \[ -\frac {{\left (2 \, {\left (4 \, a x {\left (\frac {2 \, x}{a^{2}} + \frac {5}{a^{3}}\right )} + \frac {25}{a^{3}}\right )} a x + \frac {75}{a^{3}}\right )} \sqrt {a x} \sqrt {-a x + 1} - \frac {75 \, \arcsin \left (\sqrt {a x}\right )}{a^{3}}}{64 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.04, size = 132, normalized size = 1.19 \[ -\frac {\sqrt {-a x +1}\, \left (32 \sqrt {-\left (a x -1\right ) a x}\, a^{3} x^{3} \mathrm {csgn}\relax (a )+80 \sqrt {-\left (a x -1\right ) a x}\, a^{2} x^{2} \mathrm {csgn}\relax (a )+100 \sqrt {-\left (a x -1\right ) a x}\, a x \,\mathrm {csgn}\relax (a )-75 \arctan \left (\frac {\left (2 a x -1\right ) \mathrm {csgn}\relax (a )}{2 \sqrt {-\left (a x -1\right ) a x}}\right )+150 \sqrt {-\left (a x -1\right ) a x}\, \mathrm {csgn}\relax (a )\right ) x \,\mathrm {csgn}\relax (a )}{128 \sqrt {a x}\, \sqrt {-\left (a x -1\right ) a x}\, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.97, size = 105, normalized size = 0.95 \[ -\frac {\sqrt {-a^{2} x^{2} + a x} x^{3}}{4 \, a} - \frac {5 \, \sqrt {-a^{2} x^{2} + a x} x^{2}}{8 \, a^{2}} - \frac {25 \, \sqrt {-a^{2} x^{2} + a x} x}{32 \, a^{3}} - \frac {75 \, \arcsin \left (-\frac {2 \, a^{2} x - a}{a}\right )}{128 \, a^{4}} - \frac {75 \, \sqrt {-a^{2} x^{2} + a x}}{64 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 7.78, size = 345, normalized size = 3.11 \[ \frac {75\,\mathrm {atan}\left (\frac {\sqrt {a\,x}}{\sqrt {1-a\,x}-1}\right )}{32\,a^4}-\frac {\frac {5\,\sqrt {a\,x}}{4\,\left (\sqrt {1-a\,x}-1\right )}+\frac {85\,{\left (a\,x\right )}^{3/2}}{12\,{\left (\sqrt {1-a\,x}-1\right )}^3}+\frac {33\,{\left (a\,x\right )}^{5/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^5}-\frac {33\,{\left (a\,x\right )}^{7/2}}{2\,{\left (\sqrt {1-a\,x}-1\right )}^7}-\frac {85\,{\left (a\,x\right )}^{9/2}}{12\,{\left (\sqrt {1-a\,x}-1\right )}^9}-\frac {5\,{\left (a\,x\right )}^{11/2}}{4\,{\left (\sqrt {1-a\,x}-1\right )}^{11}}}{a^4\,{\left (\frac {a\,x}{{\left (\sqrt {1-a\,x}-1\right )}^2}+1\right )}^6}-\frac {\frac {35\,\sqrt {a\,x}}{32\,\left (\sqrt {1-a\,x}-1\right )}+\frac {805\,{\left (a\,x\right )}^{3/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^3}+\frac {2681\,{\left (a\,x\right )}^{5/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^5}+\frac {5053\,{\left (a\,x\right )}^{7/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^7}-\frac {5053\,{\left (a\,x\right )}^{9/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^9}-\frac {2681\,{\left (a\,x\right )}^{11/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^{11}}-\frac {805\,{\left (a\,x\right )}^{13/2}}{96\,{\left (\sqrt {1-a\,x}-1\right )}^{13}}-\frac {35\,{\left (a\,x\right )}^{15/2}}{32\,{\left (\sqrt {1-a\,x}-1\right )}^{15}}}{a^4\,{\left (\frac {a\,x}{{\left (\sqrt {1-a\,x}-1\right )}^2}+1\right )}^8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 35.80, size = 484, normalized size = 4.36 \[ a \left (\begin {cases} - \frac {35 i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{64 a^{5}} - \frac {i x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {a x - 1}} - \frac {i x^{\frac {7}{2}}}{24 a^{\frac {3}{2}} \sqrt {a x - 1}} - \frac {7 i x^{\frac {5}{2}}}{96 a^{\frac {5}{2}} \sqrt {a x - 1}} - \frac {35 i x^{\frac {3}{2}}}{192 a^{\frac {7}{2}} \sqrt {a x - 1}} + \frac {35 i \sqrt {x}}{64 a^{\frac {9}{2}} \sqrt {a x - 1}} & \text {for}\: \left |{a x}\right | > 1 \\\frac {35 \operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{64 a^{5}} + \frac {x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {- a x + 1}} + \frac {x^{\frac {7}{2}}}{24 a^{\frac {3}{2}} \sqrt {- a x + 1}} + \frac {7 x^{\frac {5}{2}}}{96 a^{\frac {5}{2}} \sqrt {- a x + 1}} + \frac {35 x^{\frac {3}{2}}}{192 a^{\frac {7}{2}} \sqrt {- a x + 1}} - \frac {35 \sqrt {x}}{64 a^{\frac {9}{2}} \sqrt {- a x + 1}} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {5 i \operatorname {acosh}{\left (\sqrt {a} \sqrt {x} \right )}}{8 a^{4}} - \frac {i x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {a x - 1}} - \frac {i x^{\frac {5}{2}}}{12 a^{\frac {3}{2}} \sqrt {a x - 1}} - \frac {5 i x^{\frac {3}{2}}}{24 a^{\frac {5}{2}} \sqrt {a x - 1}} + \frac {5 i \sqrt {x}}{8 a^{\frac {7}{2}} \sqrt {a x - 1}} & \text {for}\: \left |{a x}\right | > 1 \\\frac {5 \operatorname {asin}{\left (\sqrt {a} \sqrt {x} \right )}}{8 a^{4}} + \frac {x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {- a x + 1}} + \frac {x^{\frac {5}{2}}}{12 a^{\frac {3}{2}} \sqrt {- a x + 1}} + \frac {5 x^{\frac {3}{2}}}{24 a^{\frac {5}{2}} \sqrt {- a x + 1}} - \frac {5 \sqrt {x}}{8 a^{\frac {7}{2}} \sqrt {- a x + 1}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________