Optimal. Leaf size=122 \[ \frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{28 (3 x+2)^2}+\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{7 (3 x+2)^3}-\frac {407 \sqrt {1-2 x} \sqrt {5 x+3}}{392 (3 x+2)}-\frac {4477 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{392 \sqrt {7}} \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 122, 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 = {96, 94, 93, 204} \begin {gather*} \frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{28 (3 x+2)^2}+\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{7 (3 x+2)^3}-\frac {407 \sqrt {1-2 x} \sqrt {5 x+3}}{392 (3 x+2)}-\frac {4477 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{392 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 93
Rule 94
Rule 96
Rule 204
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^4} \, dx &=\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{7 (2+3 x)^3}+\frac {37}{14} \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^3} \, dx\\ &=\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{7 (2+3 x)^3}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{28 (2+3 x)^2}+\frac {407}{56} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {407 \sqrt {1-2 x} \sqrt {3+5 x}}{392 (2+3 x)}+\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{7 (2+3 x)^3}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{28 (2+3 x)^2}+\frac {4477}{784} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {407 \sqrt {1-2 x} \sqrt {3+5 x}}{392 (2+3 x)}+\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{7 (2+3 x)^3}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{28 (2+3 x)^2}+\frac {4477}{392} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {407 \sqrt {1-2 x} \sqrt {3+5 x}}{392 (2+3 x)}+\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{7 (2+3 x)^3}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{28 (2+3 x)^2}-\frac {4477 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{392 \sqrt {7}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 74, normalized size = 0.61 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (3547 x^2+4902 x+1648\right )}{(3 x+2)^3}-4477 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2744} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.25, size = 106, normalized size = 0.87 \begin {gather*} -\frac {121 \sqrt {1-2 x} \left (\frac {37 (1-2 x)^2}{(5 x+3)^2}-\frac {616 (1-2 x)}{5 x+3}-1813\right )}{392 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^3}-\frac {4477 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{392 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.84, size = 101, normalized size = 0.83 \begin {gather*} -\frac {4477 \, \sqrt {7} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (3547 \, x^{2} + 4902 \, x + 1648\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{5488 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.69, size = 310, normalized size = 2.54 \begin {gather*} \frac {4477}{54880} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {121 \, \sqrt {10} {\left (37 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 24640 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {2900800 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {11603200 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{196 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 202, normalized size = 1.66 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (120879 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+241758 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+49658 \sqrt {-10 x^{2}-x +3}\, x^{2}+161172 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+68628 \sqrt {-10 x^{2}-x +3}\, x +35816 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+23072 \sqrt {-10 x^{2}-x +3}\right )}{5488 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.37, size = 121, normalized size = 0.99 \begin {gather*} \frac {4477}{5488} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {185}{294} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{7 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {111 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{196 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {1369 \, \sqrt {-10 \, x^{2} - x + 3}}{1176 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 12.76, size = 1273, normalized size = 10.43
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1 - 2 x} \sqrt {5 x + 3}}{\left (3 x + 2\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________