Optimal. Leaf size=77 \[ \frac {950 \sqrt {1-2 x}}{363 \sqrt {5 x+3}}-\frac {10 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}-\frac {18 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {104, 152, 12, 93, 204} \[ \frac {950 \sqrt {1-2 x}}{363 \sqrt {5 x+3}}-\frac {10 \sqrt {1-2 x}}{33 (5 x+3)^{3/2}}-\frac {18 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 93
Rule 104
Rule 152
Rule 204
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx &=-\frac {10 \sqrt {1-2 x}}{33 (3+5 x)^{3/2}}-\frac {2}{33} \int \frac {\frac {59}{2}-30 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {10 \sqrt {1-2 x}}{33 (3+5 x)^{3/2}}+\frac {950 \sqrt {1-2 x}}{363 \sqrt {3+5 x}}+\frac {4}{363} \int \frac {3267}{4 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {10 \sqrt {1-2 x}}{33 (3+5 x)^{3/2}}+\frac {950 \sqrt {1-2 x}}{363 \sqrt {3+5 x}}+9 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {10 \sqrt {1-2 x}}{33 (3+5 x)^{3/2}}+\frac {950 \sqrt {1-2 x}}{363 \sqrt {3+5 x}}+18 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {10 \sqrt {1-2 x}}{33 (3+5 x)^{3/2}}+\frac {950 \sqrt {1-2 x}}{363 \sqrt {3+5 x}}-\frac {18 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{\sqrt {7}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 60, normalized size = 0.78 \[ \frac {10 \sqrt {1-2 x} (475 x+274)}{363 (5 x+3)^{3/2}}-\frac {18 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{\sqrt {7}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.85, size = 86, normalized size = 1.12 \[ -\frac {3267 \, \sqrt {7} {\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 70 \, {\left (475 \, x + 274\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2541 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.96, size = 190, normalized size = 2.47 \[ \frac {9}{70} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1}{5808} \, \sqrt {10} {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {744 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {2976 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 147, normalized size = 1.91 \[ \frac {\left (81675 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+98010 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+33250 \sqrt {-10 x^{2}-x +3}\, x +29403 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+19180 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{2541 \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )} \sqrt {-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {1-2\,x}\,\left (3\,x+2\right )\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {1 - 2 x} \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________