Optimal. Leaf size=144 \[ -\frac {7396875 \sqrt {1-2 x}}{30184 \sqrt {5 x+3}}+\frac {44475 \sqrt {1-2 x}}{2744 (3 x+2) \sqrt {5 x+3}}+\frac {255 \sqrt {1-2 x}}{196 (3 x+2)^2 \sqrt {5 x+3}}+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 \sqrt {5 x+3}}+\frac {4616025 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2744 \sqrt {7}} \]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 144, 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 = {103, 151, 152, 12, 93, 204} \begin {gather*} -\frac {7396875 \sqrt {1-2 x}}{30184 \sqrt {5 x+3}}+\frac {44475 \sqrt {1-2 x}}{2744 (3 x+2) \sqrt {5 x+3}}+\frac {255 \sqrt {1-2 x}}{196 (3 x+2)^2 \sqrt {5 x+3}}+\frac {\sqrt {1-2 x}}{7 (3 x+2)^3 \sqrt {5 x+3}}+\frac {4616025 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2744 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 93
Rule 103
Rule 151
Rule 152
Rule 204
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2}} \, dx &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 \sqrt {3+5 x}}+\frac {1}{21} \int \frac {\frac {135}{2}-90 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}} \, dx\\ &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 \sqrt {3+5 x}}+\frac {255 \sqrt {1-2 x}}{196 (2+3 x)^2 \sqrt {3+5 x}}+\frac {1}{294} \int \frac {\frac {24075}{4}-7650 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}} \, dx\\ &=\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 \sqrt {3+5 x}}+\frac {255 \sqrt {1-2 x}}{196 (2+3 x)^2 \sqrt {3+5 x}}+\frac {44475 \sqrt {1-2 x}}{2744 (2+3 x) \sqrt {3+5 x}}+\frac {\int \frac {\frac {2837025}{8}-\frac {667125 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{2058}\\ &=-\frac {7396875 \sqrt {1-2 x}}{30184 \sqrt {3+5 x}}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 \sqrt {3+5 x}}+\frac {255 \sqrt {1-2 x}}{196 (2+3 x)^2 \sqrt {3+5 x}}+\frac {44475 \sqrt {1-2 x}}{2744 (2+3 x) \sqrt {3+5 x}}-\frac {\int \frac {152328825}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{11319}\\ &=-\frac {7396875 \sqrt {1-2 x}}{30184 \sqrt {3+5 x}}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 \sqrt {3+5 x}}+\frac {255 \sqrt {1-2 x}}{196 (2+3 x)^2 \sqrt {3+5 x}}+\frac {44475 \sqrt {1-2 x}}{2744 (2+3 x) \sqrt {3+5 x}}-\frac {4616025 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{5488}\\ &=-\frac {7396875 \sqrt {1-2 x}}{30184 \sqrt {3+5 x}}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 \sqrt {3+5 x}}+\frac {255 \sqrt {1-2 x}}{196 (2+3 x)^2 \sqrt {3+5 x}}+\frac {44475 \sqrt {1-2 x}}{2744 (2+3 x) \sqrt {3+5 x}}-\frac {4616025 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{2744}\\ &=-\frac {7396875 \sqrt {1-2 x}}{30184 \sqrt {3+5 x}}+\frac {\sqrt {1-2 x}}{7 (2+3 x)^3 \sqrt {3+5 x}}+\frac {255 \sqrt {1-2 x}}{196 (2+3 x)^2 \sqrt {3+5 x}}+\frac {44475 \sqrt {1-2 x}}{2744 (2+3 x) \sqrt {3+5 x}}+\frac {4616025 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2744 \sqrt {7}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 79, normalized size = 0.55 \begin {gather*} \frac {50776275 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-\frac {7 \sqrt {1-2 x} \left (199715625 x^3+395028225 x^2+260298990 x+57135248\right )}{(3 x+2)^3 \sqrt {5 x+3}}}{211288} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.26, size = 122, normalized size = 0.85 \begin {gather*} \frac {4616025 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2744 \sqrt {7}}-\frac {\sqrt {1-2 x} \left (\frac {3430000 (1-2 x)^3}{(5 x+3)^3}+\frac {111805725 (1-2 x)^2}{(5 x+3)^2}+\frac {947868600 (1-2 x)}{5 x+3}+2488050019\right )}{30184 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.90, size = 116, normalized size = 0.81 \begin {gather*} \frac {50776275 \, \sqrt {7} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\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 (199715625 \, x^{3} + 395028225 \, x^{2} + 260298990 \, x + 57135248\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{422576 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 2.55, size = 377, normalized size = 2.62 \begin {gather*} -\frac {923205}{76832} \, \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 {125}{22} \, \sqrt {10} {\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 )} - \frac {7425 \, {\left (487 \, \sqrt {10} {\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} + 217280 \, \sqrt {10} {\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} + 25693248 \, \sqrt {10} {\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 )}\right )}}{1372 \, {\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 = 250, normalized size = 1.74 \begin {gather*} -\frac {\left (6854797125 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+17822472525 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+2796018750 \sqrt {-10 x^{2}-x +3}\, x^{3}+17365486050 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+5530395150 \sqrt {-10 x^{2}-x +3}\, x^{2}+7514888700 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3644185860 \sqrt {-10 x^{2}-x +3}\, x +1218630600 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+799893472 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{422576 \left (3 x +2\right )^{3} \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{4} \sqrt {-2 \, x + 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}
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 {1}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{4} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________