Optimal. Leaf size=84 \[ \frac {7 (3 x+2)^2}{11 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {\sqrt {1-2 x} (50985 x+30443)}{12100 \sqrt {5 x+3}}-\frac {999 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{100 \sqrt {10}} \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {98, 143, 54, 216} \begin {gather*} \frac {7 (3 x+2)^2}{11 \sqrt {1-2 x} \sqrt {5 x+3}}+\frac {\sqrt {1-2 x} (50985 x+30443)}{12100 \sqrt {5 x+3}}-\frac {999 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{100 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 54
Rule 98
Rule 143
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {1}{11} \int \frac {(2+3 x) \left (89+\frac {309 x}{2}\right )}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {\sqrt {1-2 x} (30443+50985 x)}{12100 \sqrt {3+5 x}}-\frac {999}{200} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {\sqrt {1-2 x} (30443+50985 x)}{12100 \sqrt {3+5 x}}-\frac {999 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{100 \sqrt {5}}\\ &=\frac {7 (2+3 x)^2}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {\sqrt {1-2 x} (30443+50985 x)}{12100 \sqrt {3+5 x}}-\frac {999 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{100 \sqrt {10}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 70, normalized size = 0.83 \begin {gather*} \frac {-326700 x^2+824990 x-120879 \sqrt {5 x+3} \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )+612430}{121000 \sqrt {1-2 x} \sqrt {5 x+3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.13, size = 109, normalized size = 1.30 \begin {gather*} \frac {\sqrt {5 x+3} \left (-\frac {40 (1-2 x)^2}{(5 x+3)^2}+\frac {121671 (1-2 x)}{5 x+3}+34300\right )}{12100 \sqrt {1-2 x} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )}+\frac {999 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{100 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.36, size = 87, normalized size = 1.04 \begin {gather*} \frac {120879 \, \sqrt {10} {\left (10 \, x^{2} + x - 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (32670 \, x^{2} - 82499 \, x - 61243\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{242000 \, {\left (10 \, x^{2} + x - 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.28, size = 118, normalized size = 1.40 \begin {gather*} -\frac {999}{1000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (6534 \, \sqrt {5} {\left (5 \, x + 3\right )} - 121687 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{302500 \, {\left (2 \, x - 1\right )}} - \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{30250 \, \sqrt {5 \, x + 3}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{15125 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 120, normalized size = 1.43 \begin {gather*} -\frac {\sqrt {-2 x +1}\, \left (1208790 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-653400 \sqrt {-10 x^{2}-x +3}\, x^{2}+120879 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1649980 \sqrt {-10 x^{2}-x +3}\, x -362637 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1224860 \sqrt {-10 x^{2}-x +3}\right )}{242000 \left (2 x -1\right ) \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 [A] time = 1.10, size = 58, normalized size = 0.69 \begin {gather*} -\frac {27 \, x^{2}}{10 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {999}{2000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {82499 \, x}{12100 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {61243}{12100 \, \sqrt {-10 \, x^{2} - x + 3}} \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 {{\left (3\,x+2\right )}^3}{{\left (1-2\,x\right )}^{3/2}\,{\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(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________