Optimal. Leaf size=191 \[ -\frac {43214 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{8505}+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{135 (3 x+2)^{3/2}}-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}+\frac {9808 \sqrt {1-2 x} (5 x+3)^{3/2}}{945 \sqrt {3 x+2}}-\frac {43214 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{1701}+\frac {116854 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8505} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {97, 150, 154, 158, 113, 119} \[ \frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{135 (3 x+2)^{3/2}}-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{15 (3 x+2)^{5/2}}+\frac {9808 \sqrt {1-2 x} (5 x+3)^{3/2}}{945 \sqrt {3 x+2}}-\frac {43214 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{1701}-\frac {43214 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8505}+\frac {116854 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8505} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 97
Rule 113
Rule 119
Rule 150
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{7/2}} \, dx &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac {2}{15} \int \frac {\left (\frac {7}{2}-40 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{135 (2+3 x)^{3/2}}-\frac {4}{135} \int \frac {\left (241-\frac {2955 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{3/2}} \, dx\\ &=\frac {9808 \sqrt {1-2 x} (3+5 x)^{3/2}}{945 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{135 (2+3 x)^{3/2}}-\frac {8 \int \frac {\left (\frac {48285}{4}-\frac {324105 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2835}\\ &=-\frac {43214 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1701}+\frac {9808 \sqrt {1-2 x} (3+5 x)^{3/2}}{945 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{135 (2+3 x)^{3/2}}+\frac {8 \int \frac {-\frac {338655}{8}-\frac {876405 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{25515}\\ &=-\frac {43214 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1701}+\frac {9808 \sqrt {1-2 x} (3+5 x)^{3/2}}{945 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{135 (2+3 x)^{3/2}}-\frac {116854 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{8505}+\frac {237677 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{8505}\\ &=-\frac {43214 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1701}+\frac {9808 \sqrt {1-2 x} (3+5 x)^{3/2}}{945 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{15 (2+3 x)^{5/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{135 (2+3 x)^{3/2}}+\frac {116854 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8505}-\frac {43214 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8505}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.33, size = 104, normalized size = 0.54 \[ \frac {\sqrt {2} \left (829885 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )-116854 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )\right )-\frac {6 \sqrt {1-2 x} \sqrt {5 x+3} \left (47250 x^3+377793 x^2+432387 x+134497\right )}{(3 x+2)^{5/2}}}{25515} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.12, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.02, size = 319, normalized size = 1.67 \[ -\frac {\left (2835000 x^{5}+22951080 x^{4}+27359478 x^{3}-1051686 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+7468965 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+3863868 x^{2}-1402248 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+9958620 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-6975984 x -467416 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+3319540 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-2420946\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{25515 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________