Optimal. Leaf size=280 \[ -\frac {3347620 \sqrt {1-2 x} \sqrt {3+5 x}}{1702701 (2+3 x)^{7/2}}+\frac {23210828 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{5/2}}+\frac {1079936248 \sqrt {1-2 x} \sqrt {3+5 x}}{83432349 (2+3 x)^{3/2}}+\frac {75041008472 \sqrt {1-2 x} \sqrt {3+5 x}}{584026443 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}-\frac {75041008472 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{53093313 \sqrt {33}}-\frac {2257166048 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{53093313 \sqrt {33}} \]
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Rubi [A]
time = 0.08, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 157,
164, 114, 120} \begin {gather*} -\frac {2257166048 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{53093313 \sqrt {33}}-\frac {75041008472 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{53093313 \sqrt {33}}-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{39 (3 x+2)^{13/2}}+\frac {230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{1287 (3 x+2)^{11/2}}+\frac {1300 (5 x+3)^{3/2} \sqrt {1-2 x}}{891 (3 x+2)^{9/2}}+\frac {75041008472 \sqrt {5 x+3} \sqrt {1-2 x}}{584026443 \sqrt {3 x+2}}+\frac {1079936248 \sqrt {5 x+3} \sqrt {1-2 x}}{83432349 (3 x+2)^{3/2}}+\frac {23210828 \sqrt {5 x+3} \sqrt {1-2 x}}{11918907 (3 x+2)^{5/2}}-\frac {3347620 \sqrt {5 x+3} \sqrt {1-2 x}}{1702701 (3 x+2)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 99
Rule 114
Rule 120
Rule 155
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{15/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {2}{39} \int \frac {\left (-\frac {15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{13/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}-\frac {4 \int \frac {\sqrt {1-2 x} \sqrt {3+5 x} \left (-\frac {2895}{2}+\frac {1995 x}{2}\right )}{(2+3 x)^{11/2}} \, dx}{1287}\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {8 \int \frac {\left (\frac {438345}{4}-149460 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx}{34749}\\ &=-\frac {3347620 \sqrt {1-2 x} \sqrt {3+5 x}}{1702701 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {16 \int \frac {\frac {15056835}{8}-\frac {10467525 x}{4}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{5108103}\\ &=-\frac {3347620 \sqrt {1-2 x} \sqrt {3+5 x}}{1702701 (2+3 x)^{7/2}}+\frac {23210828 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {32 \int \frac {\frac {1154474415}{8}-\frac {1305609075 x}{8}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{178783605}\\ &=-\frac {3347620 \sqrt {1-2 x} \sqrt {3+5 x}}{1702701 (2+3 x)^{7/2}}+\frac {23210828 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{5/2}}+\frac {1079936248 \sqrt {1-2 x} \sqrt {3+5 x}}{83432349 (2+3 x)^{3/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {64 \int \frac {\frac {100204281585}{16}-\frac {30373206975 x}{8}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{3754455705}\\ &=-\frac {3347620 \sqrt {1-2 x} \sqrt {3+5 x}}{1702701 (2+3 x)^{7/2}}+\frac {23210828 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{5/2}}+\frac {1079936248 \sqrt {1-2 x} \sqrt {3+5 x}}{83432349 (2+3 x)^{3/2}}+\frac {75041008472 \sqrt {1-2 x} \sqrt {3+5 x}}{584026443 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {128 \int \frac {\frac {1336148092575}{16}+\frac {2110528363275 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{26281189935}\\ &=-\frac {3347620 \sqrt {1-2 x} \sqrt {3+5 x}}{1702701 (2+3 x)^{7/2}}+\frac {23210828 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{5/2}}+\frac {1079936248 \sqrt {1-2 x} \sqrt {3+5 x}}{83432349 (2+3 x)^{3/2}}+\frac {75041008472 \sqrt {1-2 x} \sqrt {3+5 x}}{584026443 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {1128583024 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{53093313}+\frac {75041008472 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{584026443}\\ &=-\frac {3347620 \sqrt {1-2 x} \sqrt {3+5 x}}{1702701 (2+3 x)^{7/2}}+\frac {23210828 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{5/2}}+\frac {1079936248 \sqrt {1-2 x} \sqrt {3+5 x}}{83432349 (2+3 x)^{3/2}}+\frac {75041008472 \sqrt {1-2 x} \sqrt {3+5 x}}{584026443 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{39 (2+3 x)^{13/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{1287 (2+3 x)^{11/2}}+\frac {1300 \sqrt {1-2 x} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}-\frac {75041008472 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{53093313 \sqrt {33}}-\frac {2257166048 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{53093313 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 9.03, size = 117, normalized size = 0.42 \begin {gather*} \frac {\frac {48 \sqrt {2-4 x} \sqrt {3+5 x} \left (2532151719515+22577209892436 x+83893544414217 x^2+166295375376786 x^3+185457331738206 x^4+110328276131100 x^5+27352447588044 x^6\right )}{(2+3 x)^{13/2}}+1200656135552 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-604764298880 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{14016634632 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(679\) vs.
\(2(208)=416\).
time = 0.10, size = 680, normalized size = 2.43
method | result | size |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {18134 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{8444007 \left (\frac {2}{3}+x \right )^{5}}-\frac {7616 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{8444007 \left (\frac {2}{3}+x \right )^{6}}+\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2302911 \left (\frac {2}{3}+x \right )^{7}}+\frac {23210828 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{321810489 \left (\frac {2}{3}+x \right )^{3}}+\frac {390100 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{137918781 \left (\frac {2}{3}+x \right )^{4}}+\frac {-\frac {750410084720}{584026443} x^{2}-\frac {75041008472}{584026443} x +\frac {75041008472}{194675481}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {1079936248 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{750891141 \left (\frac {2}{3}+x \right )^{2}}+\frac {237537438680 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{12264555303 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {375205042360 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{12264555303 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(356\) |
default | \(\frac {2 \left (27352447588044 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{6} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-13575160904184 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{6} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+109409790352176 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-54300643616736 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+182349650586960 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-90501072694560 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+162088578299520 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-80445397950720 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+820573427641320 x^{8}+81044289149760 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-40222698975360 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+3391905626697132 x^{7}+21611810439936 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-10726053060096 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+5648532752247084 x^{6}+2401312271104 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1191783673344 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+4552278771338298 x^{5}+1346576472913014 x^{4}-567661448375343 x^{3}-611345718465195 x^{2}-195598433873379 x -22789365475635\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{1752079329 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {13}{2}}}\) | \(680\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 90, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (27352447588044 \, x^{6} + 110328276131100 \, x^{5} + 185457331738206 \, x^{4} + 166295375376786 \, x^{3} + 83893544414217 \, x^{2} + 22577209892436 \, x + 2532151719515\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{584026443 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^{15/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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