Optimal. Leaf size=191 \[ -\frac {43214 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1215}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 \sqrt {2+3 x}}+\frac {788}{135} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {116854 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6075}-\frac {43214 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6075} \]
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Rubi [A]
time = 0.04, 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 = {99, 155, 159,
164, 114, 120} \begin {gather*} -\frac {43214 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6075}+\frac {116854 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6075}-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^{3/2}}+\frac {230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{27 \sqrt {3 x+2}}+\frac {788}{135} \sqrt {3 x+2} (5 x+3)^{3/2} \sqrt {1-2 x}-\frac {43214 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{1215} \end {gather*}
Antiderivative was successfully verified.
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Rule 99
Rule 114
Rule 120
Rule 155
Rule 159
Rule 164
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac {2}{9} \int \frac {\left (-\frac {15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 \sqrt {2+3 x}}-\frac {4}{27} \int \frac {\left (-210-\frac {2955 x}{2}\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 \sqrt {2+3 x}}+\frac {788}{135} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 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}{2025}\\ &=-\frac {43214 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1215}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 \sqrt {2+3 x}}+\frac {788}{135} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 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}{18225}\\ &=-\frac {43214 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1215}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 \sqrt {2+3 x}}+\frac {788}{135} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {116854 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{6075}+\frac {237677 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{6075}\\ &=-\frac {43214 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1215}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 \sqrt {2+3 x}}+\frac {788}{135} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 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 )}{6075}-\frac {43214 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6075}\\ \end {align*}
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Mathematica [A]
time = 8.68, size = 107, normalized size = 0.56 \begin {gather*} \frac {\frac {30 \sqrt {1-2 x} \sqrt {3+5 x} \left (-13231-23538 x-3906 x^2+1620 x^3\right )}{(2+3 x)^{3/2}}-116854 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+829885 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{18225} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 225, normalized size = 1.18
method | result | size |
default | \(-\frac {\left (2139093 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+350562 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1426062 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+233708 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-486000 x^{5}+1123200 x^{4}+7324380 x^{3}+4323900 x^{2}-1721490 x -1190790\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{18225 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {3}{2}}}\) | \(225\) |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {8 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{27}-\frac {1348 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1215}-\frac {22577 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{25515 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {116854 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{25515 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2156 \left (-30 x^{2}-3 x +9\right )}{729 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2187 \left (\frac {2}{3}+x \right )^{2}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(275\) |
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 = 0.16, size = 50, normalized size = 0.26 \begin {gather*} \frac {2 \, {\left (1620 \, x^{3} - 3906 \, x^{2} - 23538 \, x - 13231\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1215 \, {\left (9 \, x^{2} + 12 \, x + 4\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 = -0.01 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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