Optimal. Leaf size=187 \[ -\frac {18551}{550} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {797}{110} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {133 (2+3 x)^{5/2} \sqrt {3+5 x}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {1289089 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{500 \sqrt {33}}-\frac {9694 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{125 \sqrt {33}} \]
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Rubi [A]
time = 0.04, antiderivative size = 187, 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 {9694 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{125 \sqrt {33}}-\frac {1289089 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{500 \sqrt {33}}+\frac {\sqrt {5 x+3} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}-\frac {133 \sqrt {5 x+3} (3 x+2)^{5/2}}{33 \sqrt {1-2 x}}-\frac {797}{110} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}-\frac {18551}{550} \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2} \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 {(2+3 x)^{7/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx &=\frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {(2+3 x)^{5/2} \left (\frac {73}{2}+60 x\right )}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {133 (2+3 x)^{5/2} \sqrt {3+5 x}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {\left (-\frac {7305}{2}-\frac {11955 x}{2}\right ) (2+3 x)^{3/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {797}{110} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {133 (2+3 x)^{5/2} \sqrt {3+5 x}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}+\frac {1}{825} \int \frac {\sqrt {2+3 x} \left (\frac {1029375}{4}+\frac {834795 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {18551}{550} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {797}{110} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {133 (2+3 x)^{5/2} \sqrt {3+5 x}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {\int \frac {-\frac {36724815}{4}-\frac {58009005 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{12375}\\ &=-\frac {18551}{550} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {797}{110} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {133 (2+3 x)^{5/2} \sqrt {3+5 x}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}+\frac {4847}{125} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {1289089 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{5500}\\ &=-\frac {18551}{550} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {797}{110} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {133 (2+3 x)^{5/2} \sqrt {3+5 x}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {1289089 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{500 \sqrt {33}}-\frac {9694 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{125 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 8.11, size = 125, normalized size = 0.67 \begin {gather*} -\frac {10 \sqrt {2+3 x} \sqrt {3+5 x} \left (101763-275587 x+45342 x^2+8910 x^3\right )+1289089 \sqrt {2-4 x} (-1+2 x) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-649285 \sqrt {2-4 x} (-1+2 x) F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{16500 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 234, normalized size = 1.25
method | result | size |
default | \(-\frac {\left (1279608 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-2578178 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-639804 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+1289089 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+1336500 x^{5}+8494200 x^{4}-32188470 x^{3}-34376560 x^{2}+2799750 x +6105780\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}}{16500 \left (-1+2 x \right )^{2} \left (15 x^{2}+19 x +6\right )}\) | \(234\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {27 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{20}-\frac {411 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{50}+\frac {816107 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{23100 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1289089 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{23100 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {-\frac {89425}{88} x^{2}-\frac {339815}{264} x -\frac {17885}{44}}{\sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {343 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{96 \left (-\frac {1}{2}+x \right )^{2}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(264\) |
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 = 0.18, size = 50, normalized size = 0.27 \begin {gather*} -\frac {{\left (8910 \, x^{3} + 45342 \, x^{2} - 275587 \, x + 101763\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1650 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 (3\,x+2\right )}^{7/2}\,\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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