Optimal. Leaf size=125 \[ -\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}-\frac {62 \sqrt {1-2 x} \sqrt {2+3 x}}{165 \sqrt {3+5 x}}+\frac {62 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25 \sqrt {33}}+\frac {8 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25 \sqrt {33}} \]
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Rubi [A]
time = 0.03, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {99, 157, 164,
114, 120} \begin {gather*} \frac {8 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25 \sqrt {33}}+\frac {62 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25 \sqrt {33}}-\frac {62 \sqrt {1-2 x} \sqrt {3 x+2}}{165 \sqrt {5 x+3}}-\frac {2 \sqrt {1-2 x} \sqrt {3 x+2}}{15 (5 x+3)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 99
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{5/2}} \, dx &=-\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {-\frac {1}{2}-6 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}-\frac {62 \sqrt {1-2 x} \sqrt {2+3 x}}{165 \sqrt {3+5 x}}-\frac {4}{165} \int \frac {\frac {69}{2}+\frac {93 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}-\frac {62 \sqrt {1-2 x} \sqrt {2+3 x}}{165 \sqrt {3+5 x}}-\frac {4}{25} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {62}{275} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}-\frac {62 \sqrt {1-2 x} \sqrt {2+3 x}}{165 \sqrt {3+5 x}}+\frac {62 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25 \sqrt {33}}+\frac {8 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{25 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 3.11, size = 97, normalized size = 0.78 \begin {gather*} \frac {2}{825} \left (-\frac {5 \sqrt {1-2 x} \sqrt {2+3 x} (104+155 x)}{(3+5 x)^{3/2}}-31 \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-35 \sqrt {2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right ) \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. \(214\) vs.
\(2(93)=186\).
time = 0.09, size = 215, normalized size = 1.72
method | result | size |
default | \(\frac {2 \left (330 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-155 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+198 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-93 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-4650 x^{3}-3895 x^{2}+1030 x +1040\right ) \sqrt {1-2 x}\, \sqrt {2+3 x}}{825 \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{\frac {3}{2}}}\) | \(215\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{375 \left (x +\frac {3}{5}\right )^{2}}-\frac {62 \left (-30 x^{2}-5 x +10\right )}{825 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}-\frac {46 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{1155 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {62 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1155 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(225\) |
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 = 0.12, size = 40, normalized size = 0.32 \begin {gather*} -\frac {2 \, {\left (155 \, x + 104\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{165 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1 - 2 x} \sqrt {3 x + 2}}{\left (5 x + 3\right )^{\frac {5}{2}}}\, dx \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 {\sqrt {1-2\,x}\,\sqrt {3\,x+2}}{{\left (5\,x+3\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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