Optimal. Leaf size=125 \[ \frac {7 \sqrt {2+3 x} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {62 \sqrt {2+3 x} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}-\frac {31 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{11 \sqrt {33}}-\frac {F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{11 \sqrt {33}} \]
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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 = {100, 157, 164,
114, 120} \begin {gather*} -\frac {F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{11 \sqrt {33}}-\frac {31 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{11 \sqrt {33}}-\frac {62 \sqrt {3 x+2} \sqrt {5 x+3}}{363 \sqrt {1-2 x}}+\frac {7 \sqrt {3 x+2} \sqrt {5 x+3}}{33 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 100
Rule 114
Rule 120
Rule 157
Rule 164
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{3/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx &=\frac {7 \sqrt {2+3 x} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {\frac {121}{2}+96 x}{(1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {7 \sqrt {2+3 x} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {62 \sqrt {2+3 x} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {2 \int \frac {\frac {4137}{4}+\frac {3255 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2541}\\ &=\frac {7 \sqrt {2+3 x} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {62 \sqrt {2+3 x} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {1}{22} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {31}{121} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {7 \sqrt {2+3 x} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {62 \sqrt {2+3 x} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}-\frac {31 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{11 \sqrt {33}}-\frac {F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{11 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 6.15, size = 115, normalized size = 0.92 \begin {gather*} \frac {2 \sqrt {2+3 x} \sqrt {3+5 x} (15+124 x)-62 \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 )+29 \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 )}{726 (1-2 x)^{3/2}} \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. \(223\) vs.
\(2(93)=186\).
time = 0.10, size = 224, normalized size = 1.79
method | result | size |
default | \(-\frac {\left (66 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-124 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-33 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+62 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-3720 x^{3}-5162 x^{2}-2058 x -180\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}\, \sqrt {2+3 x}}{726 \left (-1+2 x \right )^{2} \left (15 x^{2}+19 x +6\right )}\) | \(224\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {-\frac {310}{121} x^{2}-\frac {1178}{363} x -\frac {124}{121}}{\sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {197 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5082 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {155 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{2541 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {7 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{132 \left (-\frac {1}{2}+x \right )^{2}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(225\) |
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 = 0.24, size = 40, normalized size = 0.32 \begin {gather*} \frac {{\left (124 \, x + 15\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{363 \, {\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(-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 (3\,x+2\right )}^{3/2}}{{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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