Optimal. Leaf size=160 \[ \frac {988 \sqrt {1-2 x} \sqrt {3+5 x}}{945 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{45 (2+3 x)^{3/2}}-\frac {4418}{945} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {988}{945} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {99, 155, 164,
114, 120} \begin {gather*} \frac {988}{945} \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {4418}{945} \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{45 (3 x+2)^{3/2}}-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{15 (3 x+2)^{5/2}}+\frac {988 \sqrt {1-2 x} \sqrt {5 x+3}}{945 \sqrt {3 x+2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 99
Rule 114
Rule 120
Rule 155
Rule 164
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{7/2}} \, dx &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac {2}{15} \int \frac {\left (-\frac {3}{2}-30 x\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^{5/2}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{45 (2+3 x)^{3/2}}-\frac {4}{135} \int \frac {\left (-\frac {243}{2}-\frac {735 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{3/2}} \, dx\\ &=\frac {988 \sqrt {1-2 x} \sqrt {3+5 x}}{945 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{45 (2+3 x)^{3/2}}-\frac {8 \int \frac {-\frac {5865}{2}-\frac {33135 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2835}\\ &=\frac {988 \sqrt {1-2 x} \sqrt {3+5 x}}{945 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{45 (2+3 x)^{3/2}}+\frac {4418}{945} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx-\frac {5434}{945} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {988 \sqrt {1-2 x} \sqrt {3+5 x}}{945 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{15 (2+3 x)^{5/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{45 (2+3 x)^{3/2}}-\frac {4418}{945} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {988}{945} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}
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Mathematica [A]
time = 7.67, size = 99, normalized size = 0.62 \begin {gather*} \frac {2 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (6449+20754 x+16731 x^2\right )}{(2+3 x)^{5/2}}+\sqrt {2} \left (2209 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-10360 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{2835} \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. \(307\) vs.
\(2(116)=232\).
time = 0.10, size = 308, normalized size = 1.92
method | result | size |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3645 \left (\frac {2}{3}+x \right )^{3}}-\frac {148 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1215 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {7436}{189} x^{2}-\frac {3718}{945} x +\frac {3718}{315}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {1564 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{3969 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {4418 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{3969 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(260\) |
default | \(\frac {2 \left (73359 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+19881 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+97812 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+26508 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+32604 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+8836 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+501930 x^{4}+672813 x^{3}+105153 x^{2}-167439 x -58041\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{2835 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {5}{2}}}\) | \(308\) |
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 = 0.21, size = 50, normalized size = 0.31 \begin {gather*} \frac {2 \, {\left (16731 \, x^{2} + 20754 \, x + 6449\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{945 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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 )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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