Optimal. Leaf size=222 \[ \frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac {124 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {779 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 (2+3 x)^{5/2}}-\frac {2264 \sqrt {1-2 x} \sqrt {3+5 x}}{12005 (2+3 x)^{3/2}}-\frac {3946 \sqrt {1-2 x} \sqrt {3+5 x}}{84035 \sqrt {2+3 x}}+\frac {3946 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{84035}-\frac {16732 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{84035} \]
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Rubi [A]
time = 0.06, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps
used = 8, 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 {16732 \sqrt {\frac {3}{11}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{84035}+\frac {3946 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{84035}-\frac {3946 \sqrt {1-2 x} \sqrt {5 x+3}}{84035 \sqrt {3 x+2}}-\frac {2264 \sqrt {1-2 x} \sqrt {5 x+3}}{12005 (3 x+2)^{3/2}}-\frac {779 \sqrt {1-2 x} \sqrt {5 x+3}}{1715 (3 x+2)^{5/2}}+\frac {124 \sqrt {5 x+3}}{147 \sqrt {1-2 x} (3 x+2)^{5/2}}+\frac {11 \sqrt {5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^{5/2}} \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 {(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^{7/2}} \, dx &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}-\frac {1}{21} \int \frac {-\frac {367}{2}-315 x}{(1-2 x)^{3/2} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx\\ &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac {124 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{5/2}}+\frac {2 \int \frac {\frac {59631}{4}+25575 x}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{1617}\\ &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac {124 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {779 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 (2+3 x)^{5/2}}+\frac {4 \int \frac {\frac {109857}{2}+\frac {385605 x}{4}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{56595}\\ &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac {124 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {779 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 (2+3 x)^{5/2}}-\frac {2264 \sqrt {1-2 x} \sqrt {3+5 x}}{12005 (2+3 x)^{3/2}}+\frac {8 \int \frac {\frac {682011}{8}+140085 x}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{1188495}\\ &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac {124 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {779 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 (2+3 x)^{5/2}}-\frac {2264 \sqrt {1-2 x} \sqrt {3+5 x}}{12005 (2+3 x)^{3/2}}-\frac {3946 \sqrt {1-2 x} \sqrt {3+5 x}}{84035 \sqrt {2+3 x}}+\frac {16 \int \frac {\frac {328185}{4}-\frac {976635 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{8319465}\\ &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac {124 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {779 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 (2+3 x)^{5/2}}-\frac {2264 \sqrt {1-2 x} \sqrt {3+5 x}}{12005 (2+3 x)^{3/2}}-\frac {3946 \sqrt {1-2 x} \sqrt {3+5 x}}{84035 \sqrt {2+3 x}}-\frac {3946 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{84035}+\frac {25098 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{84035}\\ &=\frac {11 \sqrt {3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^{5/2}}+\frac {124 \sqrt {3+5 x}}{147 \sqrt {1-2 x} (2+3 x)^{5/2}}-\frac {779 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 (2+3 x)^{5/2}}-\frac {2264 \sqrt {1-2 x} \sqrt {3+5 x}}{12005 (2+3 x)^{3/2}}-\frac {3946 \sqrt {1-2 x} \sqrt {3+5 x}}{84035 \sqrt {2+3 x}}+\frac {3946 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{84035}-\frac {16732 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{84035}\\ \end {align*}
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Mathematica [A]
time = 8.89, size = 108, normalized size = 0.49 \begin {gather*} \frac {2 \left (\frac {\sqrt {3+5 x} \left (43881+158902 x+2199 x^2-356292 x^3-213084 x^4\right )}{(1-2 x)^{3/2} (2+3 x)^{5/2}}+\sqrt {2} \left (-1973 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )+39620 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{252105} \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. \(397\) vs.
\(2(162)=324\).
time = 0.11, size = 398, normalized size = 1.79
method | result | size |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {22 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{7203 \left (-\frac {1}{2}+x \right )^{2}}-\frac {892 \left (-30 x^{2}-38 x -12\right )}{50421 \sqrt {\left (-\frac {1}{2}+x \right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{15435 \left (\frac {2}{3}+x \right )^{3}}+\frac {-\frac {1028}{16807} x^{2}-\frac {514}{84035} x +\frac {1542}{84035}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {52 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{15435 \left (\frac {2}{3}+x \right )^{2}}+\frac {884 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{117649 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {3946 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{352947 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(301\) |
default | \(-\frac {2 \sqrt {1-2 x}\, \left (35514 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+677646 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+29595 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+564705 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-7892 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-150588 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-7892 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-150588 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+1065420 x^{5}+2420712 x^{4}+1057881 x^{3}-801107 x^{2}-696111 x -131643\right )}{252105 \left (2+3 x \right )^{\frac {5}{2}} \left (-1+2 x \right )^{2} \sqrt {3+5 x}}\) | \(398\) |
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 = 0.25, size = 70, normalized size = 0.32 \begin {gather*} -\frac {2 \, {\left (213084 \, x^{4} + 356292 \, x^{3} - 2199 \, x^{2} - 158902 \, x - 43881\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{252105 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\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 = -0.00 \begin {gather*} \int \frac {{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{5/2}\,{\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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