Optimal. Leaf size=178 \[ -\frac {176065 \sqrt {1-2 x}}{126 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {28 \sqrt {1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac {1301 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac {117955 \sqrt {1-2 x}}{14 (3+5 x)}+\frac {813716}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-112875 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {100, 156, 162,
65, 212} \begin {gather*} \frac {117955 \sqrt {1-2 x}}{14 (5 x+3)}-\frac {176065 \sqrt {1-2 x}}{126 (5 x+3)^2}+\frac {1301 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)^2}+\frac {28 \sqrt {1-2 x}}{3 (3 x+2)^2 (5 x+3)^2}+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 (5 x+3)^2}+\frac {813716}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-112875 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 100
Rule 156
Rule 162
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^3} \, dx &=\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {1}{9} \int \frac {190-303 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^3} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {28 \sqrt {1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac {1}{126} \int \frac {27202-41160 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {28 \sqrt {1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac {1301 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac {1}{882} \int \frac {2963912-4098150 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac {176065 \sqrt {1-2 x}}{126 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {28 \sqrt {1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac {1301 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^2}-\frac {\int \frac {213252732-244026090 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx}{19404}\\ &=-\frac {176065 \sqrt {1-2 x}}{126 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {28 \sqrt {1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac {1301 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac {117955 \sqrt {1-2 x}}{14 (3+5 x)}+\frac {\int \frac {8809230276-5395025790 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{213444}\\ &=-\frac {176065 \sqrt {1-2 x}}{126 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {28 \sqrt {1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac {1301 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac {117955 \sqrt {1-2 x}}{14 (3+5 x)}-\frac {1220574}{7} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {564375}{2} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {176065 \sqrt {1-2 x}}{126 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {28 \sqrt {1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac {1301 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac {117955 \sqrt {1-2 x}}{14 (3+5 x)}+\frac {1220574}{7} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {564375}{2} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {176065 \sqrt {1-2 x}}{126 (3+5 x)^2}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 (3+5 x)^2}+\frac {28 \sqrt {1-2 x}}{3 (2+3 x)^2 (3+5 x)^2}+\frac {1301 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)^2}+\frac {117955 \sqrt {1-2 x}}{14 (3+5 x)}+\frac {813716}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-112875 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 104, normalized size = 0.58 \begin {gather*} \frac {\sqrt {1-2 x} \left (2685098+16784696 x+39307638 x^2+40874010 x^3+15923925 x^4\right )}{14 (2+3 x)^3 (3+5 x)^2}+\frac {813716}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-112875 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 103, normalized size = 0.58
method | result | size |
risch | \(-\frac {31847850 x^{5}+65824095 x^{4}+37741266 x^{3}-5738246 x^{2}-11414500 x -2685098}{14 \left (3+5 x \right )^{2} \sqrt {1-2 x}\, \left (2+3 x \right )^{3}}-\frac {112875 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}+\frac {813716 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}\) | \(86\) |
derivativedivides | \(\frac {-33625 \left (1-2 x \right )^{\frac {3}{2}}+73425 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-\frac {112875 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}-\frac {324 \left (\frac {3544 \left (1-2 x \right )^{\frac {5}{2}}}{21}-\frac {21418 \left (1-2 x \right )^{\frac {3}{2}}}{27}+\frac {25172 \sqrt {1-2 x}}{27}\right )}{\left (-4-6 x \right )^{3}}+\frac {813716 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}\) | \(103\) |
default | \(\frac {-33625 \left (1-2 x \right )^{\frac {3}{2}}+73425 \sqrt {1-2 x}}{\left (-6-10 x \right )^{2}}-\frac {112875 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{11}-\frac {324 \left (\frac {3544 \left (1-2 x \right )^{\frac {5}{2}}}{21}-\frac {21418 \left (1-2 x \right )^{\frac {3}{2}}}{27}+\frac {25172 \sqrt {1-2 x}}{27}\right )}{\left (-4-6 x \right )^{3}}+\frac {813716 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{49}\) | \(103\) |
trager | \(\frac {\left (15923925 x^{4}+40874010 x^{3}+39307638 x^{2}+16784696 x +2685098\right ) \sqrt {1-2 x}}{14 \left (2+3 x \right )^{3} \left (3+5 x \right )^{2}}+\frac {406858 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{49}-\frac {112875 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{22}\) | \(134\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 164, normalized size = 0.92 \begin {gather*} \frac {112875}{22} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {406858}{49} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {15923925 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 145443720 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 498018162 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 757678432 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 432141633 \, \sqrt {-2 \, x + 1}}{7 \, {\left (675 \, {\left (2 \, x - 1\right )}^{5} + 7695 \, {\left (2 \, x - 1\right )}^{4} + 35082 \, {\left (2 \, x - 1\right )}^{3} + 79954 \, {\left (2 \, x - 1\right )}^{2} + 182182 \, x - 49588\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.02, size = 182, normalized size = 1.02 \begin {gather*} \frac {5530875 \, \sqrt {11} \sqrt {5} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 8950876 \, \sqrt {7} \sqrt {3} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (15923925 \, x^{4} + 40874010 \, x^{3} + 39307638 \, x^{2} + 16784696 \, x + 2685098\right )} \sqrt {-2 \, x + 1}}{1078 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\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 [A]
time = 0.63, size = 151, normalized size = 0.85 \begin {gather*} \frac {112875}{22} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {406858}{49} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {25 \, {\left (1345 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2937 \, \sqrt {-2 \, x + 1}\right )}}{4 \, {\left (5 \, x + 3\right )}^{2}} + \frac {3 \, {\left (15948 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 74963 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 88102 \, \sqrt {-2 \, x + 1}\right )}}{7 \, {\left (3 \, x + 2\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 125, normalized size = 0.70 \begin {gather*} \frac {813716\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{49}-\frac {112875\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{11}+\frac {\frac {6859391\,\sqrt {1-2\,x}}{75}-\frac {108239776\,{\left (1-2\,x\right )}^{3/2}}{675}+\frac {166006054\,{\left (1-2\,x\right )}^{5/2}}{1575}-\frac {9696248\,{\left (1-2\,x\right )}^{7/2}}{315}+\frac {23591\,{\left (1-2\,x\right )}^{9/2}}{7}}{\frac {182182\,x}{675}+\frac {79954\,{\left (2\,x-1\right )}^2}{675}+\frac {3898\,{\left (2\,x-1\right )}^3}{75}+\frac {57\,{\left (2\,x-1\right )}^4}{5}+{\left (2\,x-1\right )}^5-\frac {49588}{675}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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