3.21.96 \(\int \frac {(2+3 x)^2 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx\) [2096]

Optimal. Leaf size=79 \[ \frac {65219}{32 \sqrt {1-2 x}}+\frac {144837}{32} \sqrt {1-2 x}-\frac {21439}{16} (1-2 x)^{3/2}+\frac {5711}{16} (1-2 x)^{5/2}-\frac {12675}{224} (1-2 x)^{7/2}+\frac {125}{32} (1-2 x)^{9/2} \]

[Out]

-21439/16*(1-2*x)^(3/2)+5711/16*(1-2*x)^(5/2)-12675/224*(1-2*x)^(7/2)+125/32*(1-2*x)^(9/2)+65219/32/(1-2*x)^(1
/2)+144837/32*(1-2*x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {90} \begin {gather*} \frac {125}{32} (1-2 x)^{9/2}-\frac {12675}{224} (1-2 x)^{7/2}+\frac {5711}{16} (1-2 x)^{5/2}-\frac {21439}{16} (1-2 x)^{3/2}+\frac {144837}{32} \sqrt {1-2 x}+\frac {65219}{32 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((2 + 3*x)^2*(3 + 5*x)^3)/(1 - 2*x)^(3/2),x]

[Out]

65219/(32*Sqrt[1 - 2*x]) + (144837*Sqrt[1 - 2*x])/32 - (21439*(1 - 2*x)^(3/2))/16 + (5711*(1 - 2*x)^(5/2))/16
- (12675*(1 - 2*x)^(7/2))/224 + (125*(1 - 2*x)^(9/2))/32

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin {align*} \int \frac {(2+3 x)^2 (3+5 x)^3}{(1-2 x)^{3/2}} \, dx &=\int \left (\frac {65219}{32 (1-2 x)^{3/2}}-\frac {144837}{32 \sqrt {1-2 x}}+\frac {64317}{16} \sqrt {1-2 x}-\frac {28555}{16} (1-2 x)^{3/2}+\frac {12675}{32} (1-2 x)^{5/2}-\frac {1125}{32} (1-2 x)^{7/2}\right ) \, dx\\ &=\frac {65219}{32 \sqrt {1-2 x}}+\frac {144837}{32} \sqrt {1-2 x}-\frac {21439}{16} (1-2 x)^{3/2}+\frac {5711}{16} (1-2 x)^{5/2}-\frac {12675}{224} (1-2 x)^{7/2}+\frac {125}{32} (1-2 x)^{9/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 38, normalized size = 0.48 \begin {gather*} \frac {38700-37944 x-15948 x^2-9501 x^3-4150 x^4-875 x^5}{7 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2 + 3*x)^2*(3 + 5*x)^3)/(1 - 2*x)^(3/2),x]

[Out]

(38700 - 37944*x - 15948*x^2 - 9501*x^3 - 4150*x^4 - 875*x^5)/(7*Sqrt[1 - 2*x])

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Maple [A]
time = 0.15, size = 56, normalized size = 0.71

method	result	size
gosper	\(-\frac {875 x^{5}+4150 x^{4}+9501 x^{3}+15948 x^{2}+37944 x -38700}{7 \sqrt {1-2 x}}\)	\(35\)
risch	\(-\frac {875 x^{5}+4150 x^{4}+9501 x^{3}+15948 x^{2}+37944 x -38700}{7 \sqrt {1-2 x}}\)	\(35\)
trager	\(\frac {\left (875 x^{5}+4150 x^{4}+9501 x^{3}+15948 x^{2}+37944 x -38700\right ) \sqrt {1-2 x}}{-7+14 x}\)	\(42\)
derivativedivides	\(-\frac {21439 \left (1-2 x \right )^{\frac {3}{2}}}{16}+\frac {5711 \left (1-2 x \right )^{\frac {5}{2}}}{16}-\frac {12675 \left (1-2 x \right )^{\frac {7}{2}}}{224}+\frac {125 \left (1-2 x \right )^{\frac {9}{2}}}{32}+\frac {65219}{32 \sqrt {1-2 x}}+\frac {144837 \sqrt {1-2 x}}{32}\)	\(56\)
default	\(-\frac {21439 \left (1-2 x \right )^{\frac {3}{2}}}{16}+\frac {5711 \left (1-2 x \right )^{\frac {5}{2}}}{16}-\frac {12675 \left (1-2 x \right )^{\frac {7}{2}}}{224}+\frac {125 \left (1-2 x \right )^{\frac {9}{2}}}{32}+\frac {65219}{32 \sqrt {1-2 x}}+\frac {144837 \sqrt {1-2 x}}{32}\)	\(56\)
meijerg	\(-\frac {108 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1-2 x}}\right )}{\sqrt {\pi }}+\frac {-864 \sqrt {\pi }+\frac {108 \sqrt {\pi }\, \left (-8 x +8\right )}{\sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {2763 \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-8 x^{2}-16 x +16\right )}{6 \sqrt {1-2 x}}\right )}{4 \sqrt {\pi }}+\frac {-1766 \sqrt {\pi }+\frac {883 \sqrt {\pi }\, \left (-64 x^{3}-64 x^{2}-128 x +128\right )}{64 \sqrt {1-2 x}}}{\sqrt {\pi }}-\frac {3525 \left (\frac {128 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (-160 x^{4}-128 x^{3}-128 x^{2}-256 x +256\right )}{70 \sqrt {1-2 x}}\right )}{16 \sqrt {\pi }}+\frac {-\frac {1000 \sqrt {\pi }}{7}+\frac {125 \sqrt {\pi }\, \left (-896 x^{5}-640 x^{4}-512 x^{3}-512 x^{2}-1024 x +1024\right )}{896 \sqrt {1-2 x}}}{\sqrt {\pi }}\)	\(213\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^2*(3+5*x)^3/(1-2*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-21439/16*(1-2*x)^(3/2)+5711/16*(1-2*x)^(5/2)-12675/224*(1-2*x)^(7/2)+125/32*(1-2*x)^(9/2)+65219/32/(1-2*x)^(1
/2)+144837/32*(1-2*x)^(1/2)

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Maxima [A]
time = 0.27, size = 55, normalized size = 0.70 \begin {gather*} \frac {125}{32} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {12675}{224} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {5711}{16} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {21439}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {144837}{32} \, \sqrt {-2 \, x + 1} + \frac {65219}{32 \, \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^2*(3+5*x)^3/(1-2*x)^(3/2),x, algorithm="maxima")

[Out]

125/32*(-2*x + 1)^(9/2) - 12675/224*(-2*x + 1)^(7/2) + 5711/16*(-2*x + 1)^(5/2) - 21439/16*(-2*x + 1)^(3/2) +
144837/32*sqrt(-2*x + 1) + 65219/32/sqrt(-2*x + 1)

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Fricas [A]
time = 0.80, size = 41, normalized size = 0.52 \begin {gather*} \frac {{\left (875 \, x^{5} + 4150 \, x^{4} + 9501 \, x^{3} + 15948 \, x^{2} + 37944 \, x - 38700\right )} \sqrt {-2 \, x + 1}}{7 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^2*(3+5*x)^3/(1-2*x)^(3/2),x, algorithm="fricas")

[Out]

1/7*(875*x^5 + 4150*x^4 + 9501*x^3 + 15948*x^2 + 37944*x - 38700)*sqrt(-2*x + 1)/(2*x - 1)

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Sympy [A]
time = 16.59, size = 70, normalized size = 0.89 \begin {gather*} \frac {125 \left (1 - 2 x\right )^{\frac {9}{2}}}{32} - \frac {12675 \left (1 - 2 x\right )^{\frac {7}{2}}}{224} + \frac {5711 \left (1 - 2 x\right )^{\frac {5}{2}}}{16} - \frac {21439 \left (1 - 2 x\right )^{\frac {3}{2}}}{16} + \frac {144837 \sqrt {1 - 2 x}}{32} + \frac {65219}{32 \sqrt {1 - 2 x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**2*(3+5*x)**3/(1-2*x)**(3/2),x)

[Out]

125*(1 - 2*x)**(9/2)/32 - 12675*(1 - 2*x)**(7/2)/224 + 5711*(1 - 2*x)**(5/2)/16 - 21439*(1 - 2*x)**(3/2)/16 +
144837*sqrt(1 - 2*x)/32 + 65219/(32*sqrt(1 - 2*x))

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Giac [A]
time = 1.75, size = 76, normalized size = 0.96 \begin {gather*} \frac {125}{32} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {12675}{224} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {5711}{16} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {21439}{16} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {144837}{32} \, \sqrt {-2 \, x + 1} + \frac {65219}{32 \, \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^2*(3+5*x)^3/(1-2*x)^(3/2),x, algorithm="giac")

[Out]

125/32*(2*x - 1)^4*sqrt(-2*x + 1) + 12675/224*(2*x - 1)^3*sqrt(-2*x + 1) + 5711/16*(2*x - 1)^2*sqrt(-2*x + 1)
- 21439/16*(-2*x + 1)^(3/2) + 144837/32*sqrt(-2*x + 1) + 65219/32/sqrt(-2*x + 1)

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Mupad [B]
time = 0.03, size = 55, normalized size = 0.70 \begin {gather*} \frac {65219}{32\,\sqrt {1-2\,x}}+\frac {144837\,\sqrt {1-2\,x}}{32}-\frac {21439\,{\left (1-2\,x\right )}^{3/2}}{16}+\frac {5711\,{\left (1-2\,x\right )}^{5/2}}{16}-\frac {12675\,{\left (1-2\,x\right )}^{7/2}}{224}+\frac {125\,{\left (1-2\,x\right )}^{9/2}}{32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x + 2)^2*(5*x + 3)^3)/(1 - 2*x)^(3/2),x)

[Out]

65219/(32*(1 - 2*x)^(1/2)) + (144837*(1 - 2*x)^(1/2))/32 - (21439*(1 - 2*x)^(3/2))/16 + (5711*(1 - 2*x)^(5/2))
/16 - (12675*(1 - 2*x)^(7/2))/224 + (125*(1 - 2*x)^(9/2))/32

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