3.21.27 \(\int \frac {(2+3 x) (3+5 x)^3}{\sqrt {1-2 x}} \, dx\) [2027]

Optimal. Leaf size=66 \[ -\frac {9317}{16} \sqrt {1-2 x}+\frac {2783}{8} (1-2 x)^{3/2}-\frac {561}{4} (1-2 x)^{5/2}+\frac {1675}{56} (1-2 x)^{7/2}-\frac {125}{48} (1-2 x)^{9/2} \]

[Out]

2783/8*(1-2*x)^(3/2)-561/4*(1-2*x)^(5/2)+1675/56*(1-2*x)^(7/2)-125/48*(1-2*x)^(9/2)-9317/16*(1-2*x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {78} \begin {gather*} -\frac {125}{48} (1-2 x)^{9/2}+\frac {1675}{56} (1-2 x)^{7/2}-\frac {561}{4} (1-2 x)^{5/2}+\frac {2783}{8} (1-2 x)^{3/2}-\frac {9317}{16} \sqrt {1-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-9317*Sqrt[1 - 2*x])/16 + (2783*(1 - 2*x)^(3/2))/8 - (561*(1 - 2*x)^(5/2))/4 + (1675*(1 - 2*x)^(7/2))/56 - (1
25*(1 - 2*x)^(9/2))/48

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {(2+3 x) (3+5 x)^3}{\sqrt {1-2 x}} \, dx &=\int \left (\frac {9317}{16 \sqrt {1-2 x}}-\frac {8349}{8} \sqrt {1-2 x}+\frac {2805}{4} (1-2 x)^{3/2}-\frac {1675}{8} (1-2 x)^{5/2}+\frac {375}{16} (1-2 x)^{7/2}\right ) \, dx\\ &=-\frac {9317}{16} \sqrt {1-2 x}+\frac {2783}{8} (1-2 x)^{3/2}-\frac {561}{4} (1-2 x)^{5/2}+\frac {1675}{56} (1-2 x)^{7/2}-\frac {125}{48} (1-2 x)^{9/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 33, normalized size = 0.50 \begin {gather*} -\frac {1}{21} \sqrt {1-2 x} \left (7295+6161 x+5556 x^2+3275 x^3+875 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/21*(Sqrt[1 - 2*x]*(7295 + 6161*x + 5556*x^2 + 3275*x^3 + 875*x^4))

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Maple [A]
time = 0.10, size = 47, normalized size = 0.71

method	result	size
trager	\(\left (-\frac {125}{3} x^{4}-\frac {3275}{21} x^{3}-\frac {1852}{7} x^{2}-\frac {6161}{21} x -\frac {7295}{21}\right ) \sqrt {1-2 x}\)	\(29\)
gosper	\(-\frac {\left (875 x^{4}+3275 x^{3}+5556 x^{2}+6161 x +7295\right ) \sqrt {1-2 x}}{21}\)	\(30\)
risch	\(\frac {\left (-1+2 x \right ) \left (875 x^{4}+3275 x^{3}+5556 x^{2}+6161 x +7295\right )}{21 \sqrt {1-2 x}}\)	\(35\)
derivativedivides	\(\frac {2783 \left (1-2 x \right )^{\frac {3}{2}}}{8}-\frac {561 \left (1-2 x \right )^{\frac {5}{2}}}{4}+\frac {1675 \left (1-2 x \right )^{\frac {7}{2}}}{56}-\frac {125 \left (1-2 x \right )^{\frac {9}{2}}}{48}-\frac {9317 \sqrt {1-2 x}}{16}\)	\(47\)
default	\(\frac {2783 \left (1-2 x \right )^{\frac {3}{2}}}{8}-\frac {561 \left (1-2 x \right )^{\frac {5}{2}}}{4}+\frac {1675 \left (1-2 x \right )^{\frac {7}{2}}}{56}-\frac {125 \left (1-2 x \right )^{\frac {9}{2}}}{48}-\frac {9317 \sqrt {1-2 x}}{16}\)	\(47\)
meijerg	\(-\frac {27 \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1-2 x}\right )}{\sqrt {\pi }}+\frac {117 \sqrt {\pi }-\frac {117 \sqrt {\pi }\, \left (8 x +8\right ) \sqrt {1-2 x}}{8}}{\sqrt {\pi }}-\frac {855 \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{15}\right )}{8 \sqrt {\pi }}+\frac {\frac {370 \sqrt {\pi }}{7}-\frac {185 \sqrt {\pi }\, \left (320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{448}}{\sqrt {\pi }}-\frac {375 \left (-\frac {256 \sqrt {\pi }}{315}+\frac {\sqrt {\pi }\, \left (1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{315}\right )}{32 \sqrt {\pi }}\)	\(167\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2783/8*(1-2*x)^(3/2)-561/4*(1-2*x)^(5/2)+1675/56*(1-2*x)^(7/2)-125/48*(1-2*x)^(9/2)-9317/16*(1-2*x)^(1/2)

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Maxima [A]
time = 0.33, size = 46, normalized size = 0.70 \begin {gather*} -\frac {125}{48} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {1675}{56} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {561}{4} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2783}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {9317}{16} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125/48*(-2*x + 1)^(9/2) + 1675/56*(-2*x + 1)^(7/2) - 561/4*(-2*x + 1)^(5/2) + 2783/8*(-2*x + 1)^(3/2) - 9317/
16*sqrt(-2*x + 1)

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Fricas [A]
time = 0.99, size = 29, normalized size = 0.44 \begin {gather*} -\frac {1}{21} \, {\left (875 \, x^{4} + 3275 \, x^{3} + 5556 \, x^{2} + 6161 \, x + 7295\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(875*x^4 + 3275*x^3 + 5556*x^2 + 6161*x + 7295)*sqrt(-2*x + 1)

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Sympy [A]
time = 23.80, size = 58, normalized size = 0.88 \begin {gather*} - \frac {125 \left (1 - 2 x\right )^{\frac {9}{2}}}{48} + \frac {1675 \left (1 - 2 x\right )^{\frac {7}{2}}}{56} - \frac {561 \left (1 - 2 x\right )^{\frac {5}{2}}}{4} + \frac {2783 \left (1 - 2 x\right )^{\frac {3}{2}}}{8} - \frac {9317 \sqrt {1 - 2 x}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125*(1 - 2*x)**(9/2)/48 + 1675*(1 - 2*x)**(7/2)/56 - 561*(1 - 2*x)**(5/2)/4 + 2783*(1 - 2*x)**(3/2)/8 - 9317*
sqrt(1 - 2*x)/16

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Giac [A]
time = 0.92, size = 67, normalized size = 1.02 \begin {gather*} -\frac {125}{48} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {1675}{56} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {561}{4} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2783}{8} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {9317}{16} \, \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125/48*(2*x - 1)^4*sqrt(-2*x + 1) - 1675/56*(2*x - 1)^3*sqrt(-2*x + 1) - 561/4*(2*x - 1)^2*sqrt(-2*x + 1) + 2
783/8*(-2*x + 1)^(3/2) - 9317/16*sqrt(-2*x + 1)

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Mupad [B]
time = 0.02, size = 46, normalized size = 0.70 \begin {gather*} \frac {2783\,{\left (1-2\,x\right )}^{3/2}}{8}-\frac {9317\,\sqrt {1-2\,x}}{16}-\frac {561\,{\left (1-2\,x\right )}^{5/2}}{4}+\frac {1675\,{\left (1-2\,x\right )}^{7/2}}{56}-\frac {125\,{\left (1-2\,x\right )}^{9/2}}{48} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2783*(1 - 2*x)^(3/2))/8 - (9317*(1 - 2*x)^(1/2))/16 - (561*(1 - 2*x)^(5/2))/4 + (1675*(1 - 2*x)^(7/2))/56 - (
125*(1 - 2*x)^(9/2))/48

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