3.20.58 \(\int (1-2 x)^{5/2} (2+3 x) (3+5 x)^3 \, dx\) [1958]

Optimal. Leaf size=66 \[ -\frac {1331}{16} (1-2 x)^{7/2}+\frac {2783}{24} (1-2 x)^{9/2}-\frac {255}{4} (1-2 x)^{11/2}+\frac {1675}{104} (1-2 x)^{13/2}-\frac {25}{16} (1-2 x)^{15/2} \]

[Out]

-1331/16*(1-2*x)^(7/2)+2783/24*(1-2*x)^(9/2)-255/4*(1-2*x)^(11/2)+1675/104*(1-2*x)^(13/2)-25/16*(1-2*x)^(15/2)

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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 {25}{16} (1-2 x)^{15/2}+\frac {1675}{104} (1-2 x)^{13/2}-\frac {255}{4} (1-2 x)^{11/2}+\frac {2783}{24} (1-2 x)^{9/2}-\frac {1331}{16} (1-2 x)^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1331*(1 - 2*x)^(7/2))/16 + (2783*(1 - 2*x)^(9/2))/24 - (255*(1 - 2*x)^(11/2))/4 + (1675*(1 - 2*x)^(13/2))/10
4 - (25*(1 - 2*x)^(15/2))/16

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 (1-2 x)^{5/2} (2+3 x) (3+5 x)^3 \, dx &=\int \left (\frac {9317}{16} (1-2 x)^{5/2}-\frac {8349}{8} (1-2 x)^{7/2}+\frac {2805}{4} (1-2 x)^{9/2}-\frac {1675}{8} (1-2 x)^{11/2}+\frac {375}{16} (1-2 x)^{13/2}\right ) \, dx\\ &=-\frac {1331}{16} (1-2 x)^{7/2}+\frac {2783}{24} (1-2 x)^{9/2}-\frac {255}{4} (1-2 x)^{11/2}+\frac {1675}{104} (1-2 x)^{13/2}-\frac {25}{16} (1-2 x)^{15/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 33, normalized size = 0.50 \begin {gather*} -\frac {1}{39} (1-2 x)^{7/2} \left (641+2381 x+3870 x^2+3075 x^3+975 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/39*((1 - 2*x)^(7/2)*(641 + 2381*x + 3870*x^2 + 3075*x^3 + 975*x^4))

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Maple [A]
time = 0.12, size = 47, normalized size = 0.71

method	result	size
gosper	\(-\frac {\left (975 x^{4}+3075 x^{3}+3870 x^{2}+2381 x +641\right ) \left (1-2 x \right )^{\frac {7}{2}}}{39}\)	\(30\)
trager	\(\left (200 x^{7}+\frac {4300}{13} x^{6}-\frac {30}{13} x^{5}-\frac {9917}{39} x^{4}-\frac {3299}{39} x^{3}+\frac {908}{13} x^{2}+\frac {1465}{39} x -\frac {641}{39}\right ) \sqrt {1-2 x}\)	\(44\)
derivativedivides	\(-\frac {1331 \left (1-2 x \right )^{\frac {7}{2}}}{16}+\frac {2783 \left (1-2 x \right )^{\frac {9}{2}}}{24}-\frac {255 \left (1-2 x \right )^{\frac {11}{2}}}{4}+\frac {1675 \left (1-2 x \right )^{\frac {13}{2}}}{104}-\frac {25 \left (1-2 x \right )^{\frac {15}{2}}}{16}\)	\(47\)
default	\(-\frac {1331 \left (1-2 x \right )^{\frac {7}{2}}}{16}+\frac {2783 \left (1-2 x \right )^{\frac {9}{2}}}{24}-\frac {255 \left (1-2 x \right )^{\frac {11}{2}}}{4}+\frac {1675 \left (1-2 x \right )^{\frac {13}{2}}}{104}-\frac {25 \left (1-2 x \right )^{\frac {15}{2}}}{16}\)	\(47\)
risch	\(-\frac {\left (7800 x^{7}+12900 x^{6}-90 x^{5}-9917 x^{4}-3299 x^{3}+2724 x^{2}+1465 x -641\right ) \left (-1+2 x \right )}{39 \sqrt {1-2 x}}\)	\(50\)
meijerg	\(\frac {\frac {54 \sqrt {\pi }}{7}-\frac {27 \sqrt {\pi }\, \left (-16 x^{3}+24 x^{2}-12 x +2\right ) \sqrt {1-2 x}}{7}}{\sqrt {\pi }}-\frac {5265 \left (-\frac {32 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (-448 x^{4}+608 x^{3}-240 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{945}\right )}{32 \sqrt {\pi }}+\frac {\frac {190 \sqrt {\pi }}{77}-\frac {95 \sqrt {\pi }\, \left (-4032 x^{5}+5152 x^{4}-1808 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{616}}{\sqrt {\pi }}-\frac {13875 \left (-\frac {256 \sqrt {\pi }}{45045}+\frac {2 \sqrt {\pi }\, \left (-118272 x^{6}+145152 x^{5}-47488 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{45045}\right )}{128 \sqrt {\pi }}+\frac {\frac {200 \sqrt {\pi }}{3003}-\frac {25 \sqrt {\pi }\, \left (-768768 x^{7}+916608 x^{6}-286272 x^{5}+1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{96096}}{\sqrt {\pi }}\)	\(242\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1331/16*(1-2*x)^(7/2)+2783/24*(1-2*x)^(9/2)-255/4*(1-2*x)^(11/2)+1675/104*(1-2*x)^(13/2)-25/16*(1-2*x)^(15/2)

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Maxima [A]
time = 0.27, size = 46, normalized size = 0.70 \begin {gather*} -\frac {25}{16} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} + \frac {1675}{104} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {255}{4} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {2783}{24} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {1331}{16} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/16*(-2*x + 1)^(15/2) + 1675/104*(-2*x + 1)^(13/2) - 255/4*(-2*x + 1)^(11/2) + 2783/24*(-2*x + 1)^(9/2) - 1
331/16*(-2*x + 1)^(7/2)

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Fricas [A]
time = 0.81, size = 44, normalized size = 0.67 \begin {gather*} \frac {1}{39} \, {\left (7800 \, x^{7} + 12900 \, x^{6} - 90 \, x^{5} - 9917 \, x^{4} - 3299 \, x^{3} + 2724 \, x^{2} + 1465 \, x - 641\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/39*(7800*x^7 + 12900*x^6 - 90*x^5 - 9917*x^4 - 3299*x^3 + 2724*x^2 + 1465*x - 641)*sqrt(-2*x + 1)

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Sympy [A]
time = 10.31, size = 58, normalized size = 0.88 \begin {gather*} - \frac {25 \left (1 - 2 x\right )^{\frac {15}{2}}}{16} + \frac {1675 \left (1 - 2 x\right )^{\frac {13}{2}}}{104} - \frac {255 \left (1 - 2 x\right )^{\frac {11}{2}}}{4} + \frac {2783 \left (1 - 2 x\right )^{\frac {9}{2}}}{24} - \frac {1331 \left (1 - 2 x\right )^{\frac {7}{2}}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25*(1 - 2*x)**(15/2)/16 + 1675*(1 - 2*x)**(13/2)/104 - 255*(1 - 2*x)**(11/2)/4 + 2783*(1 - 2*x)**(9/2)/24 - 1
331*(1 - 2*x)**(7/2)/16

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Giac [A]
time = 0.75, size = 81, normalized size = 1.23 \begin {gather*} \frac {25}{16} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} + \frac {1675}{104} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {255}{4} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {2783}{24} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {1331}{16} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/16*(2*x - 1)^7*sqrt(-2*x + 1) + 1675/104*(2*x - 1)^6*sqrt(-2*x + 1) + 255/4*(2*x - 1)^5*sqrt(-2*x + 1) + 27
83/24*(2*x - 1)^4*sqrt(-2*x + 1) + 1331/16*(2*x - 1)^3*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 )}^{9/2}}{24}-\frac {1331\,{\left (1-2\,x\right )}^{7/2}}{16}-\frac {255\,{\left (1-2\,x\right )}^{11/2}}{4}+\frac {1675\,{\left (1-2\,x\right )}^{13/2}}{104}-\frac {25\,{\left (1-2\,x\right )}^{15/2}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2783*(1 - 2*x)^(9/2))/24 - (1331*(1 - 2*x)^(7/2))/16 - (255*(1 - 2*x)^(11/2))/4 + (1675*(1 - 2*x)^(13/2))/104
- (25*(1 - 2*x)^(15/2))/16

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