3.1790 \(\int \sqrt {1-2 x} (2+3 x)^6 (3+5 x) \, dx\)

Optimal. Leaf size=105 \[ \frac {3645 (1-2 x)^{17/2}}{2176}-\frac {19683}{640} (1-2 x)^{15/2}+\frac {409941 (1-2 x)^{13/2}}{1664}-\frac {1580985 (1-2 x)^{11/2}}{1408}+\frac {406455}{128} (1-2 x)^{9/2}-\frac {725445}{128} (1-2 x)^{7/2}+\frac {3916031}{640} (1-2 x)^{5/2}-\frac {1294139}{384} (1-2 x)^{3/2} \]

[Out]

-1294139/384*(1-2*x)^(3/2)+3916031/640*(1-2*x)^(5/2)-725445/128*(1-2*x)^(7/2)+406455/128*(1-2*x)^(9/2)-1580985
/1408*(1-2*x)^(11/2)+409941/1664*(1-2*x)^(13/2)-19683/640*(1-2*x)^(15/2)+3645/2176*(1-2*x)^(17/2)

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Rubi [A]  time = 0.02, antiderivative size = 105, 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 = {77} \[ \frac {3645 (1-2 x)^{17/2}}{2176}-\frac {19683}{640} (1-2 x)^{15/2}+\frac {409941 (1-2 x)^{13/2}}{1664}-\frac {1580985 (1-2 x)^{11/2}}{1408}+\frac {406455}{128} (1-2 x)^{9/2}-\frac {725445}{128} (1-2 x)^{7/2}+\frac {3916031}{640} (1-2 x)^{5/2}-\frac {1294139}{384} (1-2 x)^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]*(2 + 3*x)^6*(3 + 5*x),x]

[Out]

(-1294139*(1 - 2*x)^(3/2))/384 + (3916031*(1 - 2*x)^(5/2))/640 - (725445*(1 - 2*x)^(7/2))/128 + (406455*(1 - 2
*x)^(9/2))/128 - (1580985*(1 - 2*x)^(11/2))/1408 + (409941*(1 - 2*x)^(13/2))/1664 - (19683*(1 - 2*x)^(15/2))/6
40 + (3645*(1 - 2*x)^(17/2))/2176

Rule 77

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 \sqrt {1-2 x} (2+3 x)^6 (3+5 x) \, dx &=\int \left (\frac {1294139}{128} \sqrt {1-2 x}-\frac {3916031}{128} (1-2 x)^{3/2}+\frac {5078115}{128} (1-2 x)^{5/2}-\frac {3658095}{128} (1-2 x)^{7/2}+\frac {1580985}{128} (1-2 x)^{9/2}-\frac {409941}{128} (1-2 x)^{11/2}+\frac {59049}{128} (1-2 x)^{13/2}-\frac {3645}{128} (1-2 x)^{15/2}\right ) \, dx\\ &=-\frac {1294139}{384} (1-2 x)^{3/2}+\frac {3916031}{640} (1-2 x)^{5/2}-\frac {725445}{128} (1-2 x)^{7/2}+\frac {406455}{128} (1-2 x)^{9/2}-\frac {1580985 (1-2 x)^{11/2}}{1408}+\frac {409941 (1-2 x)^{13/2}}{1664}-\frac {19683}{640} (1-2 x)^{15/2}+\frac {3645 (1-2 x)^{17/2}}{2176}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 48, normalized size = 0.46 \[ -\frac {(1-2 x)^{3/2} \left (7818525 x^7+44409222 x^6+113196204 x^5+171389520 x^4+172440720 x^3+122662080 x^2+64000896 x+23667392\right )}{36465} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - 2*x]*(2 + 3*x)^6*(3 + 5*x),x]

[Out]

-1/36465*((1 - 2*x)^(3/2)*(23667392 + 64000896*x + 122662080*x^2 + 172440720*x^3 + 171389520*x^4 + 113196204*x
^5 + 44409222*x^6 + 7818525*x^7))

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fricas [A]  time = 0.57, size = 49, normalized size = 0.47 \[ \frac {1}{36465} \, {\left (15637050 \, x^{8} + 80999919 \, x^{7} + 181983186 \, x^{6} + 229582836 \, x^{5} + 173491920 \, x^{4} + 72883440 \, x^{3} + 5339712 \, x^{2} - 16666112 \, x - 23667392\right )} \sqrt {-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6*(3+5*x)*(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/36465*(15637050*x^8 + 80999919*x^7 + 181983186*x^6 + 229582836*x^5 + 173491920*x^4 + 72883440*x^3 + 5339712*
x^2 - 16666112*x - 23667392)*sqrt(-2*x + 1)

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giac [A]  time = 1.25, size = 122, normalized size = 1.16 \[ \frac {3645}{2176} \, {\left (2 \, x - 1\right )}^{8} \sqrt {-2 \, x + 1} + \frac {19683}{640} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} + \frac {409941}{1664} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {1580985}{1408} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {406455}{128} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {725445}{128} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {3916031}{640} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {1294139}{384} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6*(3+5*x)*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

3645/2176*(2*x - 1)^8*sqrt(-2*x + 1) + 19683/640*(2*x - 1)^7*sqrt(-2*x + 1) + 409941/1664*(2*x - 1)^6*sqrt(-2*
x + 1) + 1580985/1408*(2*x - 1)^5*sqrt(-2*x + 1) + 406455/128*(2*x - 1)^4*sqrt(-2*x + 1) + 725445/128*(2*x - 1
)^3*sqrt(-2*x + 1) + 3916031/640*(2*x - 1)^2*sqrt(-2*x + 1) - 1294139/384*(-2*x + 1)^(3/2)

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maple [A]  time = 0.00, size = 45, normalized size = 0.43 \[ -\frac {\left (7818525 x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \left (-2 x +1\right )^{\frac {3}{2}}}{36465} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+2)^6*(5*x+3)*(1-2*x)^(1/2),x)

[Out]

-1/36465*(7818525*x^7+44409222*x^6+113196204*x^5+171389520*x^4+172440720*x^3+122662080*x^2+64000896*x+23667392
)*(1-2*x)^(3/2)

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maxima [A]  time = 0.45, size = 73, normalized size = 0.70 \[ \frac {3645}{2176} \, {\left (-2 \, x + 1\right )}^{\frac {17}{2}} - \frac {19683}{640} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} + \frac {409941}{1664} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {1580985}{1408} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {406455}{128} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {725445}{128} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {3916031}{640} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {1294139}{384} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6*(3+5*x)*(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

3645/2176*(-2*x + 1)^(17/2) - 19683/640*(-2*x + 1)^(15/2) + 409941/1664*(-2*x + 1)^(13/2) - 1580985/1408*(-2*x
+ 1)^(11/2) + 406455/128*(-2*x + 1)^(9/2) - 725445/128*(-2*x + 1)^(7/2) + 3916031/640*(-2*x + 1)^(5/2) - 1294
139/384*(-2*x + 1)^(3/2)

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mupad [B]  time = 1.21, size = 73, normalized size = 0.70 \[ \frac {3916031\,{\left (1-2\,x\right )}^{5/2}}{640}-\frac {1294139\,{\left (1-2\,x\right )}^{3/2}}{384}-\frac {725445\,{\left (1-2\,x\right )}^{7/2}}{128}+\frac {406455\,{\left (1-2\,x\right )}^{9/2}}{128}-\frac {1580985\,{\left (1-2\,x\right )}^{11/2}}{1408}+\frac {409941\,{\left (1-2\,x\right )}^{13/2}}{1664}-\frac {19683\,{\left (1-2\,x\right )}^{15/2}}{640}+\frac {3645\,{\left (1-2\,x\right )}^{17/2}}{2176} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(1/2)*(3*x + 2)^6*(5*x + 3),x)

[Out]

(3916031*(1 - 2*x)^(5/2))/640 - (1294139*(1 - 2*x)^(3/2))/384 - (725445*(1 - 2*x)^(7/2))/128 + (406455*(1 - 2*
x)^(9/2))/128 - (1580985*(1 - 2*x)^(11/2))/1408 + (409941*(1 - 2*x)^(13/2))/1664 - (19683*(1 - 2*x)^(15/2))/64
0 + (3645*(1 - 2*x)^(17/2))/2176

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sympy [A]  time = 3.55, size = 94, normalized size = 0.90 \[ \frac {3645 \left (1 - 2 x\right )^{\frac {17}{2}}}{2176} - \frac {19683 \left (1 - 2 x\right )^{\frac {15}{2}}}{640} + \frac {409941 \left (1 - 2 x\right )^{\frac {13}{2}}}{1664} - \frac {1580985 \left (1 - 2 x\right )^{\frac {11}{2}}}{1408} + \frac {406455 \left (1 - 2 x\right )^{\frac {9}{2}}}{128} - \frac {725445 \left (1 - 2 x\right )^{\frac {7}{2}}}{128} + \frac {3916031 \left (1 - 2 x\right )^{\frac {5}{2}}}{640} - \frac {1294139 \left (1 - 2 x\right )^{\frac {3}{2}}}{384} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**6*(3+5*x)*(1-2*x)**(1/2),x)

[Out]

3645*(1 - 2*x)**(17/2)/2176 - 19683*(1 - 2*x)**(15/2)/640 + 409941*(1 - 2*x)**(13/2)/1664 - 1580985*(1 - 2*x)*
*(11/2)/1408 + 406455*(1 - 2*x)**(9/2)/128 - 725445*(1 - 2*x)**(7/2)/128 + 3916031*(1 - 2*x)**(5/2)/640 - 1294
139*(1 - 2*x)**(3/2)/384

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