3.19.3 \(\int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^2 \, dx\) [1803]

Optimal. Leaf size=92 \[ -\frac {290521}{192} (1-2 x)^{3/2}+\frac {381073}{160} (1-2 x)^{5/2}-\frac {118993}{64} (1-2 x)^{7/2}+\frac {40453}{48} (1-2 x)^{9/2}-\frac {159111}{704} (1-2 x)^{11/2}+\frac {13905}{416} (1-2 x)^{13/2}-\frac {135}{64} (1-2 x)^{15/2} \]

[Out]

-290521/192*(1-2*x)^(3/2)+381073/160*(1-2*x)^(5/2)-118993/64*(1-2*x)^(7/2)+40453/48*(1-2*x)^(9/2)-159111/704*(
1-2*x)^(11/2)+13905/416*(1-2*x)^(13/2)-135/64*(1-2*x)^(15/2)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 92, 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 {135}{64} (1-2 x)^{15/2}+\frac {13905}{416} (1-2 x)^{13/2}-\frac {159111}{704} (1-2 x)^{11/2}+\frac {40453}{48} (1-2 x)^{9/2}-\frac {118993}{64} (1-2 x)^{7/2}+\frac {381073}{160} (1-2 x)^{5/2}-\frac {290521}{192} (1-2 x)^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]*(2 + 3*x)^4*(3 + 5*x)^2,x]

[Out]

(-290521*(1 - 2*x)^(3/2))/192 + (381073*(1 - 2*x)^(5/2))/160 - (118993*(1 - 2*x)^(7/2))/64 + (40453*(1 - 2*x)^
(9/2))/48 - (159111*(1 - 2*x)^(11/2))/704 + (13905*(1 - 2*x)^(13/2))/416 - (135*(1 - 2*x)^(15/2))/64

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 \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^2 \, dx &=\int \left (\frac {290521}{64} \sqrt {1-2 x}-\frac {381073}{32} (1-2 x)^{3/2}+\frac {832951}{64} (1-2 x)^{5/2}-\frac {121359}{16} (1-2 x)^{7/2}+\frac {159111}{64} (1-2 x)^{9/2}-\frac {13905}{32} (1-2 x)^{11/2}+\frac {2025}{64} (1-2 x)^{13/2}\right ) \, dx\\ &=-\frac {290521}{192} (1-2 x)^{3/2}+\frac {381073}{160} (1-2 x)^{5/2}-\frac {118993}{64} (1-2 x)^{7/2}+\frac {40453}{48} (1-2 x)^{9/2}-\frac {159111}{704} (1-2 x)^{11/2}+\frac {13905}{416} (1-2 x)^{13/2}-\frac {135}{64} (1-2 x)^{15/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 43, normalized size = 0.47 \begin {gather*} -\frac {(1-2 x)^{3/2} \left (734904+1895832 x+3298140 x^2+3960500 x^3+3106755 x^4+1425600 x^5+289575 x^6\right )}{2145} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - 2*x]*(2 + 3*x)^4*(3 + 5*x)^2,x]

[Out]

-1/2145*((1 - 2*x)^(3/2)*(734904 + 1895832*x + 3298140*x^2 + 3960500*x^3 + 3106755*x^4 + 1425600*x^5 + 289575*
x^6))

________________________________________________________________________________________

Maple [A]
time = 0.11, size = 65, normalized size = 0.71

method	result	size
gosper	\(-\frac {\left (289575 x^{6}+1425600 x^{5}+3106755 x^{4}+3960500 x^{3}+3298140 x^{2}+1895832 x +734904\right ) \left (1-2 x \right )^{\frac {3}{2}}}{2145}\)	\(40\)
trager	\(\left (270 x^{7}+\frac {15525}{13} x^{6}+\frac {319194}{143} x^{5}+\frac {962849}{429} x^{4}+\frac {527156}{429} x^{3}+\frac {164508}{715} x^{2}-\frac {142008}{715} x -\frac {244968}{715}\right ) \sqrt {1-2 x}\)	\(44\)
risch	\(-\frac {\left (579150 x^{7}+2561625 x^{6}+4787910 x^{5}+4814245 x^{4}+2635780 x^{3}+493524 x^{2}-426024 x -734904\right ) \left (-1+2 x \right )}{2145 \sqrt {1-2 x}}\)	\(50\)
derivativedivides	\(-\frac {290521 \left (1-2 x \right )^{\frac {3}{2}}}{192}+\frac {381073 \left (1-2 x \right )^{\frac {5}{2}}}{160}-\frac {118993 \left (1-2 x \right )^{\frac {7}{2}}}{64}+\frac {40453 \left (1-2 x \right )^{\frac {9}{2}}}{48}-\frac {159111 \left (1-2 x \right )^{\frac {11}{2}}}{704}+\frac {13905 \left (1-2 x \right )^{\frac {13}{2}}}{416}-\frac {135 \left (1-2 x \right )^{\frac {15}{2}}}{64}\)	\(65\)
default	\(-\frac {290521 \left (1-2 x \right )^{\frac {3}{2}}}{192}+\frac {381073 \left (1-2 x \right )^{\frac {5}{2}}}{160}-\frac {118993 \left (1-2 x \right )^{\frac {7}{2}}}{64}+\frac {40453 \left (1-2 x \right )^{\frac {9}{2}}}{48}-\frac {159111 \left (1-2 x \right )^{\frac {11}{2}}}{704}+\frac {13905 \left (1-2 x \right )^{\frac {13}{2}}}{416}-\frac {135 \left (1-2 x \right )^{\frac {15}{2}}}{64}\)	\(65\)
meijerg	\(\frac {48 \sqrt {\pi }-24 \sqrt {\pi }\, \left (2-4 x \right ) \sqrt {1-2 x}}{\sqrt {\pi }}-\frac {168 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (6 x +2\right )}{15}\right )}{\sqrt {\pi }}+\frac {\frac {10448 \sqrt {\pi }}{105}-\frac {1306 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (60 x^{2}+24 x +8\right )}{105}}{\sqrt {\pi }}-\frac {1353 \left (-\frac {64 \sqrt {\pi }}{315}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (280 x^{3}+120 x^{2}+48 x +16\right )}{315}\right )}{4 \sqrt {\pi }}+\frac {\frac {11208 \sqrt {\pi }}{385}-\frac {1401 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (5040 x^{4}+2240 x^{3}+960 x^{2}+384 x +128\right )}{6160}}{\sqrt {\pi }}-\frac {3915 \left (-\frac {1024 \sqrt {\pi }}{9009}+\frac {4 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (22176 x^{5}+10080 x^{4}+4480 x^{3}+1920 x^{2}+768 x +256\right )}{9009}\right )}{64 \sqrt {\pi }}+\frac {\frac {720 \sqrt {\pi }}{1001}-\frac {45 \sqrt {\pi }\, \left (1-2 x \right )^{\frac {3}{2}} \left (192192 x^{6}+88704 x^{5}+40320 x^{4}+17920 x^{3}+7680 x^{2}+3072 x +1024\right )}{64064}}{\sqrt {\pi }}\)	\(273\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^4*(3+5*x)^2*(1-2*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-290521/192*(1-2*x)^(3/2)+381073/160*(1-2*x)^(5/2)-118993/64*(1-2*x)^(7/2)+40453/48*(1-2*x)^(9/2)-159111/704*(
1-2*x)^(11/2)+13905/416*(1-2*x)^(13/2)-135/64*(1-2*x)^(15/2)

________________________________________________________________________________________

Maxima [A]
time = 0.51, size = 64, normalized size = 0.70 \begin {gather*} -\frac {135}{64} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} + \frac {13905}{416} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {159111}{704} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {40453}{48} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {118993}{64} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {381073}{160} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {290521}{192} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4*(3+5*x)^2*(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

-135/64*(-2*x + 1)^(15/2) + 13905/416*(-2*x + 1)^(13/2) - 159111/704*(-2*x + 1)^(11/2) + 40453/48*(-2*x + 1)^(
9/2) - 118993/64*(-2*x + 1)^(7/2) + 381073/160*(-2*x + 1)^(5/2) - 290521/192*(-2*x + 1)^(3/2)

________________________________________________________________________________________

Fricas [A]
time = 0.59, size = 44, normalized size = 0.48 \begin {gather*} \frac {1}{2145} \, {\left (579150 \, x^{7} + 2561625 \, x^{6} + 4787910 \, x^{5} + 4814245 \, x^{4} + 2635780 \, x^{3} + 493524 \, x^{2} - 426024 \, x - 734904\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4*(3+5*x)^2*(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/2145*(579150*x^7 + 2561625*x^6 + 4787910*x^5 + 4814245*x^4 + 2635780*x^3 + 493524*x^2 - 426024*x - 734904)*s
qrt(-2*x + 1)

________________________________________________________________________________________

Sympy [A]
time = 1.49, size = 82, normalized size = 0.89 \begin {gather*} - \frac {135 \left (1 - 2 x\right )^{\frac {15}{2}}}{64} + \frac {13905 \left (1 - 2 x\right )^{\frac {13}{2}}}{416} - \frac {159111 \left (1 - 2 x\right )^{\frac {11}{2}}}{704} + \frac {40453 \left (1 - 2 x\right )^{\frac {9}{2}}}{48} - \frac {118993 \left (1 - 2 x\right )^{\frac {7}{2}}}{64} + \frac {381073 \left (1 - 2 x\right )^{\frac {5}{2}}}{160} - \frac {290521 \left (1 - 2 x\right )^{\frac {3}{2}}}{192} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**4*(3+5*x)**2*(1-2*x)**(1/2),x)

[Out]

-135*(1 - 2*x)**(15/2)/64 + 13905*(1 - 2*x)**(13/2)/416 - 159111*(1 - 2*x)**(11/2)/704 + 40453*(1 - 2*x)**(9/2
)/48 - 118993*(1 - 2*x)**(7/2)/64 + 381073*(1 - 2*x)**(5/2)/160 - 290521*(1 - 2*x)**(3/2)/192

________________________________________________________________________________________

Giac [A]
time = 0.85, size = 106, normalized size = 1.15 \begin {gather*} \frac {135}{64} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} + \frac {13905}{416} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {159111}{704} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {40453}{48} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {118993}{64} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {381073}{160} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {290521}{192} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4*(3+5*x)^2*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

135/64*(2*x - 1)^7*sqrt(-2*x + 1) + 13905/416*(2*x - 1)^6*sqrt(-2*x + 1) + 159111/704*(2*x - 1)^5*sqrt(-2*x +
1) + 40453/48*(2*x - 1)^4*sqrt(-2*x + 1) + 118993/64*(2*x - 1)^3*sqrt(-2*x + 1) + 381073/160*(2*x - 1)^2*sqrt(
-2*x + 1) - 290521/192*(-2*x + 1)^(3/2)

________________________________________________________________________________________

Mupad [B]
time = 1.18, size = 64, normalized size = 0.70 \begin {gather*} \frac {381073\,{\left (1-2\,x\right )}^{5/2}}{160}-\frac {290521\,{\left (1-2\,x\right )}^{3/2}}{192}-\frac {118993\,{\left (1-2\,x\right )}^{7/2}}{64}+\frac {40453\,{\left (1-2\,x\right )}^{9/2}}{48}-\frac {159111\,{\left (1-2\,x\right )}^{11/2}}{704}+\frac {13905\,{\left (1-2\,x\right )}^{13/2}}{416}-\frac {135\,{\left (1-2\,x\right )}^{15/2}}{64} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(1/2)*(3*x + 2)^4*(5*x + 3)^2,x)

[Out]

(381073*(1 - 2*x)^(5/2))/160 - (290521*(1 - 2*x)^(3/2))/192 - (118993*(1 - 2*x)^(7/2))/64 + (40453*(1 - 2*x)^(
9/2))/48 - (159111*(1 - 2*x)^(11/2))/704 + (13905*(1 - 2*x)^(13/2))/416 - (135*(1 - 2*x)^(15/2))/64

________________________________________________________________________________________