3.19.59 \(\int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x) \, dx\) [1859]

Optimal. Leaf size=79 \[ -\frac {26411}{160} (1-2 x)^{5/2}+\frac {8183}{32} (1-2 x)^{7/2}-\frac {8281}{48} (1-2 x)^{9/2}+\frac {10773}{176} (1-2 x)^{11/2}-\frac {4671}{416} (1-2 x)^{13/2}+\frac {27}{32} (1-2 x)^{15/2} \]

[Out]

-26411/160*(1-2*x)^(5/2)+8183/32*(1-2*x)^(7/2)-8281/48*(1-2*x)^(9/2)+10773/176*(1-2*x)^(11/2)-4671/416*(1-2*x)
^(13/2)+27/32*(1-2*x)^(15/2)

________________________________________________________________________________________

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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {78} \begin {gather*} \frac {27}{32} (1-2 x)^{15/2}-\frac {4671}{416} (1-2 x)^{13/2}+\frac {10773}{176} (1-2 x)^{11/2}-\frac {8281}{48} (1-2 x)^{9/2}+\frac {8183}{32} (1-2 x)^{7/2}-\frac {26411}{160} (1-2 x)^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-26411*(1 - 2*x)^(5/2))/160 + (8183*(1 - 2*x)^(7/2))/32 - (8281*(1 - 2*x)^(9/2))/48 + (10773*(1 - 2*x)^(11/2)
)/176 - (4671*(1 - 2*x)^(13/2))/416 + (27*(1 - 2*x)^(15/2))/32

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)^{3/2} (2+3 x)^4 (3+5 x) \, dx &=\int \left (\frac {26411}{32} (1-2 x)^{3/2}-\frac {57281}{32} (1-2 x)^{5/2}+\frac {24843}{16} (1-2 x)^{7/2}-\frac {10773}{16} (1-2 x)^{9/2}+\frac {4671}{32} (1-2 x)^{11/2}-\frac {405}{32} (1-2 x)^{13/2}\right ) \, dx\\ &=-\frac {26411}{160} (1-2 x)^{5/2}+\frac {8183}{32} (1-2 x)^{7/2}-\frac {8281}{48} (1-2 x)^{9/2}+\frac {10773}{176} (1-2 x)^{11/2}-\frac {4671}{416} (1-2 x)^{13/2}+\frac {27}{32} (1-2 x)^{15/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 38, normalized size = 0.48 \begin {gather*} -\frac {(1-2 x)^{5/2} \left (66592+230000 x+410320 x^2+424440 x^3+240570 x^4+57915 x^5\right )}{2145} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2145*((1 - 2*x)^(5/2)*(66592 + 230000*x + 410320*x^2 + 424440*x^3 + 240570*x^4 + 57915*x^5))

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 56, normalized size = 0.71

method	result	size
gosper	\(-\frac {\left (57915 x^{5}+240570 x^{4}+424440 x^{3}+410320 x^{2}+230000 x +66592\right ) \left (1-2 x \right )^{\frac {5}{2}}}{2145}\)	\(35\)
trager	\(\left (-108 x^{7}-\frac {4428}{13} x^{6}-\frac {52893}{143} x^{5}-\frac {36818}{429} x^{4}+\frac {59368}{429} x^{3}+\frac {81104}{715} x^{2}+\frac {36368}{2145} x -\frac {66592}{2145}\right ) \sqrt {1-2 x}\)	\(44\)
risch	\(\frac {\left (231660 x^{7}+730620 x^{6}+793395 x^{5}+184090 x^{4}-296840 x^{3}-243312 x^{2}-36368 x +66592\right ) \left (-1+2 x \right )}{2145 \sqrt {1-2 x}}\)	\(50\)
derivativedivides	\(-\frac {26411 \left (1-2 x \right )^{\frac {5}{2}}}{160}+\frac {8183 \left (1-2 x \right )^{\frac {7}{2}}}{32}-\frac {8281 \left (1-2 x \right )^{\frac {9}{2}}}{48}+\frac {10773 \left (1-2 x \right )^{\frac {11}{2}}}{176}-\frac {4671 \left (1-2 x \right )^{\frac {13}{2}}}{416}+\frac {27 \left (1-2 x \right )^{\frac {15}{2}}}{32}\)	\(56\)
default	\(-\frac {26411 \left (1-2 x \right )^{\frac {5}{2}}}{160}+\frac {8183 \left (1-2 x \right )^{\frac {7}{2}}}{32}-\frac {8281 \left (1-2 x \right )^{\frac {9}{2}}}{48}+\frac {10773 \left (1-2 x \right )^{\frac {11}{2}}}{176}-\frac {4671 \left (1-2 x \right )^{\frac {13}{2}}}{416}+\frac {27 \left (1-2 x \right )^{\frac {15}{2}}}{32}\)	\(56\)
meijerg	\(-\frac {18 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (8 x^{2}-8 x +2\right ) \sqrt {1-2 x}}{15}\right )}{\sqrt {\pi }}+\frac {\frac {368 \sqrt {\pi }}{35}-\frac {46 \sqrt {\pi }\, \left (160 x^{3}-128 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{35}}{\sqrt {\pi }}-\frac {423 \left (-\frac {64 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (1120 x^{4}-800 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{945}\right )}{4 \sqrt {\pi }}+\frac {\frac {1152 \sqrt {\pi }}{385}-\frac {9 \sqrt {\pi }\, \left (26880 x^{5}-17920 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{385}}{\sqrt {\pi }}-\frac {3969 \left (-\frac {1024 \sqrt {\pi }}{45045}+\frac {4 \sqrt {\pi }\, \left (147840 x^{6}-94080 x^{5}+1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{45045}\right )}{128 \sqrt {\pi }}+\frac {\frac {72 \sqrt {\pi }}{1001}-\frac {9 \sqrt {\pi }\, \left (1537536 x^{7}-946176 x^{6}+8064 x^{5}+4480 x^{4}+2560 x^{3}+1536 x^{2}+1024 x +1024\right ) \sqrt {1-2 x}}{128128}}{\sqrt {\pi }}\)	\(275\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-26411/160*(1-2*x)^(5/2)+8183/32*(1-2*x)^(7/2)-8281/48*(1-2*x)^(9/2)+10773/176*(1-2*x)^(11/2)-4671/416*(1-2*x)
^(13/2)+27/32*(1-2*x)^(15/2)

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 55, normalized size = 0.70 \begin {gather*} \frac {27}{32} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} - \frac {4671}{416} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {10773}{176} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {8281}{48} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {8183}{32} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {26411}{160} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/32*(-2*x + 1)^(15/2) - 4671/416*(-2*x + 1)^(13/2) + 10773/176*(-2*x + 1)^(11/2) - 8281/48*(-2*x + 1)^(9/2)
+ 8183/32*(-2*x + 1)^(7/2) - 26411/160*(-2*x + 1)^(5/2)

________________________________________________________________________________________

Fricas [A]
time = 1.29, size = 44, normalized size = 0.56 \begin {gather*} -\frac {1}{2145} \, {\left (231660 \, x^{7} + 730620 \, x^{6} + 793395 \, x^{5} + 184090 \, x^{4} - 296840 \, x^{3} - 243312 \, x^{2} - 36368 \, x + 66592\right )} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2145*(231660*x^7 + 730620*x^6 + 793395*x^5 + 184090*x^4 - 296840*x^3 - 243312*x^2 - 36368*x + 66592)*sqrt(-
2*x + 1)

________________________________________________________________________________________

Sympy [A]
time = 9.70, size = 70, normalized size = 0.89 \begin {gather*} \frac {27 \left (1 - 2 x\right )^{\frac {15}{2}}}{32} - \frac {4671 \left (1 - 2 x\right )^{\frac {13}{2}}}{416} + \frac {10773 \left (1 - 2 x\right )^{\frac {11}{2}}}{176} - \frac {8281 \left (1 - 2 x\right )^{\frac {9}{2}}}{48} + \frac {8183 \left (1 - 2 x\right )^{\frac {7}{2}}}{32} - \frac {26411 \left (1 - 2 x\right )^{\frac {5}{2}}}{160} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27*(1 - 2*x)**(15/2)/32 - 4671*(1 - 2*x)**(13/2)/416 + 10773*(1 - 2*x)**(11/2)/176 - 8281*(1 - 2*x)**(9/2)/48
+ 8183*(1 - 2*x)**(7/2)/32 - 26411*(1 - 2*x)**(5/2)/160

________________________________________________________________________________________

Giac [A]
time = 1.06, size = 97, normalized size = 1.23 \begin {gather*} -\frac {27}{32} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} - \frac {4671}{416} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {10773}{176} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {8281}{48} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {8183}{32} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {26411}{160} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/32*(2*x - 1)^7*sqrt(-2*x + 1) - 4671/416*(2*x - 1)^6*sqrt(-2*x + 1) - 10773/176*(2*x - 1)^5*sqrt(-2*x + 1)
- 8281/48*(2*x - 1)^4*sqrt(-2*x + 1) - 8183/32*(2*x - 1)^3*sqrt(-2*x + 1) - 26411/160*(2*x - 1)^2*sqrt(-2*x +
1)

________________________________________________________________________________________

Mupad [B]
time = 0.03, size = 55, normalized size = 0.70 \begin {gather*} \frac {8183\,{\left (1-2\,x\right )}^{7/2}}{32}-\frac {26411\,{\left (1-2\,x\right )}^{5/2}}{160}-\frac {8281\,{\left (1-2\,x\right )}^{9/2}}{48}+\frac {10773\,{\left (1-2\,x\right )}^{11/2}}{176}-\frac {4671\,{\left (1-2\,x\right )}^{13/2}}{416}+\frac {27\,{\left (1-2\,x\right )}^{15/2}}{32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8183*(1 - 2*x)^(7/2))/32 - (26411*(1 - 2*x)^(5/2))/160 - (8281*(1 - 2*x)^(9/2))/48 + (10773*(1 - 2*x)^(11/2))
/176 - (4671*(1 - 2*x)^(13/2))/416 + (27*(1 - 2*x)^(15/2))/32

________________________________________________________________________________________