\(\int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^2 \, dx\) [1870]

Optimal result

Integrand size = 24, antiderivative size = 92 \[ \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^2 \, dx=-\frac {290521}{320} (1-2 x)^{5/2}+\frac {54439}{32} (1-2 x)^{7/2}-\frac {832951}{576} (1-2 x)^{9/2}+\frac {121359}{176} (1-2 x)^{11/2}-\frac {159111}{832} (1-2 x)^{13/2}+\frac {927}{32} (1-2 x)^{15/2}-\frac {2025 (1-2 x)^{17/2}}{1088} \]

[Out]

-290521/320*(1-2*x)^(5/2)+54439/32*(1-2*x)^(7/2)-832951/576*(1-2*x)^(9/2)+121359/176*(1-2*x)^(11/2)-159111/832
*(1-2*x)^(13/2)+927/32*(1-2*x)^(15/2)-2025/1088*(1-2*x)^(17/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {90} \[ \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^2 \, dx=-\frac {2025 (1-2 x)^{17/2}}{1088}+\frac {927}{32} (1-2 x)^{15/2}-\frac {159111}{832} (1-2 x)^{13/2}+\frac {121359}{176} (1-2 x)^{11/2}-\frac {832951}{576} (1-2 x)^{9/2}+\frac {54439}{32} (1-2 x)^{7/2}-\frac {290521}{320} (1-2 x)^{5/2} \]

[In]

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

[Out]

(-290521*(1 - 2*x)^(5/2))/320 + (54439*(1 - 2*x)^(7/2))/32 - (832951*(1 - 2*x)^(9/2))/576 + (121359*(1 - 2*x)^
(11/2))/176 - (159111*(1 - 2*x)^(13/2))/832 + (927*(1 - 2*x)^(15/2))/32 - (2025*(1 - 2*x)^(17/2))/1088

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*} \text {integral}& = \int \left (\frac {290521}{64} (1-2 x)^{3/2}-\frac {381073}{32} (1-2 x)^{5/2}+\frac {832951}{64} (1-2 x)^{7/2}-\frac {121359}{16} (1-2 x)^{9/2}+\frac {159111}{64} (1-2 x)^{11/2}-\frac {13905}{32} (1-2 x)^{13/2}+\frac {2025}{64} (1-2 x)^{15/2}\right ) \, dx \\ & = -\frac {290521}{320} (1-2 x)^{5/2}+\frac {54439}{32} (1-2 x)^{7/2}-\frac {832951}{576} (1-2 x)^{9/2}+\frac {121359}{176} (1-2 x)^{11/2}-\frac {159111}{832} (1-2 x)^{13/2}+\frac {927}{32} (1-2 x)^{15/2}-\frac {2025 (1-2 x)^{17/2}}{1088} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.47 \[ \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^2 \, dx=-\frac {(1-2 x)^{5/2} \left (13931096+53902600 x+115145660 x^2+154943820 x^3+130072635 x^4+62316540 x^5+13030875 x^6\right )}{109395} \]

[In]

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

[Out]

-1/109395*((1 - 2*x)^(5/2)*(13931096 + 53902600*x + 115145660*x^2 + 154943820*x^3 + 130072635*x^4 + 62316540*x
^5 + 13030875*x^6))

Maple [A] (verified)

Time = 0.98 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.43

method	result	size
gosper	\(-\frac {\left (1-2 x \right )^{\frac {5}{2}} \left (13030875 x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right )}{109395}\)	\(40\)
trager	\(\left (-\frac {8100}{17} x^{8}-\frac {30636}{17} x^{7}-\frac {573849}{221} x^{6}-\frac {3595584}{2431} x^{5}+\frac {5824001}{21879} x^{4}+\frac {18005684}{21879} x^{3}+\frac {14913452}{36465} x^{2}+\frac {1821784}{109395} x -\frac {13931096}{109395}\right ) \sqrt {1-2 x}\)	\(49\)
pseudoelliptic	\(-\frac {\sqrt {1-2 x}\, \left (52123500 x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}-29120005 x^{4}-90028420 x^{3}-44740356 x^{2}-1821784 x [phone]\right )}{109395}\)	\(50\)
risch	\(\frac {\left (52123500 x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}-29120005 x^{4}-90028420 x^{3}-44740356 x^{2}-1821784 x [phone]\right ) \left (-1+2 x \right )}{109395 \sqrt {1-2 x}}\)	\(55\)
derivativedivides	\(-\frac {290521 \left (1-2 x \right )^{\frac {5}{2}}}{320}+\frac {54439 \left (1-2 x \right )^{\frac {7}{2}}}{32}-\frac {832951 \left (1-2 x \right )^{\frac {9}{2}}}{576}+\frac {121359 \left (1-2 x \right )^{\frac {11}{2}}}{176}-\frac {159111 \left (1-2 x \right )^{\frac {13}{2}}}{832}+\frac {927 \left (1-2 x \right )^{\frac {15}{2}}}{32}-\frac {2025 \left (1-2 x \right )^{\frac {17}{2}}}{1088}\)	\(65\)
default	\(-\frac {290521 \left (1-2 x \right )^{\frac {5}{2}}}{320}+\frac {54439 \left (1-2 x \right )^{\frac {7}{2}}}{32}-\frac {832951 \left (1-2 x \right )^{\frac {9}{2}}}{576}+\frac {121359 \left (1-2 x \right )^{\frac {11}{2}}}{176}-\frac {159111 \left (1-2 x \right )^{\frac {13}{2}}}{832}+\frac {927 \left (1-2 x \right )^{\frac {15}{2}}}{32}-\frac {2025 \left (1-2 x \right )^{\frac {17}{2}}}{1088}\)	\(65\)
meijerg	\(-\frac {54 \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (8 x^{2}-8 x +2\right ) \sqrt {1-2 x}}{15}\right )}{\sqrt {\pi }}+\frac {\frac {192 \sqrt {\pi }}{5}-\frac {24 \sqrt {\pi }\, \left (160 x^{3}-128 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{5}}{\sqrt {\pi }}-\frac {1959 \left (-\frac {64 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (1120 x^{4}-800 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{945}\right )}{4 \sqrt {\pi }}+\frac {\frac {656 \sqrt {\pi }}{35}-\frac {41 \sqrt {\pi }\, \left (26880 x^{5}-17920 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{280}}{\sqrt {\pi }}-\frac {37827 \left (-\frac {1024 \sqrt {\pi }}{45045}+\frac {4 \sqrt {\pi }\, \left (147840 x^{6}-94080 x^{5}+1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{45045}\right )}{128 \sqrt {\pi }}+\frac {\frac {1392 \sqrt {\pi }}{1001}-\frac {87 \sqrt {\pi }\, \left (1537536 x^{7}-946176 x^{6}+8064 x^{5}+4480 x^{4}+2560 x^{3}+1536 x^{2}+1024 x +1024\right ) \sqrt {1-2 x}}{64064}}{\sqrt {\pi }}-\frac {6075 \left (-\frac {8192 \sqrt {\pi }}{765765}+\frac {4 \sqrt {\pi }\, \left (7687680 x^{8}-4612608 x^{7}+29568 x^{6}+16128 x^{5}+8960 x^{4}+5120 x^{3}+3072 x^{2}+2048 x +2048\right ) \sqrt {1-2 x}}{765765}\right )}{512 \sqrt {\pi }}\)	\(338\)

[In]

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

[Out]

-1/109395*(1-2*x)^(5/2)*(13030875*x^6+62316540*x^5+130072635*x^4+154943820*x^3+115145660*x^2+53902600*x+139310
96)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.53 \[ \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^2 \, dx=-\frac {1}{109395} \, {\left (52123500 \, x^{8} + 197142660 \, x^{7} + 284055255 \, x^{6} + 161801280 \, x^{5} - 29120005 \, x^{4} - 90028420 \, x^{3} - 44740356 \, x^{2} - 1821784 \, x + 13931096\right )} \sqrt {-2 \, x + 1} \]

[In]

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

[Out]

-1/109395*(52123500*x^8 + 197142660*x^7 + 284055255*x^6 + 161801280*x^5 - 29120005*x^4 - 90028420*x^3 - 447403
56*x^2 - 1821784*x + 13931096)*sqrt(-2*x + 1)

Sympy [A] (verification not implemented)

Time = 0.87 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89 \[ \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^2 \, dx=- \frac {2025 \left (1 - 2 x\right )^{\frac {17}{2}}}{1088} + \frac {927 \left (1 - 2 x\right )^{\frac {15}{2}}}{32} - \frac {159111 \left (1 - 2 x\right )^{\frac {13}{2}}}{832} + \frac {121359 \left (1 - 2 x\right )^{\frac {11}{2}}}{176} - \frac {832951 \left (1 - 2 x\right )^{\frac {9}{2}}}{576} + \frac {54439 \left (1 - 2 x\right )^{\frac {7}{2}}}{32} - \frac {290521 \left (1 - 2 x\right )^{\frac {5}{2}}}{320} \]

[In]

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

[Out]

-2025*(1 - 2*x)**(17/2)/1088 + 927*(1 - 2*x)**(15/2)/32 - 159111*(1 - 2*x)**(13/2)/832 + 121359*(1 - 2*x)**(11
/2)/176 - 832951*(1 - 2*x)**(9/2)/576 + 54439*(1 - 2*x)**(7/2)/32 - 290521*(1 - 2*x)**(5/2)/320

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^2 \, dx=-\frac {2025}{1088} \, {\left (-2 \, x + 1\right )}^{\frac {17}{2}} + \frac {927}{32} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} - \frac {159111}{832} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {121359}{176} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {832951}{576} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {54439}{32} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {290521}{320} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \]

[In]

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

[Out]

-2025/1088*(-2*x + 1)^(17/2) + 927/32*(-2*x + 1)^(15/2) - 159111/832*(-2*x + 1)^(13/2) + 121359/176*(-2*x + 1)
^(11/2) - 832951/576*(-2*x + 1)^(9/2) + 54439/32*(-2*x + 1)^(7/2) - 290521/320*(-2*x + 1)^(5/2)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.23 \[ \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^2 \, dx=-\frac {2025}{1088} \, {\left (2 \, x - 1\right )}^{8} \sqrt {-2 \, x + 1} - \frac {927}{32} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} - \frac {159111}{832} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {121359}{176} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {832951}{576} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {54439}{32} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {290521}{320} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} \]

[In]

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

[Out]

-2025/1088*(2*x - 1)^8*sqrt(-2*x + 1) - 927/32*(2*x - 1)^7*sqrt(-2*x + 1) - 159111/832*(2*x - 1)^6*sqrt(-2*x +
1) - 121359/176*(2*x - 1)^5*sqrt(-2*x + 1) - 832951/576*(2*x - 1)^4*sqrt(-2*x + 1) - 54439/32*(2*x - 1)^3*sqr
t(-2*x + 1) - 290521/320*(2*x - 1)^2*sqrt(-2*x + 1)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int (1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^2 \, dx=\frac {54439\,{\left (1-2\,x\right )}^{7/2}}{32}-\frac {290521\,{\left (1-2\,x\right )}^{5/2}}{320}-\frac {832951\,{\left (1-2\,x\right )}^{9/2}}{576}+\frac {121359\,{\left (1-2\,x\right )}^{11/2}}{176}-\frac {159111\,{\left (1-2\,x\right )}^{13/2}}{832}+\frac {927\,{\left (1-2\,x\right )}^{15/2}}{32}-\frac {2025\,{\left (1-2\,x\right )}^{17/2}}{1088} \]

[In]

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

[Out]

(54439*(1 - 2*x)^(7/2))/32 - (290521*(1 - 2*x)^(5/2))/320 - (832951*(1 - 2*x)^(9/2))/576 + (121359*(1 - 2*x)^(
11/2))/176 - (159111*(1 - 2*x)^(13/2))/832 + (927*(1 - 2*x)^(15/2))/32 - (2025*(1 - 2*x)^(17/2))/1088