3.20.43 \(\int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^2 \, dx\) [1943]

Optimal. Leaf size=79 \[ -\frac {5929}{32} (1-2 x)^{7/2}+\frac {91091}{288} (1-2 x)^{9/2}-\frac {39977}{176} (1-2 x)^{11/2}+\frac {17541}{208} (1-2 x)^{13/2}-\frac {513}{32} (1-2 x)^{15/2}+\frac {675}{544} (1-2 x)^{17/2} \]

[Out]

-5929/32*(1-2*x)^(7/2)+91091/288*(1-2*x)^(9/2)-39977/176*(1-2*x)^(11/2)+17541/208*(1-2*x)^(13/2)-513/32*(1-2*x
)^(15/2)+675/544*(1-2*x)^(17/2)

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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 = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {90} \begin {gather*} \frac {675}{544} (1-2 x)^{17/2}-\frac {513}{32} (1-2 x)^{15/2}+\frac {17541}{208} (1-2 x)^{13/2}-\frac {39977}{176} (1-2 x)^{11/2}+\frac {91091}{288} (1-2 x)^{9/2}-\frac {5929}{32} (1-2 x)^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5929*(1 - 2*x)^(7/2))/32 + (91091*(1 - 2*x)^(9/2))/288 - (39977*(1 - 2*x)^(11/2))/176 + (17541*(1 - 2*x)^(13
/2))/208 - (513*(1 - 2*x)^(15/2))/32 + (675*(1 - 2*x)^(17/2))/544

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 (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^2 \, dx &=\int \left (\frac {41503}{32} (1-2 x)^{5/2}-\frac {91091}{32} (1-2 x)^{7/2}+\frac {39977}{16} (1-2 x)^{9/2}-\frac {17541}{16} (1-2 x)^{11/2}+\frac {7695}{32} (1-2 x)^{13/2}-\frac {675}{32} (1-2 x)^{15/2}\right ) \, dx\\ &=-\frac {5929}{32} (1-2 x)^{7/2}+\frac {91091}{288} (1-2 x)^{9/2}-\frac {39977}{176} (1-2 x)^{11/2}+\frac {17541}{208} (1-2 x)^{13/2}-\frac {513}{32} (1-2 x)^{15/2}+\frac {675}{544} (1-2 x)^{17/2}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 38, normalized size = 0.48 \begin {gather*} -\frac {(1-2 x)^{7/2} \left (581846+2497634 x+5069475 x^2+5708637 x^3+3440151 x^4+868725 x^5\right )}{21879} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/21879*((1 - 2*x)^(7/2)*(581846 + 2497634*x + 5069475*x^2 + 5708637*x^3 + 3440151*x^4 + 868725*x^5))

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Maple [A]
time = 0.10, size = 56, normalized size = 0.71

method	result	size
gosper	\(-\frac {\left (868725 x^{5}+3440151 x^{4}+5708637 x^{3}+5069475 x^{2}+2497634 x +581846\right ) \left (1-2 x \right )^{\frac {7}{2}}}{21879}\)	\(35\)
trager	\(\left (\frac {5400}{17} x^{8}+\frac {13284}{17} x^{7}+\frac {96966}{221} x^{6}-\frac {908407}{2431} x^{5}-\frac {10040957}{21879} x^{4}-\frac {608627}{21879} x^{3}+\frac {978059}{7293} x^{2}+\frac {993442}{21879} x -\frac {581846}{21879}\right ) \sqrt {1-2 x}\)	\(49\)
risch	\(-\frac {\left (6949800 x^{8}[phone] x^{7}+9599634 x^{6}-8175663 x^{5}-10040957 x^{4}-608627 x^{3}+2934177 x^{2}+993442 x -581846\right ) \left (-1+2 x \right )}{21879 \sqrt {1-2 x}}\)	\(55\)
derivativedivides	\(-\frac {5929 \left (1-2 x \right )^{\frac {7}{2}}}{32}+\frac {91091 \left (1-2 x \right )^{\frac {9}{2}}}{288}-\frac {39977 \left (1-2 x \right )^{\frac {11}{2}}}{176}+\frac {17541 \left (1-2 x \right )^{\frac {13}{2}}}{208}-\frac {513 \left (1-2 x \right )^{\frac {15}{2}}}{32}+\frac {675 \left (1-2 x \right )^{\frac {17}{2}}}{544}\)	\(56\)
default	\(-\frac {5929 \left (1-2 x \right )^{\frac {7}{2}}}{32}+\frac {91091 \left (1-2 x \right )^{\frac {9}{2}}}{288}-\frac {39977 \left (1-2 x \right )^{\frac {11}{2}}}{176}+\frac {17541 \left (1-2 x \right )^{\frac {13}{2}}}{208}-\frac {513 \left (1-2 x \right )^{\frac {15}{2}}}{32}+\frac {675 \left (1-2 x \right )^{\frac {17}{2}}}{544}\)	\(56\)
meijerg	\(\frac {\frac {72 \sqrt {\pi }}{7}-\frac {36 \sqrt {\pi }\, \left (-16 x^{3}+24 x^{2}-12 x +2\right ) \sqrt {1-2 x}}{7}}{\sqrt {\pi }}-\frac {2115 \left (-\frac {32 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (-448 x^{4}+608 x^{3}-240 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{945}\right )}{8 \sqrt {\pi }}+\frac {\frac {3532 \sqrt {\pi }}{693}-\frac {883 \sqrt {\pi }\, \left (-4032 x^{5}+5152 x^{4}-1808 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{2772}}{\sqrt {\pi }}-\frac {41445 \left (-\frac {256 \sqrt {\pi }}{45045}+\frac {2 \sqrt {\pi }\, \left (-118272 x^{6}+145152 x^{5}-47488 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{45045}\right )}{128 \sqrt {\pi }}+\frac {\frac {384 \sqrt {\pi }}{1001}-\frac {3 \sqrt {\pi }\, \left (-768768 x^{7}+916608 x^{6}-286272 x^{5}+1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{2002}}{\sqrt {\pi }}-\frac {10125 \left (-\frac {4096 \sqrt {\pi }}{2297295}+\frac {4 \sqrt {\pi }\, \left (-9225216 x^{8}[phone] x^{7}-3252480 x^{6}+8064 x^{5}+4480 x^{4}+2560 x^{3}+1536 x^{2}+1024 x +1024\right ) \sqrt {1-2 x}}{2297295}\right )}{512 \sqrt {\pi }}\)	\(305\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5929/32*(1-2*x)^(7/2)+91091/288*(1-2*x)^(9/2)-39977/176*(1-2*x)^(11/2)+17541/208*(1-2*x)^(13/2)-513/32*(1-2*x
)^(15/2)+675/544*(1-2*x)^(17/2)

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Maxima [A]
time = 0.28, size = 55, normalized size = 0.70 \begin {gather*} \frac {675}{544} \, {\left (-2 \, x + 1\right )}^{\frac {17}{2}} - \frac {513}{32} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} + \frac {17541}{208} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {39977}{176} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {91091}{288} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {5929}{32} \, {\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*(3+5*x)^2,x, algorithm="maxima")

[Out]

675/544*(-2*x + 1)^(17/2) - 513/32*(-2*x + 1)^(15/2) + 17541/208*(-2*x + 1)^(13/2) - 39977/176*(-2*x + 1)^(11/
2) + 91091/288*(-2*x + 1)^(9/2) - 5929/32*(-2*x + 1)^(7/2)

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Fricas [A]
time = 1.08, size = 49, normalized size = 0.62 \begin {gather*} \frac {1}{21879} \, {\left (6949800 \, x^{8} + 17096508 \, x^{7} + 9599634 \, x^{6} - 8175663 \, x^{5} - 10040957 \, x^{4} - 608627 \, x^{3} + 2934177 \, x^{2} + 993442 \, x - 581846\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*(3+5*x)^2,x, algorithm="fricas")

[Out]

1/21879*(6949800*x^8 + 17096508*x^7 + 9599634*x^6 - 8175663*x^5 - 10040957*x^4 - 608627*x^3 + 2934177*x^2 + 99
3442*x - 581846)*sqrt(-2*x + 1)

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Sympy [A]
time = 12.66, size = 70, normalized size = 0.89 \begin {gather*} \frac {675 \left (1 - 2 x\right )^{\frac {17}{2}}}{544} - \frac {513 \left (1 - 2 x\right )^{\frac {15}{2}}}{32} + \frac {17541 \left (1 - 2 x\right )^{\frac {13}{2}}}{208} - \frac {39977 \left (1 - 2 x\right )^{\frac {11}{2}}}{176} + \frac {91091 \left (1 - 2 x\right )^{\frac {9}{2}}}{288} - \frac {5929 \left (1 - 2 x\right )^{\frac {7}{2}}}{32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

675*(1 - 2*x)**(17/2)/544 - 513*(1 - 2*x)**(15/2)/32 + 17541*(1 - 2*x)**(13/2)/208 - 39977*(1 - 2*x)**(11/2)/1
76 + 91091*(1 - 2*x)**(9/2)/288 - 5929*(1 - 2*x)**(7/2)/32

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Giac [A]
time = 0.57, size = 97, normalized size = 1.23 \begin {gather*} \frac {675}{544} \, {\left (2 \, x - 1\right )}^{8} \sqrt {-2 \, x + 1} + \frac {513}{32} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} + \frac {17541}{208} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {39977}{176} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {91091}{288} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {5929}{32} \, {\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*(3+5*x)^2,x, algorithm="giac")

[Out]

675/544*(2*x - 1)^8*sqrt(-2*x + 1) + 513/32*(2*x - 1)^7*sqrt(-2*x + 1) + 17541/208*(2*x - 1)^6*sqrt(-2*x + 1)
+ 39977/176*(2*x - 1)^5*sqrt(-2*x + 1) + 91091/288*(2*x - 1)^4*sqrt(-2*x + 1) + 5929/32*(2*x - 1)^3*sqrt(-2*x
+ 1)

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Mupad [B]
time = 0.03, size = 55, normalized size = 0.70 \begin {gather*} \frac {91091\,{\left (1-2\,x\right )}^{9/2}}{288}-\frac {5929\,{\left (1-2\,x\right )}^{7/2}}{32}-\frac {39977\,{\left (1-2\,x\right )}^{11/2}}{176}+\frac {17541\,{\left (1-2\,x\right )}^{13/2}}{208}-\frac {513\,{\left (1-2\,x\right )}^{15/2}}{32}+\frac {675\,{\left (1-2\,x\right )}^{17/2}}{544} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(91091*(1 - 2*x)^(9/2))/288 - (5929*(1 - 2*x)^(7/2))/32 - (39977*(1 - 2*x)^(11/2))/176 + (17541*(1 - 2*x)^(13/
2))/208 - (513*(1 - 2*x)^(15/2))/32 + (675*(1 - 2*x)^(17/2))/544

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