3.20.57 \(\int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^3 \, dx\) [1957]

Optimal. Leaf size=79 \[ -\frac {9317}{32} (1-2 x)^{7/2}+\frac {16093}{32} (1-2 x)^{9/2}-\frac {5847}{16} (1-2 x)^{11/2}+\frac {28555}{208} (1-2 x)^{13/2}-\frac {845}{32} (1-2 x)^{15/2}+\frac {1125}{544} (1-2 x)^{17/2} \]

[Out]

-9317/32*(1-2*x)^(7/2)+16093/32*(1-2*x)^(9/2)-5847/16*(1-2*x)^(11/2)+28555/208*(1-2*x)^(13/2)-845/32*(1-2*x)^(
15/2)+1125/544*(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 {1125}{544} (1-2 x)^{17/2}-\frac {845}{32} (1-2 x)^{15/2}+\frac {28555}{208} (1-2 x)^{13/2}-\frac {5847}{16} (1-2 x)^{11/2}+\frac {16093}{32} (1-2 x)^{9/2}-\frac {9317}{32} (1-2 x)^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-9317*(1 - 2*x)^(7/2))/32 + (16093*(1 - 2*x)^(9/2))/32 - (5847*(1 - 2*x)^(11/2))/16 + (28555*(1 - 2*x)^(13/2)
)/208 - (845*(1 - 2*x)^(15/2))/32 + (1125*(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)^2 (3+5 x)^3 \, dx &=\int \left (\frac {65219}{32} (1-2 x)^{5/2}-\frac {144837}{32} (1-2 x)^{7/2}+\frac {64317}{16} (1-2 x)^{9/2}-\frac {28555}{16} (1-2 x)^{11/2}+\frac {12675}{32} (1-2 x)^{13/2}-\frac {1125}{32} (1-2 x)^{15/2}\right ) \, dx\\ &=-\frac {9317}{32} (1-2 x)^{7/2}+\frac {16093}{32} (1-2 x)^{9/2}-\frac {5847}{16} (1-2 x)^{11/2}+\frac {28555}{208} (1-2 x)^{13/2}-\frac {845}{32} (1-2 x)^{15/2}+\frac {1125}{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}{221} (1-2 x)^{7/2} \left (9004+39160 x+80748 x^2+92535 x^3+56810 x^4+14625 x^5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/221*((1 - 2*x)^(7/2)*(9004 + 39160*x + 80748*x^2 + 92535*x^3 + 56810*x^4 + 14625*x^5))

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

method	result	size
gosper	\(-\frac {\left (14625 x^{5}+56810 x^{4}+92535 x^{3}+80748 x^{2}+39160 x +9004\right ) \left (1-2 x \right )^{\frac {7}{2}}}{221}\)	\(35\)
trager	\(\left (\frac {9000}{17} x^{8}+\frac {21460}{17} x^{7}+\frac {146310}{221} x^{6}-\frac {138201}{221} x^{5}-\frac {157296}{221} x^{4}-\frac {5935}{221} x^{3}+\frac {46164}{221} x^{2}+\frac {14864}{221} x -\frac {9004}{221}\right ) \sqrt {1-2 x}\)	\(49\)
risch	\(-\frac {\left (117000 x^{8}+278980 x^{7}+146310 x^{6}-138201 x^{5}-157296 x^{4}-5935 x^{3}+46164 x^{2}+14864 x -9004\right ) \left (-1+2 x \right )}{221 \sqrt {1-2 x}}\)	\(55\)
derivativedivides	\(-\frac {9317 \left (1-2 x \right )^{\frac {7}{2}}}{32}+\frac {16093 \left (1-2 x \right )^{\frac {9}{2}}}{32}-\frac {5847 \left (1-2 x \right )^{\frac {11}{2}}}{16}+\frac {28555 \left (1-2 x \right )^{\frac {13}{2}}}{208}-\frac {845 \left (1-2 x \right )^{\frac {15}{2}}}{32}+\frac {1125 \left (1-2 x \right )^{\frac {17}{2}}}{544}\)	\(56\)
default	\(-\frac {9317 \left (1-2 x \right )^{\frac {7}{2}}}{32}+\frac {16093 \left (1-2 x \right )^{\frac {9}{2}}}{32}-\frac {5847 \left (1-2 x \right )^{\frac {11}{2}}}{16}+\frac {28555 \left (1-2 x \right )^{\frac {13}{2}}}{208}-\frac {845 \left (1-2 x \right )^{\frac {15}{2}}}{32}+\frac {1125 \left (1-2 x \right )^{\frac {17}{2}}}{544}\)	\(56\)
meijerg	\(\frac {\frac {108 \sqrt {\pi }}{7}-\frac {54 \sqrt {\pi }\, \left (-16 x^{3}+24 x^{2}-12 x +2\right ) \sqrt {1-2 x}}{7}}{\sqrt {\pi }}-\frac {405 \left (-\frac {32 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (-448 x^{4}+608 x^{3}-240 x^{2}+8 x +8\right ) \sqrt {1-2 x}}{945}\right )}{\sqrt {\pi }}+\frac {\frac {614 \sqrt {\pi }}{77}-\frac {307 \sqrt {\pi }\, \left (-4032 x^{5}+5152 x^{4}-1808 x^{3}+24 x^{2}+16 x +16\right ) \sqrt {1-2 x}}{616}}{\sqrt {\pi }}-\frac {66225 \left (-\frac {256 \sqrt {\pi }}{45045}+\frac {2 \sqrt {\pi }\, \left (-118272 x^{6}+145152 x^{5}-47488 x^{4}+320 x^{3}+192 x^{2}+128 x +128\right ) \sqrt {1-2 x}}{45045}\right )}{128 \sqrt {\pi }}+\frac {\frac {1880 \sqrt {\pi }}{3003}-\frac {235 \sqrt {\pi }\, \left (-768768 x^{7}+916608 x^{6}-286272 x^{5}+1120 x^{4}+640 x^{3}+384 x^{2}+256 x +256\right ) \sqrt {1-2 x}}{96096}}{\sqrt {\pi }}-\frac {16875 \left (-\frac {4096 \sqrt {\pi }}{2297295}+\frac {4 \sqrt {\pi }\, \left (-9225216 x^{8}[phone] x^{7}-3252480 x^{6}+8064 x^{5}+4480 x^{4}+2560 x^{3}+1536 x^{2}+1024 x +1024\right ) \sqrt {1-2 x}}{2297295}\right )}{512 \sqrt {\pi }}\)	\(305\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9317/32*(1-2*x)^(7/2)+16093/32*(1-2*x)^(9/2)-5847/16*(1-2*x)^(11/2)+28555/208*(1-2*x)^(13/2)-845/32*(1-2*x)^(
15/2)+1125/544*(1-2*x)^(17/2)

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Maxima [A]
time = 0.28, size = 55, normalized size = 0.70 \begin {gather*} \frac {1125}{544} \, {\left (-2 \, x + 1\right )}^{\frac {17}{2}} - \frac {845}{32} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} + \frac {28555}{208} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {5847}{16} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {16093}{32} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {9317}{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)^2*(3+5*x)^3,x, algorithm="maxima")

[Out]

1125/544*(-2*x + 1)^(17/2) - 845/32*(-2*x + 1)^(15/2) + 28555/208*(-2*x + 1)^(13/2) - 5847/16*(-2*x + 1)^(11/2
) + 16093/32*(-2*x + 1)^(9/2) - 9317/32*(-2*x + 1)^(7/2)

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Fricas [A]
time = 0.67, size = 49, normalized size = 0.62 \begin {gather*} \frac {1}{221} \, {\left (117000 \, x^{8} + 278980 \, x^{7} + 146310 \, x^{6} - 138201 \, x^{5} - 157296 \, x^{4} - 5935 \, x^{3} + 46164 \, x^{2} + 14864 \, x - 9004\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)^2*(3+5*x)^3,x, algorithm="fricas")

[Out]

1/221*(117000*x^8 + 278980*x^7 + 146310*x^6 - 138201*x^5 - 157296*x^4 - 5935*x^3 + 46164*x^2 + 14864*x - 9004)
*sqrt(-2*x + 1)

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Sympy [A]
time = 12.04, size = 70, normalized size = 0.89 \begin {gather*} \frac {1125 \left (1 - 2 x\right )^{\frac {17}{2}}}{544} - \frac {845 \left (1 - 2 x\right )^{\frac {15}{2}}}{32} + \frac {28555 \left (1 - 2 x\right )^{\frac {13}{2}}}{208} - \frac {5847 \left (1 - 2 x\right )^{\frac {11}{2}}}{16} + \frac {16093 \left (1 - 2 x\right )^{\frac {9}{2}}}{32} - \frac {9317 \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)**2*(3+5*x)**3,x)

[Out]

1125*(1 - 2*x)**(17/2)/544 - 845*(1 - 2*x)**(15/2)/32 + 28555*(1 - 2*x)**(13/2)/208 - 5847*(1 - 2*x)**(11/2)/1
6 + 16093*(1 - 2*x)**(9/2)/32 - 9317*(1 - 2*x)**(7/2)/32

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Giac [A]
time = 0.62, size = 97, normalized size = 1.23 \begin {gather*} \frac {1125}{544} \, {\left (2 \, x - 1\right )}^{8} \sqrt {-2 \, x + 1} + \frac {845}{32} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} + \frac {28555}{208} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {5847}{16} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {16093}{32} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {9317}{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)^2*(3+5*x)^3,x, algorithm="giac")

[Out]

1125/544*(2*x - 1)^8*sqrt(-2*x + 1) + 845/32*(2*x - 1)^7*sqrt(-2*x + 1) + 28555/208*(2*x - 1)^6*sqrt(-2*x + 1)
+ 5847/16*(2*x - 1)^5*sqrt(-2*x + 1) + 16093/32*(2*x - 1)^4*sqrt(-2*x + 1) + 9317/32*(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 {16093\,{\left (1-2\,x\right )}^{9/2}}{32}-\frac {9317\,{\left (1-2\,x\right )}^{7/2}}{32}-\frac {5847\,{\left (1-2\,x\right )}^{11/2}}{16}+\frac {28555\,{\left (1-2\,x\right )}^{13/2}}{208}-\frac {845\,{\left (1-2\,x\right )}^{15/2}}{32}+\frac {1125\,{\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)^2*(5*x + 3)^3,x)

[Out]

(16093*(1 - 2*x)^(9/2))/32 - (9317*(1 - 2*x)^(7/2))/32 - (5847*(1 - 2*x)^(11/2))/16 + (28555*(1 - 2*x)^(13/2))
/208 - (845*(1 - 2*x)^(15/2))/32 + (1125*(1 - 2*x)^(17/2))/544

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