3.22.57 \(\int \frac {(2+3 x)^5 (3+5 x)^3}{(1-2 x)^{5/2}} \, dx\) [2157]

Optimal. Leaf size=118 \[ \frac {22370117}{768 (1-2 x)^{3/2}}-\frac {39220335}{128 \sqrt {1-2 x}}-\frac {60160485}{128} \sqrt {1-2 x}+\frac {52725715}{384} (1-2 x)^{3/2}-\frac {2887773}{64} (1-2 x)^{5/2}+\frac {10121229}{896} (1-2 x)^{7/2}-\frac {246315}{128} (1-2 x)^{9/2}+\frac {277425 (1-2 x)^{11/2}}{1408}-\frac {30375 (1-2 x)^{13/2}}{3328} \]

[Out]

22370117/768/(1-2*x)^(3/2)[phone]/384*(1-2*x)^(3/2)-2887773/64*(1-2*x)^(5/2)[phone]/896*(1-2*x)^(7/2)-2463
15/128*(1-2*x)^(9/2)+277425/1408*(1-2*x)^(11/2)-30375/3328*(1-2*x)^(13/2)-39220335/128/(1-2*x)^(1/2)-60160485/
128*(1-2*x)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 118, 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 {30375 (1-2 x)^{13/2}}{3328}+\frac {277425 (1-2 x)^{11/2}}{1408}-\frac {246315}{128} (1-2 x)^{9/2}+\frac {10121229}{896} (1-2 x)^{7/2}-\frac {2887773}{64} (1-2 x)^{5/2}+\frac {52725715}{384} (1-2 x)^{3/2}-\frac {60160485}{128} \sqrt {1-2 x}-\frac {39220335}{128 \sqrt {1-2 x}}+\frac {22370117}{768 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

22370117/(768*(1 - 2*x)^(3/2)) - 39220335/(128*Sqrt[1 - 2*x]) - (60160485*Sqrt[1 - 2*x])/128 + (52725715*(1 -
2*x)^(3/2))/384 - (2887773*(1 - 2*x)^(5/2))/64 + (10121229*(1 - 2*x)^(7/2))/896 - (246315*(1 - 2*x)^(9/2))/128
+ (277425*(1 - 2*x)^(11/2))/1408 - (30375*(1 - 2*x)^(13/2))/3328

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 \frac {(2+3 x)^5 (3+5 x)^3}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac {22370117}{256 (1-2 x)^{5/2}}-\frac {39220335}{128 (1-2 x)^{3/2}}+\frac {60160485}{128 \sqrt {1-2 x}}-\frac {52725715}{128} \sqrt {1-2 x}+\frac {14438865}{64} (1-2 x)^{3/2}-\frac {10121229}{128} (1-2 x)^{5/2}+\frac {2216835}{128} (1-2 x)^{7/2}-\frac {277425}{128} (1-2 x)^{9/2}+\frac {30375}{256} (1-2 x)^{11/2}\right ) \, dx\\ &=\frac {22370117}{768 (1-2 x)^{3/2}}-\frac {39220335}{128 \sqrt {1-2 x}}-\frac {60160485}{128} \sqrt {1-2 x}+\frac {52725715}{384} (1-2 x)^{3/2}-\frac {2887773}{64} (1-2 x)^{5/2}+\frac {10121229}{896} (1-2 x)^{7/2}-\frac {246315}{128} (1-2 x)^{9/2}+\frac {277425 (1-2 x)^{11/2}}{1408}-\frac {30375 (1-2 x)^{13/2}}{3328}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 53, normalized size = 0.45 \begin {gather*} -\frac {1938557272-5818266408 x+2892917004 x^2+905206628 x^3+540496701 x^4+324478899 x^5+153878760 x^6+47670525 x^7+7016625 x^8}{3003 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3003*(1938557272 - 5818266408*x + 2892917004*x^2 + 905206628*x^3 + 540496701*x^4 + 324478899*x^5 + 15387876
0*x^6 + 47670525*x^7 + 7016625*x^8)/(1 - 2*x)^(3/2)

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Maple [A]
time = 0.13, size = 83, normalized size = 0.70

method	result	size
gosper	\(-\frac {7016625 x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}-5818266408 x [phone]}{3003 \left (1-2 x \right )^{\frac {3}{2}}}\)	\(50\)
trager	\(-\frac {\left (7016625 x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}-5818266408 x [phone]\right ) \sqrt {1-2 x}}{3003 \left (-1+2 x \right )^{2}}\)	\(57\)
risch	\(\frac {7016625 x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}-5818266408 x [phone]}{3003 \left (-1+2 x \right ) \sqrt {1-2 x}}\)	\(57\)
derivativedivides	\(\frac {22370117}{768 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {52725715 \left (1-2 x \right )^{\frac {3}{2}}}{384}-\frac {2887773 \left (1-2 x \right )^{\frac {5}{2}}}{64}+\frac {10121229 \left (1-2 x \right )^{\frac {7}{2}}}{896}-\frac {246315 \left (1-2 x \right )^{\frac {9}{2}}}{128}+\frac {277425 \left (1-2 x \right )^{\frac {11}{2}}}{1408}-\frac {30375 \left (1-2 x \right )^{\frac {13}{2}}}{3328}-\frac {39220335}{128 \sqrt {1-2 x}}-\frac {60160485 \sqrt {1-2 x}}{128}\)	\(83\)
default	\(\frac {22370117}{768 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {52725715 \left (1-2 x \right )^{\frac {3}{2}}}{384}-\frac {2887773 \left (1-2 x \right )^{\frac {5}{2}}}{64}+\frac {10121229 \left (1-2 x \right )^{\frac {7}{2}}}{896}-\frac {246315 \left (1-2 x \right )^{\frac {9}{2}}}{128}+\frac {277425 \left (1-2 x \right )^{\frac {11}{2}}}{1408}-\frac {30375 \left (1-2 x \right )^{\frac {13}{2}}}{3328}-\frac {39220335}{128 \sqrt {1-2 x}}-\frac {60160485 \sqrt {1-2 x}}{128}\)	\(83\)
meijerg	\(-\frac {576 \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }}{2 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {3600 \sqrt {\pi }-\frac {450 \sqrt {\pi }\, \left (-24 x +8\right )}{\left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {9840 \left (-4 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (24 x^{2}-48 x +16\right )}{4 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{\sqrt {\pi }}+\frac {\frac {368720 \sqrt {\pi }}{3}-\frac {23045 \sqrt {\pi }\, \left (64 x^{3}+192 x^{2}-384 x +128\right )}{24 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {59945 \left (-\frac {64 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (96 x^{4}+128 x^{3}+384 x^{2}-768 x +256\right )}{20 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{4 \sqrt {\pi }}+\frac {\frac {1197096 \sqrt {\pi }}{7}-\frac {149637 \sqrt {\pi }\, \left (384 x^{5}+384 x^{4}+512 x^{3}+1536 x^{2}-3072 x +1024\right )}{896 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {116685 \left (-\frac {512 \sqrt {\pi }}{21}+\frac {\sqrt {\pi }\, \left (896 x^{6}+768 x^{5}+768 x^{4}+1024 x^{3}+3072 x^{2}-6144 x +2048\right )}{84 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{32 \sqrt {\pi }}+\frac {25200 \sqrt {\pi }-\frac {1575 \sqrt {\pi }\, \left (18432 x^{7}+14336 x^{6}+12288 x^{5}+12288 x^{4}+16384 x^{3}+49152 x^{2}-98304 x +32768\right )}{2048 \left (1-2 x \right )^{\frac {3}{2}}}}{\sqrt {\pi }}-\frac {10125 \left (-\frac {16384 \sqrt {\pi }}{429}+\frac {\sqrt {\pi }\, \left (50688 x^{8}+36864 x^{7}+28672 x^{6}+24576 x^{5}+24576 x^{4}+32768 x^{3}+98304 x^{2}-196608 x +65536\right )}{1716 \left (1-2 x \right )^{\frac {3}{2}}}\right )}{128 \sqrt {\pi }}\)	\(387\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

22370117/768/(1-2*x)^(3/2)[phone]/384*(1-2*x)^(3/2)-2887773/64*(1-2*x)^(5/2)[phone]/896*(1-2*x)^(7/2)-2463
15/128*(1-2*x)^(9/2)+277425/1408*(1-2*x)^(11/2)-30375/3328*(1-2*x)^(13/2)-39220335/128/(1-2*x)^(1/2)-60160485/
128*(1-2*x)^(1/2)

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Maxima [A]
time = 0.27, size = 78, normalized size = 0.66 \begin {gather*} -\frac {30375}{3328} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} + \frac {277425}{1408} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {246315}{128} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {10121229}{896} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {2887773}{64} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {52725715}{384} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {60160485}{128} \, \sqrt {-2 \, x + 1} + \frac {290521 \, {\left (1620 \, x - 733\right )}}{768 \, {\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)^5*(3+5*x)^3/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

-30375/3328*(-2*x + 1)^(13/2) + 277425/1408*(-2*x + 1)^(11/2) - 246315/128*(-2*x + 1)^(9/2) + 10121229/896*(-2
*x + 1)^(7/2) - 2887773/64*(-2*x + 1)^(5/2) + 52725715/384*(-2*x + 1)^(3/2) - 60160485/128*sqrt(-2*x + 1) + 29
0521/768*(1620*x - 733)/(-2*x + 1)^(3/2)

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Fricas [A]
time = 0.83, size = 61, normalized size = 0.52 \begin {gather*} -\frac {{\left (7016625 \, x^{8} + 47670525 \, x^{7} + 153878760 \, x^{6} + 324478899 \, x^{5} + 540496701 \, x^{4} + 905206628 \, x^{3} + 2892917004 \, x^{2} - 5818266408 \, x + 1938557272\right )} \sqrt {-2 \, x + 1}}{3003 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3003*(7016625*x^8 + 47670525*x^7 + 153878760*x^6 + 324478899*x^5 + 540496701*x^4 + 905206628*x^3 + 28929170
04*x^2 - 5818266408*x + 1938557272)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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Sympy [A]
time = 26.65, size = 105, normalized size = 0.89 \begin {gather*} - \frac {30375 \left (1 - 2 x\right )^{\frac {13}{2}}}{3328} + \frac {277425 \left (1 - 2 x\right )^{\frac {11}{2}}}{1408} - \frac {246315 \left (1 - 2 x\right )^{\frac {9}{2}}}{128} + \frac {10121229 \left (1 - 2 x\right )^{\frac {7}{2}}}{896} - \frac {2887773 \left (1 - 2 x\right )^{\frac {5}{2}}}{64} + \frac {52725715 \left (1 - 2 x\right )^{\frac {3}{2}}}{384} - \frac {60160485 \sqrt {1 - 2 x}}{128} - \frac {39220335}{128 \sqrt {1 - 2 x}} + \frac {22370117}{768 \left (1 - 2 x\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-30375*(1 - 2*x)**(13/2)/3328 + 277425*(1 - 2*x)**(11/2)/1408 - 246315*(1 - 2*x)**(9/2)/128 + 10121229*(1 - 2*
x)**(7/2)/896 - 2887773*(1 - 2*x)**(5/2)/64 + 52725715*(1 - 2*x)**(3/2)/384 - 60160485*sqrt(1 - 2*x)/128 - 392
20335/(128*sqrt(1 - 2*x)) + 22370117/(768*(1 - 2*x)**(3/2))

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Giac [A]
time = 1.40, size = 120, normalized size = 1.02 \begin {gather*} -\frac {30375}{3328} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} - \frac {277425}{1408} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {246315}{128} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {10121229}{896} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {2887773}{64} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {52725715}{384} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {60160485}{128} \, \sqrt {-2 \, x + 1} - \frac {290521 \, {\left (1620 \, x - 733\right )}}{768 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-30375/3328*(2*x - 1)^6*sqrt(-2*x + 1) - 277425/1408*(2*x - 1)^5*sqrt(-2*x + 1) - 246315/128*(2*x - 1)^4*sqrt(
-2*x + 1) - 10121229/896*(2*x - 1)^3*sqrt(-2*x + 1) - 2887773/64*(2*x - 1)^2*sqrt(-2*x + 1) + 52725715/384*(-2
*x + 1)^(3/2) - 60160485/128*sqrt(-2*x + 1) - 290521/768*(1620*x - 733)/((2*x - 1)*sqrt(-2*x + 1))

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Mupad [B]
time = 0.03, size = 77, normalized size = 0.65 \begin {gather*} \frac {\frac {39220335\,x}{64}-\frac {212951893}{768}}{{\left (1-2\,x\right )}^{3/2}}-\frac {60160485\,\sqrt {1-2\,x}}{128}+\frac {52725715\,{\left (1-2\,x\right )}^{3/2}}{384}-\frac {2887773\,{\left (1-2\,x\right )}^{5/2}}{64}+\frac {10121229\,{\left (1-2\,x\right )}^{7/2}}{896}-\frac {246315\,{\left (1-2\,x\right )}^{9/2}}{128}+\frac {277425\,{\left (1-2\,x\right )}^{11/2}}{1408}-\frac {30375\,{\left (1-2\,x\right )}^{13/2}}{3328} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((39220335*x)/64 - 212951893/768)/(1 - 2*x)^(3/2) - (60160485*(1 - 2*x)^(1/2))/128 + (52725715*(1 - 2*x)^(3/2)
)/384 - (2887773*(1 - 2*x)^(5/2))/64 + (10121229*(1 - 2*x)^(7/2))/896 - (246315*(1 - 2*x)^(9/2))/128 + (277425
*(1 - 2*x)^(11/2))/1408 - (30375*(1 - 2*x)^(13/2))/3328

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