3.21.18 \(\int \frac {(2+3 x)^5 (3+5 x)^3}{(1-2 x)^{5/2}} \, dx\)

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

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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 = {88} \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 88

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 {7016625 x^8+47670525 x^7+153878760 x^6+324478899 x^5+540496701 x^4+905206628 x^3+2892917004 x^2-5818266408 x+1938557272}{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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IntegrateAlgebraic [A]  time = 0.03, size = 85, normalized size = 0.72 \begin {gather*} \frac {-7016625 (1-2 x)^8+151474050 (1-2 x)^7-1479367890 (1-2 x)^6+8684014482 (1-2 x)^5-34687929276 (1-2 x)^4+105556881430 (1-2 x)^3-361323872910 (1-2 x)^2-235557332010 (1-2 x)[phone]}{768768 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(22392487117 - 235557332010*(1 - 2*x) - 361323872910*(1 - 2*x)^2 + 105556881430*(1 - 2*x)^3 - 34687929276*(1 -
2*x)^4 + 8684014482*(1 - 2*x)^5 - 1479367890*(1 - 2*x)^6 + 151474050*(1 - 2*x)^7 - 7016625*(1 - 2*x)^8)/(7687
68*(1 - 2*x)^(3/2))

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fricas [A]  time = 1.21, 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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giac [A]  time = 1.24, 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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maple [A]  time = 0.00, size = 50, normalized size = 0.42 \begin {gather*} -\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 (-2 x +1\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3003*(7016625*x^8+47670525*x^7+153878760*x^6+324478899*x^5+540496701*x^4+905206628*x^3+2892917004*x^2-58182
66408*x+1938557272)/(-2*x+1)^(3/2)

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maxima [A]  time = 0.45, 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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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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sympy [A]  time = 53.47, 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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