3.2146 \(\int \frac {(2+3 x)^5 (3+5 x)^2}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=105 \[ \frac {6075 (1-2 x)^{11/2}}{1408}-\frac {10845}{128} (1-2 x)^{9/2}+\frac {672003}{896} (1-2 x)^{7/2}-\frac {514017}{128} (1-2 x)^{5/2}+\frac {1965635}{128} (1-2 x)^{3/2}-\frac {8117095}{128} \sqrt {1-2 x}-\frac {6206585}{128 \sqrt {1-2 x}}+\frac {2033647}{384 (1-2 x)^{3/2}} \]

[Out]

2033647/384/(1-2*x)^(3/2)+1965635/128*(1-2*x)^(3/2)-514017/128*(1-2*x)^(5/2)+672003/896*(1-2*x)^(7/2)-10845/12
8*(1-2*x)^(9/2)+6075/1408*(1-2*x)^(11/2)-6206585/128/(1-2*x)^(1/2)-8117095/128*(1-2*x)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 105, 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} \[ \frac {6075 (1-2 x)^{11/2}}{1408}-\frac {10845}{128} (1-2 x)^{9/2}+\frac {672003}{896} (1-2 x)^{7/2}-\frac {514017}{128} (1-2 x)^{5/2}+\frac {1965635}{128} (1-2 x)^{3/2}-\frac {8117095}{128} \sqrt {1-2 x}-\frac {6206585}{128 \sqrt {1-2 x}}+\frac {2033647}{384 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2033647/(384*(1 - 2*x)^(3/2)) - 6206585/(128*Sqrt[1 - 2*x]) - (8117095*Sqrt[1 - 2*x])/128 + (1965635*(1 - 2*x)
^(3/2))/128 - (514017*(1 - 2*x)^(5/2))/128 + (672003*(1 - 2*x)^(7/2))/896 - (10845*(1 - 2*x)^(9/2))/128 + (607
5*(1 - 2*x)^(11/2))/1408

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)^2}{(1-2 x)^{5/2}} \, dx &=\int \left (\frac {2033647}{128 (1-2 x)^{5/2}}-\frac {6206585}{128 (1-2 x)^{3/2}}+\frac {8117095}{128 \sqrt {1-2 x}}-\frac {5896905}{128} \sqrt {1-2 x}+\frac {2570085}{128} (1-2 x)^{3/2}-\frac {672003}{128} (1-2 x)^{5/2}+\frac {97605}{128} (1-2 x)^{7/2}-\frac {6075}{128} (1-2 x)^{9/2}\right ) \, dx\\ &=\frac {2033647}{384 (1-2 x)^{3/2}}-\frac {6206585}{128 \sqrt {1-2 x}}-\frac {8117095}{128} \sqrt {1-2 x}+\frac {1965635}{128} (1-2 x)^{3/2}-\frac {514017}{128} (1-2 x)^{5/2}+\frac {672003}{896} (1-2 x)^{7/2}-\frac {10845}{128} (1-2 x)^{9/2}+\frac {6075 (1-2 x)^{11/2}}{1408}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 48, normalized size = 0.46 \[ -\frac {127575 x^7+806085 x^6+2456001 x^5+5121279 x^4+9702012 x^3+32450916 x^2-65622552 x+21852008}{231 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/231*(21852008 - 65622552*x + 32450916*x^2 + 9702012*x^3 + 5121279*x^4 + 2456001*x^5 + 806085*x^6 + 127575*x
^7)/(1 - 2*x)^(3/2)

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fricas [A]  time = 0.86, size = 56, normalized size = 0.53 \[ -\frac {{\left (127575 \, x^{7} + 806085 \, x^{6} + 2456001 \, x^{5} + 5121279 \, x^{4} + 9702012 \, x^{3} + 32450916 \, x^{2} - 65622552 \, x + 21852008\right )} \sqrt {-2 \, x + 1}}{231 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/231*(127575*x^7 + 806085*x^6 + 2456001*x^5 + 5121279*x^4 + 9702012*x^3 + 32450916*x^2 - 65622552*x + 218520
08)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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giac [A]  time = 1.26, size = 104, normalized size = 0.99 \[ -\frac {6075}{1408} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} - \frac {10845}{128} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} - \frac {672003}{896} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {514017}{128} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {1965635}{128} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {8117095}{128} \, \sqrt {-2 \, x + 1} - \frac {26411 \, {\left (705 \, x - 314\right )}}{192 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6075/1408*(2*x - 1)^5*sqrt(-2*x + 1) - 10845/128*(2*x - 1)^4*sqrt(-2*x + 1) - 672003/896*(2*x - 1)^3*sqrt(-2*
x + 1) - 514017/128*(2*x - 1)^2*sqrt(-2*x + 1) + 1965635/128*(-2*x + 1)^(3/2) - 8117095/128*sqrt(-2*x + 1) - 2
6411/192*(705*x - 314)/((2*x - 1)*sqrt(-2*x + 1))

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maple [A]  time = 0.00, size = 45, normalized size = 0.43 \[ -\frac {127575 x^{7}+806085 x^{6}+2456001 x^{5}+5121279 x^{4}+9702012 x^{3}[phone] x^{2}-65622552 x [phone]}{231 \left (-2 x +1\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/231*(127575*x^7+806085*x^6+2456001*x^5+5121279*x^4+9702012*x^3+32450916*x^2-65622552*x+21852008)/(-2*x+1)^(
3/2)

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maxima [A]  time = 0.52, size = 69, normalized size = 0.66 \[ \frac {6075}{1408} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - \frac {10845}{128} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + \frac {672003}{896} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {514017}{128} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {1965635}{128} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {8117095}{128} \, \sqrt {-2 \, x + 1} + \frac {26411 \, {\left (705 \, x - 314\right )}}{192 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6075/1408*(-2*x + 1)^(11/2) - 10845/128*(-2*x + 1)^(9/2) + 672003/896*(-2*x + 1)^(7/2) - 514017/128*(-2*x + 1)
^(5/2) + 1965635/128*(-2*x + 1)^(3/2) - 8117095/128*sqrt(-2*x + 1) + 26411/192*(705*x - 314)/(-2*x + 1)^(3/2)

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mupad [B]  time = 0.03, size = 68, normalized size = 0.65 \[ \frac {\frac {6206585\,x}{64}-\frac {4146527}{96}}{{\left (1-2\,x\right )}^{3/2}}-\frac {8117095\,\sqrt {1-2\,x}}{128}+\frac {1965635\,{\left (1-2\,x\right )}^{3/2}}{128}-\frac {514017\,{\left (1-2\,x\right )}^{5/2}}{128}+\frac {672003\,{\left (1-2\,x\right )}^{7/2}}{896}-\frac {10845\,{\left (1-2\,x\right )}^{9/2}}{128}+\frac {6075\,{\left (1-2\,x\right )}^{11/2}}{1408} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((6206585*x)/64 - 4146527/96)/(1 - 2*x)^(3/2) - (8117095*(1 - 2*x)^(1/2))/128 + (1965635*(1 - 2*x)^(3/2))/128
- (514017*(1 - 2*x)^(5/2))/128 + (672003*(1 - 2*x)^(7/2))/896 - (10845*(1 - 2*x)^(9/2))/128 + (6075*(1 - 2*x)^
(11/2))/1408

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sympy [A]  time = 44.49, size = 94, normalized size = 0.90 \[ \frac {6075 \left (1 - 2 x\right )^{\frac {11}{2}}}{1408} - \frac {10845 \left (1 - 2 x\right )^{\frac {9}{2}}}{128} + \frac {672003 \left (1 - 2 x\right )^{\frac {7}{2}}}{896} - \frac {514017 \left (1 - 2 x\right )^{\frac {5}{2}}}{128} + \frac {1965635 \left (1 - 2 x\right )^{\frac {3}{2}}}{128} - \frac {8117095 \sqrt {1 - 2 x}}{128} - \frac {6206585}{128 \sqrt {1 - 2 x}} + \frac {2033647}{384 \left (1 - 2 x\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6075*(1 - 2*x)**(11/2)/1408 - 10845*(1 - 2*x)**(9/2)/128 + 672003*(1 - 2*x)**(7/2)/896 - 514017*(1 - 2*x)**(5/
2)/128 + 1965635*(1 - 2*x)**(3/2)/128 - 8117095*sqrt(1 - 2*x)/128 - 6206585/(128*sqrt(1 - 2*x)) + 2033647/(384
*(1 - 2*x)**(3/2))

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