3.1817 \(\int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3 \, dx\)

Optimal. Leaf size=79 \[ \frac {1125}{416} (1-2 x)^{13/2}-\frac {12675}{352} (1-2 x)^{11/2}+\frac {28555}{144} (1-2 x)^{9/2}-\frac {64317}{112} (1-2 x)^{7/2}+\frac {144837}{160} (1-2 x)^{5/2}-\frac {65219}{96} (1-2 x)^{3/2} \]

[Out]

-65219/96*(1-2*x)^(3/2)+144837/160*(1-2*x)^(5/2)-64317/112*(1-2*x)^(7/2)+28555/144*(1-2*x)^(9/2)-12675/352*(1-
2*x)^(11/2)+1125/416*(1-2*x)^(13/2)

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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 = {88} \[ \frac {1125}{416} (1-2 x)^{13/2}-\frac {12675}{352} (1-2 x)^{11/2}+\frac {28555}{144} (1-2 x)^{9/2}-\frac {64317}{112} (1-2 x)^{7/2}+\frac {144837}{160} (1-2 x)^{5/2}-\frac {65219}{96} (1-2 x)^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-65219*(1 - 2*x)^(3/2))/96 + (144837*(1 - 2*x)^(5/2))/160 - (64317*(1 - 2*x)^(7/2))/112 + (28555*(1 - 2*x)^(9
/2))/144 - (12675*(1 - 2*x)^(11/2))/352 + (1125*(1 - 2*x)^(13/2))/416

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 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^3 \, dx &=\int \left (\frac {65219}{32} \sqrt {1-2 x}-\frac {144837}{32} (1-2 x)^{3/2}+\frac {64317}{16} (1-2 x)^{5/2}-\frac {28555}{16} (1-2 x)^{7/2}+\frac {12675}{32} (1-2 x)^{9/2}-\frac {1125}{32} (1-2 x)^{11/2}\right ) \, dx\\ &=-\frac {65219}{96} (1-2 x)^{3/2}+\frac {144837}{160} (1-2 x)^{5/2}-\frac {64317}{112} (1-2 x)^{7/2}+\frac {28555}{144} (1-2 x)^{9/2}-\frac {12675}{352} (1-2 x)^{11/2}+\frac {1125}{416} (1-2 x)^{13/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 0.48 \[ -\frac {(1-2 x)^{3/2} \left (3898125 x^5+16206750 x^4+29300075 x^3+30337080 x^2+19918608 x+8261156\right )}{45045} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/45045*((1 - 2*x)^(3/2)*(8261156 + 19918608*x + 30337080*x^2 + 29300075*x^3 + 16206750*x^4 + 3898125*x^5))

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fricas [A]  time = 0.76, size = 39, normalized size = 0.49 \[ \frac {1}{45045} \, {\left (7796250 \, x^{6} + 28515375 \, x^{5} + 42393400 \, x^{4} + 31374085 \, x^{3} + 9500136 \, x^{2} - 3396296 \, x - 8261156\right )} \sqrt {-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/45045*(7796250*x^6 + 28515375*x^5 + 42393400*x^4 + 31374085*x^3 + 9500136*x^2 - 3396296*x - 8261156)*sqrt(-2
*x + 1)

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giac [A]  time = 1.21, size = 90, normalized size = 1.14 \[ \frac {1125}{416} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {12675}{352} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {28555}{144} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {64317}{112} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {144837}{160} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - \frac {65219}{96} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1125/416*(2*x - 1)^6*sqrt(-2*x + 1) + 12675/352*(2*x - 1)^5*sqrt(-2*x + 1) + 28555/144*(2*x - 1)^4*sqrt(-2*x +
1) + 64317/112*(2*x - 1)^3*sqrt(-2*x + 1) + 144837/160*(2*x - 1)^2*sqrt(-2*x + 1) - 65219/96*(-2*x + 1)^(3/2)

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maple [A]  time = 0.00, size = 35, normalized size = 0.44 \[ -\frac {\left (3898125 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x +8261156\right ) \left (-2 x +1\right )^{\frac {3}{2}}}{45045} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/45045*(3898125*x^5+16206750*x^4+29300075*x^3+30337080*x^2+19918608*x+8261156)*(-2*x+1)^(3/2)

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maxima [A]  time = 0.45, size = 55, normalized size = 0.70 \[ \frac {1125}{416} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {12675}{352} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {28555}{144} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {64317}{112} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {144837}{160} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - \frac {65219}{96} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1125/416*(-2*x + 1)^(13/2) - 12675/352*(-2*x + 1)^(11/2) + 28555/144*(-2*x + 1)^(9/2) - 64317/112*(-2*x + 1)^(
7/2) + 144837/160*(-2*x + 1)^(5/2) - 65219/96*(-2*x + 1)^(3/2)

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mupad [B]  time = 0.03, size = 55, normalized size = 0.70 \[ \frac {144837\,{\left (1-2\,x\right )}^{5/2}}{160}-\frac {65219\,{\left (1-2\,x\right )}^{3/2}}{96}-\frac {64317\,{\left (1-2\,x\right )}^{7/2}}{112}+\frac {28555\,{\left (1-2\,x\right )}^{9/2}}{144}-\frac {12675\,{\left (1-2\,x\right )}^{11/2}}{352}+\frac {1125\,{\left (1-2\,x\right )}^{13/2}}{416} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(144837*(1 - 2*x)^(5/2))/160 - (65219*(1 - 2*x)^(3/2))/96 - (64317*(1 - 2*x)^(7/2))/112 + (28555*(1 - 2*x)^(9/
2))/144 - (12675*(1 - 2*x)^(11/2))/352 + (1125*(1 - 2*x)^(13/2))/416

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sympy [A]  time = 2.83, size = 70, normalized size = 0.89 \[ \frac {1125 \left (1 - 2 x\right )^{\frac {13}{2}}}{416} - \frac {12675 \left (1 - 2 x\right )^{\frac {11}{2}}}{352} + \frac {28555 \left (1 - 2 x\right )^{\frac {9}{2}}}{144} - \frac {64317 \left (1 - 2 x\right )^{\frac {7}{2}}}{112} + \frac {144837 \left (1 - 2 x\right )^{\frac {5}{2}}}{160} - \frac {65219 \left (1 - 2 x\right )^{\frac {3}{2}}}{96} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1125*(1 - 2*x)**(13/2)/416 - 12675*(1 - 2*x)**(11/2)/352 + 28555*(1 - 2*x)**(9/2)/144 - 64317*(1 - 2*x)**(7/2)
/112 + 144837*(1 - 2*x)**(5/2)/160 - 65219*(1 - 2*x)**(3/2)/96

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