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

Optimal. Leaf size=66 \[ -\frac {25}{16} (1-2 x)^{15/2}+\frac {1675}{104} (1-2 x)^{13/2}-\frac {255}{4} (1-2 x)^{11/2}+\frac {2783}{24} (1-2 x)^{9/2}-\frac {1331}{16} (1-2 x)^{7/2} \]

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Rubi [A]  time = 0.01, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {77} \begin {gather*} -\frac {25}{16} (1-2 x)^{15/2}+\frac {1675}{104} (1-2 x)^{13/2}-\frac {255}{4} (1-2 x)^{11/2}+\frac {2783}{24} (1-2 x)^{9/2}-\frac {1331}{16} (1-2 x)^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1331*(1 - 2*x)^(7/2))/16 + (2783*(1 - 2*x)^(9/2))/24 - (255*(1 - 2*x)^(11/2))/4 + (1675*(1 - 2*x)^(13/2))/10
4 - (25*(1 - 2*x)^(15/2))/16

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int (1-2 x)^{5/2} (2+3 x) (3+5 x)^3 \, dx &=\int \left (\frac {9317}{16} (1-2 x)^{5/2}-\frac {8349}{8} (1-2 x)^{7/2}+\frac {2805}{4} (1-2 x)^{9/2}-\frac {1675}{8} (1-2 x)^{11/2}+\frac {375}{16} (1-2 x)^{13/2}\right ) \, dx\\ &=-\frac {1331}{16} (1-2 x)^{7/2}+\frac {2783}{24} (1-2 x)^{9/2}-\frac {255}{4} (1-2 x)^{11/2}+\frac {1675}{104} (1-2 x)^{13/2}-\frac {25}{16} (1-2 x)^{15/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 33, normalized size = 0.50 \begin {gather*} -\frac {1}{39} (1-2 x)^{7/2} \left (975 x^4+3075 x^3+3870 x^2+2381 x+641\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/39*((1 - 2*x)^(7/2)*(641 + 2381*x + 3870*x^2 + 3075*x^3 + 975*x^4))

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IntegrateAlgebraic [A]  time = 0.02, size = 49, normalized size = 0.74 \begin {gather*} -\frac {1}{624} \left (975 (1-2 x)^4-10050 (1-2 x)^3+39780 (1-2 x)^2-72358 (1-2 x)+51909\right ) (1-2 x)^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/624*((51909 - 72358*(1 - 2*x) + 39780*(1 - 2*x)^2 - 10050*(1 - 2*x)^3 + 975*(1 - 2*x)^4)*(1 - 2*x)^(7/2))

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fricas [A]  time = 0.97, size = 44, normalized size = 0.67 \begin {gather*} \frac {1}{39} \, {\left (7800 \, x^{7} + 12900 \, x^{6} - 90 \, x^{5} - 9917 \, x^{4} - 3299 \, x^{3} + 2724 \, x^{2} + 1465 \, x - 641\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)*(3+5*x)^3,x, algorithm="fricas")

[Out]

1/39*(7800*x^7 + 12900*x^6 - 90*x^5 - 9917*x^4 - 3299*x^3 + 2724*x^2 + 1465*x - 641)*sqrt(-2*x + 1)

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giac [A]  time = 1.04, size = 81, normalized size = 1.23 \begin {gather*} \frac {25}{16} \, {\left (2 \, x - 1\right )}^{7} \sqrt {-2 \, x + 1} + \frac {1675}{104} \, {\left (2 \, x - 1\right )}^{6} \sqrt {-2 \, x + 1} + \frac {255}{4} \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + \frac {2783}{24} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {1331}{16} \, {\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)*(3+5*x)^3,x, algorithm="giac")

[Out]

25/16*(2*x - 1)^7*sqrt(-2*x + 1) + 1675/104*(2*x - 1)^6*sqrt(-2*x + 1) + 255/4*(2*x - 1)^5*sqrt(-2*x + 1) + 27
83/24*(2*x - 1)^4*sqrt(-2*x + 1) + 1331/16*(2*x - 1)^3*sqrt(-2*x + 1)

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maple [A]  time = 0.00, size = 30, normalized size = 0.45 \begin {gather*} -\frac {\left (975 x^{4}+3075 x^{3}+3870 x^{2}+2381 x +641\right ) \left (-2 x +1\right )^{\frac {7}{2}}}{39} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/39*(975*x^4+3075*x^3+3870*x^2+2381*x+641)*(-2*x+1)^(7/2)

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maxima [A]  time = 0.58, size = 46, normalized size = 0.70 \begin {gather*} -\frac {25}{16} \, {\left (-2 \, x + 1\right )}^{\frac {15}{2}} + \frac {1675}{104} \, {\left (-2 \, x + 1\right )}^{\frac {13}{2}} - \frac {255}{4} \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} + \frac {2783}{24} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {1331}{16} \, {\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)*(3+5*x)^3,x, algorithm="maxima")

[Out]

-25/16*(-2*x + 1)^(15/2) + 1675/104*(-2*x + 1)^(13/2) - 255/4*(-2*x + 1)^(11/2) + 2783/24*(-2*x + 1)^(9/2) - 1
331/16*(-2*x + 1)^(7/2)

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mupad [B]  time = 0.02, size = 46, normalized size = 0.70 \begin {gather*} \frac {2783\,{\left (1-2\,x\right )}^{9/2}}{24}-\frac {1331\,{\left (1-2\,x\right )}^{7/2}}{16}-\frac {255\,{\left (1-2\,x\right )}^{11/2}}{4}+\frac {1675\,{\left (1-2\,x\right )}^{13/2}}{104}-\frac {25\,{\left (1-2\,x\right )}^{15/2}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2783*(1 - 2*x)^(9/2))/24 - (1331*(1 - 2*x)^(7/2))/16 - (255*(1 - 2*x)^(11/2))/4 + (1675*(1 - 2*x)^(13/2))/104
- (25*(1 - 2*x)^(15/2))/16

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sympy [A]  time = 16.93, size = 58, normalized size = 0.88 \begin {gather*} - \frac {25 \left (1 - 2 x\right )^{\frac {15}{2}}}{16} + \frac {1675 \left (1 - 2 x\right )^{\frac {13}{2}}}{104} - \frac {255 \left (1 - 2 x\right )^{\frac {11}{2}}}{4} + \frac {2783 \left (1 - 2 x\right )^{\frac {9}{2}}}{24} - \frac {1331 \left (1 - 2 x\right )^{\frac {7}{2}}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25*(1 - 2*x)**(15/2)/16 + 1675*(1 - 2*x)**(13/2)/104 - 255*(1 - 2*x)**(11/2)/4 + 2783*(1 - 2*x)**(9/2)/24 - 1
331*(1 - 2*x)**(7/2)/16

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