3.1944 \(\int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^2 \, dx\)

Optimal. Leaf size=66 \[ -\frac{15}{16} (1-2 x)^{15/2}+\frac{255}{26} (1-2 x)^{13/2}-\frac{3467}{88} (1-2 x)^{11/2}+\frac{1309}{18} (1-2 x)^{9/2}-\frac{847}{16} (1-2 x)^{7/2} \]

[Out]

(-847*(1 - 2*x)^(7/2))/16 + (1309*(1 - 2*x)^(9/2))/18 - (3467*(1 - 2*x)^(11/2))/88 + (255*(1 - 2*x)^(13/2))/26
- (15*(1 - 2*x)^(15/2))/16

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Rubi [A]  time = 0.0143686, antiderivative size = 66, normalized size of antiderivative = 1., 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} \[ -\frac{15}{16} (1-2 x)^{15/2}+\frac{255}{26} (1-2 x)^{13/2}-\frac{3467}{88} (1-2 x)^{11/2}+\frac{1309}{18} (1-2 x)^{9/2}-\frac{847}{16} (1-2 x)^{7/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-847*(1 - 2*x)^(7/2))/16 + (1309*(1 - 2*x)^(9/2))/18 - (3467*(1 - 2*x)^(11/2))/88 + (255*(1 - 2*x)^(13/2))/26
- (15*(1 - 2*x)^(15/2))/16

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 (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^2 \, dx &=\int \left (\frac{5929}{16} (1-2 x)^{5/2}-\frac{1309}{2} (1-2 x)^{7/2}+\frac{3467}{8} (1-2 x)^{9/2}-\frac{255}{2} (1-2 x)^{11/2}+\frac{225}{16} (1-2 x)^{13/2}\right ) \, dx\\ &=-\frac{847}{16} (1-2 x)^{7/2}+\frac{1309}{18} (1-2 x)^{9/2}-\frac{3467}{88} (1-2 x)^{11/2}+\frac{255}{26} (1-2 x)^{13/2}-\frac{15}{16} (1-2 x)^{15/2}\\ \end{align*}

Mathematica [A]  time = 0.0164471, size = 33, normalized size = 0.5 \[ -\frac{(1-2 x)^{7/2} \left (19305 x^4+62370 x^3+80307 x^2+50450 x+13826\right )}{1287} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(7/2)*(13826 + 50450*x + 80307*x^2 + 62370*x^3 + 19305*x^4))/1287

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Maple [A]  time = 0.003, size = 30, normalized size = 0.5 \begin{align*} -{\frac{19305\,{x}^{4}+62370\,{x}^{3}+80307\,{x}^{2}+50450\,x+13826}{1287} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1287*(19305*x^4+62370*x^3+80307*x^2+50450*x+13826)*(1-2*x)^(7/2)

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Maxima [A]  time = 2.07266, size = 62, normalized size = 0.94 \begin{align*} -\frac{15}{16} \,{\left (-2 \, x + 1\right )}^{\frac{15}{2}} + \frac{255}{26} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - \frac{3467}{88} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{1309}{18} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{847}{16} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-15/16*(-2*x + 1)^(15/2) + 255/26*(-2*x + 1)^(13/2) - 3467/88*(-2*x + 1)^(11/2) + 1309/18*(-2*x + 1)^(9/2) - 8
47/16*(-2*x + 1)^(7/2)

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Fricas [A]  time = 1.44721, size = 155, normalized size = 2.35 \begin{align*} \frac{1}{1287} \,{\left (154440 \, x^{7} + 267300 \, x^{6} + 9846 \, x^{5} - 205169 \, x^{4} - 75320 \, x^{3} + 56481 \, x^{2} + 32506 \, x - 13826\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1287*(154440*x^7 + 267300*x^6 + 9846*x^5 - 205169*x^4 - 75320*x^3 + 56481*x^2 + 32506*x - 13826)*sqrt(-2*x +
1)

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Sympy [A]  time = 13.3218, size = 58, normalized size = 0.88 \begin{align*} - \frac{15 \left (1 - 2 x\right )^{\frac{15}{2}}}{16} + \frac{255 \left (1 - 2 x\right )^{\frac{13}{2}}}{26} - \frac{3467 \left (1 - 2 x\right )^{\frac{11}{2}}}{88} + \frac{1309 \left (1 - 2 x\right )^{\frac{9}{2}}}{18} - \frac{847 \left (1 - 2 x\right )^{\frac{7}{2}}}{16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-15*(1 - 2*x)**(15/2)/16 + 255*(1 - 2*x)**(13/2)/26 - 3467*(1 - 2*x)**(11/2)/88 + 1309*(1 - 2*x)**(9/2)/18 - 8
47*(1 - 2*x)**(7/2)/16

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Giac [A]  time = 2.79938, size = 109, normalized size = 1.65 \begin{align*} \frac{15}{16} \,{\left (2 \, x - 1\right )}^{7} \sqrt{-2 \, x + 1} + \frac{255}{26} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + \frac{3467}{88} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{1309}{18} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{847}{16} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

15/16*(2*x - 1)^7*sqrt(-2*x + 1) + 255/26*(2*x - 1)^6*sqrt(-2*x + 1) + 3467/88*(2*x - 1)^5*sqrt(-2*x + 1) + 13
09/18*(2*x - 1)^4*sqrt(-2*x + 1) + 847/16*(2*x - 1)^3*sqrt(-2*x + 1)