3.1883 \(\int (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^3 \, dx\)

Optimal. Leaf size=92 \[ -\frac{3375 (1-2 x)^{17/2}}{1088}+\frac{765}{16} (1-2 x)^{15/2}-\frac{260055}{832} (1-2 x)^{13/2}+\frac{98209}{88} (1-2 x)^{11/2}-\frac{444983}{192} (1-2 x)^{9/2}+\frac{43197}{16} (1-2 x)^{7/2}-\frac{456533}{320} (1-2 x)^{5/2} \]

[Out]

(-456533*(1 - 2*x)^(5/2))/320 + (43197*(1 - 2*x)^(7/2))/16 - (444983*(1 - 2*x)^(9/2))/192 + (98209*(1 - 2*x)^(
11/2))/88 - (260055*(1 - 2*x)^(13/2))/832 + (765*(1 - 2*x)^(15/2))/16 - (3375*(1 - 2*x)^(17/2))/1088

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Rubi [A]  time = 0.0164627, antiderivative size = 92, 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} \[ -\frac{3375 (1-2 x)^{17/2}}{1088}+\frac{765}{16} (1-2 x)^{15/2}-\frac{260055}{832} (1-2 x)^{13/2}+\frac{98209}{88} (1-2 x)^{11/2}-\frac{444983}{192} (1-2 x)^{9/2}+\frac{43197}{16} (1-2 x)^{7/2}-\frac{456533}{320} (1-2 x)^{5/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-456533*(1 - 2*x)^(5/2))/320 + (43197*(1 - 2*x)^(7/2))/16 - (444983*(1 - 2*x)^(9/2))/192 + (98209*(1 - 2*x)^(
11/2))/88 - (260055*(1 - 2*x)^(13/2))/832 + (765*(1 - 2*x)^(15/2))/16 - (3375*(1 - 2*x)^(17/2))/1088

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)^{3/2} (2+3 x)^3 (3+5 x)^3 \, dx &=\int \left (\frac{456533}{64} (1-2 x)^{3/2}-\frac{302379}{16} (1-2 x)^{5/2}+\frac{1334949}{64} (1-2 x)^{7/2}-\frac{98209}{8} (1-2 x)^{9/2}+\frac{260055}{64} (1-2 x)^{11/2}-\frac{11475}{16} (1-2 x)^{13/2}+\frac{3375}{64} (1-2 x)^{15/2}\right ) \, dx\\ &=-\frac{456533}{320} (1-2 x)^{5/2}+\frac{43197}{16} (1-2 x)^{7/2}-\frac{444983}{192} (1-2 x)^{9/2}+\frac{98209}{88} (1-2 x)^{11/2}-\frac{260055}{832} (1-2 x)^{13/2}+\frac{765}{16} (1-2 x)^{15/2}-\frac{3375 (1-2 x)^{17/2}}{1088}\\ \end{align*}

Mathematica [A]  time = 0.0203421, size = 43, normalized size = 0.47 \[ -\frac{(1-2 x)^{5/2} \left (7239375 x^6+34073325 x^5+70032600 x^4+82215885 x^3+60296725 x^2+27917090 x+7158706\right )}{36465} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(5/2)*(7158706 + 27917090*x + 60296725*x^2 + 82215885*x^3 + 70032600*x^4 + 34073325*x^5 + 7239375*
x^6))/36465

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Maple [A]  time = 0.003, size = 40, normalized size = 0.4 \begin{align*} -{\frac{7239375\,{x}^{6}[phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+7158706}{36465} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36465*(7239375*x^6+34073325*x^5+70032600*x^4+82215885*x^3+60296725*x^2+27917090*x+7158706)*(1-2*x)^(5/2)

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Maxima [A]  time = 1.14006, size = 86, normalized size = 0.93 \begin{align*} -\frac{3375}{1088} \,{\left (-2 \, x + 1\right )}^{\frac{17}{2}} + \frac{765}{16} \,{\left (-2 \, x + 1\right )}^{\frac{15}{2}} - \frac{260055}{832} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} + \frac{98209}{88} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{444983}{192} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{43197}{16} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{456533}{320} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3375/1088*(-2*x + 1)^(17/2) + 765/16*(-2*x + 1)^(15/2) - 260055/832*(-2*x + 1)^(13/2) + 98209/88*(-2*x + 1)^(
11/2) - 444983/192*(-2*x + 1)^(9/2) + 43197/16*(-2*x + 1)^(7/2) - 456533/320*(-2*x + 1)^(5/2)

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Fricas [A]  time = 1.35002, size = 207, normalized size = 2.25 \begin{align*} -\frac{1}{36465} \,{\left (28957500 \, x^{8} + 107335800 \, x^{7} + 151076475 \, x^{6} + 82806465 \, x^{5} - 17644040 \, x^{4} - 47302655 \, x^{3} - 22736811 \, x^{2} - 717734 \, x + 7158706\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36465*(28957500*x^8 + 107335800*x^7 + 151076475*x^6 + 82806465*x^5 - 17644040*x^4 - 47302655*x^3 - 22736811
*x^2 - 717734*x + 7158706)*sqrt(-2*x + 1)

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Sympy [A]  time = 15.8586, size = 82, normalized size = 0.89 \begin{align*} - \frac{3375 \left (1 - 2 x\right )^{\frac{17}{2}}}{1088} + \frac{765 \left (1 - 2 x\right )^{\frac{15}{2}}}{16} - \frac{260055 \left (1 - 2 x\right )^{\frac{13}{2}}}{832} + \frac{98209 \left (1 - 2 x\right )^{\frac{11}{2}}}{88} - \frac{444983 \left (1 - 2 x\right )^{\frac{9}{2}}}{192} + \frac{43197 \left (1 - 2 x\right )^{\frac{7}{2}}}{16} - \frac{456533 \left (1 - 2 x\right )^{\frac{5}{2}}}{320} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3375*(1 - 2*x)**(17/2)/1088 + 765*(1 - 2*x)**(15/2)/16 - 260055*(1 - 2*x)**(13/2)/832 + 98209*(1 - 2*x)**(11/
2)/88 - 444983*(1 - 2*x)**(9/2)/192 + 43197*(1 - 2*x)**(7/2)/16 - 456533*(1 - 2*x)**(5/2)/320

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Giac [A]  time = 1.7957, size = 153, normalized size = 1.66 \begin{align*} -\frac{3375}{1088} \,{\left (2 \, x - 1\right )}^{8} \sqrt{-2 \, x + 1} - \frac{765}{16} \,{\left (2 \, x - 1\right )}^{7} \sqrt{-2 \, x + 1} - \frac{260055}{832} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} - \frac{98209}{88} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{444983}{192} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{43197}{16} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{456533}{320} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3375/1088*(2*x - 1)^8*sqrt(-2*x + 1) - 765/16*(2*x - 1)^7*sqrt(-2*x + 1) - 260055/832*(2*x - 1)^6*sqrt(-2*x +
1) - 98209/88*(2*x - 1)^5*sqrt(-2*x + 1) - 444983/192*(2*x - 1)^4*sqrt(-2*x + 1) - 43197/16*(2*x - 1)^3*sqrt(
-2*x + 1) - 456533/320*(2*x - 1)^2*sqrt(-2*x + 1)