3.1872 \(\int (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2 \, dx\)

Optimal. Leaf size=66 \[ -\frac{225}{208} (1-2 x)^{13/2}+\frac{255}{22} (1-2 x)^{11/2}-\frac{3467}{72} (1-2 x)^{9/2}+\frac{187}{2} (1-2 x)^{7/2}-\frac{5929}{80} (1-2 x)^{5/2} \]

[Out]

(-5929*(1 - 2*x)^(5/2))/80 + (187*(1 - 2*x)^(7/2))/2 - (3467*(1 - 2*x)^(9/2))/72 + (255*(1 - 2*x)^(11/2))/22 -
(225*(1 - 2*x)^(13/2))/208

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Rubi [A]  time = 0.013116, 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} \[ -\frac{225}{208} (1-2 x)^{13/2}+\frac{255}{22} (1-2 x)^{11/2}-\frac{3467}{72} (1-2 x)^{9/2}+\frac{187}{2} (1-2 x)^{7/2}-\frac{5929}{80} (1-2 x)^{5/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5929*(1 - 2*x)^(5/2))/80 + (187*(1 - 2*x)^(7/2))/2 - (3467*(1 - 2*x)^(9/2))/72 + (255*(1 - 2*x)^(11/2))/22 -
(225*(1 - 2*x)^(13/2))/208

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)^2 (3+5 x)^2 \, dx &=\int \left (\frac{5929}{16} (1-2 x)^{3/2}-\frac{1309}{2} (1-2 x)^{5/2}+\frac{3467}{8} (1-2 x)^{7/2}-\frac{255}{2} (1-2 x)^{9/2}+\frac{225}{16} (1-2 x)^{11/2}\right ) \, dx\\ &=-\frac{5929}{80} (1-2 x)^{5/2}+\frac{187}{2} (1-2 x)^{7/2}-\frac{3467}{72} (1-2 x)^{9/2}+\frac{255}{22} (1-2 x)^{11/2}-\frac{225}{208} (1-2 x)^{13/2}\\ \end{align*}

Mathematica [A]  time = 0.0153248, size = 33, normalized size = 0.5 \[ -\frac{(1-2 x)^{5/2} \left (111375 x^4+373950 x^3+511465 x^2+355730 x+117478\right )}{6435} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(5/2)*(117478 + 355730*x + 511465*x^2 + 373950*x^3 + 111375*x^4))/6435

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Maple [A]  time = 0.004, size = 30, normalized size = 0.5 \begin{align*} -{\frac{111375\,{x}^{4}+373950\,{x}^{3}+511465\,{x}^{2}+355730\,x+117478}{6435} \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)^2*(3+5*x)^2,x)

[Out]

-1/6435*(111375*x^4+373950*x^3+511465*x^2+355730*x+117478)*(1-2*x)^(5/2)

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Maxima [A]  time = 1.03509, size = 62, normalized size = 0.94 \begin{align*} -\frac{225}{208} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} + \frac{255}{22} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{3467}{72} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{187}{2} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{5929}{80} \,{\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)^2*(3+5*x)^2,x, algorithm="maxima")

[Out]

-225/208*(-2*x + 1)^(13/2) + 255/22*(-2*x + 1)^(11/2) - 3467/72*(-2*x + 1)^(9/2) + 187/2*(-2*x + 1)^(7/2) - 59
29/80*(-2*x + 1)^(5/2)

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Fricas [A]  time = 1.35767, size = 149, normalized size = 2.26 \begin{align*} -\frac{1}{6435} \,{\left (445500 \, x^{6} + 1050300 \, x^{5} + 661435 \, x^{4} - 248990 \, x^{3} - 441543 \, x^{2} - 114182 \, x + 117478\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)^2*(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/6435*(445500*x^6 + 1050300*x^5 + 661435*x^4 - 248990*x^3 - 441543*x^2 - 114182*x + 117478)*sqrt(-2*x + 1)

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Sympy [A]  time = 9.99064, size = 58, normalized size = 0.88 \begin{align*} - \frac{225 \left (1 - 2 x\right )^{\frac{13}{2}}}{208} + \frac{255 \left (1 - 2 x\right )^{\frac{11}{2}}}{22} - \frac{3467 \left (1 - 2 x\right )^{\frac{9}{2}}}{72} + \frac{187 \left (1 - 2 x\right )^{\frac{7}{2}}}{2} - \frac{5929 \left (1 - 2 x\right )^{\frac{5}{2}}}{80} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-225*(1 - 2*x)**(13/2)/208 + 255*(1 - 2*x)**(11/2)/22 - 3467*(1 - 2*x)**(9/2)/72 + 187*(1 - 2*x)**(7/2)/2 - 59
29*(1 - 2*x)**(5/2)/80

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Giac [A]  time = 1.93236, size = 109, normalized size = 1.65 \begin{align*} -\frac{225}{208} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} - \frac{255}{22} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{3467}{72} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{187}{2} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{5929}{80} \,{\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)^2*(3+5*x)^2,x, algorithm="giac")

[Out]

-225/208*(2*x - 1)^6*sqrt(-2*x + 1) - 255/22*(2*x - 1)^5*sqrt(-2*x + 1) - 3467/72*(2*x - 1)^4*sqrt(-2*x + 1) -
187/2*(2*x - 1)^3*sqrt(-2*x + 1) - 5929/80*(2*x - 1)^2*sqrt(-2*x + 1)