3.1957 \(\int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^3 \, dx\)

Optimal. Leaf size=79 \[ \frac{1125}{544} (1-2 x)^{17/2}-\frac{845}{32} (1-2 x)^{15/2}+\frac{28555}{208} (1-2 x)^{13/2}-\frac{5847}{16} (1-2 x)^{11/2}+\frac{16093}{32} (1-2 x)^{9/2}-\frac{9317}{32} (1-2 x)^{7/2} \]

[Out]

(-9317*(1 - 2*x)^(7/2))/32 + (16093*(1 - 2*x)^(9/2))/32 - (5847*(1 - 2*x)^(11/2))/16 + (28555*(1 - 2*x)^(13/2)
)/208 - (845*(1 - 2*x)^(15/2))/32 + (1125*(1 - 2*x)^(17/2))/544

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Rubi [A]  time = 0.0158904, antiderivative size = 79, 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} \[ \frac{1125}{544} (1-2 x)^{17/2}-\frac{845}{32} (1-2 x)^{15/2}+\frac{28555}{208} (1-2 x)^{13/2}-\frac{5847}{16} (1-2 x)^{11/2}+\frac{16093}{32} (1-2 x)^{9/2}-\frac{9317}{32} (1-2 x)^{7/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-9317*(1 - 2*x)^(7/2))/32 + (16093*(1 - 2*x)^(9/2))/32 - (5847*(1 - 2*x)^(11/2))/16 + (28555*(1 - 2*x)^(13/2)
)/208 - (845*(1 - 2*x)^(15/2))/32 + (1125*(1 - 2*x)^(17/2))/544

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)^3 \, dx &=\int \left (\frac{65219}{32} (1-2 x)^{5/2}-\frac{144837}{32} (1-2 x)^{7/2}+\frac{64317}{16} (1-2 x)^{9/2}-\frac{28555}{16} (1-2 x)^{11/2}+\frac{12675}{32} (1-2 x)^{13/2}-\frac{1125}{32} (1-2 x)^{15/2}\right ) \, dx\\ &=-\frac{9317}{32} (1-2 x)^{7/2}+\frac{16093}{32} (1-2 x)^{9/2}-\frac{5847}{16} (1-2 x)^{11/2}+\frac{28555}{208} (1-2 x)^{13/2}-\frac{845}{32} (1-2 x)^{15/2}+\frac{1125}{544} (1-2 x)^{17/2}\\ \end{align*}

Mathematica [A]  time = 0.0189067, size = 38, normalized size = 0.48 \[ -\frac{1}{221} (1-2 x)^{7/2} \left (14625 x^5+56810 x^4+92535 x^3+80748 x^2+39160 x+9004\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((1 - 2*x)^(7/2)*(9004 + 39160*x + 80748*x^2 + 92535*x^3 + 56810*x^4 + 14625*x^5))/221

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Maple [A]  time = 0.002, size = 35, normalized size = 0.4 \begin{align*} -{\frac{14625\,{x}^{5}+56810\,{x}^{4}+92535\,{x}^{3}+80748\,{x}^{2}+39160\,x+9004}{221} \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)^3,x)

[Out]

-1/221*(14625*x^5+56810*x^4+92535*x^3+80748*x^2+39160*x+9004)*(1-2*x)^(7/2)

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Maxima [A]  time = 3.33019, size = 74, normalized size = 0.94 \begin{align*} \frac{1125}{544} \,{\left (-2 \, x + 1\right )}^{\frac{17}{2}} - \frac{845}{32} \,{\left (-2 \, x + 1\right )}^{\frac{15}{2}} + \frac{28555}{208} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - \frac{5847}{16} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{16093}{32} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{9317}{32} \,{\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)^3,x, algorithm="maxima")

[Out]

1125/544*(-2*x + 1)^(17/2) - 845/32*(-2*x + 1)^(15/2) + 28555/208*(-2*x + 1)^(13/2) - 5847/16*(-2*x + 1)^(11/2
) + 16093/32*(-2*x + 1)^(9/2) - 9317/32*(-2*x + 1)^(7/2)

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Fricas [A]  time = 1.34214, size = 171, normalized size = 2.16 \begin{align*} \frac{1}{221} \,{\left (117000 \, x^{8} + 278980 \, x^{7} + 146310 \, x^{6} - 138201 \, x^{5} - 157296 \, x^{4} - 5935 \, x^{3} + 46164 \, x^{2} + 14864 \, x - 9004\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)^3,x, algorithm="fricas")

[Out]

1/221*(117000*x^8 + 278980*x^7 + 146310*x^6 - 138201*x^5 - 157296*x^4 - 5935*x^3 + 46164*x^2 + 14864*x - 9004)
*sqrt(-2*x + 1)

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Sympy [A]  time = 16.1935, size = 70, normalized size = 0.89 \begin{align*} \frac{1125 \left (1 - 2 x\right )^{\frac{17}{2}}}{544} - \frac{845 \left (1 - 2 x\right )^{\frac{15}{2}}}{32} + \frac{28555 \left (1 - 2 x\right )^{\frac{13}{2}}}{208} - \frac{5847 \left (1 - 2 x\right )^{\frac{11}{2}}}{16} + \frac{16093 \left (1 - 2 x\right )^{\frac{9}{2}}}{32} - \frac{9317 \left (1 - 2 x\right )^{\frac{7}{2}}}{32} \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)**3,x)

[Out]

1125*(1 - 2*x)**(17/2)/544 - 845*(1 - 2*x)**(15/2)/32 + 28555*(1 - 2*x)**(13/2)/208 - 5847*(1 - 2*x)**(11/2)/1
6 + 16093*(1 - 2*x)**(9/2)/32 - 9317*(1 - 2*x)**(7/2)/32

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Giac [A]  time = 2.73069, size = 131, normalized size = 1.66 \begin{align*} \frac{1125}{544} \,{\left (2 \, x - 1\right )}^{8} \sqrt{-2 \, x + 1} + \frac{845}{32} \,{\left (2 \, x - 1\right )}^{7} \sqrt{-2 \, x + 1} + \frac{28555}{208} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + \frac{5847}{16} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{16093}{32} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{9317}{32} \,{\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)^3,x, algorithm="giac")

[Out]

1125/544*(2*x - 1)^8*sqrt(-2*x + 1) + 845/32*(2*x - 1)^7*sqrt(-2*x + 1) + 28555/208*(2*x - 1)^6*sqrt(-2*x + 1)
+ 5847/16*(2*x - 1)^5*sqrt(-2*x + 1) + 16093/32*(2*x - 1)^4*sqrt(-2*x + 1) + 9317/32*(2*x - 1)^3*sqrt(-2*x +
1)