3.1815 \(\int \sqrt{1-2 x} (2+3 x)^4 (3+5 x)^3 \, dx\)

Optimal. Leaf size=105 \[ \frac{10125 (1-2 x)^{17/2}}{2176}-\frac{10755}{128} (1-2 x)^{15/2}+\frac{1101465 (1-2 x)^{13/2}}{1664}-\frac{4177401 (1-2 x)^{11/2}}{1408}+\frac{9504551 (1-2 x)^{9/2}}{1152}-\frac{1853313}{128} (1-2 x)^{7/2}+\frac{9836211}{640} (1-2 x)^{5/2}-\frac{3195731}{384} (1-2 x)^{3/2} \]

[Out]

(-3195731*(1 - 2*x)^(3/2))/384 + (9836211*(1 - 2*x)^(5/2))/640 - (1853313*(1 - 2*x)^(7/2))/128 + (9504551*(1 -
2*x)^(9/2))/1152 - (4177401*(1 - 2*x)^(11/2))/1408 + (1101465*(1 - 2*x)^(13/2))/1664 - (10755*(1 - 2*x)^(15/2
))/128 + (10125*(1 - 2*x)^(17/2))/2176

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Rubi [A]  time = 0.0191593, antiderivative size = 105, 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} \[ \frac{10125 (1-2 x)^{17/2}}{2176}-\frac{10755}{128} (1-2 x)^{15/2}+\frac{1101465 (1-2 x)^{13/2}}{1664}-\frac{4177401 (1-2 x)^{11/2}}{1408}+\frac{9504551 (1-2 x)^{9/2}}{1152}-\frac{1853313}{128} (1-2 x)^{7/2}+\frac{9836211}{640} (1-2 x)^{5/2}-\frac{3195731}{384} (1-2 x)^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]*(2 + 3*x)^4*(3 + 5*x)^3,x]

[Out]

(-3195731*(1 - 2*x)^(3/2))/384 + (9836211*(1 - 2*x)^(5/2))/640 - (1853313*(1 - 2*x)^(7/2))/128 + (9504551*(1 -
2*x)^(9/2))/1152 - (4177401*(1 - 2*x)^(11/2))/1408 + (1101465*(1 - 2*x)^(13/2))/1664 - (10755*(1 - 2*x)^(15/2
))/128 + (10125*(1 - 2*x)^(17/2))/2176

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 \sqrt{1-2 x} (2+3 x)^4 (3+5 x)^3 \, dx &=\int \left (\frac{3195731}{128} \sqrt{1-2 x}-\frac{9836211}{128} (1-2 x)^{3/2}+\frac{12973191}{128} (1-2 x)^{5/2}-\frac{9504551}{128} (1-2 x)^{7/2}+\frac{4177401}{128} (1-2 x)^{9/2}-\frac{1101465}{128} (1-2 x)^{11/2}+\frac{161325}{128} (1-2 x)^{13/2}-\frac{10125}{128} (1-2 x)^{15/2}\right ) \, dx\\ &=-\frac{3195731}{384} (1-2 x)^{3/2}+\frac{9836211}{640} (1-2 x)^{5/2}-\frac{1853313}{128} (1-2 x)^{7/2}+\frac{9504551 (1-2 x)^{9/2}}{1152}-\frac{4177401 (1-2 x)^{11/2}}{1408}+\frac{1101465 (1-2 x)^{13/2}}{1664}-\frac{10755}{128} (1-2 x)^{15/2}+\frac{10125 (1-2 x)^{17/2}}{2176}\\ \end{align*}

Mathematica [A]  time = 0.0204795, size = 48, normalized size = 0.46 \[ -\frac{(1-2 x)^{3/2} \left (65154375 x^7+360231300 x^6+894452625 x^5+1320982290 x^4+1299289000 x^3+906777120 x^2+466679856 x+171312832\right )}{109395} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - 2*x]*(2 + 3*x)^4*(3 + 5*x)^3,x]

[Out]

-((1 - 2*x)^(3/2)*(171312832 + 466679856*x + 906777120*x^2 + 1299289000*x^3 + 1320982290*x^4 + 894452625*x^5 +
360231300*x^6 + 65154375*x^7))/109395

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Maple [A]  time = 0.003, size = 45, normalized size = 0.4 \begin{align*} -{\frac{65154375\,{x}^{7}[phone]\,{x}^{6}[phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+171312832}{109395} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^4*(3+5*x)^3*(1-2*x)^(1/2),x)

[Out]

-1/109395*(65154375*x^7+360231300*x^6+894452625*x^5+1320982290*x^4+1299289000*x^3+906777120*x^2+466679856*x+17
1312832)*(1-2*x)^(3/2)

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Maxima [A]  time = 1.86946, size = 99, normalized size = 0.94 \begin{align*} \frac{10125}{2176} \,{\left (-2 \, x + 1\right )}^{\frac{17}{2}} - \frac{10755}{128} \,{\left (-2 \, x + 1\right )}^{\frac{15}{2}} + \frac{1101465}{1664} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} - \frac{4177401}{1408} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} + \frac{9504551}{1152} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - \frac{1853313}{128} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + \frac{9836211}{640} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{3195731}{384} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4*(3+5*x)^3*(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

10125/2176*(-2*x + 1)^(17/2) - 10755/128*(-2*x + 1)^(15/2) + 1101465/1664*(-2*x + 1)^(13/2) - 4177401/1408*(-2
*x + 1)^(11/2) + 9504551/1152*(-2*x + 1)^(9/2) - 1853313/128*(-2*x + 1)^(7/2) + 9836211/640*(-2*x + 1)^(5/2) -
3195731/384*(-2*x + 1)^(3/2)

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Fricas [A]  time = 1.58162, size = 223, normalized size = 2.12 \begin{align*} \frac{1}{109395} \,{\left (130308750 \, x^{8} + 655308225 \, x^{7} + 1428673950 \, x^{6} + 1747511955 \, x^{5} + 1277595710 \, x^{4} + 514265240 \, x^{3} + 26582592 \, x^{2} - 124054192 \, x - 171312832\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4*(3+5*x)^3*(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/109395*(130308750*x^8 + 655308225*x^7 + 1428673950*x^6 + 1747511955*x^5 + 1277595710*x^4 + 514265240*x^3 + 2
6582592*x^2 - 124054192*x - 171312832)*sqrt(-2*x + 1)

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Sympy [A]  time = 2.56541, size = 94, normalized size = 0.9 \begin{align*} \frac{10125 \left (1 - 2 x\right )^{\frac{17}{2}}}{2176} - \frac{10755 \left (1 - 2 x\right )^{\frac{15}{2}}}{128} + \frac{1101465 \left (1 - 2 x\right )^{\frac{13}{2}}}{1664} - \frac{4177401 \left (1 - 2 x\right )^{\frac{11}{2}}}{1408} + \frac{9504551 \left (1 - 2 x\right )^{\frac{9}{2}}}{1152} - \frac{1853313 \left (1 - 2 x\right )^{\frac{7}{2}}}{128} + \frac{9836211 \left (1 - 2 x\right )^{\frac{5}{2}}}{640} - \frac{3195731 \left (1 - 2 x\right )^{\frac{3}{2}}}{384} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**4*(3+5*x)**3*(1-2*x)**(1/2),x)

[Out]

10125*(1 - 2*x)**(17/2)/2176 - 10755*(1 - 2*x)**(15/2)/128 + 1101465*(1 - 2*x)**(13/2)/1664 - 4177401*(1 - 2*x
)**(11/2)/1408 + 9504551*(1 - 2*x)**(9/2)/1152 - 1853313*(1 - 2*x)**(7/2)/128 + 9836211*(1 - 2*x)**(5/2)/640 -
3195731*(1 - 2*x)**(3/2)/384

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Giac [A]  time = 2.07102, size = 165, normalized size = 1.57 \begin{align*} \frac{10125}{2176} \,{\left (2 \, x - 1\right )}^{8} \sqrt{-2 \, x + 1} + \frac{10755}{128} \,{\left (2 \, x - 1\right )}^{7} \sqrt{-2 \, x + 1} + \frac{1101465}{1664} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} + \frac{4177401}{1408} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} + \frac{9504551}{1152} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + \frac{1853313}{128} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + \frac{9836211}{640} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{3195731}{384} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4*(3+5*x)^3*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

10125/2176*(2*x - 1)^8*sqrt(-2*x + 1) + 10755/128*(2*x - 1)^7*sqrt(-2*x + 1) + 1101465/1664*(2*x - 1)^6*sqrt(-
2*x + 1) + 4177401/1408*(2*x - 1)^5*sqrt(-2*x + 1) + 9504551/1152*(2*x - 1)^4*sqrt(-2*x + 1) + 1853313/128*(2*
x - 1)^3*sqrt(-2*x + 1) + 9836211/640*(2*x - 1)^2*sqrt(-2*x + 1) - 3195731/384*(-2*x + 1)^(3/2)