3.2024 \(\int \frac{(2+3 x)^4 (3+5 x)^3}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=105 \[ \frac{675}{128} (1-2 x)^{15/2}-\frac{161325 (1-2 x)^{13/2}}{1664}+\frac{1101465 (1-2 x)^{11/2}}{1408}-\frac{1392467}{384} (1-2 x)^{9/2}+\frac{1357793}{128} (1-2 x)^{7/2}-\frac{12973191}{640} (1-2 x)^{5/2}+\frac{3278737}{128} (1-2 x)^{3/2}-\frac{3195731}{128} \sqrt{1-2 x} \]

[Out]

(-3195731*Sqrt[1 - 2*x])/128 + (3278737*(1 - 2*x)^(3/2))/128 - (12973191*(1 - 2*x)^(5/2))/640 + (1357793*(1 -
2*x)^(7/2))/128 - (1392467*(1 - 2*x)^(9/2))/384 + (1101465*(1 - 2*x)^(11/2))/1408 - (161325*(1 - 2*x)^(13/2))/
1664 + (675*(1 - 2*x)^(15/2))/128

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Rubi [A]  time = 0.0183953, 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} \[ \frac{675}{128} (1-2 x)^{15/2}-\frac{161325 (1-2 x)^{13/2}}{1664}+\frac{1101465 (1-2 x)^{11/2}}{1408}-\frac{1392467}{384} (1-2 x)^{9/2}+\frac{1357793}{128} (1-2 x)^{7/2}-\frac{12973191}{640} (1-2 x)^{5/2}+\frac{3278737}{128} (1-2 x)^{3/2}-\frac{3195731}{128} \sqrt{1-2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3195731*Sqrt[1 - 2*x])/128 + (3278737*(1 - 2*x)^(3/2))/128 - (12973191*(1 - 2*x)^(5/2))/640 + (1357793*(1 -
2*x)^(7/2))/128 - (1392467*(1 - 2*x)^(9/2))/384 + (1101465*(1 - 2*x)^(11/2))/1408 - (161325*(1 - 2*x)^(13/2))/
1664 + (675*(1 - 2*x)^(15/2))/128

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 \frac{(2+3 x)^4 (3+5 x)^3}{\sqrt{1-2 x}} \, dx &=\int \left (\frac{3195731}{128 \sqrt{1-2 x}}-\frac{9836211}{128} \sqrt{1-2 x}+\frac{12973191}{128} (1-2 x)^{3/2}-\frac{9504551}{128} (1-2 x)^{5/2}+\frac{4177401}{128} (1-2 x)^{7/2}-\frac{1101465}{128} (1-2 x)^{9/2}+\frac{161325}{128} (1-2 x)^{11/2}-\frac{10125}{128} (1-2 x)^{13/2}\right ) \, dx\\ &=-\frac{3195731}{128} \sqrt{1-2 x}+\frac{3278737}{128} (1-2 x)^{3/2}-\frac{12973191}{640} (1-2 x)^{5/2}+\frac{1357793}{128} (1-2 x)^{7/2}-\frac{1392467}{384} (1-2 x)^{9/2}+\frac{1101465 (1-2 x)^{11/2}}{1408}-\frac{161325 (1-2 x)^{13/2}}{1664}+\frac{675}{128} (1-2 x)^{15/2}\\ \end{align*}

Mathematica [A]  time = 0.017817, size = 48, normalized size = 0.46 \[ -\frac{\sqrt{1-2 x} \left (1447875 x^7+8241750 x^6+21369825 x^5+33786160 x^4+37260640 x^3+31962552 x^2+24706048 x+25632688\right )}{2145} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*(25632688 + 24706048*x + 31962552*x^2 + 37260640*x^3 + 33786160*x^4 + 21369825*x^5 + 8241750*x
^6 + 1447875*x^7))/2145

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Maple [A]  time = 0.004, size = 45, normalized size = 0.4 \begin{align*} -{\frac{1447875\,{x}^{7}+8241750\,{x}^{6}[phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+25632688}{2145}\sqrt{1-2\,x}} \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]

-1/2145*(1447875*x^7+8241750*x^6+21369825*x^5+33786160*x^4+37260640*x^3+31962552*x^2+24706048*x+25632688)*(1-2
*x)^(1/2)

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Maxima [A]  time = 1.18511, size = 99, normalized size = 0.94 \begin{align*} \frac{675}{128} \,{\left (-2 \, x + 1\right )}^{\frac{15}{2}} - \frac{161325}{1664} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} + \frac{1101465}{1408} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{1392467}{384} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{1357793}{128} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{12973191}{640} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{3278737}{128} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{3195731}{128} \, \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="maxima")

[Out]

675/128*(-2*x + 1)^(15/2) - 161325/1664*(-2*x + 1)^(13/2) + 1101465/1408*(-2*x + 1)^(11/2) - 1392467/384*(-2*x
+ 1)^(9/2) + 1357793/128*(-2*x + 1)^(7/2) - 12973191/640*(-2*x + 1)^(5/2) + 3278737/128*(-2*x + 1)^(3/2) - 31
95731/128*sqrt(-2*x + 1)

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Fricas [A]  time = 1.59887, size = 184, normalized size = 1.75 \begin{align*} -\frac{1}{2145} \,{\left (1447875 \, x^{7} + 8241750 \, x^{6} + 21369825 \, x^{5} + 33786160 \, x^{4} + 37260640 \, x^{3} + 31962552 \, x^{2} + 24706048 \, x + 25632688\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]

-1/2145*(1447875*x^7 + 8241750*x^6 + 21369825*x^5 + 33786160*x^4 + 37260640*x^3 + 31962552*x^2 + 24706048*x +
25632688)*sqrt(-2*x + 1)

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Sympy [A]  time = 69.9577, size = 94, normalized size = 0.9 \begin{align*} \frac{675 \left (1 - 2 x\right )^{\frac{15}{2}}}{128} - \frac{161325 \left (1 - 2 x\right )^{\frac{13}{2}}}{1664} + \frac{1101465 \left (1 - 2 x\right )^{\frac{11}{2}}}{1408} - \frac{1392467 \left (1 - 2 x\right )^{\frac{9}{2}}}{384} + \frac{1357793 \left (1 - 2 x\right )^{\frac{7}{2}}}{128} - \frac{12973191 \left (1 - 2 x\right )^{\frac{5}{2}}}{640} + \frac{3278737 \left (1 - 2 x\right )^{\frac{3}{2}}}{128} - \frac{3195731 \sqrt{1 - 2 x}}{128} \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]

675*(1 - 2*x)**(15/2)/128 - 161325*(1 - 2*x)**(13/2)/1664 + 1101465*(1 - 2*x)**(11/2)/1408 - 1392467*(1 - 2*x)
**(9/2)/384 + 1357793*(1 - 2*x)**(7/2)/128 - 12973191*(1 - 2*x)**(5/2)/640 + 3278737*(1 - 2*x)**(3/2)/128 - 31
95731*sqrt(1 - 2*x)/128

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Giac [A]  time = 2.5427, size = 155, normalized size = 1.48 \begin{align*} -\frac{675}{128} \,{\left (2 \, x - 1\right )}^{7} \sqrt{-2 \, x + 1} - \frac{161325}{1664} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} - \frac{1101465}{1408} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{1392467}{384} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{1357793}{128} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{12973191}{640} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{3278737}{128} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{3195731}{128} \, \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="giac")

[Out]

-675/128*(2*x - 1)^7*sqrt(-2*x + 1) - 161325/1664*(2*x - 1)^6*sqrt(-2*x + 1) - 1101465/1408*(2*x - 1)^5*sqrt(-
2*x + 1) - 1392467/384*(2*x - 1)^4*sqrt(-2*x + 1) - 1357793/128*(2*x - 1)^3*sqrt(-2*x + 1) - 12973191/640*(2*x
- 1)^2*sqrt(-2*x + 1) + 3278737/128*(-2*x + 1)^(3/2) - 3195731/128*sqrt(-2*x + 1)