3.2012 \(\int \frac{(2+3 x)^5 (3+5 x)^2}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=105 \[ \frac{405}{128} (1-2 x)^{15/2}-\frac{97605 (1-2 x)^{13/2}}{1664}+\frac{672003 (1-2 x)^{11/2}}{1408}-\frac{285565}{128} (1-2 x)^{9/2}+\frac{842415}{128} (1-2 x)^{7/2}-\frac{1623419}{128} (1-2 x)^{5/2}+\frac{6206585}{384} (1-2 x)^{3/2}-\frac{2033647}{128} \sqrt{1-2 x} \]

[Out]

(-2033647*Sqrt[1 - 2*x])/128 + (6206585*(1 - 2*x)^(3/2))/384 - (1623419*(1 - 2*x)^(5/2))/128 + (842415*(1 - 2*
x)^(7/2))/128 - (285565*(1 - 2*x)^(9/2))/128 + (672003*(1 - 2*x)^(11/2))/1408 - (97605*(1 - 2*x)^(13/2))/1664
+ (405*(1 - 2*x)^(15/2))/128

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Rubi [A]  time = 0.0184158, 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} \[ \frac{405}{128} (1-2 x)^{15/2}-\frac{97605 (1-2 x)^{13/2}}{1664}+\frac{672003 (1-2 x)^{11/2}}{1408}-\frac{285565}{128} (1-2 x)^{9/2}+\frac{842415}{128} (1-2 x)^{7/2}-\frac{1623419}{128} (1-2 x)^{5/2}+\frac{6206585}{384} (1-2 x)^{3/2}-\frac{2033647}{128} \sqrt{1-2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2033647*Sqrt[1 - 2*x])/128 + (6206585*(1 - 2*x)^(3/2))/384 - (1623419*(1 - 2*x)^(5/2))/128 + (842415*(1 - 2*
x)^(7/2))/128 - (285565*(1 - 2*x)^(9/2))/128 + (672003*(1 - 2*x)^(11/2))/1408 - (97605*(1 - 2*x)^(13/2))/1664
+ (405*(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)^5 (3+5 x)^2}{\sqrt{1-2 x}} \, dx &=\int \left (\frac{2033647}{128 \sqrt{1-2 x}}-\frac{6206585}{128} \sqrt{1-2 x}+\frac{8117095}{128} (1-2 x)^{3/2}-\frac{5896905}{128} (1-2 x)^{5/2}+\frac{2570085}{128} (1-2 x)^{7/2}-\frac{672003}{128} (1-2 x)^{9/2}+\frac{97605}{128} (1-2 x)^{11/2}-\frac{6075}{128} (1-2 x)^{13/2}\right ) \, dx\\ &=-\frac{2033647}{128} \sqrt{1-2 x}+\frac{6206585}{384} (1-2 x)^{3/2}-\frac{1623419}{128} (1-2 x)^{5/2}+\frac{842415}{128} (1-2 x)^{7/2}-\frac{285565}{128} (1-2 x)^{9/2}+\frac{672003 (1-2 x)^{11/2}}{1408}-\frac{97605 (1-2 x)^{13/2}}{1664}+\frac{405}{128} (1-2 x)^{15/2}\\ \end{align*}

Mathematica [A]  time = 0.0189129, size = 48, normalized size = 0.46 \[ -\frac{1}{429} \sqrt{1-2 x} \left (173745 x^7+1002375 x^6+2632743 x^5+4212525 x^4+4694340 x^3+4058988 x^2+3152152 x+3275704\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*(3275704 + 3152152*x + 4058988*x^2 + 4694340*x^3 + 4212525*x^4 + 2632743*x^5 + 1002375*x^6 + 1
73745*x^7))/429

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Maple [A]  time = 0.004, size = 45, normalized size = 0.4 \begin{align*} -{\frac{173745\,{x}^{7}+1002375\,{x}^{6}+2632743\,{x}^{5}+4212525\,{x}^{4}+4694340\,{x}^{3}+4058988\,{x}^{2}+3152152\,x+3275704}{429}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/429*(173745*x^7+1002375*x^6+2632743*x^5+4212525*x^4+4694340*x^3+4058988*x^2+3152152*x+3275704)*(1-2*x)^(1/2
)

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Maxima [A]  time = 1.13129, size = 99, normalized size = 0.94 \begin{align*} \frac{405}{128} \,{\left (-2 \, x + 1\right )}^{\frac{15}{2}} - \frac{97605}{1664} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} + \frac{672003}{1408} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{285565}{128} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{842415}{128} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{1623419}{128} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{6206585}{384} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{2033647}{128} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

405/128*(-2*x + 1)^(15/2) - 97605/1664*(-2*x + 1)^(13/2) + 672003/1408*(-2*x + 1)^(11/2) - 285565/128*(-2*x +
1)^(9/2) + 842415/128*(-2*x + 1)^(7/2) - 1623419/128*(-2*x + 1)^(5/2) + 6206585/384*(-2*x + 1)^(3/2) - 2033647
/128*sqrt(-2*x + 1)

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Fricas [A]  time = 1.3, size = 173, normalized size = 1.65 \begin{align*} -\frac{1}{429} \,{\left (173745 \, x^{7} + 1002375 \, x^{6} + 2632743 \, x^{5} + 4212525 \, x^{4} + 4694340 \, x^{3} + 4058988 \, x^{2} + 3152152 \, x + 3275704\right )} \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/429*(173745*x^7 + 1002375*x^6 + 2632743*x^5 + 4212525*x^4 + 4694340*x^3 + 4058988*x^2 + 3152152*x + 3275704
)*sqrt(-2*x + 1)

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Sympy [A]  time = 69.6894, size = 94, normalized size = 0.9 \begin{align*} \frac{405 \left (1 - 2 x\right )^{\frac{15}{2}}}{128} - \frac{97605 \left (1 - 2 x\right )^{\frac{13}{2}}}{1664} + \frac{672003 \left (1 - 2 x\right )^{\frac{11}{2}}}{1408} - \frac{285565 \left (1 - 2 x\right )^{\frac{9}{2}}}{128} + \frac{842415 \left (1 - 2 x\right )^{\frac{7}{2}}}{128} - \frac{1623419 \left (1 - 2 x\right )^{\frac{5}{2}}}{128} + \frac{6206585 \left (1 - 2 x\right )^{\frac{3}{2}}}{384} - \frac{2033647 \sqrt{1 - 2 x}}{128} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

405*(1 - 2*x)**(15/2)/128 - 97605*(1 - 2*x)**(13/2)/1664 + 672003*(1 - 2*x)**(11/2)/1408 - 285565*(1 - 2*x)**(
9/2)/128 + 842415*(1 - 2*x)**(7/2)/128 - 1623419*(1 - 2*x)**(5/2)/128 + 6206585*(1 - 2*x)**(3/2)/384 - 2033647
*sqrt(1 - 2*x)/128

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Giac [A]  time = 1.31901, size = 155, normalized size = 1.48 \begin{align*} -\frac{405}{128} \,{\left (2 \, x - 1\right )}^{7} \sqrt{-2 \, x + 1} - \frac{97605}{1664} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} - \frac{672003}{1408} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{285565}{128} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{842415}{128} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{1623419}{128} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{6206585}{384} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{2033647}{128} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-405/128*(2*x - 1)^7*sqrt(-2*x + 1) - 97605/1664*(2*x - 1)^6*sqrt(-2*x + 1) - 672003/1408*(2*x - 1)^5*sqrt(-2*
x + 1) - 285565/128*(2*x - 1)^4*sqrt(-2*x + 1) - 842415/128*(2*x - 1)^3*sqrt(-2*x + 1) - 1623419/128*(2*x - 1)
^2*sqrt(-2*x + 1) + 6206585/384*(-2*x + 1)^(3/2) - 2033647/128*sqrt(-2*x + 1)