3.2002 \(\int \frac{(2+3 x)^4 (3+5 x)}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=79 \[ \frac{405}{352} (1-2 x)^{11/2}-\frac{519}{32} (1-2 x)^{9/2}+\frac{1539}{16} (1-2 x)^{7/2}-\frac{24843}{80} (1-2 x)^{5/2}+\frac{57281}{96} (1-2 x)^{3/2}-\frac{26411}{32} \sqrt{1-2 x} \]

[Out]

(-26411*Sqrt[1 - 2*x])/32 + (57281*(1 - 2*x)^(3/2))/96 - (24843*(1 - 2*x)^(5/2))/80 + (1539*(1 - 2*x)^(7/2))/1
6 - (519*(1 - 2*x)^(9/2))/32 + (405*(1 - 2*x)^(11/2))/352

________________________________________________________________________________________

Rubi [A]  time = 0.0143568, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {77} \[ \frac{405}{352} (1-2 x)^{11/2}-\frac{519}{32} (1-2 x)^{9/2}+\frac{1539}{16} (1-2 x)^{7/2}-\frac{24843}{80} (1-2 x)^{5/2}+\frac{57281}{96} (1-2 x)^{3/2}-\frac{26411}{32} \sqrt{1-2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-26411*Sqrt[1 - 2*x])/32 + (57281*(1 - 2*x)^(3/2))/96 - (24843*(1 - 2*x)^(5/2))/80 + (1539*(1 - 2*x)^(7/2))/1
6 - (519*(1 - 2*x)^(9/2))/32 + (405*(1 - 2*x)^(11/2))/352

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{(2+3 x)^4 (3+5 x)}{\sqrt{1-2 x}} \, dx &=\int \left (\frac{26411}{32 \sqrt{1-2 x}}-\frac{57281}{32} \sqrt{1-2 x}+\frac{24843}{16} (1-2 x)^{3/2}-\frac{10773}{16} (1-2 x)^{5/2}+\frac{4671}{32} (1-2 x)^{7/2}-\frac{405}{32} (1-2 x)^{9/2}\right ) \, dx\\ &=-\frac{26411}{32} \sqrt{1-2 x}+\frac{57281}{96} (1-2 x)^{3/2}-\frac{24843}{80} (1-2 x)^{5/2}+\frac{1539}{16} (1-2 x)^{7/2}-\frac{519}{32} (1-2 x)^{9/2}+\frac{405}{352} (1-2 x)^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.0132592, size = 38, normalized size = 0.48 \[ -\frac{1}{165} \sqrt{1-2 x} \left (6075 x^5+27630 x^4+56520 x^3+71136 x^2+67664 x+75584\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*(75584 + 67664*x + 71136*x^2 + 56520*x^3 + 27630*x^4 + 6075*x^5))/165

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 35, normalized size = 0.4 \begin{align*} -{\frac{6075\,{x}^{5}+27630\,{x}^{4}+56520\,{x}^{3}+71136\,{x}^{2}+67664\,x+75584}{165}\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)/(1-2*x)^(1/2),x)

[Out]

-1/165*(6075*x^5+27630*x^4+56520*x^3+71136*x^2+67664*x+75584)*(1-2*x)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.28215, size = 74, normalized size = 0.94 \begin{align*} \frac{405}{352} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{519}{32} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{1539}{16} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{24843}{80} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{57281}{96} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{26411}{32} \, \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)/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

405/352*(-2*x + 1)^(11/2) - 519/32*(-2*x + 1)^(9/2) + 1539/16*(-2*x + 1)^(7/2) - 24843/80*(-2*x + 1)^(5/2) + 5
7281/96*(-2*x + 1)^(3/2) - 26411/32*sqrt(-2*x + 1)

________________________________________________________________________________________

Fricas [A]  time = 1.21031, size = 119, normalized size = 1.51 \begin{align*} -\frac{1}{165} \,{\left (6075 \, x^{5} + 27630 \, x^{4} + 56520 \, x^{3} + 71136 \, x^{2} + 67664 \, x + 75584\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)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/165*(6075*x^5 + 27630*x^4 + 56520*x^3 + 71136*x^2 + 67664*x + 75584)*sqrt(-2*x + 1)

________________________________________________________________________________________

Sympy [A]  time = 41.7553, size = 70, normalized size = 0.89 \begin{align*} \frac{405 \left (1 - 2 x\right )^{\frac{11}{2}}}{352} - \frac{519 \left (1 - 2 x\right )^{\frac{9}{2}}}{32} + \frac{1539 \left (1 - 2 x\right )^{\frac{7}{2}}}{16} - \frac{24843 \left (1 - 2 x\right )^{\frac{5}{2}}}{80} + \frac{57281 \left (1 - 2 x\right )^{\frac{3}{2}}}{96} - \frac{26411 \sqrt{1 - 2 x}}{32} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

405*(1 - 2*x)**(11/2)/352 - 519*(1 - 2*x)**(9/2)/32 + 1539*(1 - 2*x)**(7/2)/16 - 24843*(1 - 2*x)**(5/2)/80 + 5
7281*(1 - 2*x)**(3/2)/96 - 26411*sqrt(1 - 2*x)/32

________________________________________________________________________________________

Giac [A]  time = 2.68147, size = 112, normalized size = 1.42 \begin{align*} -\frac{405}{352} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{519}{32} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{1539}{16} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{24843}{80} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{57281}{96} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{26411}{32} \, \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)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-405/352*(2*x - 1)^5*sqrt(-2*x + 1) - 519/32*(2*x - 1)^4*sqrt(-2*x + 1) - 1539/16*(2*x - 1)^3*sqrt(-2*x + 1) -
24843/80*(2*x - 1)^2*sqrt(-2*x + 1) + 57281/96*(-2*x + 1)^(3/2) - 26411/32*sqrt(-2*x + 1)