3.2541 \(\int \frac{(2+3 x)^2 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=138 \[ \frac{9}{80} \sqrt{1-2 x} (5 x+3)^{7/2}+\frac{49 (5 x+3)^{7/2}}{22 \sqrt{1-2 x}}+\frac{25397 \sqrt{1-2 x} (5 x+3)^{5/2}}{3520}+\frac{25397}{512} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{838101 \sqrt{1-2 x} \sqrt{5 x+3}}{2048}-\frac{9219111 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2048 \sqrt{10}} \]

[Out]

(838101*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2048 + (25397*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/512 + (25397*Sqrt[1 - 2*x]*(
3 + 5*x)^(5/2))/3520 + (49*(3 + 5*x)^(7/2))/(22*Sqrt[1 - 2*x]) + (9*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/80 - (92191
11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2048*Sqrt[10])

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Rubi [A]  time = 0.0422166, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {89, 80, 50, 54, 216} \[ \frac{9}{80} \sqrt{1-2 x} (5 x+3)^{7/2}+\frac{49 (5 x+3)^{7/2}}{22 \sqrt{1-2 x}}+\frac{25397 \sqrt{1-2 x} (5 x+3)^{5/2}}{3520}+\frac{25397}{512} \sqrt{1-2 x} (5 x+3)^{3/2}+\frac{838101 \sqrt{1-2 x} \sqrt{5 x+3}}{2048}-\frac{9219111 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{2048 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(838101*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2048 + (25397*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/512 + (25397*Sqrt[1 - 2*x]*(
3 + 5*x)^(5/2))/3520 + (49*(3 + 5*x)^(7/2))/(22*Sqrt[1 - 2*x]) + (9*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/80 - (92191
11*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2048*Sqrt[10])

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{(2+3 x)^2 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx &=\frac{49 (3+5 x)^{7/2}}{22 \sqrt{1-2 x}}-\frac{1}{22} \int \frac{(3+5 x)^{5/2} \left (\frac{1833}{2}+99 x\right )}{\sqrt{1-2 x}} \, dx\\ &=\frac{49 (3+5 x)^{7/2}}{22 \sqrt{1-2 x}}+\frac{9}{80} \sqrt{1-2 x} (3+5 x)^{7/2}-\frac{76191 \int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx}{1760}\\ &=\frac{25397 \sqrt{1-2 x} (3+5 x)^{5/2}}{3520}+\frac{49 (3+5 x)^{7/2}}{22 \sqrt{1-2 x}}+\frac{9}{80} \sqrt{1-2 x} (3+5 x)^{7/2}-\frac{25397}{128} \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\\ &=\frac{25397}{512} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{25397 \sqrt{1-2 x} (3+5 x)^{5/2}}{3520}+\frac{49 (3+5 x)^{7/2}}{22 \sqrt{1-2 x}}+\frac{9}{80} \sqrt{1-2 x} (3+5 x)^{7/2}-\frac{838101 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx}{1024}\\ &=\frac{838101 \sqrt{1-2 x} \sqrt{3+5 x}}{2048}+\frac{25397}{512} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{25397 \sqrt{1-2 x} (3+5 x)^{5/2}}{3520}+\frac{49 (3+5 x)^{7/2}}{22 \sqrt{1-2 x}}+\frac{9}{80} \sqrt{1-2 x} (3+5 x)^{7/2}-\frac{9219111 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{4096}\\ &=\frac{838101 \sqrt{1-2 x} \sqrt{3+5 x}}{2048}+\frac{25397}{512} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{25397 \sqrt{1-2 x} (3+5 x)^{5/2}}{3520}+\frac{49 (3+5 x)^{7/2}}{22 \sqrt{1-2 x}}+\frac{9}{80} \sqrt{1-2 x} (3+5 x)^{7/2}-\frac{9219111 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{2048 \sqrt{5}}\\ &=\frac{838101 \sqrt{1-2 x} \sqrt{3+5 x}}{2048}+\frac{25397}{512} \sqrt{1-2 x} (3+5 x)^{3/2}+\frac{25397 \sqrt{1-2 x} (3+5 x)^{5/2}}{3520}+\frac{49 (3+5 x)^{7/2}}{22 \sqrt{1-2 x}}+\frac{9}{80} \sqrt{1-2 x} (3+5 x)^{7/2}-\frac{9219111 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{2048 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0406542, size = 74, normalized size = 0.54 \[ \frac{9219111 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (57600 x^4+243520 x^3+517096 x^2+966014 x-1405233\right )}{20480 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(-1405233 + 966014*x + 517096*x^2 + 243520*x^3 + 57600*x^4) + 9219111*Sqrt[10 - 20*x]*ArcSi
n[Sqrt[5/11]*Sqrt[1 - 2*x]])/(20480*Sqrt[1 - 2*x])

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Maple [A]  time = 0.013, size = 140, normalized size = 1. \begin{align*} -{\frac{1}{81920\,x-40960} \left ( -1152000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-4870400\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-10341920\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-9219111\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -19320280\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/40960*(-1152000*x^4*(-10*x^2-x+3)^(1/2)-4870400*x^3*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/
11)*x-10341920*x^2*(-10*x^2-x+3)^(1/2)-9219111*10^(1/2)*arcsin(20/11*x+1/11)-19320280*x*(-10*x^2-x+3)^(1/2)+28
104660*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2*x-1)/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.75005, size = 147, normalized size = 1.07 \begin{align*} -\frac{1125 \, x^{5}}{8 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{21725 \, x^{4}}{32 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{414505 \, x^{3}}{256 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{3190679 \, x^{2}}{1024 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{9219111}{40960} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{4128123 \, x}{2048 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{4215699}{2048 \, \sqrt{-10 \, x^{2} - x + 3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1125/8*x^5/sqrt(-10*x^2 - x + 3) - 21725/32*x^4/sqrt(-10*x^2 - x + 3) - 414505/256*x^3/sqrt(-10*x^2 - x + 3)
- 3190679/1024*x^2/sqrt(-10*x^2 - x + 3) + 9219111/40960*sqrt(10)*arcsin(-20/11*x - 1/11) + 4128123/2048*x/sqr
t(-10*x^2 - x + 3) + 4215699/2048/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.77008, size = 297, normalized size = 2.15 \begin{align*} \frac{9219111 \, \sqrt{10}{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \,{\left (57600 \, x^{4} + 243520 \, x^{3} + 517096 \, x^{2} + 966014 \, x - 1405233\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{40960 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/40960*(9219111*sqrt(10)*(2*x - 1)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x -
3)) + 20*(57600*x^4 + 243520*x^3 + 517096*x^2 + 966014*x - 1405233)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 2.47542, size = 131, normalized size = 0.95 \begin{align*} -\frac{9219111}{20480} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (2 \,{\left (4 \,{\left (8 \,{\left (36 \, \sqrt{5}{\left (5 \, x + 3\right )} + 329 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 25397 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 1396835 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 46095555 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{256000 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9219111/20480*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/256000*(2*(4*(8*(36*sqrt(5)*(5*x + 3) + 329*sq
rt(5))*(5*x + 3) + 25397*sqrt(5))*(5*x + 3) + 1396835*sqrt(5))*(5*x + 3) - 46095555*sqrt(5))*sqrt(5*x + 3)*sqr
t(-10*x + 5)/(2*x - 1)