3.2550 \(\int \frac{(2+3 x)^5}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=142 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^4}{11 \sqrt{1-2 x}}+\frac{939}{880} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^3+\frac{76587 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{17600}+\frac{21 \sqrt{1-2 x} \sqrt{5 x+3} (7645620 x+18424549)}{2816000}-\frac{291096141 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{256000 \sqrt{10}} \]

[Out]

(76587*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/17600 + (939*Sqrt[1 - 2*x]*(2 + 3*x)^3*Sqrt[3 + 5*x])/880 + (7
*(2 + 3*x)^4*Sqrt[3 + 5*x])/(11*Sqrt[1 - 2*x]) + (21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(18424549 + 7645620*x))/28160
00 - (291096141*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(256000*Sqrt[10])

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Rubi [A]  time = 0.0440307, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {98, 153, 147, 54, 216} \[ \frac{7 \sqrt{5 x+3} (3 x+2)^4}{11 \sqrt{1-2 x}}+\frac{939}{880} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^3+\frac{76587 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{17600}+\frac{21 \sqrt{1-2 x} \sqrt{5 x+3} (7645620 x+18424549)}{2816000}-\frac{291096141 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{256000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(76587*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/17600 + (939*Sqrt[1 - 2*x]*(2 + 3*x)^3*Sqrt[3 + 5*x])/880 + (7
*(2 + 3*x)^4*Sqrt[3 + 5*x])/(11*Sqrt[1 - 2*x]) + (21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(18424549 + 7645620*x))/28160
00 - (291096141*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(256000*Sqrt[10])

Rule 98

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

Rule 153

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

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)^5}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx &=\frac{7 (2+3 x)^4 \sqrt{3+5 x}}{11 \sqrt{1-2 x}}-\frac{1}{11} \int \frac{(2+3 x)^3 \left (285+\frac{939 x}{2}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{939}{880} \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}+\frac{7 (2+3 x)^4 \sqrt{3+5 x}}{11 \sqrt{1-2 x}}+\frac{1}{440} \int \frac{\left (-35007-\frac{229761 x}{4}\right ) (2+3 x)^2}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{76587 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{17600}+\frac{939}{880} \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}+\frac{7 (2+3 x)^4 \sqrt{3+5 x}}{11 \sqrt{1-2 x}}-\frac{\int \frac{(2+3 x) \left (\frac{12307617}{4}+\frac{40139505 x}{8}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{13200}\\ &=\frac{76587 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{17600}+\frac{939}{880} \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}+\frac{7 (2+3 x)^4 \sqrt{3+5 x}}{11 \sqrt{1-2 x}}+\frac{21 \sqrt{1-2 x} \sqrt{3+5 x} (18424549+7645620 x)}{2816000}-\frac{291096141 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{512000}\\ &=\frac{76587 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{17600}+\frac{939}{880} \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}+\frac{7 (2+3 x)^4 \sqrt{3+5 x}}{11 \sqrt{1-2 x}}+\frac{21 \sqrt{1-2 x} \sqrt{3+5 x} (18424549+7645620 x)}{2816000}-\frac{291096141 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{256000 \sqrt{5}}\\ &=\frac{76587 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{17600}+\frac{939}{880} \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}+\frac{7 (2+3 x)^4 \sqrt{3+5 x}}{11 \sqrt{1-2 x}}+\frac{21 \sqrt{1-2 x} \sqrt{3+5 x} (18424549+7645620 x)}{2816000}-\frac{291096141 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{256000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.104969, size = 74, normalized size = 0.52 \[ \frac{3202057551 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (17107200 x^4+76887360 x^3+171939240 x^2+332129358 x-488641609\right )}{28160000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(-488641609 + 332129358*x + 171939240*x^2 + 76887360*x^3 + 17107200*x^4) + 3202057551*Sqrt[
10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(28160000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.013, size = 140, normalized size = 1. \begin{align*} -{\frac{1}{112640000\,x-56320000} \left ( -342144000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-1537747200\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-3438784800\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-3202057551\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -6642587160\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,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)^5/(1-2*x)^(3/2)/(3+5*x)^(1/2),x)

[Out]

-1/56320000*(-342144000*x^4*(-10*x^2-x+3)^(1/2)-1537747200*x^3*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(
20/11*x+1/11)*x-3438784800*x^2*(-10*x^2-x+3)^(1/2)-3202057551*10^(1/2)*arcsin(20/11*x+1/11)-6642587160*x*(-10*
x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(2*x-1)/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 2.80041, size = 134, normalized size = 0.94 \begin{align*} \frac{243}{80} \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + \frac{24273}{1600} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} - \frac{291096141}{5120000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{487863}{12800} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{19975419}{256000} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{16807 \, \sqrt{-10 \, x^{2} - x + 3}}{176 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

243/80*sqrt(-10*x^2 - x + 3)*x^3 + 24273/1600*sqrt(-10*x^2 - x + 3)*x^2 - 291096141/5120000*sqrt(5)*sqrt(2)*ar
csin(20/11*x + 1/11) + 487863/12800*sqrt(-10*x^2 - x + 3)*x + 19975419/256000*sqrt(-10*x^2 - x + 3) - 16807/17
6*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 1.75298, size = 323, normalized size = 2.27 \begin{align*} \frac{3202057551 \, \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 (17107200 \, x^{4} + 76887360 \, x^{3} + 171939240 \, x^{2} + 332129358 \, x - 488641609\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{56320000 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/56320000*(3202057551*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*(17107200*x^4 + 76887360*x^3 + 171939240*x^2 + 332129358*x - 488641609)*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)**5/(1-2*x)**(3/2)/(3+5*x)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 2.07281, size = 131, normalized size = 0.92 \begin{align*} -\frac{291096141}{2560000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (198 \,{\left (12 \,{\left (8 \,{\left (36 \, \sqrt{5}{\left (5 \, x + 3\right )} + 377 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 29669 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 4900505 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 16010291851 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{352000000 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-291096141/2560000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/352000000*(198*(12*(8*(36*sqrt(5)*(5*x + 3
) + 377*sqrt(5))*(5*x + 3) + 29669*sqrt(5))*(5*x + 3) + 4900505*sqrt(5))*(5*x + 3) - 16010291851*sqrt(5))*sqrt
(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)