3.2608 \(\int \frac{(2+3 x)^5}{(1-2 x)^{5/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=142 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^4}{33 (1-2 x)^{3/2}}-\frac{2051 \sqrt{5 x+3} (3 x+2)^3}{726 \sqrt{1-2 x}}-\frac{23909 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{4840}-\frac{\sqrt{1-2 x} \sqrt{5 x+3} (50124540 x+120791143)}{774400}+\frac{8261577 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6400 \sqrt{10}} \]

[Out]

(-23909*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/4840 - (2051*(2 + 3*x)^3*Sqrt[3 + 5*x])/(726*Sqrt[1 - 2*x]) +
(7*(2 + 3*x)^4*Sqrt[3 + 5*x])/(33*(1 - 2*x)^(3/2)) - (Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(120791143 + 50124540*x))/7
74400 + (8261577*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(6400*Sqrt[10])

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Rubi [A]  time = 0.0424436, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {98, 150, 153, 147, 54, 216} \[ \frac{7 \sqrt{5 x+3} (3 x+2)^4}{33 (1-2 x)^{3/2}}-\frac{2051 \sqrt{5 x+3} (3 x+2)^3}{726 \sqrt{1-2 x}}-\frac{23909 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{4840}-\frac{\sqrt{1-2 x} \sqrt{5 x+3} (50124540 x+120791143)}{774400}+\frac{8261577 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6400 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-23909*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/4840 - (2051*(2 + 3*x)^3*Sqrt[3 + 5*x])/(726*Sqrt[1 - 2*x]) +
(7*(2 + 3*x)^4*Sqrt[3 + 5*x])/(33*(1 - 2*x)^(3/2)) - (Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(120791143 + 50124540*x))/7
74400 + (8261577*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(6400*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 150

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

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)^{5/2} \sqrt{3+5 x}} \, dx &=\frac{7 (2+3 x)^4 \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}-\frac{1}{33} \int \frac{(2+3 x)^3 \left (281+\frac{927 x}{2}\right )}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx\\ &=-\frac{2051 (2+3 x)^3 \sqrt{3+5 x}}{726 \sqrt{1-2 x}}+\frac{7 (2+3 x)^4 \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}-\frac{1}{363} \int \frac{\left (-32787-\frac{215181 x}{4}\right ) (2+3 x)^2}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{23909 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{4840}-\frac{2051 (2+3 x)^3 \sqrt{3+5 x}}{726 \sqrt{1-2 x}}+\frac{7 (2+3 x)^4 \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}+\frac{\int \frac{(2+3 x) \left (\frac{11526957}{4}+\frac{37593405 x}{8}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{10890}\\ &=-\frac{23909 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{4840}-\frac{2051 (2+3 x)^3 \sqrt{3+5 x}}{726 \sqrt{1-2 x}}+\frac{7 (2+3 x)^4 \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}-\frac{\sqrt{1-2 x} \sqrt{3+5 x} (120791143+50124540 x)}{774400}+\frac{8261577 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{12800}\\ &=-\frac{23909 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{4840}-\frac{2051 (2+3 x)^3 \sqrt{3+5 x}}{726 \sqrt{1-2 x}}+\frac{7 (2+3 x)^4 \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}-\frac{\sqrt{1-2 x} \sqrt{3+5 x} (120791143+50124540 x)}{774400}+\frac{8261577 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{6400 \sqrt{5}}\\ &=-\frac{23909 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{4840}-\frac{2051 (2+3 x)^3 \sqrt{3+5 x}}{726 \sqrt{1-2 x}}+\frac{7 (2+3 x)^4 \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}-\frac{\sqrt{1-2 x} \sqrt{3+5 x} (120791143+50124540 x)}{774400}+\frac{8261577 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{6400 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0746912, size = 79, normalized size = 0.56 \[ \frac{2998952451 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (18817920 x^4+101146320 x^3+359461476 x^2-1261070176 x+452899509\right )}{23232000 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(452899509 - 1261070176*x + 359461476*x^2 + 101146320*x^3 + 18817920*x^4) + 2998952451*Sqrt
[10 - 20*x]*(-1 + 2*x)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(23232000*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.013, size = 154, normalized size = 1.1 \begin{align*}{\frac{1}{46464000\, \left ( 2\,x-1 \right ) ^{2}} \left ( -376358400\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-2022926400\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-11995809804\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-7189229520\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-9057990180\,\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)^(5/2)/(3+5*x)^(1/2),x)

[Out]

1/46464000*(-376358400*x^4*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x^2-2022926400*x^3*(-
10*x^2-x+3)^(1/2)-11995809804*10^(1/2)*arcsin(20/11*x+1/11)*x-7189229520*x^2*(-10*x^2-x+3)^(1/2)[phone]*10
^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)-9057990180*(-10*x^2-x+3)^(1/2))*(3+5*x)^(1/2)*(1
-2*x)^(1/2)/(2*x-1)^2/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 1.61691, size = 146, normalized size = 1.03 \begin{align*} -\frac{81}{40} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + \frac{8261577}{128000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{4131}{320} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{326943}{6400} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{16807 \, \sqrt{-10 \, x^{2} - x + 3}}{528 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{1020425 \, \sqrt{-10 \, x^{2} - x + 3}}{5808 \,{\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)^(5/2)/(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

-81/40*sqrt(-10*x^2 - x + 3)*x^2 + 8261577/128000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 4131/320*sqrt(-10*x
^2 - x + 3)*x - 326943/6400*sqrt(-10*x^2 - x + 3) + 16807/528*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) + 102042
5/5808*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 1.55128, size = 348, normalized size = 2.45 \begin{align*} -\frac{2998952451 \, \sqrt{10}{\left (4 \, x^{2} - 4 \, 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 (18817920 \, x^{4} + 101146320 \, x^{3} + 359461476 \, x^{2} - 1261070176 \, x + 452899509\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{46464000 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/46464000*(2998952451*sqrt(10)*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1
)/(10*x^2 + x - 3)) + 20*(18817920*x^4 + 101146320*x^3 + 359461476*x^2 - 1261070176*x + 452899509)*sqrt(5*x +
3)*sqrt(-2*x + 1))/(4*x^2 - 4*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)**(5/2)/(3+5*x)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 2.28538, size = 131, normalized size = 0.92 \begin{align*} \frac{8261577}{64000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (9801 \,{\left (12 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 119 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 27809 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 9996528778 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 164942367909 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1452000000 \,{\left (2 \, x - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8261577/64000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/1452000000*(4*(9801*(12*(8*sqrt(5)*(5*x + 3) +
119*sqrt(5))*(5*x + 3) + 27809*sqrt(5))*(5*x + 3) - 9996528778*sqrt(5))*(5*x + 3) + 164942367909*sqrt(5))*sqrt
(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2