3.2424 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^4}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=179 \[ -\frac{\sqrt{5 x+3} (11603280 x+12923401) (1-2 x)^{7/2}}{22400000}-\frac{3}{70} (3 x+2)^3 \sqrt{5 x+3} (1-2 x)^{7/2}-\frac{271 (3 x+2)^2 \sqrt{5 x+3} (1-2 x)^{7/2}}{2800}+\frac{9526549 \sqrt{5 x+3} (1-2 x)^{5/2}}{96000000}+\frac{104792039 \sqrt{5 x+3} (1-2 x)^{3/2}}{384000000}+\frac{1152712429 \sqrt{5 x+3} \sqrt{1-2 x}}{1280000000}+\frac{12679836719 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1280000000 \sqrt{10}} \]

[Out]

(1152712429*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1280000000 + (104792039*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/384000000 + (9
526549*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/96000000 - (271*(1 - 2*x)^(7/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/2800 - (3*(1
- 2*x)^(7/2)*(2 + 3*x)^3*Sqrt[3 + 5*x])/70 - ((1 - 2*x)^(7/2)*Sqrt[3 + 5*x]*(12923401 + 11603280*x))/22400000
+ (12679836719*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1280000000*Sqrt[10])

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Rubi [A]  time = 0.0551209, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {100, 153, 147, 50, 54, 216} \[ -\frac{\sqrt{5 x+3} (11603280 x+12923401) (1-2 x)^{7/2}}{22400000}-\frac{3}{70} (3 x+2)^3 \sqrt{5 x+3} (1-2 x)^{7/2}-\frac{271 (3 x+2)^2 \sqrt{5 x+3} (1-2 x)^{7/2}}{2800}+\frac{9526549 \sqrt{5 x+3} (1-2 x)^{5/2}}{96000000}+\frac{104792039 \sqrt{5 x+3} (1-2 x)^{3/2}}{384000000}+\frac{1152712429 \sqrt{5 x+3} \sqrt{1-2 x}}{1280000000}+\frac{12679836719 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1280000000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1152712429*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1280000000 + (104792039*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/384000000 + (9
526549*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/96000000 - (271*(1 - 2*x)^(7/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/2800 - (3*(1
- 2*x)^(7/2)*(2 + 3*x)^3*Sqrt[3 + 5*x])/70 - ((1 - 2*x)^(7/2)*Sqrt[3 + 5*x]*(12923401 + 11603280*x))/22400000
+ (12679836719*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1280000000*Sqrt[10])

Rule 100

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

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 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{(1-2 x)^{5/2} (2+3 x)^4}{\sqrt{3+5 x}} \, dx &=-\frac{3}{70} (1-2 x)^{7/2} (2+3 x)^3 \sqrt{3+5 x}-\frac{1}{70} \int \frac{\left (-250-\frac{813 x}{2}\right ) (1-2 x)^{5/2} (2+3 x)^2}{\sqrt{3+5 x}} \, dx\\ &=-\frac{271 (1-2 x)^{7/2} (2+3 x)^2 \sqrt{3+5 x}}{2800}-\frac{3}{70} (1-2 x)^{7/2} (2+3 x)^3 \sqrt{3+5 x}+\frac{\int \frac{(1-2 x)^{5/2} (2+3 x) \left (\frac{44553}{2}+\frac{145041 x}{4}\right )}{\sqrt{3+5 x}} \, dx}{4200}\\ &=-\frac{271 (1-2 x)^{7/2} (2+3 x)^2 \sqrt{3+5 x}}{2800}-\frac{3}{70} (1-2 x)^{7/2} (2+3 x)^3 \sqrt{3+5 x}-\frac{(1-2 x)^{7/2} \sqrt{3+5 x} (12923401+11603280 x)}{22400000}+\frac{9526549 \int \frac{(1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx}{6400000}\\ &=\frac{9526549 (1-2 x)^{5/2} \sqrt{3+5 x}}{96000000}-\frac{271 (1-2 x)^{7/2} (2+3 x)^2 \sqrt{3+5 x}}{2800}-\frac{3}{70} (1-2 x)^{7/2} (2+3 x)^3 \sqrt{3+5 x}-\frac{(1-2 x)^{7/2} \sqrt{3+5 x} (12923401+11603280 x)}{22400000}+\frac{104792039 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{38400000}\\ &=\frac{104792039 (1-2 x)^{3/2} \sqrt{3+5 x}}{384000000}+\frac{9526549 (1-2 x)^{5/2} \sqrt{3+5 x}}{96000000}-\frac{271 (1-2 x)^{7/2} (2+3 x)^2 \sqrt{3+5 x}}{2800}-\frac{3}{70} (1-2 x)^{7/2} (2+3 x)^3 \sqrt{3+5 x}-\frac{(1-2 x)^{7/2} \sqrt{3+5 x} (12923401+11603280 x)}{22400000}+\frac{1152712429 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{256000000}\\ &=\frac{1152712429 \sqrt{1-2 x} \sqrt{3+5 x}}{1280000000}+\frac{104792039 (1-2 x)^{3/2} \sqrt{3+5 x}}{384000000}+\frac{9526549 (1-2 x)^{5/2} \sqrt{3+5 x}}{96000000}-\frac{271 (1-2 x)^{7/2} (2+3 x)^2 \sqrt{3+5 x}}{2800}-\frac{3}{70} (1-2 x)^{7/2} (2+3 x)^3 \sqrt{3+5 x}-\frac{(1-2 x)^{7/2} \sqrt{3+5 x} (12923401+11603280 x)}{22400000}+\frac{12679836719 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{2560000000}\\ &=\frac{1152712429 \sqrt{1-2 x} \sqrt{3+5 x}}{1280000000}+\frac{104792039 (1-2 x)^{3/2} \sqrt{3+5 x}}{384000000}+\frac{9526549 (1-2 x)^{5/2} \sqrt{3+5 x}}{96000000}-\frac{271 (1-2 x)^{7/2} (2+3 x)^2 \sqrt{3+5 x}}{2800}-\frac{3}{70} (1-2 x)^{7/2} (2+3 x)^3 \sqrt{3+5 x}-\frac{(1-2 x)^{7/2} \sqrt{3+5 x} (12923401+11603280 x)}{22400000}+\frac{12679836719 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{1280000000 \sqrt{5}}\\ &=\frac{1152712429 \sqrt{1-2 x} \sqrt{3+5 x}}{1280000000}+\frac{104792039 (1-2 x)^{3/2} \sqrt{3+5 x}}{384000000}+\frac{9526549 (1-2 x)^{5/2} \sqrt{3+5 x}}{96000000}-\frac{271 (1-2 x)^{7/2} (2+3 x)^2 \sqrt{3+5 x}}{2800}-\frac{3}{70} (1-2 x)^{7/2} (2+3 x)^3 \sqrt{3+5 x}-\frac{(1-2 x)^{7/2} \sqrt{3+5 x} (12923401+11603280 x)}{22400000}+\frac{12679836719 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{1280000000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0600142, size = 89, normalized size = 0.5 \[ \frac{-10 \sqrt{5 x+3} \left (497664000000 x^7+374630400000 x^6-607578624000 x^5-403491052800 x^4+322052135360 x^3+174481885240 x^2-100668418342 x+920643741\right )-266276571099 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{268800000000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(920643741 - 100668418342*x + 174481885240*x^2 + 322052135360*x^3 - 403491052800*x^4 - 6075
78624000*x^5 + 374630400000*x^6 + 497664000000*x^7) - 266276571099*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 -
2*x]])/(268800000000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.01, size = 155, normalized size = 0.9 \begin{align*}{\frac{1}{537600000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 4976640000000\,\sqrt{-10\,{x}^{2}-x+3}{x}^{6}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-2958474240000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-5514147648000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{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}-18412874820\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/537600000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(4976640000000*(-10*x^2-x+3)^(1/2)*x^6+6234624000000*x^5*(-10*x^2-x
+3)^(1/2)-2958474240000*x^4*(-10*x^2-x+3)^(1/2)-5514147648000*x^3*(-10*x^2-x+3)^(1/2)[phone]*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)-18412874820*(-10*x
^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 2.00511, size = 170, normalized size = 0.95 \begin{align*} \frac{324}{35} \, \sqrt{-10 \, x^{2} - x + 3} x^{6} + \frac{4059}{350} \, \sqrt{-10 \, x^{2} - x + 3} x^{5} - \frac{192609}{35000} \, \sqrt{-10 \, x^{2} - x + 3} x^{4} - \frac{28719519}{2800000} \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + \frac{144827353}{168000000} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + \frac{4941356543}{1344000000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{12679836719}{25600000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{306881247}{8960000000} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^4/(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

324/35*sqrt(-10*x^2 - x + 3)*x^6 + 4059/350*sqrt(-10*x^2 - x + 3)*x^5 - 192609/35000*sqrt(-10*x^2 - x + 3)*x^4
- 28719519/2800000*sqrt(-10*x^2 - x + 3)*x^3 + 144827353/168000000*sqrt(-10*x^2 - x + 3)*x^2 + 4941356543/134
4000000*sqrt(-10*x^2 - x + 3)*x - 12679836719/25600000000*sqrt(10)*arcsin(-20/11*x - 1/11) - 306881247/8960000
000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.81771, size = 378, normalized size = 2.11 \begin{align*} \frac{1}{26880000000} \,{\left (248832000000 \, x^{6} + 311731200000 \, x^{5} - 147923712000 \, x^{4} - 275707382400 \, x^{3} + 23172376480 \, x^{2} + 98827130860 \, x - 920643741\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{12679836719}{25600000000} \, \sqrt{10} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^4/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

1/26880000000*(248832000000*x^6 + 311731200000*x^5 - 147923712000*x^4 - 275707382400*x^3 + 23172376480*x^2 + 9
8827130860*x - 920643741)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 12679836719/25600000000*sqrt(10)*arctan(1/20*sqrt(10)
*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

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

[Out]

Timed out

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Giac [B]  time = 2.60289, size = 602, normalized size = 3.36 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^4/(3+5*x)^(1/2),x, algorithm="giac")

[Out]

27/448000000000*sqrt(5)*(2*(4*(8*(4*(16*(20*(120*x - 443)*(5*x + 3) + 94933)*(5*x + 3) - 7838433)*(5*x + 3) +
98794353)*(5*x + 3) - 1568443065)*(5*x + 3) + 8438816295)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 17534989395*sqrt(2)*
arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 9/640000000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 311)*(5*x + 3) + 46071)*(5
*x + 3) - 775911)*(5*x + 3) + 15385695)*(5*x + 3) - 99422145)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 220189365*sqrt(2
)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 27/320000000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*
x + 3) - 506185)*(5*x + 3) + 4031895)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcsin(1/11*sqrt(22)*sq
rt(5*x + 3))) - 11/400000*sqrt(5)*(2*(4*(8*(60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqr
t(-10*x + 5) - 184305*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 13/15000*sqrt(5)*(2*(4*(40*x - 59)*(5*x +
3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 2/125*sqrt(5)*
(2*(20*x - 23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 8/25*sqrt(5)
*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))