3.2425 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^3}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=150 \[ -\frac{\sqrt{5 x+3} (47280 x+52951) (1-2 x)^{7/2}}{160000}-\frac{1}{20} (3 x+2)^2 \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{276493 \sqrt{5 x+3} (1-2 x)^{5/2}}{4800000}+\frac{3041423 \sqrt{5 x+3} (1-2 x)^{3/2}}{19200000}+\frac{33455653 \sqrt{5 x+3} \sqrt{1-2 x}}{64000000}+\frac{368012183 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{64000000 \sqrt{10}} \]

[Out]

(33455653*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/64000000 + (3041423*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/19200000 + (276493*(
1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/4800000 - ((1 - 2*x)^(7/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/20 - ((1 - 2*x)^(7/2)*Sqrt
[3 + 5*x]*(52951 + 47280*x))/160000 + (368012183*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(64000000*Sqrt[10])

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Rubi [A]  time = 0.0426855, antiderivative size = 150, 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 = {100, 147, 50, 54, 216} \[ -\frac{\sqrt{5 x+3} (47280 x+52951) (1-2 x)^{7/2}}{160000}-\frac{1}{20} (3 x+2)^2 \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{276493 \sqrt{5 x+3} (1-2 x)^{5/2}}{4800000}+\frac{3041423 \sqrt{5 x+3} (1-2 x)^{3/2}}{19200000}+\frac{33455653 \sqrt{5 x+3} \sqrt{1-2 x}}{64000000}+\frac{368012183 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{64000000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(33455653*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/64000000 + (3041423*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/19200000 + (276493*(
1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/4800000 - ((1 - 2*x)^(7/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/20 - ((1 - 2*x)^(7/2)*Sqrt
[3 + 5*x]*(52951 + 47280*x))/160000 + (368012183*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(64000000*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 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)^3}{\sqrt{3+5 x}} \, dx &=-\frac{1}{20} (1-2 x)^{7/2} (2+3 x)^2 \sqrt{3+5 x}-\frac{1}{60} \int \frac{\left (-183-\frac{591 x}{2}\right ) (1-2 x)^{5/2} (2+3 x)}{\sqrt{3+5 x}} \, dx\\ &=-\frac{1}{20} (1-2 x)^{7/2} (2+3 x)^2 \sqrt{3+5 x}-\frac{(1-2 x)^{7/2} \sqrt{3+5 x} (52951+47280 x)}{160000}+\frac{276493 \int \frac{(1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx}{320000}\\ &=\frac{276493 (1-2 x)^{5/2} \sqrt{3+5 x}}{4800000}-\frac{1}{20} (1-2 x)^{7/2} (2+3 x)^2 \sqrt{3+5 x}-\frac{(1-2 x)^{7/2} \sqrt{3+5 x} (52951+47280 x)}{160000}+\frac{3041423 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{1920000}\\ &=\frac{3041423 (1-2 x)^{3/2} \sqrt{3+5 x}}{19200000}+\frac{276493 (1-2 x)^{5/2} \sqrt{3+5 x}}{4800000}-\frac{1}{20} (1-2 x)^{7/2} (2+3 x)^2 \sqrt{3+5 x}-\frac{(1-2 x)^{7/2} \sqrt{3+5 x} (52951+47280 x)}{160000}+\frac{33455653 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{12800000}\\ &=\frac{33455653 \sqrt{1-2 x} \sqrt{3+5 x}}{64000000}+\frac{3041423 (1-2 x)^{3/2} \sqrt{3+5 x}}{19200000}+\frac{276493 (1-2 x)^{5/2} \sqrt{3+5 x}}{4800000}-\frac{1}{20} (1-2 x)^{7/2} (2+3 x)^2 \sqrt{3+5 x}-\frac{(1-2 x)^{7/2} \sqrt{3+5 x} (52951+47280 x)}{160000}+\frac{368012183 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{128000000}\\ &=\frac{33455653 \sqrt{1-2 x} \sqrt{3+5 x}}{64000000}+\frac{3041423 (1-2 x)^{3/2} \sqrt{3+5 x}}{19200000}+\frac{276493 (1-2 x)^{5/2} \sqrt{3+5 x}}{4800000}-\frac{1}{20} (1-2 x)^{7/2} (2+3 x)^2 \sqrt{3+5 x}-\frac{(1-2 x)^{7/2} \sqrt{3+5 x} (52951+47280 x)}{160000}+\frac{368012183 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{64000000 \sqrt{5}}\\ &=\frac{33455653 \sqrt{1-2 x} \sqrt{3+5 x}}{64000000}+\frac{3041423 (1-2 x)^{3/2} \sqrt{3+5 x}}{19200000}+\frac{276493 (1-2 x)^{5/2} \sqrt{3+5 x}}{4800000}-\frac{1}{20} (1-2 x)^{7/2} (2+3 x)^2 \sqrt{3+5 x}-\frac{(1-2 x)^{7/2} \sqrt{3+5 x} (52951+47280 x)}{160000}+\frac{368012183 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{64000000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.139108, size = 84, normalized size = 0.56 \[ \frac{-10 \sqrt{5 x+3} \left (1382400000 x^6-13824000 x^5-1797292800 x^4+261623360 x^3+903127240 x^2-254844442 x-39899709\right )-1104036549 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1920000000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(-39899709 - 254844442*x + 903127240*x^2 + 261623360*x^3 - 1797292800*x^4 - 13824000*x^5 +
1382400000*x^6) - 1104036549*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1920000000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.008, size = 138, normalized size = 0.9 \begin{align*}{\frac{1}{3840000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 13824000000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-14586048000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-4676790400\,{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}[phone]\,\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)^3/(3+5*x)^(1/2),x)

[Out]

1/3840000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(13824000000*x^5*(-10*x^2-x+3)^(1/2)[phone]*x^4*(-10*x^2-x+3)^(1/
2)-14586048000*x^3*(-10*x^2-x+3)^(1/2)-4676790400*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)[phone]*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 2.39134, size = 147, normalized size = 0.98 \begin{align*} \frac{18}{5} \, \sqrt{-10 \, x^{2} - x + 3} x^{5} + \frac{441}{250} \, \sqrt{-10 \, x^{2} - x + 3} x^{4} - \frac{75969}{20000} \, \sqrt{-10 \, x^{2} - x + 3} x^{3} - \frac{1461497}{1200000} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + \frac{16732193}{9600000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{368012183}{1280000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{13299903}{64000000} \, \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)^3/(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

18/5*sqrt(-10*x^2 - x + 3)*x^5 + 441/250*sqrt(-10*x^2 - x + 3)*x^4 - 75969/20000*sqrt(-10*x^2 - x + 3)*x^3 - 1
461497/1200000*sqrt(-10*x^2 - x + 3)*x^2 + 16732193/9600000*sqrt(-10*x^2 - x + 3)*x - 368012183/1280000000*sqr
t(10)*arcsin(-20/11*x - 1/11) + 13299903/64000000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.78735, size = 327, normalized size = 2.18 \begin{align*} \frac{1}{192000000} \,{\left (691200000 \, x^{5} + 338688000 \, x^{4} - 729302400 \, x^{3} - 233839520 \, x^{2} + 334643860 \, x + 39899709\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{368012183}{1280000000} \, \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)^3/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

1/192000000*(691200000*x^5 + 338688000*x^4 - 729302400*x^3 - 233839520*x^2 + 334643860*x + 39899709)*sqrt(5*x
+ 3)*sqrt(-2*x + 1) - 368012183/1280000000*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)**3/(3+5*x)**(1/2),x)

[Out]

Timed out

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Giac [B]  time = 2.1218, size = 481, normalized size = 3.21 \begin{align*} \frac{9}{3200000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (100 \, x - 311\right )}{\left (5 \, x + 3\right )} + 46071\right )}{\left (5 \, x + 3\right )} - 775911\right )}{\left (5 \, x + 3\right )} + 15385695\right )}{\left (5 \, x + 3\right )} - 99422145\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 220189365 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{9}{80000000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (12 \,{\left (80 \, x - 203\right )}{\left (5 \, x + 3\right )} + 19073\right )}{\left (5 \, x + 3\right )} - 506185\right )}{\left (5 \, x + 3\right )} + 4031895\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 10392195 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{3}{640000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 119\right )}{\left (5 \, x + 3\right )} + 6163\right )}{\left (5 \, x + 3\right )} - 66189\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 184305 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{29}{60000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 59\right )}{\left (5 \, x + 3\right )} + 1293\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 4785 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{500} \, \sqrt{5}{\left (2 \,{\left (20 \, x - 23\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 143 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{4}{25} \, \sqrt{5}{\left (11 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + 2 \, \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/3200000000*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)))
+ 9/80000000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) - 506185)*(5*x + 3) + 4031895)*sq
rt(5*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 3/640000*sqrt(5)*(2*(4*(
8*(60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 184305*sqrt(2)*arcsin(1/11
*sqrt(22)*sqrt(5*x + 3))) - 29/60000*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))) + 1/500*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))) + 4/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*
x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))