3.25.26 \(\int \frac {(1-2 x)^{5/2} (2+3 x)^2}{\sqrt {3+5 x}} \, dx\) [2426]

Optimal. Leaf size=143 \[ \frac {593747 \sqrt {1-2 x} \sqrt {3+5 x}}{1600000}+\frac {53977 (1-2 x)^{3/2} \sqrt {3+5 x}}{480000}+\frac {4907 (1-2 x)^{5/2} \sqrt {3+5 x}}{120000}-\frac {369 (1-2 x)^{7/2} \sqrt {3+5 x}}{4000}-\frac {3}{50} (1-2 x)^{7/2} (2+3 x) \sqrt {3+5 x}+\frac {6531217 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1600000 \sqrt {10}} \]

[Out]

6531217/16000000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+53977/480000*(1-2*x)^(3/2)*(3+5*x)^(1/2)+4907/12
0000*(1-2*x)^(5/2)*(3+5*x)^(1/2)-369/4000*(1-2*x)^(7/2)*(3+5*x)^(1/2)-3/50*(1-2*x)^(7/2)*(2+3*x)*(3+5*x)^(1/2)
+593747/1600000*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 143, normalized size of antiderivative = 1.00, 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 = {92, 81, 52, 56, 222} \begin {gather*} \frac {6531217 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1600000 \sqrt {10}}-\frac {3}{50} (3 x+2) \sqrt {5 x+3} (1-2 x)^{7/2}-\frac {369 \sqrt {5 x+3} (1-2 x)^{7/2}}{4000}+\frac {4907 \sqrt {5 x+3} (1-2 x)^{5/2}}{120000}+\frac {53977 \sqrt {5 x+3} (1-2 x)^{3/2}}{480000}+\frac {593747 \sqrt {5 x+3} \sqrt {1-2 x}}{1600000} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(593747*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1600000 + (53977*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/480000 + (4907*(1 - 2*x)^
(5/2)*Sqrt[3 + 5*x])/120000 - (369*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/4000 - (3*(1 - 2*x)^(7/2)*(2 + 3*x)*Sqrt[3 +
5*x])/50 + (6531217*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1600000*Sqrt[10])

Rule 52

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 56

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 81

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 92

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a + b*x
)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 222

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)^2}{\sqrt {3+5 x}} \, dx &=-\frac {3}{50} (1-2 x)^{7/2} (2+3 x) \sqrt {3+5 x}-\frac {1}{50} \int \frac {\left (-116-\frac {369 x}{2}\right ) (1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {369 (1-2 x)^{7/2} \sqrt {3+5 x}}{4000}-\frac {3}{50} (1-2 x)^{7/2} (2+3 x) \sqrt {3+5 x}+\frac {4907 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{8000}\\ &=\frac {4907 (1-2 x)^{5/2} \sqrt {3+5 x}}{120000}-\frac {369 (1-2 x)^{7/2} \sqrt {3+5 x}}{4000}-\frac {3}{50} (1-2 x)^{7/2} (2+3 x) \sqrt {3+5 x}+\frac {53977 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{48000}\\ &=\frac {53977 (1-2 x)^{3/2} \sqrt {3+5 x}}{480000}+\frac {4907 (1-2 x)^{5/2} \sqrt {3+5 x}}{120000}-\frac {369 (1-2 x)^{7/2} \sqrt {3+5 x}}{4000}-\frac {3}{50} (1-2 x)^{7/2} (2+3 x) \sqrt {3+5 x}+\frac {593747 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{320000}\\ &=\frac {593747 \sqrt {1-2 x} \sqrt {3+5 x}}{1600000}+\frac {53977 (1-2 x)^{3/2} \sqrt {3+5 x}}{480000}+\frac {4907 (1-2 x)^{5/2} \sqrt {3+5 x}}{120000}-\frac {369 (1-2 x)^{7/2} \sqrt {3+5 x}}{4000}-\frac {3}{50} (1-2 x)^{7/2} (2+3 x) \sqrt {3+5 x}+\frac {6531217 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{3200000}\\ &=\frac {593747 \sqrt {1-2 x} \sqrt {3+5 x}}{1600000}+\frac {53977 (1-2 x)^{3/2} \sqrt {3+5 x}}{480000}+\frac {4907 (1-2 x)^{5/2} \sqrt {3+5 x}}{120000}-\frac {369 (1-2 x)^{7/2} \sqrt {3+5 x}}{4000}-\frac {3}{50} (1-2 x)^{7/2} (2+3 x) \sqrt {3+5 x}+\frac {6531217 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1600000 \sqrt {5}}\\ &=\frac {593747 \sqrt {1-2 x} \sqrt {3+5 x}}{1600000}+\frac {53977 (1-2 x)^{3/2} \sqrt {3+5 x}}{480000}+\frac {4907 (1-2 x)^{5/2} \sqrt {3+5 x}}{120000}-\frac {369 (1-2 x)^{7/2} \sqrt {3+5 x}}{4000}-\frac {3}{50} (1-2 x)^{7/2} (2+3 x) \sqrt {3+5 x}+\frac {6531217 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1600000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]
time = 0.21, size = 83, normalized size = 0.58 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (4495473+17644875 x-1848740 x^2-37935200 x^3+9648000 x^4+34560000 x^5\right )-19593651 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{48000000 \sqrt {3+5 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[1 - 2*x]*(4495473 + 17644875*x - 1848740*x^2 - 37935200*x^3 + 9648000*x^4 + 34560000*x^5) - 19593651*
Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(48000000*Sqrt[3 + 5*x])

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Maple [A]
time = 0.11, size = 121, normalized size = 0.85

method	result	size
risch	\(-\frac {\left (6912000 x^{4}-2217600 x^{3}-6256480 x^{2}+3384140 x +1498491\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{4800000 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {6531217 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{32000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(108\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (138240000 x^{4} \sqrt {-10 x^{2}-x +3}-44352000 x^{3} \sqrt {-10 x^{2}-x +3}-125129600 x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{96000000 \sqrt {-10 x^{2}-x +3}}\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(138240000*x^4*(-10*x^2-x+3)^(1/2)-44352000*x^3*(-10*x^2-x+3)^(1/2)-125
129600*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 = 0.50, size = 92, normalized size = 0.64 \begin {gather*} \frac {36}{25} \, \sqrt {-10 \, x^{2} - x + 3} x^{4} - \frac {231}{500} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {39103}{30000} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + \frac {169207}{240000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {6531217}{32000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {499497}{1600000} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

36/25*sqrt(-10*x^2 - x + 3)*x^4 - 231/500*sqrt(-10*x^2 - x + 3)*x^3 - 39103/30000*sqrt(-10*x^2 - x + 3)*x^2 +
169207/240000*sqrt(-10*x^2 - x + 3)*x - 6531217/32000000*sqrt(10)*arcsin(-20/11*x - 1/11) + 499497/1600000*sqr
t(-10*x^2 - x + 3)

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Fricas [A]
time = 0.50, size = 77, normalized size = 0.54 \begin {gather*} \frac {1}{4800000} \, {\left (6912000 \, x^{4} - 2217600 \, x^{3} - 6256480 \, x^{2} + 3384140 \, x + 1498491\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {6531217}{32000000} \, \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4800000*(6912000*x^4 - 2217600*x^3 - 6256480*x^2 + 3384140*x + 1498491)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 65312
17/32000000*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 [A]
time = 183.34, size = 556, normalized size = 3.89 \begin {gather*} - \frac {49 \sqrt {2} \left (\begin {cases} \frac {1331 \sqrt {5} \cdot \left (\frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} + \frac {3 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{1936} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{16}\right )}{625} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{4} + \frac {21 \sqrt {2} \left (\begin {cases} \frac {14641 \sqrt {5} \cdot \left (\frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{3993} + \frac {7 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{3872} + \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{1874048} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {35 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{128}\right )}{3125} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{2} - \frac {9 \sqrt {2} \left (\begin {cases} \frac {161051 \sqrt {5} \left (- \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {5}{2}} \left (10 x + 6\right )^{\frac {5}{2}}}{322102} + \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{2662} + \frac {15 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{7744} + \frac {5 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{3748096} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {63 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{256}\right )}{15625} & \text {for}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-49*sqrt(2)*Piecewise((1331*sqrt(5)*(5*sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)/7986 + 3*sqrt(5)*sqrt(1 - 2*
x)*sqrt(10*x + 6)*(20*x + 1)/1936 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)/22 + 5*asin(sqrt(55)*sqrt(1 - 2*x)/11
)/16)/625, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/4 + 21*sqrt(2)*Piecewise((14641*sqrt
(5)*(5*sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)/3993 + 7*sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/387
2 + sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(12100*x - 2000*(1 - 2*x)**3 + 6600*(1 - 2*x)**2 - 4719)/1874048 - sq
rt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)/22 + 35*asin(sqrt(55)*sqrt(1 - 2*x)/11)/128)/3125, (sqrt(1 - 2*x) > -sqrt(5
5)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/2 - 9*sqrt(2)*Piecewise((161051*sqrt(5)*(-5*sqrt(5)*(1 - 2*x)**(5/2)*(1
0*x + 6)**(5/2)/322102 + 5*sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)/2662 + 15*sqrt(5)*sqrt(1 - 2*x)*sqrt(10*
x + 6)*(20*x + 1)/7744 + 5*sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(12100*x - 2000*(1 - 2*x)**3 + 6600*(1 - 2*x)*
*2 - 4719)/3748096 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)/22 + 63*asin(sqrt(55)*sqrt(1 - 2*x)/11)/256)/15625,
(sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/4

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (104) = 208\).
time = 0.58, size = 275, normalized size = 1.92 \begin {gather*} \frac {3}{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 {1}{800000} \, \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 {23}{120000} \, \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 {2}{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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/80000000*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)*sqrt(5*x + 3))) + 1/800000*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*sq
rt(22)*sqrt(5*x + 3))) - 23/120000*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))) + 2/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x
+ 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^2}{\sqrt {5\,x+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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