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

Optimal. Leaf size=135 \[ -\frac {3}{50} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^3-\frac {987 \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^2}{4000}-\frac {21 \sqrt {1-2 x} (5 x+3)^{3/2} (92040 x+194923)}{640000}-\frac {97032047 \sqrt {1-2 x} \sqrt {5 x+3}}{2560000}+\frac {1067352517 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2560000 \sqrt {10}} \]

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Rubi [A]  time = 0.04, antiderivative size = 135, normalized size of antiderivative = 1.00, 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 = {100, 153, 147, 50, 54, 216} \begin {gather*} -\frac {3}{50} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^3-\frac {987 \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^2}{4000}-\frac {21 \sqrt {1-2 x} (5 x+3)^{3/2} (92040 x+194923)}{640000}-\frac {97032047 \sqrt {1-2 x} \sqrt {5 x+3}}{2560000}+\frac {1067352517 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2560000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-97032047*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/2560000 - (987*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(3/2))/4000 - (3*Sq
rt[1 - 2*x]*(2 + 3*x)^3*(3 + 5*x)^(3/2))/50 - (21*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)*(194923 + 92040*x))/640000 + (
1067352517*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(2560000*Sqrt[10])

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 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 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 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)^4 \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx &=-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}-\frac {1}{50} \int \frac {\left (-308-\frac {987 x}{2}\right ) (2+3 x)^2 \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {987 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{4000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}+\frac {\int \frac {(2+3 x) \sqrt {3+5 x} \left (\frac {75929}{2}+\frac {241605 x}{4}\right )}{\sqrt {1-2 x}} \, dx}{2000}\\ &=-\frac {987 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{4000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}-\frac {21 \sqrt {1-2 x} (3+5 x)^{3/2} (194923+92040 x)}{640000}+\frac {97032047 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{1280000}\\ &=-\frac {97032047 \sqrt {1-2 x} \sqrt {3+5 x}}{2560000}-\frac {987 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{4000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}-\frac {21 \sqrt {1-2 x} (3+5 x)^{3/2} (194923+92040 x)}{640000}+\frac {1067352517 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{5120000}\\ &=-\frac {97032047 \sqrt {1-2 x} \sqrt {3+5 x}}{2560000}-\frac {987 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{4000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}-\frac {21 \sqrt {1-2 x} (3+5 x)^{3/2} (194923+92040 x)}{640000}+\frac {1067352517 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{2560000 \sqrt {5}}\\ &=-\frac {97032047 \sqrt {1-2 x} \sqrt {3+5 x}}{2560000}-\frac {987 \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}}{4000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}-\frac {21 \sqrt {1-2 x} (3+5 x)^{3/2} (194923+92040 x)}{640000}+\frac {1067352517 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{2560000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 88, normalized size = 0.65 \begin {gather*} -\frac {\sqrt {1-2 x} \left (10 \sqrt {2 x-1} \sqrt {5 x+3} \left (20736000 x^4+82339200 x^3+146144160 x^2+163168620 x+157419203\right )[phone] \sqrt {10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{25600000 \sqrt {2 x-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/25600000*(Sqrt[1 - 2*x]*(10*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(157419203 + 163168620*x + 146144160*x^2 + 8233920
0*x^3 + 20736000*x^4) + 1067352517*Sqrt[10]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]]))/Sqrt[-1 + 2*x]

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IntegrateAlgebraic [A]  time = 0.67, size = 147, normalized size = 1.09 \begin {gather*} \frac {\sqrt {11-2 (5 x+3)} \left (-165888 \sqrt {5} (5 x+3)^{9/2}-1302912 \sqrt {5} (5 x+3)^{7/2}-8544672 \sqrt {5} (5 x+3)^{5/2}-58806060 \sqrt {5} (5 x+3)^{3/2}-485160235 \sqrt {5} \sqrt {5 x+3}\right )}{64000000}-\frac {1067352517 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{1280000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[11 - 2*(3 + 5*x)]*(-485160235*Sqrt[5]*Sqrt[3 + 5*x] - 58806060*Sqrt[5]*(3 + 5*x)^(3/2) - 8544672*Sqrt[5]
*(3 + 5*x)^(5/2) - 1302912*Sqrt[5]*(3 + 5*x)^(7/2) - 165888*Sqrt[5]*(3 + 5*x)^(9/2)))/64000000 - (1067352517*A
rcTan[(Sqrt[2]*Sqrt[3 + 5*x])/(Sqrt[11] - Sqrt[11 - 2*(3 + 5*x)])])/(1280000*Sqrt[10])

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fricas [A]  time = 1.97, size = 77, normalized size = 0.57 \begin {gather*} -\frac {1}{2560000} \, {\left (20736000 \, x^{4} + 82339200 \, x^{3} + 146144160 \, x^{2} + 163168620 \, x + 157419203\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {1067352517}{51200000} \, \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((2+3*x)^4*(3+5*x)^(1/2)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/2560000*(20736000*x^4 + 82339200*x^3 + 146144160*x^2 + 163168620*x + 157419203)*sqrt(5*x + 3)*sqrt(-2*x + 1
) - 1067352517/51200000*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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giac [A]  time = 1.30, size = 72, normalized size = 0.53 \begin {gather*} -\frac {1}{128000000} \, \sqrt {5} {\left (2 \, {\left (12 \, {\left (24 \, {\left (12 \, {\left (240 \, x + 521\right )} {\left (5 \, x + 3\right )} + 29669\right )} {\left (5 \, x + 3\right )} + 4900505\right )} {\left (5 \, x + 3\right )} + 485160235\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 5336762585 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/128000000*sqrt(5)*(2*(12*(24*(12*(240*x + 521)*(5*x + 3) + 29669)*(5*x + 3) + 4900505)*(5*x + 3) + 48516023
5)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 5336762585*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))

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maple [A]  time = 0.02, size = 121, normalized size = 0.90 \begin {gather*} \frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (-414720000 \sqrt {-10 x^{2}-x +3}\, x^{4}-1646784000 \sqrt {-10 x^{2}-x +3}\, x^{3}-2922883200 \sqrt {-10 x^{2}-x +3}\, x^{2}-3263372400 \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-3148384060 \sqrt {-10 x^{2}-x +3}\right )}{51200000 \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/51200000*(5*x+3)^(1/2)*(-2*x+1)^(1/2)*(-414720000*(-10*x^2-x+3)^(1/2)*x^4-1646784000*(-10*x^2-x+3)^(1/2)*x^3
-2922883200*(-10*x^2-x+3)^(1/2)*x^2+1067352517*10^(1/2)*arcsin(20/11*x+1/11)-3263372400*(-10*x^2-x+3)^(1/2)*x-
3148384060*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.21, size = 90, normalized size = 0.67 \begin {gather*} \frac {81}{100} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {25083}{8000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {1067352517}{51200000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {180423}{32000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {8640723}{128000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {200720723}{2560000} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

81/100*(-10*x^2 - x + 3)^(3/2)*x^2 + 25083/8000*(-10*x^2 - x + 3)^(3/2)*x + 1067352517/51200000*sqrt(5)*sqrt(2
)*arcsin(20/11*x + 1/11) + 180423/32000*(-10*x^2 - x + 3)^(3/2) - 8640723/128000*sqrt(-10*x^2 - x + 3)*x - 200
720723/2560000*sqrt(-10*x^2 - x + 3)

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mupad [B]  time = 14.33, size = 882, normalized size = 6.53

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1067352517*10^(1/2)*atan((10^(1/2)*((1 - 2*x)^(1/2) - 1))/(2*(3^(1/2) - (5*x + 3)^(1/2)))))/12800000 - ((8215
92517*((1 - 2*x)^(1/2) - 1))/(24414062500*(3^(1/2) - (5*x + 3)^(1/2))) - (1047293669*((1 - 2*x)^(1/2) - 1)^3)/
(9765625000*(3^(1/2) - (5*x + 3)^(1/2))^3) - (47930155877*((1 - 2*x)^(1/2) - 1)^5)/(4882812500*(3^(1/2) - (5*x
+ 3)^(1/2))^5) - (3319241183*((1 - 2*x)^(1/2) - 1)^7)/(390625000*(3^(1/2) - (5*x + 3)^(1/2))^7) - (7192921169
*((1 - 2*x)^(1/2) - 1)^9)/(312500000*(3^(1/2) - (5*x + 3)^(1/2))^9) + (7192921169*((1 - 2*x)^(1/2) - 1)^11)/(1
25000000*(3^(1/2) - (5*x + 3)^(1/2))^11) + (3319241183*((1 - 2*x)^(1/2) - 1)^13)/(25000000*(3^(1/2) - (5*x + 3
)^(1/2))^13) + (47930155877*((1 - 2*x)^(1/2) - 1)^15)/(50000000*(3^(1/2) - (5*x + 3)^(1/2))^15) + (1047293669*
((1 - 2*x)^(1/2) - 1)^17)/(16000000*(3^(1/2) - (5*x + 3)^(1/2))^17) - (821592517*((1 - 2*x)^(1/2) - 1)^19)/(64
00000*(3^(1/2) - (5*x + 3)^(1/2))^19) + (753664*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(9765625*(3^(1/2) - (5*x + 3)
^(1/2))^2) + (1359872*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^4) + (54837248*3^(1
/2)*((1 - 2*x)^(1/2) - 1)^6)/(1953125*(3^(1/2) - (5*x + 3)^(1/2))^6) + (768075776*3^(1/2)*((1 - 2*x)^(1/2) - 1
)^8)/(9765625*(3^(1/2) - (5*x + 3)^(1/2))^8) + (721424384*3^(1/2)*((1 - 2*x)^(1/2) - 1)^10)/(1953125*(3^(1/2)
- (5*x + 3)^(1/2))^10) + (192018944*3^(1/2)*((1 - 2*x)^(1/2) - 1)^12)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^12)
+ (3427328*3^(1/2)*((1 - 2*x)^(1/2) - 1)^14)/(3125*(3^(1/2) - (5*x + 3)^(1/2))^14) + (21248*3^(1/2)*((1 - 2*x)
^(1/2) - 1)^16)/(25*(3^(1/2) - (5*x + 3)^(1/2))^16) + (2944*3^(1/2)*((1 - 2*x)^(1/2) - 1)^18)/(25*(3^(1/2) - (
5*x + 3)^(1/2))^18))/((1024*((1 - 2*x)^(1/2) - 1)^2)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^2) + (2304*((1 - 2*x)
^(1/2) - 1)^4)/(78125*(3^(1/2) - (5*x + 3)^(1/2))^4) + (3072*((1 - 2*x)^(1/2) - 1)^6)/(15625*(3^(1/2) - (5*x +
3)^(1/2))^6) + (2688*((1 - 2*x)^(1/2) - 1)^8)/(3125*(3^(1/2) - (5*x + 3)^(1/2))^8) + (8064*((1 - 2*x)^(1/2) -
1)^10)/(3125*(3^(1/2) - (5*x + 3)^(1/2))^10) + (672*((1 - 2*x)^(1/2) - 1)^12)/(125*(3^(1/2) - (5*x + 3)^(1/2)
)^12) + (192*((1 - 2*x)^(1/2) - 1)^14)/(25*(3^(1/2) - (5*x + 3)^(1/2))^14) + (36*((1 - 2*x)^(1/2) - 1)^16)/(5*
(3^(1/2) - (5*x + 3)^(1/2))^16) + (4*((1 - 2*x)^(1/2) - 1)^18)/(3^(1/2) - (5*x + 3)^(1/2))^18 + ((1 - 2*x)^(1/
2) - 1)^20/(3^(1/2) - (5*x + 3)^(1/2))^20 + 1024/9765625)

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sympy [A]  time = 59.70, size = 665, normalized size = 4.93 \begin {gather*} \frac {2 \sqrt {5} \left (\begin {cases} \frac {11 \sqrt {2} \left (- \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {\operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{2}\right )}{4} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{3125} + \frac {24 \sqrt {5} \left (\begin {cases} \frac {121 \sqrt {2} \left (\frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{968} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{8}\right )}{8} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{3125} + \frac {108 \sqrt {5} \left (\begin {cases} \frac {1331 \sqrt {2} \left (\frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} + \frac {3 \sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{1936} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{16}\right )}{16} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{3125} + \frac {216 \sqrt {5} \left (\begin {cases} \frac {14641 \sqrt {2} \left (\frac {2 \sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} + \frac {7 \sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{3872} + \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{1874048} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {35 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{128}\right )}{32} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{3125} + \frac {162 \sqrt {5} \left (\begin {cases} \frac {161051 \sqrt {2} \left (- \frac {2 \sqrt {2} \left (5 - 10 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{805255} + \frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{1331} + \frac {15 \sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{7744} + \frac {5 \sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{3748096} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {63 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{256}\right )}{64} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{3125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(5)*Piecewise((11*sqrt(2)*(-sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + asin(sqrt(22)*sqrt(5*x + 3)/11)/2)
/4, (x >= -3/5) & (x < 1/2)))/3125 + 24*sqrt(5)*Piecewise((121*sqrt(2)*(sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqr
t(5*x + 3)/968 - sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + 3*asin(sqrt(22)*sqrt(5*x + 3)/11)/8)/8, (x >= -3/5)
& (x < 1/2)))/3125 + 108*sqrt(5)*Piecewise((1331*sqrt(2)*(sqrt(2)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/3993 + 3
*sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/1936 - sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + 5*asin(sqrt
(22)*sqrt(5*x + 3)/11)/16)/16, (x >= -3/5) & (x < 1/2)))/3125 + 216*sqrt(5)*Piecewise((14641*sqrt(2)*(2*sqrt(2
)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/3993 + 7*sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/3872 + sqrt(2)*
sqrt(5 - 10*x)*sqrt(5*x + 3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 - 5929)/1874048 - sqrt(2)*sqrt(5
- 10*x)*sqrt(5*x + 3)/22 + 35*asin(sqrt(22)*sqrt(5*x + 3)/11)/128)/32, (x >= -3/5) & (x < 1/2)))/3125 + 162*s
qrt(5)*Piecewise((161051*sqrt(2)*(-2*sqrt(2)*(5 - 10*x)**(5/2)*(5*x + 3)**(5/2)/805255 + sqrt(2)*(5 - 10*x)**(
3/2)*(5*x + 3)**(3/2)/1331 + 15*sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/7744 + 5*sqrt(2)*sqrt(5 - 10*
x)*sqrt(5*x + 3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 - 5929)/3748096 - sqrt(2)*sqrt(5 - 10*x)*sqr
t(5*x + 3)/22 + 63*asin(sqrt(22)*sqrt(5*x + 3)/11)/256)/64, (x >= -3/5) & (x < 1/2)))/3125

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