3.23.86 \(\int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2} \, dx\) [2286]

Optimal. Leaf size=201 \[ \frac {180773237579 \sqrt {1-2 x} \sqrt {3+5 x}}{1310720000}-\frac {16433930689 (1-2 x)^{3/2} \sqrt {3+5 x}}{131072000}-\frac {1493993699 (1-2 x)^{3/2} (3+5 x)^{3/2}}{49152000}-\frac {135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac {1988505613369 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1310720000 \sqrt {10}} \]

[Out]

-1493993699/49152000*(1-2*x)^(3/2)*(3+5*x)^(3/2)-135817609/20480000*(1-2*x)^(3/2)*(3+5*x)^(5/2)-1419/11200*(1-
2*x)^(3/2)*(2+3*x)^2*(3+5*x)^(7/2)-3/80*(1-2*x)^(3/2)*(2+3*x)^3*(3+5*x)^(7/2)-3/1280000*(1-2*x)^(3/2)*(3+5*x)^
(7/2)*(899099+522420*x)[phone]/13107200000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-16433930689/131
072000*(1-2*x)^(3/2)*(3+5*x)^(1/2)[phone]/1310720000*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {102, 158, 152, 52, 56, 222} \begin {gather*} \frac {1988505613369 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1310720000 \sqrt {10}}-\frac {3}{80} (1-2 x)^{3/2} (3 x+2)^3 (5 x+3)^{7/2}-\frac {1419 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{7/2}}{11200}-\frac {3 (1-2 x)^{3/2} (522420 x+899099) (5 x+3)^{7/2}}{1280000}-\frac {135817609 (1-2 x)^{3/2} (5 x+3)^{5/2}}{20480000}-\frac {1493993699 (1-2 x)^{3/2} (5 x+3)^{3/2}}{49152000}-\frac {16433930689 (1-2 x)^{3/2} \sqrt {5 x+3}}{131072000}+\frac {180773237579 \sqrt {1-2 x} \sqrt {5 x+3}}{1310720000} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(180773237579*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1310720000 - (16433930689*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/131072000
- (1493993699*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/49152000 - (135817609*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/20480000
- (1419*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(7/2))/11200 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)^3*(3 + 5*x)^(7/2))/
80 - (3*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2)*(899099 + 522420*x))/1280000 + (1988505613369*ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]])/(1310720000*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 102

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 152

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 158

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 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 \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2} \, dx &=-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {1}{80} \int \left (-452-\frac {1419 x}{2}\right ) \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2} \, dx\\ &=-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}+\frac {\int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \left (\frac {176225}{2}+\frac {548541 x}{4}\right ) \, dx}{5600}\\ &=-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac {135817609 \int \sqrt {1-2 x} (3+5 x)^{5/2} \, dx}{2560000}\\ &=-\frac {135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac {1493993699 \int \sqrt {1-2 x} (3+5 x)^{3/2} \, dx}{8192000}\\ &=-\frac {1493993699 (1-2 x)^{3/2} (3+5 x)^{3/2}}{49152000}-\frac {135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac {16433930689 \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx}{32768000}\\ &=-\frac {16433930689 (1-2 x)^{3/2} \sqrt {3+5 x}}{131072000}-\frac {1493993699 (1-2 x)^{3/2} (3+5 x)^{3/2}}{49152000}-\frac {135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac {180773237579 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{262144000}\\ &=\frac {180773237579 \sqrt {1-2 x} \sqrt {3+5 x}}{1310720000}-\frac {16433930689 (1-2 x)^{3/2} \sqrt {3+5 x}}{131072000}-\frac {1493993699 (1-2 x)^{3/2} (3+5 x)^{3/2}}{49152000}-\frac {135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac {1988505613369 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{2621440000}\\ &=\frac {180773237579 \sqrt {1-2 x} \sqrt {3+5 x}}{1310720000}-\frac {16433930689 (1-2 x)^{3/2} \sqrt {3+5 x}}{131072000}-\frac {1493993699 (1-2 x)^{3/2} (3+5 x)^{3/2}}{49152000}-\frac {135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac {1988505613369 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1310720000 \sqrt {5}}\\ &=\frac {180773237579 \sqrt {1-2 x} \sqrt {3+5 x}}{1310720000}-\frac {16433930689 (1-2 x)^{3/2} \sqrt {3+5 x}}{131072000}-\frac {1493993699 (1-2 x)^{3/2} (3+5 x)^{3/2}}{49152000}-\frac {135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac {1988505613369 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1310720000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 98, normalized size = 0.49 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (-17919913416273-41841631696875 x+690018024740 x^2+131786487855200 x^3+337325400912000 x^4+464971057920000 x^5+381250022400000 x^6+175094784000000 x^7+34836480000000 x^8\right )-41758617880749 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{275251200000 \sqrt {3+5 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[1 - 2*x]*(-17919913416273 - 41841631696875*x + 690018024740*x^2 + 131786487855200*x^3 + 3373254009120
00*x^4 + 464971057920000*x^5 + 381250022400000*x^6 + 175094784000000*x^7 + 34836480000000*x^8) - 4175861788074
9*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(275251200000*Sqrt[3 + 5*x])

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Maple [A]
time = 0.12, size = 172, normalized size = 0.86

method	result	size
risch	\(-\frac {\left (6967296000000 x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}-3991703112140 x -5973304472091\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{27525120000 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {1988505613369 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{26214400000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(123\)
default	\(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (139345920000000 \sqrt {-10 x^{2}-x +3}\, x^{7}[phone] \sqrt {-10 x^{2}-x +3}\, x^{6}+1154937139200000 x^{5} \sqrt {-10 x^{2}-x +3}+1166921948160000 x^{4} \sqrt {-10 x^{2}-x +3}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-79834062242800 x \sqrt {-10 x^{2}-x +3}-119466089441820 \sqrt {-10 x^{2}-x +3}\right )}{550502400000 \sqrt {-10 x^{2}-x +3}}\)	\(172\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/550502400000*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(139345920000000*(-10*x^2-x+3)^(1/2)*x^7+616771584000000*(-10*x^2-x
+3)^(1/2)*x^6+1154937139200000*x^5*(-10*x^2-x+3)^(1/2)+1166921948160000*x^4*(-10*x^2-x+3)^(1/2)[phone]
0*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)
-79834062242800*x*(-10*x^2-x+3)^(1/2)-119466089441820*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]
time = 0.52, size = 138, normalized size = 0.69 \begin {gather*} -\frac {405}{16} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{5} - \frac {49059}{448} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} - \frac {739881}{3584} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {80346831}{358400} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {4513921183}{28672000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {26326737569}{344064000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {16433930689}{65536000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1988505613369}{26214400000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {16433930689}{1310720000} \, \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)^(5/2)*(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

-405/16*(-10*x^2 - x + 3)^(3/2)*x^5 - 49059/448*(-10*x^2 - x + 3)^(3/2)*x^4 - 739881/3584*(-10*x^2 - x + 3)^(3
/2)*x^3 - 80346831/358400*(-10*x^2 - x + 3)^(3/2)*x^2 - 4513921183/28672000*(-10*x^2 - x + 3)^(3/2)*x - 263267
37569/344064000*(-10*x^2 - x + 3)^(3/2) + 16433930689/65536000*sqrt(-10*x^2 - x + 3)*x - 1988505613369/2621440
0000*sqrt(10)*arcsin(-20/11*x - 1/11) + 16433930689/1310720000*sqrt(-10*x^2 - x + 3)

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Fricas [A]
time = 1.09, size = 92, normalized size = 0.46 \begin {gather*} \frac {1}{27525120000} \, {\left (6967296000000 \, x^{7} + 30838579200000 \, x^{6} + 57746856960000 \, x^{5} + 58346097408000 \, x^{4} + 32457421737600 \, x^{3} + 6882844528480 \, x^{2} - 3991703112140 \, x - 5973304472091\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {1988505613369}{26214400000} \, \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)^(5/2)*(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/27525120000*(6967296000000*x^7 + 30838579200000*x^6 + 57746856960000*x^5 + 58346097408000*x^4 + 324574217376
00*x^3 + 6882844528480*x^2 - 3991703112140*x - 5973304472091)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1988505613369/262
14400000*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 = 173.82, size = 1379, normalized size = 6.86 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-290521*sqrt(2)*Piecewise((121*sqrt(5)*(-sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/121 + asin(sqrt(55)*s
qrt(1 - 2*x)/11))/200, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/128 + 381073*sqrt(2)*Pie
cewise((1331*sqrt(5)*(-5*sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)/7986 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6
)*(20*x + 1)/1936 + asin(sqrt(55)*sqrt(1 - 2*x)/11)/16)/125, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) <
sqrt(55)/5)))/64 - 832951*sqrt(2)*Piecewise((14641*sqrt(5)*(-5*sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)/7986
- sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/3872 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(12100*x - 2000
*(1 - 2*x)**3 + 6600*(1 - 2*x)**2 - 4719)/1874048 + 5*asin(sqrt(55)*sqrt(1 - 2*x)/11)/128)/625, (sqrt(1 - 2*x)
> -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/128 + 121359*sqrt(2)*Piecewise((161051*sqrt(5)*(5*sqrt(5)*(1
- 2*x)**(5/2)*(10*x + 6)**(5/2)/322102 - 5*sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)/7986 - sqrt(5)*sqrt(1 -
2*x)*sqrt(10*x + 6)*(20*x + 1)/7744 - 3*sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(12100*x - 2000*(1 - 2*x)**3 + 66
00*(1 - 2*x)**2 - 4719)/3748096 + 7*asin(sqrt(55)*sqrt(1 - 2*x)/11)/256)/3125, (sqrt(1 - 2*x) > -sqrt(55)/5) &
(sqrt(1 - 2*x) < sqrt(55)/5)))/32 - 159111*sqrt(2)*Piecewise((1771561*sqrt(5)*(5*sqrt(5)*(1 - 2*x)**(5/2)*(10
*x + 6)**(5/2)/161051 + 5*sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)*(20*x + 1)**3/170069856 - 5*sqrt(5)*(1 -
2*x)**(3/2)*(10*x + 6)**(3/2)/7986 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/15488 - 13*sqrt(5)*sqrt(1
- 2*x)*sqrt(10*x + 6)*(12100*x - 2000*(1 - 2*x)**3 + 6600*(1 - 2*x)**2 - 4719)/14992384 + 21*asin(sqrt(55)*sq
rt(1 - 2*x)/11)/1024)/15625, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/128 + 13905*sqrt(2
)*Piecewise((19487171*sqrt(5)*(-125*sqrt(5)*(1 - 2*x)**(7/2)*(10*x + 6)**(7/2)/272820394 + 15*sqrt(5)*(1 - 2*x
)**(5/2)*(10*x + 6)**(5/2)/322102 + 25*sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)*(20*x + 1)**3/340139712 - 5*
sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)/7986 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/30976 - 25*s
qrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(12100*x - 2000*(1 - 2*x)**3 + 6600*(1 - 2*x)**2 - 4719)/29984768 + 33*asi
n(sqrt(55)*sqrt(1 - 2*x)/11)/2048)/78125, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/64 -
2025*sqrt(2)*Piecewise((214358881*sqrt(5)*(-375*sqrt(5)*(1 - 2*x)**(7/2)*(10*x + 6)**(7/2)/272820394 + 10*sqrt
(5)*(1 - 2*x)**(5/2)*(10*x + 6)**(5/2)/161051 + 5*sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)*(20*x + 1)**3/425
17464 - 5*sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)/7986 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/61
952 - 91*sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(12100*x - 2000*(1 - 2*x)**3 + 6600*(1 - 2*x)**2 - 4719)/1199390
72 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(744055620*x - 160000000*(1 - 2*x)**7 + 1232000000*(1 - 2*x)**6 - 37
75200000*(1 - 2*x)**5 + 5856400000*(1 - 2*x)**4 - 4831530000*(1 - 2*x)**3 + 2029242600*(1 - 2*x)**2 - 35254063
9)/7024111812608 + 429*asin(sqrt(55)*sqrt(1 - 2*x)/11)/32768)/390625, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1
- 2*x) < sqrt(55)/5)))/128

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 545 vs. \(2 (150) = 300\).
time = 1.30, size = 545, normalized size = 2.71 \begin {gather*} \frac {27}{458752000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (4 \, {\left (24 \, {\left (140 \, x - 599\right )} {\left (5 \, x + 3\right )} + 175163\right )} {\left (5 \, x + 3\right )} - 4295993\right )} {\left (5 \, x + 3\right )} + 265620213\right )} {\left (5 \, x + 3\right )} - 2676516549\right )} {\left (5 \, x + 3\right )} + 35390483373\right )} {\left (5 \, x + 3\right )} - 164483997363\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 309625826895 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {603}{71680000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (20 \, {\left (120 \, x - 443\right )} {\left (5 \, x + 3\right )} + 94933\right )} {\left (5 \, x + 3\right )} - 7838433\right )} {\left (5 \, x + 3\right )} + 98794353\right )} {\left (5 \, x + 3\right )} - 1568443065\right )} {\left (5 \, x + 3\right )} + 8438816295\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 17534989395 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {5769}{2560000000} \, \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 {30649}{320000000} \, \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 {1831}{300000} \, \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 {933}{5000} \, \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 {297}{125} \, \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 {216}{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((2+3*x)^4*(3+5*x)^(5/2)*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

27/458752000000*sqrt(5)*(2*(4*(8*(4*(16*(4*(24*(140*x - 599)*(5*x + 3) + 175163)*(5*x + 3) - 4295993)*(5*x + 3
) + 265620213)*(5*x + 3) - 2676516549)*(5*x + 3) + 35390483373)*(5*x + 3) - 164483997363)*sqrt(5*x + 3)*sqrt(-
10*x + 5) - 309625826895*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 603/71680000000*sqrt(5)*(2*(4*(8*(4*(1
6*(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))) +
5769/2560000000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 311)*(5*x + 3) + 46071)*(5*x + 3) - 775911)*(5*x + 3) + 1538
5695)*(5*x + 3) - 99422145)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 220189365*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x +
3))) + 30649/320000000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) - 506185)*(5*x + 3) + 40
31895)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1831/300000*sqr
t(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))) + 933/5000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqr
t(-10*x + 5) + 4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 297/125*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))) + 216/25*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 = -0.00 \begin {gather*} \int \sqrt {1-2\,x}\,{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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