3.24.62 \(\int \frac {(1-2 x)^{3/2} (2+3 x)^2}{\sqrt {3+5 x}} \, dx\) [2362]

Optimal. Leaf size=121 \[ \frac {9933 \sqrt {1-2 x} \sqrt {3+5 x}}{32000}+\frac {301 (1-2 x)^{3/2} \sqrt {3+5 x}}{3200}-\frac {119}{800} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {3}{40} (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x}+\frac {109263 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{32000 \sqrt {10}} \]

[Out]

109263/320000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+301/3200*(1-2*x)^(3/2)*(3+5*x)^(1/2)-119/800*(1-2*x
)^(5/2)*(3+5*x)^(1/2)-3/40*(1-2*x)^(5/2)*(2+3*x)*(3+5*x)^(1/2)+9933/32000*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {109263 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{32000 \sqrt {10}}-\frac {3}{40} (3 x+2) \sqrt {5 x+3} (1-2 x)^{5/2}-\frac {119}{800} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {301 \sqrt {5 x+3} (1-2 x)^{3/2}}{3200}+\frac {9933 \sqrt {5 x+3} \sqrt {1-2 x}}{32000} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(9933*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/32000 + (301*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/3200 - (119*(1 - 2*x)^(5/2)*Sqr
t[3 + 5*x])/800 - (3*(1 - 2*x)^(5/2)*(2 + 3*x)*Sqrt[3 + 5*x])/40 + (109263*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(
32000*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)^{3/2} (2+3 x)^2}{\sqrt {3+5 x}} \, dx &=-\frac {3}{40} (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x}-\frac {1}{40} \int \frac {\left (-112-\frac {357 x}{2}\right ) (1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {119}{800} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {3}{40} (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x}+\frac {301}{320} \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx\\ &=\frac {301 (1-2 x)^{3/2} \sqrt {3+5 x}}{3200}-\frac {119}{800} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {3}{40} (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x}+\frac {9933 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{6400}\\ &=\frac {9933 \sqrt {1-2 x} \sqrt {3+5 x}}{32000}+\frac {301 (1-2 x)^{3/2} \sqrt {3+5 x}}{3200}-\frac {119}{800} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {3}{40} (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x}+\frac {109263 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{64000}\\ &=\frac {9933 \sqrt {1-2 x} \sqrt {3+5 x}}{32000}+\frac {301 (1-2 x)^{3/2} \sqrt {3+5 x}}{3200}-\frac {119}{800} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {3}{40} (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x}+\frac {109263 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{32000 \sqrt {5}}\\ &=\frac {9933 \sqrt {1-2 x} \sqrt {3+5 x}}{32000}+\frac {301 (1-2 x)^{3/2} \sqrt {3+5 x}}{3200}-\frac {119}{800} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {3}{40} (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x}+\frac {109263 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{32000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 78, normalized size = 0.64 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (10149+91975 x+96780 x^2-133600 x^3-144000 x^4\right )-109263 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{320000 \sqrt {3+5 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[1 - 2*x]*(10149 + 91975*x + 96780*x^2 - 133600*x^3 - 144000*x^4) - 109263*Sqrt[30 + 50*x]*ArcTan[Sqrt
[5/2 - 5*x]/Sqrt[3 + 5*x]])/(320000*Sqrt[3 + 5*x])

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Maple [A]
time = 0.11, size = 104, normalized size = 0.86

method	result	size
risch	\(\frac {\left (28800 x^{3}+9440 x^{2}-25020 x -3383\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{32000 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {109263 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{640000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(103\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (-576000 x^{3} \sqrt {-10 x^{2}-x +3}-188800 x^{2} \sqrt {-10 x^{2}-x +3}+109263 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+500400 x \sqrt {-10 x^{2}-x +3}+67660 \sqrt {-10 x^{2}-x +3}\right )}{640000 \sqrt {-10 x^{2}-x +3}}\)	\(104\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/640000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-576000*x^3*(-10*x^2-x+3)^(1/2)-188800*x^2*(-10*x^2-x+3)^(1/2)+109263*10
^(1/2)*arcsin(20/11*x+1/11)+500400*x*(-10*x^2-x+3)^(1/2)+67660*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]
time = 0.53, size = 75, normalized size = 0.62 \begin {gather*} -\frac {9}{10} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {59}{200} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + \frac {1251}{1600} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {109263}{640000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {3383}{32000} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/10*sqrt(-10*x^2 - x + 3)*x^3 - 59/200*sqrt(-10*x^2 - x + 3)*x^2 + 1251/1600*sqrt(-10*x^2 - x + 3)*x - 10926
3/640000*sqrt(10)*arcsin(-20/11*x - 1/11) + 3383/32000*sqrt(-10*x^2 - x + 3)

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Fricas [A]
time = 1.28, size = 72, normalized size = 0.60 \begin {gather*} -\frac {1}{32000} \, {\left (28800 \, x^{3} + 9440 \, x^{2} - 25020 \, x - 3383\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {109263}{640000} \, \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)^(3/2)*(2+3*x)^2/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

-1/32000*(28800*x^3 + 9440*x^2 - 25020*x - 3383)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 109263/640000*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 = 86.59, size = 456, normalized size = 3.77 \begin {gather*} - \frac {49 \sqrt {2} \left (\begin {cases} \frac {121 \sqrt {5} \left (\frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \cdot \left (20 x + 1\right )}{968} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{8}\right )}{125} & \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 {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 )}{2} - \frac {9 \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 )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-49*sqrt(2)*Piecewise((121*sqrt(5)*(sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/968 - sqrt(5)*sqrt(1 - 2*x
)*sqrt(10*x + 6)/22 + 3*asin(sqrt(55)*sqrt(1 - 2*x)/11)/8)/125, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x)
< sqrt(55)/5)))/4 + 21*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)))/2 - 9*sqrt(2)*Pi
ecewise((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)/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 - sqrt(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(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. 203 vs. \(2 (88) = 176\).
time = 1.16, size = 203, normalized size = 1.68 \begin {gather*} -\frac {3}{1600000} \, \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 {1}{8000} \, \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)^(3/2)*(2+3*x)^2/(3+5*x)^(1/2),x, algorithm="giac")

[Out]

-3/1600000*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))) - 1/8000*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)*sq
rt(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)*arcs
in(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 )}^{3/2}\,{\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)^(3/2)*(3*x + 2)^2)/(5*x + 3)^(1/2),x)

[Out]

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

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