3.2302 \(\int \frac {\sqrt {1-2 x} (2+3 x)^2}{\sqrt {3+5 x}} \, dx\)

Optimal. Leaf size=99 \[ -\frac {1}{10} (3 x+2) \sqrt {5 x+3} (1-2 x)^{3/2}-\frac {23}{80} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {277}{800} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {3047 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{800 \sqrt {10}} \]

[Out]

3047/8000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-23/80*(1-2*x)^(3/2)*(3+5*x)^(1/2)-1/10*(1-2*x)^(3/2)*(2
+3*x)*(3+5*x)^(1/2)+277/800*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {90, 80, 50, 54, 216} \[ -\frac {1}{10} (3 x+2) \sqrt {5 x+3} (1-2 x)^{3/2}-\frac {23}{80} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {277}{800} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {3047 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{800 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(277*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/800 - (23*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/80 - ((1 - 2*x)^(3/2)*(2 + 3*x)*Sqr
t[3 + 5*x])/10 + (3047*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(800*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 80

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 90

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 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 {\sqrt {1-2 x} (2+3 x)^2}{\sqrt {3+5 x}} \, dx &=-\frac {1}{10} (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}-\frac {1}{30} \int \frac {\left (-108-\frac {345 x}{2}\right ) \sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {23}{80} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}+\frac {277}{160} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx\\ &=\frac {277}{800} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {23}{80} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}+\frac {3047 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1600}\\ &=\frac {277}{800} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {23}{80} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}+\frac {3047 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{800 \sqrt {5}}\\ &=\frac {277}{800} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {23}{80} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x}+\frac {3047 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{800 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 69, normalized size = 0.70 \[ \frac {3047 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (960 x^3+600 x^2-766 x+113\right )}{8000 \sqrt {1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(113 - 766*x + 600*x^2 + 960*x^3) + 3047*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]
])/(8000*Sqrt[1 - 2*x])

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fricas [A]  time = 0.91, size = 67, normalized size = 0.68 \[ \frac {1}{800} \, {\left (480 \, x^{2} + 540 \, x - 113\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {3047}{16000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/800*(480*x^2 + 540*x - 113)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 3047/16000*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 = 1.20, size = 140, normalized size = 1.41 \[ \frac {3}{40000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {3}{500} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {2}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/40000*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*s
qrt(22)*sqrt(5*x + 3))) + 3/500*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)*sqr
t(-10*x + 5))

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maple [A]  time = 0.01, size = 87, normalized size = 0.88 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (9600 \sqrt {-10 x^{2}-x +3}\, x^{2}+10800 \sqrt {-10 x^{2}-x +3}\, x +3047 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-2260 \sqrt {-10 x^{2}-x +3}\right )}{16000 \sqrt {-10 x^{2}-x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(9600*(-10*x^2-x+3)^(1/2)*x^2+3047*10^(1/2)*arcsin(20/11*x+1/11)+10800*(-
10*x^2-x+3)^(1/2)*x-2260*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.30, size = 58, normalized size = 0.59 \[ \frac {3047}{16000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {3}{50} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {123}{200} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {31}{800} \, \sqrt {-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3047/16000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 3/50*(-10*x^2 - x + 3)^(3/2) + 123/200*sqrt(-10*x^2 - x +
3)*x + 31/800*sqrt(-10*x^2 - x + 3)

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mupad [B]  time = 7.77, size = 534, normalized size = 5.39 \[ \frac {3047\,\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {1-2\,x}-1\right )}{2\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )}{4000}-\frac {\frac {20253\,{\left (\sqrt {1-2\,x}-1\right )}^3}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {6706\,\left (\sqrt {1-2\,x}-1\right )}{390625\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {9913\,{\left (\sqrt {1-2\,x}-1\right )}^5}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}+\frac {9913\,{\left (\sqrt {1-2\,x}-1\right )}^7}{31250\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}-\frac {20253\,{\left (\sqrt {1-2\,x}-1\right )}^9}{5000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}+\frac {3353\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{2000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}+\frac {512\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}-\frac {6784\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {49536\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}-\frac {1696\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^8}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {32\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}}{\frac {192\,{\left (\sqrt {1-2\,x}-1\right )}^2}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {48\,{\left (\sqrt {1-2\,x}-1\right )}^4}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {32\,{\left (\sqrt {1-2\,x}-1\right )}^6}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {12\,{\left (\sqrt {1-2\,x}-1\right )}^8}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {12\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^{12}}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {64}{15625}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3047*10^(1/2)*atan((10^(1/2)*((1 - 2*x)^(1/2) - 1))/(2*(3^(1/2) - (5*x + 3)^(1/2)))))/4000 - ((20253*((1 - 2*
x)^(1/2) - 1)^3)/(78125*(3^(1/2) - (5*x + 3)^(1/2))^3) - (6706*((1 - 2*x)^(1/2) - 1))/(390625*(3^(1/2) - (5*x
+ 3)^(1/2))) - (9913*((1 - 2*x)^(1/2) - 1)^5)/(78125*(3^(1/2) - (5*x + 3)^(1/2))^5) + (9913*((1 - 2*x)^(1/2) -
1)^7)/(31250*(3^(1/2) - (5*x + 3)^(1/2))^7) - (20253*((1 - 2*x)^(1/2) - 1)^9)/(5000*(3^(1/2) - (5*x + 3)^(1/2
))^9) + (3353*((1 - 2*x)^(1/2) - 1)^11)/(2000*(3^(1/2) - (5*x + 3)^(1/2))^11) + (512*3^(1/2)*((1 - 2*x)^(1/2)
- 1)^2)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^2) - (6784*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(15625*(3^(1/2) - (5*x
+ 3)^(1/2))^4) + (49536*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^6) - (1696*3^(1/2)
*((1 - 2*x)^(1/2) - 1)^8)/(625*(3^(1/2) - (5*x + 3)^(1/2))^8) + (32*3^(1/2)*((1 - 2*x)^(1/2) - 1)^10)/(25*(3^(
1/2) - (5*x + 3)^(1/2))^10))/((192*((1 - 2*x)^(1/2) - 1)^2)/(3125*(3^(1/2) - (5*x + 3)^(1/2))^2) + (48*((1 - 2
*x)^(1/2) - 1)^4)/(125*(3^(1/2) - (5*x + 3)^(1/2))^4) + (32*((1 - 2*x)^(1/2) - 1)^6)/(25*(3^(1/2) - (5*x + 3)^
(1/2))^6) + (12*((1 - 2*x)^(1/2) - 1)^8)/(5*(3^(1/2) - (5*x + 3)^(1/2))^8) + (12*((1 - 2*x)^(1/2) - 1)^10)/(5*
(3^(1/2) - (5*x + 3)^(1/2))^10) + ((1 - 2*x)^(1/2) - 1)^12/(3^(1/2) - (5*x + 3)^(1/2))^12 + 64/15625)

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sympy [A]  time = 19.96, size = 291, normalized size = 2.94 \[ - \frac {49 \sqrt {2} \left (\begin {cases} \frac {11 \sqrt {5} \left (- \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {\operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{2}\right )}{25} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{4} + \frac {21 \sqrt {2} \left (\begin {cases} \frac {121 \sqrt {5} \left (\frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{968} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{8}\right )}{125} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{2} - \frac {9 \sqrt {2} \left (\begin {cases} \frac {1331 \sqrt {5} \left (\frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} + \frac {3 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{1936} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{16}\right )}{625} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-49*sqrt(2)*Piecewise((11*sqrt(5)*(-sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)/22 + asin(sqrt(55)*sqrt(1 - 2*x)/11)/
2)/25, (x <= 1/2) & (x > -3/5)))/4 + 21*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, (x <= 1/2)
& (x > -3/5)))/2 - 9*sqrt(2)*Piecewise((1331*sqrt(5)*(5*sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)/7986 + 3*s
qrt(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, (x <= 1/2) & (x > -3/5)))/4

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