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

Optimal. Leaf size=94 \[ -\frac {1}{10} (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac {37}{80} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {407}{800} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {4477 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{800 \sqrt {10}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {80, 50, 54, 216} \begin {gather*} -\frac {1}{10} (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac {37}{80} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {407}{800} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {4477 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{800 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(407*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/800 - (37*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/80 - ((1 - 2*x)^(3/2)*(3 + 5*x)^(3/
2))/10 + (4477*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 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 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x} \, dx &=-\frac {1}{10} (1-2 x)^{3/2} (3+5 x)^{3/2}+\frac {37}{20} \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx\\ &=-\frac {37}{80} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{3/2} (3+5 x)^{3/2}+\frac {407}{160} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx\\ &=\frac {407}{800} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {37}{80} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{3/2} (3+5 x)^{3/2}+\frac {4477 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1600}\\ &=\frac {407}{800} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {37}{80} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{3/2} (3+5 x)^{3/2}+\frac {4477 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{800 \sqrt {5}}\\ &=\frac {407}{800} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {37}{80} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{3/2} (3+5 x)^{3/2}+\frac {4477 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{800 \sqrt {10}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 69, normalized size = 0.73 \begin {gather*} \frac {4477 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (1600 x^3+840 x^2-1226 x+203\right )}{8000 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(203 - 1226*x + 840*x^2 + 1600*x^3) + 4477*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*
x]])/(8000*Sqrt[1 - 2*x])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.16, size = 109, normalized size = 1.16 \begin {gather*} -\frac {121 \sqrt {1-2 x} \left (\frac {925 (1-2 x)^2}{(5 x+3)^2}+\frac {880 (1-2 x)}{5 x+3}-148\right )}{800 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^3}-\frac {4477 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{800 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-121*Sqrt[1 - 2*x]*(-148 + (925*(1 - 2*x)^2)/(3 + 5*x)^2 + (880*(1 - 2*x))/(3 + 5*x)))/(800*Sqrt[3 + 5*x]*(2
+ (5*(1 - 2*x))/(3 + 5*x))^3) - (4477*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/(800*Sqrt[10])

________________________________________________________________________________________

fricas [A]  time = 1.38, size = 67, normalized size = 0.71 \begin {gather*} \frac {1}{800} \, {\left (800 \, x^{2} + 820 \, x - 203\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {4477}{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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/800*(800*x^2 + 820*x - 203)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 4477/16000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x +
1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

________________________________________________________________________________________

giac [B]  time = 1.12, size = 140, normalized size = 1.49 \begin {gather*} \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 {19}{2000} \, \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 {3}{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)*(1-2*x)^(1/2)*(3+5*x)^(1/2),x, algorithm="giac")

[Out]

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*sq
rt(22)*sqrt(5*x + 3))) + 19/2000*sqrt(5)*(2*(20*x - 23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/1
1*sqrt(22)*sqrt(5*x + 3))) + 3/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sq
rt(-10*x + 5))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 87, normalized size = 0.93 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (16000 \sqrt {-10 x^{2}-x +3}\, x^{2}+16400 \sqrt {-10 x^{2}-x +3}\, x +4477 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-4060 \sqrt {-10 x^{2}-x +3}\right )}{16000 \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+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)*(16000*(-10*x^2-x+3)^(1/2)*x^2+4477*10^(1/2)*arcsin(20/11*x+1/11)+16400*(
-10*x^2-x+3)^(1/2)*x-4060*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 1.20, size = 55, normalized size = 0.59 \begin {gather*} -\frac {1}{10} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {37}{40} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {4477}{16000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {37}{800} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*(-10*x^2 - x + 3)^(3/2) + 37/40*sqrt(-10*x^2 - x + 3)*x - 4477/16000*sqrt(10)*arcsin(-20/11*x - 1/11) +
37/800*sqrt(-10*x^2 - x + 3)

________________________________________________________________________________________

mupad [B]  time = 7.48, size = 588, normalized size = 6.26 \begin {gather*} 2\,\sqrt {1-2\,x}\,\sqrt {5\,x+3}\,\left (\frac {x}{2}+\frac {1}{40}\right )-\frac {363\,\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 {7543\,{\left (\sqrt {1-2\,x}-1\right )}^3}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {726\,\left (\sqrt {1-2\,x}-1\right )}{390625\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {19023\,{\left (\sqrt {1-2\,x}-1\right )}^5}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}+\frac {19023\,{\left (\sqrt {1-2\,x}-1\right )}^7}{6250\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}-\frac {7543\,{\left (\sqrt {1-2\,x}-1\right )}^9}{5000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}+\frac {363\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{2000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}+\frac {1152\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}-\frac {11136\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {15936\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}-\frac {2784\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^8}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {72\,\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}}-\frac {\sqrt {2}\,\sqrt {5}\,\ln \left (x+\frac {1}{20}-\frac {\sqrt {10}\,\sqrt {1-2\,x}\,\sqrt {5\,x+3}\,1{}\mathrm {i}}{10}\right )\,121{}\mathrm {i}}{400} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(1 - 2*x)^(1/2)*(5*x + 3)^(1/2)*(x/2 + 1/40) - (363*10^(1/2)*atan((10^(1/2)*((1 - 2*x)^(1/2) - 1))/(2*(3^(1/
2) - (5*x + 3)^(1/2)))))/4000 - (2^(1/2)*5^(1/2)*log(x - (10^(1/2)*(1 - 2*x)^(1/2)*(5*x + 3)^(1/2)*1i)/10 + 1/
20)*121i)/400 - ((7543*((1 - 2*x)^(1/2) - 1)^3)/(78125*(3^(1/2) - (5*x + 3)^(1/2))^3) - (726*((1 - 2*x)^(1/2)
- 1))/(390625*(3^(1/2) - (5*x + 3)^(1/2))) - (19023*((1 - 2*x)^(1/2) - 1)^5)/(15625*(3^(1/2) - (5*x + 3)^(1/2)
)^5) + (19023*((1 - 2*x)^(1/2) - 1)^7)/(6250*(3^(1/2) - (5*x + 3)^(1/2))^7) - (7543*((1 - 2*x)^(1/2) - 1)^9)/(
5000*(3^(1/2) - (5*x + 3)^(1/2))^9) + (363*((1 - 2*x)^(1/2) - 1)^11)/(2000*(3^(1/2) - (5*x + 3)^(1/2))^11) + (
1152*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^2) - (11136*3^(1/2)*((1 - 2*x)^(1/2)
- 1)^4)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^4) + (15936*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(3125*(3^(1/2) - (5*x
+ 3)^(1/2))^6) - (2784*3^(1/2)*((1 - 2*x)^(1/2) - 1)^8)/(625*(3^(1/2) - (5*x + 3)^(1/2))^8) + (72*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)

________________________________________________________________________________________

sympy [A]  time = 127.45, size = 168, normalized size = 1.79 \begin {gather*} - \frac {7 \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 )}{121} + \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}\right )}{200} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{4} + \frac {3 \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 {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{1936} + \frac {\operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{16}\right )}{125} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7*sqrt(2)*Piecewise((121*sqrt(5)*(-sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/121 + asin(sqrt(55)*sqrt(1
- 2*x)/11))/200, (x <= 1/2) & (x > -3/5)))/4 + 3*sqrt(2)*Piecewise((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)/1
1)/16)/125, (x <= 1/2) & (x > -3/5)))/4

________________________________________________________________________________________