3.24.3 \(\int \frac {\sqrt {1-2 x} (2+3 x)}{\sqrt {3+5 x}} \, dx\) [2303]

Optimal. Leaf size=72 \[ \frac {41}{200} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {3}{20} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {451 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{200 \sqrt {10}} \]

[Out]

451/2000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-3/20*(1-2*x)^(3/2)*(3+5*x)^(1/2)+41/200*(1-2*x)^(1/2)*(3
+5*x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {81, 52, 56, 222} \begin {gather*} \frac {451 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{200 \sqrt {10}}-\frac {3}{20} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {41}{200} \sqrt {5 x+3} \sqrt {1-2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(41*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/200 - (3*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/20 + (451*ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]])/(200*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 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 {\sqrt {1-2 x} (2+3 x)}{\sqrt {3+5 x}} \, dx &=-\frac {3}{20} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {41}{40} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx\\ &=\frac {41}{200} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {3}{20} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {451}{400} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {41}{200} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {3}{20} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {451 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{200 \sqrt {5}}\\ &=\frac {41}{200} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {3}{20} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {451 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{200 \sqrt {10}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.14, size = 68, normalized size = 0.94 \begin {gather*} \frac {2 \sqrt {5-10 x} \sqrt {3+5 x} (11+60 x)+451 i \sqrt {2} \log \left (\sqrt {5-10 x}-i \sqrt {6+10 x}\right )}{400 \sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[5 - 10*x]*Sqrt[3 + 5*x]*(11 + 60*x) + (451*I)*Sqrt[2]*Log[Sqrt[5 - 10*x] - I*Sqrt[6 + 10*x]])/(400*Sqr
t[5])

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Maple [A]
time = 0.10, size = 70, normalized size = 0.97

method	result	size
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (451 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1200 x \sqrt {-10 x^{2}-x +3}+220 \sqrt {-10 x^{2}-x +3}\right )}{4000 \sqrt {-10 x^{2}-x +3}}\)	\(70\)
risch	\(-\frac {\left (11+60 x \right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{200 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {451 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{4000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(451*10^(1/2)*arcsin(20/11*x+1/11)+1200*x*(-10*x^2-x+3)^(1/2)+220*(-10*x^2-
x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]
time = 0.60, size = 44, normalized size = 0.61 \begin {gather*} \frac {451}{4000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {3}{10} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {11}{200} \, \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]

451/4000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 3/10*sqrt(-10*x^2 - x + 3)*x + 11/200*sqrt(-10*x^2 - x + 3)

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Fricas [A]
time = 0.77, size = 62, normalized size = 0.86 \begin {gather*} \frac {1}{200} \, {\left (60 \, x + 11\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {451}{4000} \, \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]

1/200*(60*x + 11)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 451/4000*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 = 4.19, size = 206, normalized size = 2.86 \begin {gather*} - \frac {7 \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}\: \sqrt {1 - 2 x} > - \frac {\sqrt {55}}{5} \wedge \sqrt {1 - 2 x} < \frac {\sqrt {55}}{5} \end {cases}\right )}{2} + \frac {3 \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 )}{2} \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((11*sqrt(5)*(-sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)/22 + asin(sqrt(55)*sqrt(1 - 2*x)/11)/2
)/25, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/2 + 3*sqrt(2)*Piecewise((121*sqrt(5)*(sqr
t(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)*s
qrt(1 - 2*x)/11)/8)/125, (sqrt(1 - 2*x) > -sqrt(55)/5) & (sqrt(1 - 2*x) < sqrt(55)/5)))/2

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Giac [A]
time = 1.44, size = 86, normalized size = 1.19 \begin {gather*} \frac {3}{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 {1}{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]

3/2000*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)))
+ 1/25*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 [B]
time = 12.27, size = 550, normalized size = 7.64 \begin {gather*} \frac {\frac {4\,{\left (\sqrt {1-2\,x}-1\right )}^3}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {8\,\left (\sqrt {1-2\,x}-1\right )}{125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {32\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}}{\frac {4\,{\left (\sqrt {1-2\,x}-1\right )}^2}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^4}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {4}{25}}-\frac {\frac {2427\,{\left (\sqrt {1-2\,x}-1\right )}^3}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {858\,\left (\sqrt {1-2\,x}-1\right )}{15625\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {2427\,{\left (\sqrt {1-2\,x}-1\right )}^5}{1250\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}+\frac {429\,{\left (\sqrt {1-2\,x}-1\right )}^7}{500\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}+\frac {96\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {672\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {24\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}}{\frac {32\,{\left (\sqrt {1-2\,x}-1\right )}^2}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {24\,{\left (\sqrt {1-2\,x}-1\right )}^4}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {8\,{\left (\sqrt {1-2\,x}-1\right )}^6}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^8}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {16}{625}}-\frac {429\,\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 )}{1000}+\frac {22\,\sqrt {2}\,\sqrt {5}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {5}\,\left (\sqrt {1-2\,x}-1\right )}{2\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )}{25} \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]

((4*((1 - 2*x)^(1/2) - 1)^3)/(25*(3^(1/2) - (5*x + 3)^(1/2))^3) - (8*((1 - 2*x)^(1/2) - 1))/(125*(3^(1/2) - (5
*x + 3)^(1/2))) + (32*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2))/((4*((1 - 2*x)^(1/2
) - 1)^2)/(5*(3^(1/2) - (5*x + 3)^(1/2))^2) + ((1 - 2*x)^(1/2) - 1)^4/(3^(1/2) - (5*x + 3)^(1/2))^4 + 4/25) -
((2427*((1 - 2*x)^(1/2) - 1)^3)/(3125*(3^(1/2) - (5*x + 3)^(1/2))^3) - (858*((1 - 2*x)^(1/2) - 1))/(15625*(3^(
1/2) - (5*x + 3)^(1/2))) - (2427*((1 - 2*x)^(1/2) - 1)^5)/(1250*(3^(1/2) - (5*x + 3)^(1/2))^5) + (429*((1 - 2*
x)^(1/2) - 1)^7)/(500*(3^(1/2) - (5*x + 3)^(1/2))^7) + (96*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(625*(3^(1/2) - (5
*x + 3)^(1/2))^2) + (672*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(625*(3^(1/2) - (5*x + 3)^(1/2))^4) + (24*3^(1/2)*((
1 - 2*x)^(1/2) - 1)^6)/(25*(3^(1/2) - (5*x + 3)^(1/2))^6))/((32*((1 - 2*x)^(1/2) - 1)^2)/(125*(3^(1/2) - (5*x
+ 3)^(1/2))^2) + (24*((1 - 2*x)^(1/2) - 1)^4)/(25*(3^(1/2) - (5*x + 3)^(1/2))^4) + (8*((1 - 2*x)^(1/2) - 1)^6)
/(5*(3^(1/2) - (5*x + 3)^(1/2))^6) + ((1 - 2*x)^(1/2) - 1)^8/(3^(1/2) - (5*x + 3)^(1/2))^8 + 16/625) - (429*10
^(1/2)*atan((10^(1/2)*((1 - 2*x)^(1/2) - 1))/(2*(3^(1/2) - (5*x + 3)^(1/2)))))/1000 + (22*2^(1/2)*5^(1/2)*atan
((2^(1/2)*5^(1/2)*((1 - 2*x)^(1/2) - 1))/(2*(3^(1/2) - (5*x + 3)^(1/2)))))/25

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