3.2470 \(\int \frac {(2+3 x) (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\)

Optimal. Leaf size=94 \[ -\frac {1}{10} \sqrt {1-2 x} (5 x+3)^{5/2}-\frac {59}{80} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {1947}{320} \sqrt {1-2 x} \sqrt {5 x+3}+\frac {21417 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{320 \sqrt {10}} \]

[Out]

21417/3200*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-59/80*(3+5*x)^(3/2)*(1-2*x)^(1/2)-1/10*(3+5*x)^(5/2)*(
1-2*x)^(1/2)-1947/320*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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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} \[ -\frac {1}{10} \sqrt {1-2 x} (5 x+3)^{5/2}-\frac {59}{80} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {1947}{320} \sqrt {1-2 x} \sqrt {5 x+3}+\frac {21417 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{320 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1947*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/320 - (59*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/80 - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/
2))/10 + (21417*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(320*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 \frac {(2+3 x) (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx &=-\frac {1}{10} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {59}{20} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {59}{80} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {1}{10} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {1947}{160} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {1947}{320} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {59}{80} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {1}{10} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {21417}{640} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {1947}{320} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {59}{80} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {1}{10} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {21417 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{320 \sqrt {5}}\\ &=-\frac {1947}{320} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {59}{80} \sqrt {1-2 x} (3+5 x)^{3/2}-\frac {1}{10} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {21417 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{320 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 78, normalized size = 0.83 \[ -\frac {\sqrt {1-2 x} \left (10 \sqrt {2 x-1} \sqrt {5 x+3} \left (800 x^2+2140 x+2943\right )+21417 \sqrt {10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{3200 \sqrt {2 x-1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3200*(Sqrt[1 - 2*x]*(10*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(2943 + 2140*x + 800*x^2) + 21417*Sqrt[10]*ArcSinh[Sqr
t[5/11]*Sqrt[-1 + 2*x]]))/Sqrt[-1 + 2*x]

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fricas [A]  time = 0.83, size = 67, normalized size = 0.71 \[ -\frac {1}{320} \, {\left (800 \, x^{2} + 2140 \, x + 2943\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {21417}{6400} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/320*(800*x^2 + 2140*x + 2943)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 21417/6400*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 = 1.32, size = 54, normalized size = 0.57 \[ -\frac {1}{3200} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x + 83\right )} {\left (5 \, x + 3\right )} + 1947\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 21417 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3200*sqrt(5)*(2*(4*(40*x + 83)*(5*x + 3) + 1947)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 21417*sqrt(2)*arcsin(1/11*
sqrt(22)*sqrt(5*x + 3)))

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maple [A]  time = 0.01, size = 87, normalized size = 0.93 \[ \frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (-16000 \sqrt {-10 x^{2}-x +3}\, x^{2}-42800 \sqrt {-10 x^{2}-x +3}\, x +21417 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-58860 \sqrt {-10 x^{2}-x +3}\right )}{6400 \sqrt {-10 x^{2}-x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6400*(5*x+3)^(1/2)*(-2*x+1)^(1/2)*(-16000*(-10*x^2-x+3)^(1/2)*x^2+21417*10^(1/2)*arcsin(20/11*x+1/11)-42800*
(-10*x^2-x+3)^(1/2)*x-58860*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.19, size = 58, normalized size = 0.62 \[ -\frac {5}{2} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {107}{16} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {21417}{6400} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {2943}{320} \, \sqrt {-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/2*sqrt(-10*x^2 - x + 3)*x^2 - 107/16*sqrt(-10*x^2 - x + 3)*x - 21417/6400*sqrt(10)*arcsin(-20/11*x - 1/11)
- 2943/320*sqrt(-10*x^2 - x + 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (3\,x+2\right )\,{\left (5\,x+3\right )}^{3/2}}{\sqrt {1-2\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 80.23, size = 224, normalized size = 2.38 \[ \frac {2 \sqrt {5} \left (\begin {cases} \frac {121 \sqrt {2} \left (\frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{968} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{8}\right )}{8} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{25} + \frac {6 \sqrt {5} \left (\begin {cases} \frac {1331 \sqrt {2} \left (\frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} + \frac {3 \sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{1936} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{16}\right )}{16} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(5)*Piecewise((121*sqrt(2)*(sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/968 - sqrt(2)*sqrt(5 - 10*x
)*sqrt(5*x + 3)/22 + 3*asin(sqrt(22)*sqrt(5*x + 3)/11)/8)/8, (x >= -3/5) & (x < 1/2)))/25 + 6*sqrt(5)*Piecewis
e((1331*sqrt(2)*(sqrt(2)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/3993 + 3*sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5
*x + 3)/1936 - sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + 5*asin(sqrt(22)*sqrt(5*x + 3)/11)/16)/16, (x >= -3/5)
& (x < 1/2)))/25

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