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

Optimal. Leaf size=116 \[ -\frac {3}{40} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac {181}{480} (1-2 x)^{3/2} (5 x+3)^{3/2}-\frac {1991 (1-2 x)^{3/2} \sqrt {5 x+3}}{1280}+\frac {21901 \sqrt {1-2 x} \sqrt {5 x+3}}{12800}+\frac {240911 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{12800 \sqrt {10}} \]

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Rubi [A]  time = 0.03, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {3}{40} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac {181}{480} (1-2 x)^{3/2} (5 x+3)^{3/2}-\frac {1991 (1-2 x)^{3/2} \sqrt {5 x+3}}{1280}+\frac {21901 \sqrt {1-2 x} \sqrt {5 x+3}}{12800}+\frac {240911 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{12800 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(21901*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/12800 - (1991*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/1280 - (181*(1 - 2*x)^(3/2)*(
3 + 5*x)^(3/2))/480 - (3*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/40 + (240911*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1280
0*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) (3+5 x)^{3/2} \, dx &=-\frac {3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {181}{80} \int \sqrt {1-2 x} (3+5 x)^{3/2} \, dx\\ &=-\frac {181}{480} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {1991}{320} \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx\\ &=-\frac {1991 (1-2 x)^{3/2} \sqrt {3+5 x}}{1280}-\frac {181}{480} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {21901 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{2560}\\ &=\frac {21901 \sqrt {1-2 x} \sqrt {3+5 x}}{12800}-\frac {1991 (1-2 x)^{3/2} \sqrt {3+5 x}}{1280}-\frac {181}{480} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {240911 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{25600}\\ &=\frac {21901 \sqrt {1-2 x} \sqrt {3+5 x}}{12800}-\frac {1991 (1-2 x)^{3/2} \sqrt {3+5 x}}{1280}-\frac {181}{480} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {240911 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{12800 \sqrt {5}}\\ &=\frac {21901 \sqrt {1-2 x} \sqrt {3+5 x}}{12800}-\frac {1991 (1-2 x)^{3/2} \sqrt {3+5 x}}{1280}-\frac {181}{480} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {240911 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{12800 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 74, normalized size = 0.64 \begin {gather*} \frac {722733 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (288000 x^4+347200 x^3-46840 x^2-226154 x+63387\right )}{384000 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(63387 - 226154*x - 46840*x^2 + 347200*x^3 + 288000*x^4) + 722733*Sqrt[-10 + 20*x]*ArcSinh[
Sqrt[5/11]*Sqrt[-1 + 2*x]])/(384000*Sqrt[1 - 2*x])

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IntegrateAlgebraic [A]  time = 0.18, size = 125, normalized size = 1.08 \begin {gather*} -\frac {1331 \sqrt {1-2 x} \left (\frac {67875 (1-2 x)^3}{(5 x+3)^3}+\frac {99550 (1-2 x)^2}{(5 x+3)^2}+\frac {49780 (1-2 x)}{5 x+3}-4344\right )}{38400 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^4}-\frac {240911 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{12800 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1331*Sqrt[1 - 2*x]*(-4344 + (67875*(1 - 2*x)^3)/(3 + 5*x)^3 + (99550*(1 - 2*x)^2)/(3 + 5*x)^2 + (49780*(1 -
2*x))/(3 + 5*x)))/(38400*Sqrt[3 + 5*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^4) - (240911*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2
*x])/Sqrt[3 + 5*x]])/(12800*Sqrt[10])

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fricas [A]  time = 1.40, size = 72, normalized size = 0.62 \begin {gather*} \frac {1}{38400} \, {\left (144000 \, x^{3} + 245600 \, x^{2} + 99380 \, x - 63387\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {240911}{256000} \, \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)*(3+5*x)^(3/2)*(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/38400*(144000*x^3 + 245600*x^2 + 99380*x - 63387)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 240911/256000*sqrt(10)*arct
an(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 [B]  time = 1.00, size = 203, normalized size = 1.75 \begin {gather*} \frac {1}{128000} \, \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 {7}{6000} \, \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 {87}{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 {9}{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)*(3+5*x)^(3/2)*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

1/128000*sqrt(5)*(2*(4*(8*(60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 18
4305*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 7/6000*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))) + 87/2000*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))) + 9/25*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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maple [A]  time = 0.01, size = 104, normalized size = 0.90 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (2880000 \sqrt {-10 x^{2}-x +3}\, x^{3}+4912000 \sqrt {-10 x^{2}-x +3}\, x^{2}+1987600 \sqrt {-10 x^{2}-x +3}\, x +722733 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-1267740 \sqrt {-10 x^{2}-x +3}\right )}{768000 \sqrt {-10 x^{2}-x +3}} \end {gather*}

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]

1/768000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(2880000*(-10*x^2-x+3)^(1/2)*x^3+4912000*(-10*x^2-x+3)^(1/2)*x^2+722733*
10^(1/2)*arcsin(20/11*x+1/11)+1987600*(-10*x^2-x+3)^(1/2)*x-1267740*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.27, size = 70, normalized size = 0.60 \begin {gather*} -\frac {3}{8} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {289}{480} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {1991}{640} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {240911}{256000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {1991}{12800} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

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]

-3/8*(-10*x^2 - x + 3)^(3/2)*x - 289/480*(-10*x^2 - x + 3)^(3/2) + 1991/640*sqrt(-10*x^2 - x + 3)*x - 240911/2
56000*sqrt(10)*arcsin(-20/11*x - 1/11) + 1991/12800*sqrt(-10*x^2 - x + 3)

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

[Out]

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

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sympy [A]  time = 37.59, size = 314, normalized size = 2.71 \begin {gather*} - \frac {77 \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 )}{8} + \frac {17 \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 )}{2} - \frac {15 \sqrt {2} \left (\begin {cases} \frac {14641 \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 )}{3872} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{1874048} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{128}\right )}{625} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{8} \end {gather*}

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]

-77*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)))/8 + 17*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)
/11)/16)/125, (x <= 1/2) & (x > -3/5)))/2 - 15*sqrt(2)*Piecewise((14641*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)/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 + 5*asin(sqrt(55)*sqrt(1 - 2*x)/11)/128
)/625, (x <= 1/2) & (x > -3/5)))/8

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