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

Optimal. Leaf size=138 \[ -\frac {3}{50} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac {119}{800} \sqrt {5 x+3} (1-2 x)^{7/2}+\frac {1309 \sqrt {5 x+3} (1-2 x)^{5/2}}{24000}+\frac {14399 \sqrt {5 x+3} (1-2 x)^{3/2}}{96000}+\frac {158389 \sqrt {5 x+3} \sqrt {1-2 x}}{320000}+\frac {1742279 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{320000 \sqrt {10}} \]

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Rubi [A]  time = 0.04, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 7, 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}{50} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac {119}{800} \sqrt {5 x+3} (1-2 x)^{7/2}+\frac {1309 \sqrt {5 x+3} (1-2 x)^{5/2}}{24000}+\frac {14399 \sqrt {5 x+3} (1-2 x)^{3/2}}{96000}+\frac {158389 \sqrt {5 x+3} \sqrt {1-2 x}}{320000}+\frac {1742279 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{320000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(158389*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/320000 + (14399*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/96000 + (1309*(1 - 2*x)^(5
/2)*Sqrt[3 + 5*x])/24000 - (119*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/800 - (3*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/50 +
(1742279*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(320000*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 (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x} \, dx &=-\frac {3}{50} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac {119}{100} \int (1-2 x)^{5/2} \sqrt {3+5 x} \, dx\\ &=-\frac {119}{800} (1-2 x)^{7/2} \sqrt {3+5 x}-\frac {3}{50} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac {1309 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{1600}\\ &=\frac {1309 (1-2 x)^{5/2} \sqrt {3+5 x}}{24000}-\frac {119}{800} (1-2 x)^{7/2} \sqrt {3+5 x}-\frac {3}{50} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac {14399 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{9600}\\ &=\frac {14399 (1-2 x)^{3/2} \sqrt {3+5 x}}{96000}+\frac {1309 (1-2 x)^{5/2} \sqrt {3+5 x}}{24000}-\frac {119}{800} (1-2 x)^{7/2} \sqrt {3+5 x}-\frac {3}{50} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac {158389 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{64000}\\ &=\frac {158389 \sqrt {1-2 x} \sqrt {3+5 x}}{320000}+\frac {14399 (1-2 x)^{3/2} \sqrt {3+5 x}}{96000}+\frac {1309 (1-2 x)^{5/2} \sqrt {3+5 x}}{24000}-\frac {119}{800} (1-2 x)^{7/2} \sqrt {3+5 x}-\frac {3}{50} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac {1742279 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{640000}\\ &=\frac {158389 \sqrt {1-2 x} \sqrt {3+5 x}}{320000}+\frac {14399 (1-2 x)^{3/2} \sqrt {3+5 x}}{96000}+\frac {1309 (1-2 x)^{5/2} \sqrt {3+5 x}}{24000}-\frac {119}{800} (1-2 x)^{7/2} \sqrt {3+5 x}-\frac {3}{50} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac {1742279 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{320000 \sqrt {5}}\\ &=\frac {158389 \sqrt {1-2 x} \sqrt {3+5 x}}{320000}+\frac {14399 (1-2 x)^{3/2} \sqrt {3+5 x}}{96000}+\frac {1309 (1-2 x)^{5/2} \sqrt {3+5 x}}{24000}-\frac {119}{800} (1-2 x)^{7/2} \sqrt {3+5 x}-\frac {3}{50} (1-2 x)^{7/2} (3+5 x)^{3/2}+\frac {1742279 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{320000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 79, normalized size = 0.57 \begin {gather*} \frac {10 \sqrt {5 x+3} \left (-4608000 x^5+4166400 x^4+2768320 x^3-4066120 x^2+396346 x+355917\right )+5226837 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{9600000 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(355917 + 396346*x - 4066120*x^2 + 2768320*x^3 + 4166400*x^4 - 4608000*x^5) + 5226837*Sqrt[-
10 + 20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(9600000*Sqrt[1 - 2*x])

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IntegrateAlgebraic [A]  time = 0.24, size = 141, normalized size = 1.02 \begin {gather*} -\frac {14641 \sqrt {1-2 x} \left (\frac {223125 (1-2 x)^4}{(5 x+3)^4}+\frac {288500 (1-2 x)^3}{(5 x+3)^3}-\frac {304640 (1-2 x)^2}{(5 x+3)^2}-\frac {66640 (1-2 x)}{5 x+3}-5712\right )}{960000 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^5}-\frac {1742279 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{320000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-14641*Sqrt[1 - 2*x]*(-5712 + (223125*(1 - 2*x)^4)/(3 + 5*x)^4 + (288500*(1 - 2*x)^3)/(3 + 5*x)^3 - (304640*(
1 - 2*x)^2)/(3 + 5*x)^2 - (66640*(1 - 2*x))/(3 + 5*x)))/(960000*Sqrt[3 + 5*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^5)
- (1742279*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/(320000*Sqrt[10])

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fricas [A]  time = 1.40, size = 77, normalized size = 0.56 \begin {gather*} \frac {1}{960000} \, {\left (2304000 \, x^{4} - 931200 \, x^{3} - 1849760 \, x^{2} + 1108180 \, x + 355917\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {1742279}{6400000} \, \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((1-2*x)^(5/2)*(2+3*x)*(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

1/960000*(2304000*x^4 - 931200*x^3 - 1849760*x^2 + 1108180*x + 355917)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1742279/
6400000*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 [B]  time = 1.33, size = 275, normalized size = 1.99 \begin {gather*} \frac {1}{16000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1}{600000} \, \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 {37}{120000} \, \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 {1}{400} \, \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((1-2*x)^(5/2)*(2+3*x)*(3+5*x)^(1/2),x, algorithm="giac")

[Out]

1/16000000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) - 506185)*(5*x + 3) + 4031895)*sqrt(
5*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/600000*sqrt(5)*(2*(4*(8*(
60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 184305*sqrt(2)*arcsin(1/11*sq
rt(22)*sqrt(5*x + 3))) - 37/120000*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))) - 1/400*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))) + 3/25*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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maple [A]  time = 0.01, size = 121, normalized size = 0.88 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (46080000 \sqrt {-10 x^{2}-x +3}\, x^{4}-18624000 \sqrt {-10 x^{2}-x +3}\, x^{3}-36995200 \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {-10 x^{2}-x +3}\, x +5226837 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+7118340 \sqrt {-10 x^{2}-x +3}\right )}{19200000 \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/19200000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(46080000*(-10*x^2-x+3)^(1/2)*x^4-18624000*(-10*x^2-x+3)^(1/2)*x^3-369
95200*(-10*x^2-x+3)^(1/2)*x^2+5226837*10^(1/2)*arcsin(20/11*x+1/11)[phone]*(-10*x^2-x+3)^(1/2)*x+7118340*(-1
0*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.26, size = 87, normalized size = 0.63 \begin {gather*} -\frac {6}{25} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {121}{1000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {1303}{12000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {14399}{16000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1742279}{6400000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {14399}{320000} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6/25*(-10*x^2 - x + 3)^(3/2)*x^2 + 121/1000*(-10*x^2 - x + 3)^(3/2)*x + 1303/12000*(-10*x^2 - x + 3)^(3/2) +
14399/16000*sqrt(-10*x^2 - x + 3)*x - 1742279/6400000*sqrt(10)*arcsin(-20/11*x - 1/11) + 14399/320000*sqrt(-10
*x^2 - x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 75.64, size = 490, normalized size = 3.55 \begin {gather*} \frac {242 \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}}{121} + \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}\right )}{32} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{3125} + \frac {638 \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 {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{1936} + \frac {\operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{16}\right )}{8} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{3125} - \frac {256 \sqrt {5} \left (\begin {cases} \frac {14641 \sqrt {2} \left (- \frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} - \frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{3872} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{1874048} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{128}\right )}{16} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{3125} + \frac {24 \sqrt {5} \left (\begin {cases} \frac {161051 \sqrt {2} \left (\frac {2 \sqrt {2} \left (5 - 10 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{805255} - \frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} - \frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{7744} - \frac {3 \sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{3748096} + \frac {7 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{256}\right )}{32} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{3125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

242*sqrt(5)*Piecewise((121*sqrt(2)*(-sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/121 + asin(sqrt(22)*sqrt
(5*x + 3)/11))/32, (x >= -3/5) & (x < 1/2)))/3125 + 638*sqrt(5)*Piecewise((1331*sqrt(2)*(-sqrt(2)*(5 - 10*x)**
(3/2)*(5*x + 3)**(3/2)/3993 - sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/1936 + asin(sqrt(22)*sqrt(5*x +
3)/11)/16)/8, (x >= -3/5) & (x < 1/2)))/3125 - 256*sqrt(5)*Piecewise((14641*sqrt(2)*(-sqrt(2)*(5 - 10*x)**(3/
2)*(5*x + 3)**(3/2)/3993 - sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/3872 - sqrt(2)*sqrt(5 - 10*x)*sqrt
(5*x + 3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 - 5929)/1874048 + 5*asin(sqrt(22)*sqrt(5*x + 3)/11)
/128)/16, (x >= -3/5) & (x < 1/2)))/3125 + 24*sqrt(5)*Piecewise((161051*sqrt(2)*(2*sqrt(2)*(5 - 10*x)**(5/2)*(
5*x + 3)**(5/2)/805255 - sqrt(2)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/3993 - sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*
sqrt(5*x + 3)/7744 - 3*sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 -
5929)/3748096 + 7*asin(sqrt(22)*sqrt(5*x + 3)/11)/256)/32, (x >= -3/5) & (x < 1/2)))/3125

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