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

Optimal. Leaf size=116 \[ -\frac {3}{40} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac {23}{96} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {253 \sqrt {5 x+3} (1-2 x)^{3/2}}{1920}+\frac {2783 \sqrt {5 x+3} \sqrt {1-2 x}}{6400}+\frac {30613 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{6400 \sqrt {10}} \]

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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} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac {23}{96} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {253 \sqrt {5 x+3} (1-2 x)^{3/2}}{1920}+\frac {2783 \sqrt {5 x+3} \sqrt {1-2 x}}{6400}+\frac {30613 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{6400 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2783*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/6400 + (253*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/1920 - (23*(1 - 2*x)^(5/2)*Sqrt[
3 + 5*x])/96 - (3*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/40 + (30613*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(6400*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)^{3/2} (2+3 x) \sqrt {3+5 x} \, dx &=-\frac {3}{40} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac {23}{16} \int (1-2 x)^{3/2} \sqrt {3+5 x} \, dx\\ &=-\frac {23}{96} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {3}{40} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac {253}{192} \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx\\ &=\frac {253 (1-2 x)^{3/2} \sqrt {3+5 x}}{1920}-\frac {23}{96} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {3}{40} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac {2783 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{1280}\\ &=\frac {2783 \sqrt {1-2 x} \sqrt {3+5 x}}{6400}+\frac {253 (1-2 x)^{3/2} \sqrt {3+5 x}}{1920}-\frac {23}{96} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {3}{40} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac {30613 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{12800}\\ &=\frac {2783 \sqrt {1-2 x} \sqrt {3+5 x}}{6400}+\frac {253 (1-2 x)^{3/2} \sqrt {3+5 x}}{1920}-\frac {23}{96} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {3}{40} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac {30613 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{6400 \sqrt {5}}\\ &=\frac {2783 \sqrt {1-2 x} \sqrt {3+5 x}}{6400}+\frac {253 (1-2 x)^{3/2} \sqrt {3+5 x}}{1920}-\frac {23}{96} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {3}{40} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac {30613 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{6400 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 74, normalized size = 0.64 \begin {gather*} \frac {10 \sqrt {5 x+3} \left (57600 x^4-15040 x^3-53720 x^2+19502 x+1959\right )+91839 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{192000 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(1959 + 19502*x - 53720*x^2 - 15040*x^3 + 57600*x^4) + 91839*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5
/11]*Sqrt[-1 + 2*x]])/(192000*Sqrt[1 - 2*x])

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IntegrateAlgebraic [A]  time = 0.19, size = 125, normalized size = 1.08 \begin {gather*} -\frac {1331 \sqrt {1-2 x} \left (\frac {8625 (1-2 x)^3}{(5 x+3)^3}+\frac {10090 (1-2 x)^2}{(5 x+3)^2}-\frac {5060 (1-2 x)}{5 x+3}-552\right )}{19200 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^4}-\frac {30613 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{6400 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1331*Sqrt[1 - 2*x]*(-552 + (8625*(1 - 2*x)^3)/(3 + 5*x)^3 + (10090*(1 - 2*x)^2)/(3 + 5*x)^2 - (5060*(1 - 2*x
))/(3 + 5*x)))/(19200*Sqrt[3 + 5*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^4) - (30613*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])
/Sqrt[3 + 5*x]])/(6400*Sqrt[10])

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fricas [A]  time = 0.87, size = 72, normalized size = 0.62 \begin {gather*} -\frac {1}{19200} \, {\left (28800 \, x^{3} + 6880 \, x^{2} - 23420 \, x - 1959\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {30613}{128000} \, \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)^(3/2)*(2+3*x)*(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

-1/19200*(28800*x^3 + 6880*x^2 - 23420*x - 1959)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 30613/128000*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 = 1.08, size = 203, normalized size = 1.75 \begin {gather*} -\frac {1}{320000} \, \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 {23}{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 {7}{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((1-2*x)^(3/2)*(2+3*x)*(3+5*x)^(1/2),x, algorithm="giac")

[Out]

-1/320000*sqrt(5)*(2*(4*(8*(60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 1
84305*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 23/120000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqr
t(5*x + 3)*sqrt(-10*x + 5) + 4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 7/2000*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))) + 3/25*sqrt(5)*(11*sqrt(2)*a
rcsin(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 (-576000 \sqrt {-10 x^{2}-x +3}\, x^{3}-137600 \sqrt {-10 x^{2}-x +3}\, x^{2}+468400 \sqrt {-10 x^{2}-x +3}\, x +91839 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+39180 \sqrt {-10 x^{2}-x +3}\right )}{384000 \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(-576000*(-10*x^2-x+3)^(1/2)*x^3-137600*(-10*x^2-x+3)^(1/2)*x^2+91839*10
^(1/2)*arcsin(20/11*x+1/11)+468400*(-10*x^2-x+3)^(1/2)*x+39180*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.21, size = 70, normalized size = 0.60 \begin {gather*} \frac {3}{20} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {1}{48} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {253}{320} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {30613}{128000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {253}{6400} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/20*(-10*x^2 - x + 3)^(3/2)*x + 1/48*(-10*x^2 - x + 3)^(3/2) + 253/320*sqrt(-10*x^2 - x + 3)*x - 30613/128000
*sqrt(10)*arcsin(-20/11*x - 1/11) + 253/6400*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 )}^{3/2}\,\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)^(3/2)*(3*x + 2)*(5*x + 3)^(1/2),x)

[Out]

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

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sympy [A]  time = 38.59, size = 316, normalized size = 2.72 \begin {gather*} \frac {22 \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 )}{625} + \frac {62 \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 )}{625} - \frac {12 \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 )}{625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

22*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)))/625 + 62*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)))/625 - 12*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)))/625

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