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

Optimal. Leaf size=110 \[ \frac {\sqrt {5 x+3} (3 x+2)^3}{\sqrt {1-2 x}}+\frac {7}{4} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2+\frac {\sqrt {1-2 x} \sqrt {5 x+3} (73380 x+176833)}{3200}-\frac {1463447 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{3200 \sqrt {10}} \]

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Rubi [A]  time = 0.03, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {97, 153, 147, 54, 216} \begin {gather*} \frac {\sqrt {5 x+3} (3 x+2)^3}{\sqrt {1-2 x}}+\frac {7}{4} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2+\frac {\sqrt {1-2 x} \sqrt {5 x+3} (73380 x+176833)}{3200}-\frac {1463447 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{3200 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/4 + ((2 + 3*x)^3*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] + (Sqrt[1 - 2*x]*Sqr
t[3 + 5*x]*(176833 + 73380*x))/3200 - (1463447*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(3200*Sqrt[10])

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 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 153

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]

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 \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx &=\frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}}-\int \frac {(2+3 x)^2 \left (32+\frac {105 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {7}{4} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {1}{30} \int \frac {\left (-\frac {5625}{2}-\frac {18345 x}{4}\right ) (2+3 x)}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {7}{4} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} \sqrt {3+5 x} (176833+73380 x)}{3200}-\frac {1463447 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{6400}\\ &=\frac {7}{4} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} \sqrt {3+5 x} (176833+73380 x)}{3200}-\frac {1463447 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{3200 \sqrt {5}}\\ &=\frac {7}{4} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} \sqrt {3+5 x} (176833+73380 x)}{3200}-\frac {1463447 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{3200 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 83, normalized size = 0.75 \begin {gather*} \frac {-10 \sqrt {2 x-1} \sqrt {5 x+3} \left (14400 x^3+57960 x^2+142686 x-224833\right )-1463447 \sqrt {10} (2 x-1) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{32000 \sqrt {-(1-2 x)^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-10*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(-224833 + 142686*x + 57960*x^2 + 14400*x^3) - 1463447*Sqrt[10]*(-1 + 2*x)*A
rcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(32000*Sqrt[-(1 - 2*x)^2])

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IntegrateAlgebraic [A]  time = 0.17, size = 125, normalized size = 1.14 \begin {gather*} \frac {\sqrt {5 x+3} \left (\frac {36586175 (1-2 x)^3}{(5 x+3)^3}+\frac {39025680 (1-2 x)^2}{(5 x+3)^2}+\frac {12883812 (1-2 x)}{5 x+3}+1097600\right )}{3200 \sqrt {1-2 x} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^3}+\frac {1463447 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{3200 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[3 + 5*x]*(1097600 + (36586175*(1 - 2*x)^3)/(3 + 5*x)^3 + (39025680*(1 - 2*x)^2)/(3 + 5*x)^2 + (12883812*
(1 - 2*x))/(3 + 5*x)))/(3200*Sqrt[1 - 2*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^3) + (1463447*ArcTan[(Sqrt[5/2]*Sqrt[
1 - 2*x])/Sqrt[3 + 5*x]])/(3200*Sqrt[10])

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fricas [A]  time = 1.35, size = 86, normalized size = 0.78 \begin {gather*} \frac {1463447 \, \sqrt {10} {\left (2 \, x - 1\right )} \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 ) + 20 \, {\left (14400 \, x^{3} + 57960 \, x^{2} + 142686 \, x - 224833\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{64000 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64000*(1463447*sqrt(10)*(2*x - 1)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x -
3)) + 20*(14400*x^3 + 57960*x^2 + 142686*x - 224833)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)

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giac [A]  time = 1.01, size = 84, normalized size = 0.76 \begin {gather*} -\frac {1463447}{32000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (18 \, {\left (4 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} + 89 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 4927 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 1463447 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{80000 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1463447/32000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/80000*(18*(4*(8*sqrt(5)*(5*x + 3) + 89*sqrt(5)
)*(5*x + 3) + 4927*sqrt(5))*(5*x + 3) - 1463447*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)

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maple [A]  time = 0.01, size = 123, normalized size = 1.12 \begin {gather*} -\frac {\left (-288000 \sqrt {-10 x^{2}-x +3}\, x^{3}-1159200 \sqrt {-10 x^{2}-x +3}\, x^{2}+2926894 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-2853720 \sqrt {-10 x^{2}-x +3}\, x -1463447 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+4496660 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{64000 \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64000*(-288000*(-10*x^2-x+3)^(1/2)*x^3+2926894*10^(1/2)*x*arcsin(20/11*x+1/11)-1159200*(-10*x^2-x+3)^(1/2)*
x^2-1463447*10^(1/2)*arcsin(20/11*x+1/11)-2853720*(-10*x^2-x+3)^(1/2)*x+4496660*(-10*x^2-x+3)^(1/2))*(-2*x+1)^
(1/2)*(5*x+3)^(1/2)/(2*x-1)/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.20, size = 79, normalized size = 0.72 \begin {gather*} -\frac {1463447}{64000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {9}{40} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {1593}{160} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {89793}{3200} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {343 \, \sqrt {-10 \, x^{2} - x + 3}}{8 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1463447/64000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 9/40*(-10*x^2 - x + 3)^(3/2) + 1593/160*sqrt(-10*x^2 -
x + 3)*x + 89793/3200*sqrt(-10*x^2 - x + 3) - 343/8*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3 x + 2\right )^{3} \sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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