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

Optimal. Leaf size=94 \[ \frac {7 (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}+\frac {173}{88} \sqrt {1-2 x} (5 x+3)^{3/2}+\frac {519}{32} \sqrt {1-2 x} \sqrt {5 x+3}-\frac {5709 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{32 \sqrt {10}} \]

________________________________________________________________________________________

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 = {78, 50, 54, 216} \begin {gather*} \frac {7 (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}+\frac {173}{88} \sqrt {1-2 x} (5 x+3)^{3/2}+\frac {519}{32} \sqrt {1-2 x} \sqrt {5 x+3}-\frac {5709 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{32 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(519*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/32 + (173*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/88 + (7*(3 + 5*x)^(5/2))/(11*Sqrt[1
- 2*x]) - (5709*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(32*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 78

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

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}}{(1-2 x)^{3/2}} \, dx &=\frac {7 (3+5 x)^{5/2}}{11 \sqrt {1-2 x}}-\frac {173}{22} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=\frac {173}{88} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {7 (3+5 x)^{5/2}}{11 \sqrt {1-2 x}}-\frac {519}{16} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=\frac {519}{32} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {173}{88} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {7 (3+5 x)^{5/2}}{11 \sqrt {1-2 x}}-\frac {5709}{64} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {519}{32} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {173}{88} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {7 (3+5 x)^{5/2}}{11 \sqrt {1-2 x}}-\frac {5709 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{32 \sqrt {5}}\\ &=\frac {519}{32} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {173}{88} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {7 (3+5 x)^{5/2}}{11 \sqrt {1-2 x}}-\frac {5709 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{32 \sqrt {10}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 78, normalized size = 0.83 \begin {gather*} \frac {-10 \sqrt {2 x-1} \sqrt {5 x+3} \left (120 x^2+490 x-891\right )-5709 \sqrt {10} (2 x-1) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{320 \sqrt {-(1-2 x)^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-10*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(-891 + 490*x + 120*x^2) - 5709*Sqrt[10]*(-1 + 2*x)*ArcSinh[Sqrt[5/11]*Sqrt[
-1 + 2*x]])/(320*Sqrt[-(1 - 2*x)^2])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.42, size = 116, normalized size = 1.23 \begin {gather*} \frac {\sqrt {11-2 (5 x+3)} \left (24 (5 x+3)^{5/2}+346 (5 x+3)^{3/2}-5709 \sqrt {5 x+3}\right )}{32 \sqrt {5} (2 (5 x+3)-11)}+\frac {5709 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{16 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[11 - 2*(3 + 5*x)]*(-5709*Sqrt[3 + 5*x] + 346*(3 + 5*x)^(3/2) + 24*(3 + 5*x)^(5/2)))/(32*Sqrt[5]*(-11 + 2
*(3 + 5*x))) + (5709*ArcTan[(Sqrt[2]*Sqrt[3 + 5*x])/(Sqrt[11] - Sqrt[11 - 2*(3 + 5*x)])])/(16*Sqrt[10])

________________________________________________________________________________________

fricas [A]  time = 1.28, size = 81, normalized size = 0.86 \begin {gather*} \frac {5709 \, \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 (120 \, x^{2} + 490 \, x - 891\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{640 \, {\left (2 \, x - 1\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)^(3/2),x, algorithm="fricas")

[Out]

1/640*(5709*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*(120*x^2 + 490*x - 891)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)

________________________________________________________________________________________

giac [A]  time = 0.86, size = 71, normalized size = 0.76 \begin {gather*} -\frac {5709}{320} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (2 \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} + 173 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 5709 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{800 \, {\left (2 \, x - 1\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)^(3/2),x, algorithm="giac")

[Out]

-5709/320*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/800*(2*(12*sqrt(5)*(5*x + 3) + 173*sqrt(5))*(5*x +
3) - 5709*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 106, normalized size = 1.13 \begin {gather*} -\frac {\left (-2400 \sqrt {-10 x^{2}-x +3}\, x^{2}+11418 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-9800 \sqrt {-10 x^{2}-x +3}\, x -5709 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+17820 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{640 \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)*(5*x+3)^(3/2)/(-2*x+1)^(3/2),x)

[Out]

-1/640*(11418*10^(1/2)*x*arcsin(20/11*x+1/11)-2400*(-10*x^2-x+3)^(1/2)*x^2-5709*10^(1/2)*arcsin(20/11*x+1/11)-
9800*(-10*x^2-x+3)^(1/2)*x+17820*(-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)

________________________________________________________________________________________

maxima [A]  time = 1.04, size = 97, normalized size = 1.03 \begin {gather*} -\frac {5709}{640} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {99}{32} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {7 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{4 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{8 \, {\left (2 \, x - 1\right )}} - \frac {231 \, \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+5*x)^(3/2)/(1-2*x)^(3/2),x, algorithm="maxima")

[Out]

-5709/640*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 99/32*sqrt(-10*x^2 - x + 3) - 7/4*(-10*x^2 - x + 3)^(3/2)/(
4*x^2 - 4*x + 1) - 3/8*(-10*x^2 - x + 3)^(3/2)/(2*x - 1) - 231/8*sqrt(-10*x^2 - x + 3)/(2*x - 1)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________