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

Optimal. Leaf size=116 \[ \frac {3}{20} \sqrt {1-2 x} (5 x+3)^{5/2}+\frac {49 (5 x+3)^{5/2}}{22 \sqrt {1-2 x}}+\frac {14057 \sqrt {1-2 x} (5 x+3)^{3/2}}{1760}+\frac {42171}{640} \sqrt {1-2 x} \sqrt {5 x+3}-\frac {463881 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{640 \sqrt {10}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {89, 80, 50, 54, 216} \begin {gather*} \frac {3}{20} \sqrt {1-2 x} (5 x+3)^{5/2}+\frac {49 (5 x+3)^{5/2}}{22 \sqrt {1-2 x}}+\frac {14057 \sqrt {1-2 x} (5 x+3)^{3/2}}{1760}+\frac {42171}{640} \sqrt {1-2 x} \sqrt {5 x+3}-\frac {463881 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{640 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(42171*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/640 + (14057*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/1760 + (49*(3 + 5*x)^(5/2))/(2
2*Sqrt[1 - 2*x]) + (3*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/20 - (463881*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(640*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 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

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)^2 (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx &=\frac {49 (3+5 x)^{5/2}}{22 \sqrt {1-2 x}}-\frac {1}{22} \int \frac {(3+5 x)^{3/2} \left (\frac {1343}{2}+99 x\right )}{\sqrt {1-2 x}} \, dx\\ &=\frac {49 (3+5 x)^{5/2}}{22 \sqrt {1-2 x}}+\frac {3}{20} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {14057}{440} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=\frac {14057 \sqrt {1-2 x} (3+5 x)^{3/2}}{1760}+\frac {49 (3+5 x)^{5/2}}{22 \sqrt {1-2 x}}+\frac {3}{20} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {42171}{320} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=\frac {42171}{640} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {14057 \sqrt {1-2 x} (3+5 x)^{3/2}}{1760}+\frac {49 (3+5 x)^{5/2}}{22 \sqrt {1-2 x}}+\frac {3}{20} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {463881 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1280}\\ &=\frac {42171}{640} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {14057 \sqrt {1-2 x} (3+5 x)^{3/2}}{1760}+\frac {49 (3+5 x)^{5/2}}{22 \sqrt {1-2 x}}+\frac {3}{20} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {463881 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{640 \sqrt {5}}\\ &=\frac {42171}{640} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {14057 \sqrt {1-2 x} (3+5 x)^{3/2}}{1760}+\frac {49 (3+5 x)^{5/2}}{22 \sqrt {1-2 x}}+\frac {3}{20} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {463881 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{640 \sqrt {10}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 83, normalized size = 0.72 \begin {gather*} \frac {-10 \sqrt {2 x-1} \sqrt {5 x+3} \left (4800 x^3+18840 x^2+45538 x-71199\right )-463881 \sqrt {10} (2 x-1) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{6400 \sqrt {-(1-2 x)^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-10*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(-71199 + 45538*x + 18840*x^2 + 4800*x^3) - 463881*Sqrt[10]*(-1 + 2*x)*ArcSi
nh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(6400*Sqrt[-(1 - 2*x)^2])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.53, size = 127, normalized size = 1.09 \begin {gather*} \frac {\sqrt {11-2 (5 x+3)} \left (192 (5 x+3)^{7/2}+2040 (5 x+3)^{5/2}+28114 (5 x+3)^{3/2}-463881 \sqrt {5 x+3}\right )}{640 \sqrt {5} (2 (5 x+3)-11)}+\frac {463881 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{320 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[11 - 2*(3 + 5*x)]*(-463881*Sqrt[3 + 5*x] + 28114*(3 + 5*x)^(3/2) + 2040*(3 + 5*x)^(5/2) + 192*(3 + 5*x)^
(7/2)))/(640*Sqrt[5]*(-11 + 2*(3 + 5*x))) + (463881*ArcTan[(Sqrt[2]*Sqrt[3 + 5*x])/(Sqrt[11] - Sqrt[11 - 2*(3
+ 5*x)])])/(320*Sqrt[10])

________________________________________________________________________________________

fricas [A]  time = 1.17, size = 86, normalized size = 0.74 \begin {gather*} \frac {463881 \, \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 (4800 \, x^{3} + 18840 \, x^{2} + 45538 \, x - 71199\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{12800 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12800*(463881*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*(4800*x^3 + 18840*x^2 + 45538*x - 71199)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)

________________________________________________________________________________________

giac [A]  time = 1.02, size = 84, normalized size = 0.72 \begin {gather*} -\frac {463881}{6400} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (2 \, {\left (12 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} + 85 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 14057 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 463881 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{16000 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-463881/6400*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/16000*(2*(12*(8*sqrt(5)*(5*x + 3) + 85*sqrt(5))*
(5*x + 3) + 14057*sqrt(5))*(5*x + 3) - 463881*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 123, normalized size = 1.06 \begin {gather*} -\frac {\left (-96000 \sqrt {-10 x^{2}-x +3}\, x^{3}-376800 \sqrt {-10 x^{2}-x +3}\, x^{2}+927762 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-910760 \sqrt {-10 x^{2}-x +3}\, x -463881 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1423980 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{12800 \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)^2*(5*x+3)^(3/2)/(-2*x+1)^(3/2),x)

[Out]

-1/12800*(-96000*(-10*x^2-x+3)^(1/2)*x^3+927762*10^(1/2)*x*arcsin(20/11*x+1/11)-376800*(-10*x^2-x+3)^(1/2)*x^2
-463881*10^(1/2)*arcsin(20/11*x+1/11)-910760*(-10*x^2-x+3)^(1/2)*x+1423980*(-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 [C]  time = 1.33, size = 154, normalized size = 1.33 \begin {gather*} -\frac {23793}{640} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {11979}{12800} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x - \frac {21}{11}\right ) - \frac {3}{8} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {99}{32} \, \sqrt {10 \, x^{2} - 21 \, x + 8} x - \frac {2079}{640} \, \sqrt {10 \, x^{2} - 21 \, x + 8} + \frac {693}{32} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {49 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{8 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac {21 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{8 \, {\left (2 \, x - 1\right )}} - \frac {1617 \, \sqrt {-10 \, x^{2} - x + 3}}{16 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-23793/640*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 11979/12800*I*sqrt(5)*sqrt(2)*arcsin(20/11*x - 21/11) - 3/
8*(-10*x^2 - x + 3)^(3/2) + 99/32*sqrt(10*x^2 - 21*x + 8)*x - 2079/640*sqrt(10*x^2 - 21*x + 8) + 693/32*sqrt(-
10*x^2 - x + 3) - 49/8*(-10*x^2 - x + 3)^(3/2)/(4*x^2 - 4*x + 1) - 21/8*(-10*x^2 - x + 3)^(3/2)/(2*x - 1) - 16
17/16*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 )}^2\,{\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)^2*(5*x + 3)^(3/2))/(1 - 2*x)^(3/2),x)

[Out]

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

[Out]

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

________________________________________________________________________________________