\(\int x^3 (2+3 x)^{3/2} \sqrt {1+4 x} \, dx\) [438]

Optimal result

Integrand size = 22, antiderivative size = 160 \[ \int x^3 (2+3 x)^{3/2} \sqrt {1+4 x} \, dx=\frac {213575 \sqrt {2+3 x} \sqrt {1+4 x}}{42467328}+\frac {42715 (2+3 x)^{3/2} \sqrt {1+4 x}}{15925248}-\frac {8543 (2+3 x)^{5/2} \sqrt {1+4 x}}{995328}+\frac {323 (2+3 x)^{5/2} (1+4 x)^{3/2}}{18432}-\frac {3}{320} (2+3 x)^{7/2} (1+4 x)^{3/2}+\frac {1}{648} (2+3 x)^{9/2} (1+4 x)^{3/2}+\frac {1067875 \text {arcsinh}\left (\sqrt {\frac {3}{5}} \sqrt {1+4 x}\right )}{84934656 \sqrt {3}} \] Output:

213575/42467328*(2+3*x)^(1/2)*(1+4*x)^(1/2)+42715/15925248*(2+3*x)^(3/2)*( 
1+4*x)^(1/2)-8543/995328*(2+3*x)^(5/2)*(1+4*x)^(1/2)+323/18432*(2+3*x)^(5/ 
2)*(1+4*x)^(3/2)-3/320*(2+3*x)^(7/2)*(1+4*x)^(3/2)+1/648*(2+3*x)^(9/2)*(1+ 
4*x)^(3/2)+1067875/254803968*arcsinh(1/5*15^(1/2)*(1+4*x)^(1/2))*3^(1/2)
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.56 \[ \int x^3 (2+3 x)^{3/2} \sqrt {1+4 x} \, dx=\frac {6 \sqrt {1+4 x} \left (-1763226-465655 x-430680 x^2+2689920 x^3+201692160 x^4+496336896 x^5+318504960 x^6\right )+5339375 \sqrt {6+9 x} \text {arctanh}\left (\frac {\sqrt {3+12 x}}{2 \sqrt {2+3 x}}\right )}{1274019840 \sqrt {2+3 x}} \] Input:

Integrate[x^3*(2 + 3*x)^(3/2)*Sqrt[1 + 4*x],x]
 

Output:

(6*Sqrt[1 + 4*x]*(-1763226 - 465655*x - 430680*x^2 + 2689920*x^3 + 2016921 
60*x^4 + 496336896*x^5 + 318504960*x^6) + 5339375*Sqrt[6 + 9*x]*ArcTanh[Sq 
rt[3 + 12*x]/(2*Sqrt[2 + 3*x])])/(1274019840*Sqrt[2 + 3*x])
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {111, 27, 164, 60, 60, 60, 64, 222}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^3 (3 x+2)^{3/2} \sqrt {4 x+1} \, dx\)

\(\Big \downarrow \) 111

\(\displaystyle \frac {1}{72} \int -\frac {1}{2} x (3 x+2)^{3/2} \sqrt {4 x+1} (83 x+8)dx+\frac {1}{72} x^2 (4 x+1)^{3/2} (3 x+2)^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{72} x^2 (3 x+2)^{5/2} (4 x+1)^{3/2}-\frac {1}{144} \int x (3 x+2)^{3/2} \sqrt {4 x+1} (83 x+8)dx\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {1}{144} \left (\frac {(4103-7968 x) (3 x+2)^{5/2} (4 x+1)^{3/2}}{5760}-\frac {8543}{768} \int (3 x+2)^{3/2} \sqrt {4 x+1}dx\right )+\frac {1}{72} x^2 (4 x+1)^{3/2} (3 x+2)^{5/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{144} \left (\frac {(4103-7968 x) (3 x+2)^{5/2} (4 x+1)^{3/2}}{5760}-\frac {8543}{768} \left (\frac {1}{9} (3 x+2)^{5/2} \sqrt {4 x+1}-\frac {5}{18} \int \frac {(3 x+2)^{3/2}}{\sqrt {4 x+1}}dx\right )\right )+\frac {1}{72} x^2 (4 x+1)^{3/2} (3 x+2)^{5/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{144} \left (\frac {(4103-7968 x) (3 x+2)^{5/2} (4 x+1)^{3/2}}{5760}-\frac {8543}{768} \left (\frac {1}{9} (3 x+2)^{5/2} \sqrt {4 x+1}-\frac {5}{18} \left (\frac {15}{16} \int \frac {\sqrt {3 x+2}}{\sqrt {4 x+1}}dx+\frac {1}{8} \sqrt {4 x+1} (3 x+2)^{3/2}\right )\right )\right )+\frac {1}{72} x^2 (4 x+1)^{3/2} (3 x+2)^{5/2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{144} \left (\frac {(4103-7968 x) (3 x+2)^{5/2} (4 x+1)^{3/2}}{5760}-\frac {8543}{768} \left (\frac {1}{9} (3 x+2)^{5/2} \sqrt {4 x+1}-\frac {5}{18} \left (\frac {15}{16} \left (\frac {5}{8} \int \frac {1}{\sqrt {3 x+2} \sqrt {4 x+1}}dx+\frac {1}{4} \sqrt {3 x+2} \sqrt {4 x+1}\right )+\frac {1}{8} \sqrt {4 x+1} (3 x+2)^{3/2}\right )\right )\right )+\frac {1}{72} x^2 (4 x+1)^{3/2} (3 x+2)^{5/2}\)

\(\Big \downarrow \) 64

\(\displaystyle \frac {1}{144} \left (\frac {(4103-7968 x) (3 x+2)^{5/2} (4 x+1)^{3/2}}{5760}-\frac {8543}{768} \left (\frac {1}{9} (3 x+2)^{5/2} \sqrt {4 x+1}-\frac {5}{18} \left (\frac {15}{16} \left (\frac {5}{16} \int \frac {1}{\sqrt {\frac {3}{4} (4 x+1)+\frac {5}{4}}}d\sqrt {4 x+1}+\frac {1}{4} \sqrt {3 x+2} \sqrt {4 x+1}\right )+\frac {1}{8} \sqrt {4 x+1} (3 x+2)^{3/2}\right )\right )\right )+\frac {1}{72} x^2 (4 x+1)^{3/2} (3 x+2)^{5/2}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {1}{144} \left (\frac {(4103-7968 x) (3 x+2)^{5/2} (4 x+1)^{3/2}}{5760}-\frac {8543}{768} \left (\frac {1}{9} (3 x+2)^{5/2} \sqrt {4 x+1}-\frac {5}{18} \left (\frac {15}{16} \left (\frac {5 \text {arcsinh}\left (\sqrt {\frac {3}{5}} \sqrt {4 x+1}\right )}{8 \sqrt {3}}+\frac {1}{4} \sqrt {3 x+2} \sqrt {4 x+1}\right )+\frac {1}{8} \sqrt {4 x+1} (3 x+2)^{3/2}\right )\right )\right )+\frac {1}{72} x^2 (4 x+1)^{3/2} (3 x+2)^{5/2}\)

Input:

Int[x^3*(2 + 3*x)^(3/2)*Sqrt[1 + 4*x],x]
 

Output:

(x^2*(2 + 3*x)^(5/2)*(1 + 4*x)^(3/2))/72 + (((4103 - 7968*x)*(2 + 3*x)^(5/ 
2)*(1 + 4*x)^(3/2))/5760 - (8543*(((2 + 3*x)^(5/2)*Sqrt[1 + 4*x])/9 - (5*( 
((2 + 3*x)^(3/2)*Sqrt[1 + 4*x])/8 + (15*((Sqrt[2 + 3*x]*Sqrt[1 + 4*x])/4 + 
 (5*ArcSinh[Sqrt[3/5]*Sqrt[1 + 4*x]])/(8*Sqrt[3])))/16))/18))/768)/144
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 60

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] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 64

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp 
[2/b   Subst[Int[1/Sqrt[c - a*(d/b) + d*(x^2/b)], x], x, Sqrt[a + b*x]], x] 
 /; FreeQ[{a, b, c, d}, x] && GtQ[c - a*(d/b), 0] && ( !GtQ[a - c*(b/d), 0] 
 || PosQ[b])
 

rule 111

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

rule 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> 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] + Simp[(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 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.62

method	result	size
risch	\(\frac {\left (106168320 x^{5}[phone] x^{4}+4119552 x^{3}-1849728 x^{2}+1089592 x -881613\right ) \sqrt {2+3 x}\, \sqrt {1+4 x}}{212336640}+\frac {1067875 \ln \left (\frac {\left (\frac {11}{2}+12 x \right ) \sqrt {12}}{12}+\sqrt {12 x^{2}+11 x +2}\right ) \sqrt {12}\, \sqrt {\left (2+3 x \right ) \left (1+4 x \right )}}{1019215872 \sqrt {2+3 x}\, \sqrt {1+4 x}}\)	\(99\)
default	\(\frac {\sqrt {2+3 x}\, \sqrt {1+4 x}\, \left (1274019840 x^{5} \sqrt {12 x^{2}+11 x +2}[phone] x^{4} \sqrt {12 x^{2}+11 x +2}[phone] x^{3} \sqrt {12 x^{2}+11 x +2}-22196736 x^{2} \sqrt {12 x^{2}+11 x +2}+5339375 \ln \left (\frac {11 \sqrt {3}}{12}+2 \sqrt {3}\, x +\sqrt {12 x^{2}+11 x +2}\right ) \sqrt {3}[phone] \sqrt {12 x^{2}+11 x +2}\, x -10579356 \sqrt {12 x^{2}+11 x +2}\right )}{2548039680 \sqrt {12 x^{2}+11 x +2}}\)	\(157\)

Input:

int(x^3*(2+3*x)^(3/2)*(1+4*x)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/212336640*(106168320*x^5+94666752*x^4+4119552*x^3-1849728*x^2+1089592*x- 
881613)*(2+3*x)^(1/2)*(1+4*x)^(1/2)+1067875/1019215872*ln(1/12*(11/2+12*x) 
*12^(1/2)+(12*x^2+11*x+2)^(1/2))*12^(1/2)*((2+3*x)*(1+4*x))^(1/2)/(2+3*x)^ 
(1/2)/(1+4*x)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.51 \[ \int x^3 (2+3 x)^{3/2} \sqrt {1+4 x} \, dx=\frac {1}{212336640} \, {\left (106168320 \, x^{5} + 94666752 \, x^{4} + 4119552 \, x^{3} - 1849728 \, x^{2} + 1089592 \, x - 881613\right )} \sqrt {4 \, x + 1} \sqrt {3 \, x + 2} + \frac {1067875}{1019215872} \, \sqrt {3} \log \left (8 \, \sqrt {3} {\left (24 \, x + 11\right )} \sqrt {4 \, x + 1} \sqrt {3 \, x + 2} + 1152 \, x^{2} + 1056 \, x + 217\right ) \] Input:

integrate(x^3*(2+3*x)^(3/2)*(1+4*x)^(1/2),x, algorithm="fricas")
 

Output:

1/212336640*(106168320*x^5 + 94666752*x^4 + 4119552*x^3 - 1849728*x^2 + 10 
89592*x - 881613)*sqrt(4*x + 1)*sqrt(3*x + 2) + 1067875/1019215872*sqrt(3) 
*log(8*sqrt(3)*(24*x + 11)*sqrt(4*x + 1)*sqrt(3*x + 2) + 1152*x^2 + 1056*x 
 + 217)
 

Sympy [F]

\[ \int x^3 (2+3 x)^{3/2} \sqrt {1+4 x} \, dx=\int x^{3} \left (3 x + 2\right )^{\frac {3}{2}} \sqrt {4 x + 1}\, dx \] Input:

integrate(x**3*(2+3*x)**(3/2)*(1+4*x)**(1/2),x)
 

Output:

Integral(x**3*(3*x + 2)**(3/2)*sqrt(4*x + 1), x)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.76 \[ \int x^3 (2+3 x)^{3/2} \sqrt {1+4 x} \, dx=\frac {1}{24} \, {\left (12 \, x^{2} + 11 \, x + 2\right )}^{\frac {3}{2}} x^{3} - \frac {1}{960} \, {\left (12 \, x^{2} + 11 \, x + 2\right )}^{\frac {3}{2}} x^{2} - \frac {403}{92160} \, {\left (12 \, x^{2} + 11 \, x + 2\right )}^{\frac {3}{2}} x + \frac {22933}{6635520} \, {\left (12 \, x^{2} + 11 \, x + 2\right )}^{\frac {3}{2}} - \frac {42715}{1769472} \, \sqrt {12 \, x^{2} + 11 \, x + 2} x + \frac {1067875}{509607936} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {12 \, x^{2} + 11 \, x + 2} + 24 \, x + 11\right ) - \frac {469865}{42467328} \, \sqrt {12 \, x^{2} + 11 \, x + 2} \] Input:

integrate(x^3*(2+3*x)^(3/2)*(1+4*x)^(1/2),x, algorithm="maxima")
 

Output:

1/24*(12*x^2 + 11*x + 2)^(3/2)*x^3 - 1/960*(12*x^2 + 11*x + 2)^(3/2)*x^2 - 
 403/92160*(12*x^2 + 11*x + 2)^(3/2)*x + 22933/6635520*(12*x^2 + 11*x + 2) 
^(3/2) - 42715/1769472*sqrt(12*x^2 + 11*x + 2)*x + 1067875/509607936*sqrt( 
3)*log(4*sqrt(3)*sqrt(12*x^2 + 11*x + 2) + 24*x + 11) - 469865/42467328*sq 
rt(12*x^2 + 11*x + 2)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.08 \[ \int x^3 (2+3 x)^{3/2} \sqrt {1+4 x} \, dx=\frac {1}{14155776} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (24 \, x - 29\right )} {\left (4 \, x + 1\right )} + 645\right )} {\left (4 \, x + 1\right )} - 3685\right )} {\left (4 \, x + 1\right )} - 28835\right )} {\left (4 \, x + 1\right )} + 448303\right )} \sqrt {4 \, x + 1} \sqrt {3 \, x + 2} + \frac {11}{53084160} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (96 \, x - 91\right )} {\left (4 \, x + 1\right )} + 3545\right )} {\left (4 \, x + 1\right )} - 10865\right )} {\left (4 \, x + 1\right )} - 239435\right )} \sqrt {4 \, x + 1} \sqrt {3 \, x + 2} + \frac {1}{221184} \, {\left (2 \, {\left (12 \, {\left (72 \, x - 49\right )} {\left (4 \, x + 1\right )} + 811\right )} {\left (4 \, x + 1\right )} + 2857\right )} \sqrt {4 \, x + 1} \sqrt {3 \, x + 2} - \frac {1067875}{254803968} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {4 \, x + 1} + 2 \, \sqrt {3 \, x + 2}\right ) \] Input:

integrate(x^3*(2+3*x)^(3/2)*(1+4*x)^(1/2),x, algorithm="giac")
 

Output:

1/14155776*(2*(12*(6*(8*(24*x - 29)*(4*x + 1) + 645)*(4*x + 1) - 3685)*(4* 
x + 1) - 28835)*(4*x + 1) + 448303)*sqrt(4*x + 1)*sqrt(3*x + 2) + 11/53084 
160*(2*(12*(18*(96*x - 91)*(4*x + 1) + 3545)*(4*x + 1) - 10865)*(4*x + 1) 
- 239435)*sqrt(4*x + 1)*sqrt(3*x + 2) + 1/221184*(2*(12*(72*x - 49)*(4*x + 
 1) + 811)*(4*x + 1) + 2857)*sqrt(4*x + 1)*sqrt(3*x + 2) - 1067875/2548039 
68*sqrt(3)*log(-sqrt(3)*sqrt(4*x + 1) + 2*sqrt(3*x + 2))
 

Mupad [F(-1)]

Timed out. \[ \int x^3 (2+3 x)^{3/2} \sqrt {1+4 x} \, dx=\int x^3\,{\left (3\,x+2\right )}^{3/2}\,\sqrt {4\,x+1} \,d x \] Input:

int(x^3*(3*x + 2)^(3/2)*(4*x + 1)^(1/2),x)
 

Output:

int(x^3*(3*x + 2)^(3/2)*(4*x + 1)^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.79 \[ \int x^3 (2+3 x)^{3/2} \sqrt {1+4 x} \, dx=\frac {\sqrt {3 x +2}\, \sqrt {4 x +1}\, x^{5}}{2}+\frac {107 \sqrt {3 x +2}\, \sqrt {4 x +1}\, x^{4}}{240}+\frac {149 \sqrt {3 x +2}\, \sqrt {4 x +1}\, x^{3}}{7680}-\frac {4817 \sqrt {3 x +2}\, \sqrt {4 x +1}\, x^{2}}{552960}+\frac {136199 \sqrt {3 x +2}\, \sqrt {4 x +1}\, x}{26542080}-\frac {97957 \sqrt {3 x +2}\, \sqrt {4 x +1}}{23592960}+\frac {1067875 \sqrt {3}\, \mathrm {log}\left (\frac {\sqrt {4 x +1}\, \sqrt {3}+2 \sqrt {3 x +2}}{\sqrt {5}}\right )}{254803968} \] Input:

int(x^3*(2+3*x)^(3/2)*(1+4*x)^(1/2),x)
 

Output:

(637009920*sqrt(3*x + 2)*sqrt(4*x + 1)*x**5 + 568000512*sqrt(3*x + 2)*sqrt 
(4*x + 1)*x**4 + 24717312*sqrt(3*x + 2)*sqrt(4*x + 1)*x**3 - 11098368*sqrt 
(3*x + 2)*sqrt(4*x + 1)*x**2 + 6537552*sqrt(3*x + 2)*sqrt(4*x + 1)*x - 528 
9678*sqrt(3*x + 2)*sqrt(4*x + 1) + 5339375*sqrt(3)*log((sqrt(4*x + 1)*sqrt 
(3) + 2*sqrt(3*x + 2))/sqrt(5)))/1274019840