\(\int (-3 x^{3/5}+x^{3/2})^2 (-\frac {x^{2/3}}{3}+4 x^{3/2}) \, dx\) [209]

Optimal result

Integrand size = 33, antiderivative size = 55 \[ \int \left (-3 x^{3/5}+x^{3/2}\right )^2 \left (-\frac {x^{2/3}}{3}+4 x^{3/2}\right ) \, dx=-\frac {45 x^{43/15}}{43}+\frac {360 x^{37/10}}{37}+\frac {60 x^{113/30}}{113}-\frac {120 x^{23/5}}{23}-\frac {x^{14/3}}{14}+\frac {8 x^{11/2}}{11} \] Output:

-45/43*x^(43/15)+360/37*x^(37/10)+60/113*x^(113/30)-120/23*x^(23/5)-1/14*x 
^(14/3)+8/11*x^(11/2)
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \left (-3 x^{3/5}+x^{3/2}\right )^2 \left (-\frac {x^{2/3}}{3}+4 x^{3/2}\right ) \, dx=-\frac {45 x^{43/15}}{43}+\frac {360 x^{37/10}}{37}+\frac {60 x^{113/30}}{113}-\frac {120 x^{23/5}}{23}-\frac {x^{14/3}}{14}+\frac {8 x^{11/2}}{11} \] Input:

Integrate[(-3*x^(3/5) + x^(3/2))^2*(-1/3*x^(2/3) + 4*x^(3/2)),x]
 

Output:

(-45*x^(43/15))/43 + (360*x^(37/10))/37 + (60*x^(113/30))/113 - (120*x^(23 
/5))/23 - x^(14/3)/14 + (8*x^(11/2))/11
 

Rubi [A] (warning: unable to verify)

Time = 0.55 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {2027, 10, 27, 2035, 7267, 25, 2360, 2009}

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.

\(\displaystyle \int \left (x^{3/2}-3 x^{3/5}\right )^2 \left (4 x^{3/2}-\frac {x^{2/3}}{3}\right ) \, dx\)

\(\Big \downarrow \) 2027

\(\displaystyle \int \left (x^{9/10}-3\right )^2 x^{6/5} \left (4 x^{3/2}-\frac {x^{2/3}}{3}\right )dx\)

\(\Big \downarrow \) 10

\(\displaystyle \int -\frac {1}{3} \left (1-12 x^{5/6}\right ) \left (3-x^{9/10}\right )^2 x^{28/15}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{3} \int \left (1-12 x^{5/6}\right ) \left (3-x^{9/10}\right )^2 x^{28/15}dx\)

\(\Big \downarrow \) 2035

\(\displaystyle -5 \int \left (1-12 x^{5/6}\right ) \left (3-x^{9/10}\right )^2 x^{14/5}d\sqrt [15]{x}\)

\(\Big \downarrow \) 7267

\(\displaystyle 10 \int -x^{17/3} \left (1-12 x^{5/3}\right ) \left (3-x^{9/5}\right )^2d\sqrt [30]{x}\)

\(\Big \downarrow \) 25

\(\displaystyle -10 \int x^{17/3} \left (1-12 x^{5/3}\right ) \left (3-x^{9/5}\right )^2d\sqrt [30]{x}\)

\(\Big \downarrow \) 2360

\(\displaystyle -10 \int \left (-12 x^{164/15}+x^{139/15}+72 x^{137/15}-6 x^{112/15}-108 x^{22/3}+9 x^{17/3}\right )d\sqrt [30]{x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 10 \left (-\frac {x^{28/3}}{140}-\frac {12 x^{46/5}}{23}+\frac {6 x^{113/15}}{113}+\frac {36 x^{37/5}}{37}-\frac {9 x^{86/15}}{86}+\frac {4 x^{11}}{55}\right )\)

Input:

Int[(-3*x^(3/5) + x^(3/2))^2*(-1/3*x^(2/3) + 4*x^(3/2)),x]
 

Output:

10*((-9*x^(86/15))/86 + (36*x^(37/5))/37 + (6*x^(113/15))/113 - (12*x^(46/ 
5))/23 - x^(28/3)/140 + (4*x^11)/55)
 

Defintions of rubi rules used

rule 10

Int[(u_.)*((e_.)*(x_))^(m_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x 
_Symbol] :> Simp[1/e^(p*r)   Int[u*(e*x)^(m + p*r)*(a + b*x^(s - r))^p, x], 
 x] /; FreeQ[{a, b, e, m, r, s}, x] && IntegerQ[p] && (IntegerQ[p*r] || GtQ 
[e, 0]) && PosQ[s - r]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2027

Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ 
(p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & 
& PosQ[s - r] &&  !(EqQ[p, 1] && EqQ[u, 1])
 

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2360

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> 
Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, 
n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])
 

rule 7267

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
 

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.58

method	result	size
derivativedivides	\(-\frac {45 x^{\frac {43}{15}}}{43}+\frac {360 x^{\frac {37}{10}}}{37}+\frac {60 x^{\frac {113}{30}}}{113}-\frac {120 x^{\frac {23}{5}}}{23}-\frac {x^{\frac {14}{3}}}{14}+\frac {8 x^{\frac {11}{2}}}{11}\)	\(32\)
default	\(-\frac {45 x^{\frac {43}{15}}}{43}+\frac {360 x^{\frac {37}{10}}}{37}+\frac {60 x^{\frac {113}{30}}}{113}-\frac {120 x^{\frac {23}{5}}}{23}-\frac {x^{\frac {14}{3}}}{14}+\frac {8 x^{\frac {11}{2}}}{11}\)	\(32\)
orering	\(\text {Expression too large to display}\)	\(1068\)

Input:

int((-3*x^(3/5)+x^(3/2))^2*(-1/3*x^(2/3)+4*x^(3/2)),x,method=_RETURNVERBOS 
E)
 

Output:

-45/43*x^(43/15)+360/37*x^(37/10)+60/113*x^(113/30)-120/23*x^(23/5)-1/14*x 
^(14/3)+8/11*x^(11/2)
 

Fricas [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.56 \[ \int \left (-3 x^{3/5}+x^{3/2}\right )^2 \left (-\frac {x^{2/3}}{3}+4 x^{3/2}\right ) \, dx=\frac {8}{11} \, x^{\frac {11}{2}} - \frac {1}{14} \, x^{\frac {14}{3}} - \frac {120}{23} \, x^{\frac {23}{5}} + \frac {60}{113} \, x^{\frac {113}{30}} + \frac {360}{37} \, x^{\frac {37}{10}} - \frac {45}{43} \, x^{\frac {43}{15}} \] Input:

integrate((-3*x^(3/5)+x^(3/2))^2*(-1/3*x^(2/3)+4*x^(3/2)),x, algorithm="fr 
icas")
 

Output:

8/11*x^(11/2) - 1/14*x^(14/3) - 120/23*x^(23/5) + 60/113*x^(113/30) + 360/ 
37*x^(37/10) - 45/43*x^(43/15)
 

Sympy [A] (verification not implemented)

Time = 1.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87 \[ \int \left (-3 x^{3/5}+x^{3/2}\right )^2 \left (-\frac {x^{2/3}}{3}+4 x^{3/2}\right ) \, dx=\frac {60 x^{\frac {113}{30}}}{113} - \frac {45 x^{\frac {43}{15}}}{43} + \frac {360 x^{\frac {37}{10}}}{37} - \frac {120 x^{\frac {23}{5}}}{23} - \frac {x^{\frac {14}{3}}}{14} + \frac {8 x^{\frac {11}{2}}}{11} \] Input:

integrate((-3*x**(3/5)+x**(3/2))**2*(-1/3*x**(2/3)+4*x**(3/2)),x)
 

Output:

60*x**(113/30)/113 - 45*x**(43/15)/43 + 360*x**(37/10)/37 - 120*x**(23/5)/ 
23 - x**(14/3)/14 + 8*x**(11/2)/11
 

Maxima [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.56 \[ \int \left (-3 x^{3/5}+x^{3/2}\right )^2 \left (-\frac {x^{2/3}}{3}+4 x^{3/2}\right ) \, dx=\frac {8}{11} \, x^{\frac {11}{2}} - \frac {1}{14} \, x^{\frac {14}{3}} - \frac {120}{23} \, x^{\frac {23}{5}} + \frac {60}{113} \, x^{\frac {113}{30}} + \frac {360}{37} \, x^{\frac {37}{10}} - \frac {45}{43} \, x^{\frac {43}{15}} \] Input:

integrate((-3*x^(3/5)+x^(3/2))^2*(-1/3*x^(2/3)+4*x^(3/2)),x, algorithm="ma 
xima")
 

Output:

8/11*x^(11/2) - 1/14*x^(14/3) - 120/23*x^(23/5) + 60/113*x^(113/30) + 360/ 
37*x^(37/10) - 45/43*x^(43/15)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.56 \[ \int \left (-3 x^{3/5}+x^{3/2}\right )^2 \left (-\frac {x^{2/3}}{3}+4 x^{3/2}\right ) \, dx=\frac {8}{11} \, x^{\frac {11}{2}} - \frac {1}{14} \, x^{\frac {14}{3}} - \frac {120}{23} \, x^{\frac {23}{5}} + \frac {60}{113} \, x^{\frac {113}{30}} + \frac {360}{37} \, x^{\frac {37}{10}} - \frac {45}{43} \, x^{\frac {43}{15}} \] Input:

integrate((-3*x^(3/5)+x^(3/2))^2*(-1/3*x^(2/3)+4*x^(3/2)),x, algorithm="gi 
ac")
 

Output:

8/11*x^(11/2) - 1/14*x^(14/3) - 120/23*x^(23/5) + 60/113*x^(113/30) + 360/ 
37*x^(37/10) - 45/43*x^(43/15)
 

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.56 \[ \int \left (-3 x^{3/5}+x^{3/2}\right )^2 \left (-\frac {x^{2/3}}{3}+4 x^{3/2}\right ) \, dx=\frac {8\,x^{11/2}}{11}-\frac {x^{14/3}}{14}-\frac {120\,x^{23/5}}{23}+\frac {360\,x^{37/10}}{37}-\frac {45\,x^{43/15}}{43}+\frac {60\,x^{113/30}}{113} \] Input:

int(-(x^(3/2) - 3*x^(3/5))^2*(x^(2/3)/3 - 4*x^(3/2)),x)
 

Output:

(8*x^(11/2))/11 - x^(14/3)/14 - (120*x^(23/5))/23 + (360*x^(37/10))/37 - ( 
45*x^(43/15))/43 + (60*x^(113/30))/113
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.69 \[ \int \left (-3 x^{3/5}+x^{3/2}\right )^2 \left (-\frac {x^{2/3}}{3}+4 x^{3/2}\right ) \, dx=\frac {x^{2} \left (338119320 x^{\frac {53}{30}}-666409590 x^{\frac {13}{15}}[phone] x^{\frac {17}{10}}-3322389840 x^{\frac {13}{5}}-45485099 x^{\frac {8}{3}}[phone] \sqrt {x}\, x^{3}\right )}{636791386} \] Input:

int((-3*x^(3/5)+x^(3/2))^2*(-1/3*x^(2/3)+4*x^(3/2)),x)
 

Output:

(x**2*(338119320*x**(23/30)*x - 666409590*x**(13/15) + 6195808080*x**(7/10 
)*x - 3322389840*x**(3/5)*x**2 - 45485099*x**(2/3)*x**2 + 463121008*sqrt(x 
)*x**3))/636791386