\(\int \frac {1}{(-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3)^{7/2}} \, dx\) [99]

Optimal result

Integrand size = 32, antiderivative size = 241 \[ \int \frac {1}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{7/2}} \, dx=-\frac {b+c x}{480 c \left (-64+(b+c x)^3\right )^{5/2}}+\frac {13 (b+c x)}{276480 c \left (-64+(b+c x)^3\right )^{3/2}}-\frac {91 (b+c x)}{53084160 c \sqrt {-64+(b+c x)^3}}+\frac {91 \sqrt {2-\sqrt {3}} (4-b-c x) \sqrt {\frac {16+4 (b+c x)+(b+c x)^2}{\left (4-4 \sqrt {3}-b-c x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {4+4 \sqrt {3}-b-c x}{4-4 \sqrt {3}-b-c x}\right ),-7+4 \sqrt {3}\right )}{106168320 \sqrt [4]{3} c \sqrt {-\frac {4-b-c x}{\left (4-4 \sqrt {3}-b-c x\right )^2}} \sqrt {-64+(b+c x)^3}} \] Output:

-1/480*(c*x+b)/c/(-64+(c*x+b)^3)^(5/2)+13/276480*(c*x+b)/c/(-64+(c*x+b)^3) 
^(3/2)-91/53084160*(c*x+b)/c/(-64+(c*x+b)^3)^(1/2)+91/318504960*3^(3/4)*(1 
/2*6^(1/2)-1/2*2^(1/2))*(-c*x-b+4)*((16+4*c*x+4*b+(c*x+b)^2)/(4-4*3^(1/2)- 
b-c*x)^2)^(1/2)*EllipticF((4+4*3^(1/2)-b-c*x)/(4-4*3^(1/2)-b-c*x),2*I-I*3^ 
(1/2))/c/(-(-c*x-b+4)/(4-4*3^(1/2)-b-c*x)^2)^(1/2)/(-64+(c*x+b)^3)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.92 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.19 \[ \int \frac {1}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{7/2}} \, dx=\frac {-2 (b+c x) \left (643072+91 b^6+546 b^5 c x+1365 b^4 c^2 x^2-14144 c^3 x^3+91 c^6 x^6+78 b c^2 x^2 \left (-544+7 c^3 x^3\right )+39 b^2 c x \left (-1088+35 c^3 x^3\right )+52 b^3 \left (-272+35 c^3 x^3\right )\right )+91 i \sqrt {2} \sqrt {-\frac {i (-4+b+c x)}{3 i+\sqrt {3}}} \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^2 \sqrt {16+b^2+4 c x+c^2 x^2+2 b (2+c x)} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2 i+2 \sqrt {3}+i b+i c x}}{2 \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )}{106168320 c \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{5/2}} \] Input:

Integrate[(-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^2 + c^3*x^3)^(-7/2),x]
 

Output:

(-2*(b + c*x)*(643072 + 91*b^6 + 546*b^5*c*x + 1365*b^4*c^2*x^2 - 14144*c^ 
3*x^3 + 91*c^6*x^6 + 78*b*c^2*x^2*(-544 + 7*c^3*x^3) + 39*b^2*c*x*(-1088 + 
 35*c^3*x^3) + 52*b^3*(-272 + 35*c^3*x^3)) + (91*I)*Sqrt[2]*Sqrt[((-I)*(-4 
 + b + c*x))/(3*I + Sqrt[3])]*(-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^2 + c^3*x 
^3)^2*Sqrt[16 + b^2 + 4*c*x + c^2*x^2 + 2*b*(2 + c*x)]*EllipticF[ArcSin[Sq 
rt[2*I + 2*Sqrt[3] + I*b + I*c*x]/(2*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3] 
)])/(106168320*c*(-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^2 + c^3*x^3)^(5/2))
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.28, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2458, 749, 749, 749, 760}

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.

\(\displaystyle \int \frac {1}{\left (b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3-64\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 2458

\(\displaystyle \int \frac {1}{\left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{7/2}}d\left (\frac {b}{c}+x\right )\)

\(\Big \downarrow \) 749

\(\displaystyle -\frac {13}{960} \int \frac {1}{\left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{5/2}}d\left (\frac {b}{c}+x\right )-\frac {\frac {b}{c}+x}{480 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{5/2}}\)

\(\Big \downarrow \) 749

\(\displaystyle -\frac {13}{960} \left (-\frac {7}{576} \int \frac {1}{\left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}}d\left (\frac {b}{c}+x\right )-\frac {\frac {b}{c}+x}{288 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}}\right )-\frac {\frac {b}{c}+x}{480 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{5/2}}\)

\(\Big \downarrow \) 749

\(\displaystyle -\frac {13}{960} \left (-\frac {7}{576} \left (-\frac {1}{192} \int \frac {1}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )-\frac {\frac {b}{c}+x}{96 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}\right )-\frac {\frac {b}{c}+x}{288 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}}\right )-\frac {\frac {b}{c}+x}{480 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{5/2}}\)

\(\Big \downarrow \) 760

\(\displaystyle -\frac {13}{960} \left (-\frac {7}{576} \left (\frac {\sqrt {2-\sqrt {3}} \left (4-c \left (\frac {b}{c}+x\right )\right ) \sqrt {\frac {c^2 \left (\frac {b}{c}+x\right )^2+4 c \left (\frac {b}{c}+x\right )+16}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}\right ),-7+4 \sqrt {3}\right )}{192 \sqrt [4]{3} c \sqrt {-\frac {4-c \left (\frac {b}{c}+x\right )}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}-\frac {\frac {b}{c}+x}{96 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}\right )-\frac {\frac {b}{c}+x}{288 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}}\right )-\frac {\frac {b}{c}+x}{480 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{5/2}}\)

Input:

Int[(-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^2 + c^3*x^3)^(-7/2),x]
 

Output:

-1/480*(b/c + x)/(-64 + c^3*(b/c + x)^3)^(5/2) - (13*(-1/288*(b/c + x)/(-6 
4 + c^3*(b/c + x)^3)^(3/2) - (7*(-1/96*(b/c + x)/Sqrt[-64 + c^3*(b/c + x)^ 
3] + (Sqrt[2 - Sqrt[3]]*(4 - c*(b/c + x))*Sqrt[(16 + 4*c*(b/c + x) + c^2*( 
b/c + x)^2)/(4*(1 - Sqrt[3]) - c*(b/c + x))^2]*EllipticF[ArcSin[(4*(1 + Sq 
rt[3]) - c*(b/c + x))/(4*(1 - Sqrt[3]) - c*(b/c + x))], -7 + 4*Sqrt[3]])/( 
192*3^(1/4)*c*Sqrt[-((4 - c*(b/c + x))/(4*(1 - Sqrt[3]) - c*(b/c + x))^2)] 
*Sqrt[-64 + c^3*(b/c + x)^3])))/576))/960
 

Defintions of rubi rules used

rule 749

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 
 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1))   Int[(a + b*x^ 
n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (Inte 
gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
 

rule 760

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- 
s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) 
*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x 
] && NegQ[a]
 

rule 2458

Int[(Pn_)^(p_.), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1]/(Exp 
on[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x 
- S, x]^p, x], x, x + S] /; BinomialQ[Pn /. x -> x - S, x] || (IntegerQ[Exp 
on[Pn, x]/2] && TrinomialQ[Pn /. x -> x - S, x])] /; FreeQ[p, x] && PolyQ[P 
n, x] && GtQ[Expon[Pn, x], 2] && NeQ[Coeff[Pn, x, Expon[Pn, x] - 1], 0]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 491 vs. \(2 (208 ) = 416\).

Time = 0.41 (sec) , antiderivative size = 492, normalized size of antiderivative = 2.04

method	result	size
default	\(\frac {\left (-\frac {x}{480 c^{9}}-\frac {b}{480 c^{10}}\right ) \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}}{\left (x^{3}+\frac {3 b \,x^{2}}{c}+\frac {3 b^{2} x}{c^{2}}+\frac {b^{3}-64}{c^{3}}\right )^{3}}+\frac {\left (\frac {13 x}{276480 c^{6}}+\frac {13 b}{276480 c^{7}}\right ) \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}}{\left (x^{3}+\frac {3 b \,x^{2}}{c}+\frac {3 b^{2} x}{c^{2}}+\frac {b^{3}-64}{c^{3}}\right )^{2}}-\frac {2 c^{3} \left (\frac {91 x}{106168320 c^{3}}+\frac {91 b}{106168320 c^{4}}\right )}{\sqrt {\left (x^{3}+\frac {3 b \,x^{2}}{c}+\frac {3 b^{2} x}{c^{2}}+\frac {b^{3}-64}{c^{3}}\right ) c^{3}}}-\frac {91 \left (\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}\right ) \sqrt {\frac {x +\frac {b -4}{c}}{\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}}}\, \sqrt {\frac {x -\frac {-b -2+2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2+2 i \sqrt {3}}{c}}}\, \sqrt {\frac {x -\frac {-b -2-2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2-2 i \sqrt {3}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {b -4}{c}}{\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}}}, \sqrt {\frac {-\frac {b -4}{c}-\frac {-b -2-2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2+2 i \sqrt {3}}{c}}}\right )}{53084160 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}}\)	\(492\)
elliptic	\(\frac {\left (-\frac {x}{480 c^{9}}-\frac {b}{480 c^{10}}\right ) \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}}{\left (x^{3}+\frac {3 b \,x^{2}}{c}+\frac {3 b^{2} x}{c^{2}}+\frac {b^{3}-64}{c^{3}}\right )^{3}}+\frac {\left (\frac {13 x}{276480 c^{6}}+\frac {13 b}{276480 c^{7}}\right ) \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}}{\left (x^{3}+\frac {3 b \,x^{2}}{c}+\frac {3 b^{2} x}{c^{2}}+\frac {b^{3}-64}{c^{3}}\right )^{2}}-\frac {2 c^{3} \left (\frac {91 x}{106168320 c^{3}}+\frac {91 b}{106168320 c^{4}}\right )}{\sqrt {\left (x^{3}+\frac {3 b \,x^{2}}{c}+\frac {3 b^{2} x}{c^{2}}+\frac {b^{3}-64}{c^{3}}\right ) c^{3}}}-\frac {91 \left (\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}\right ) \sqrt {\frac {x +\frac {b -4}{c}}{\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}}}\, \sqrt {\frac {x -\frac {-b -2+2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2+2 i \sqrt {3}}{c}}}\, \sqrt {\frac {x -\frac {-b -2-2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2-2 i \sqrt {3}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {b -4}{c}}{\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}}}, \sqrt {\frac {-\frac {b -4}{c}-\frac {-b -2-2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2+2 i \sqrt {3}}{c}}}\right )}{53084160 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}}\)	\(492\)

Input:

int(1/(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(7/2),x,method=_RETURNVERBOSE 
)
 

Output:

(-1/480/c^9*x-1/480*b/c^10)*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(1/2)/( 
x^3+3*b/c*x^2+3*b^2/c^2*x+(b^3-64)/c^3)^3+(13/276480/c^6*x+13/276480*b/c^7 
)*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(1/2)/(x^3+3*b/c*x^2+3*b^2/c^2*x+ 
(b^3-64)/c^3)^2-2*c^3*(91/106168320*x/c^3+91/106168320*b/c^4)/((x^3+3*b/c* 
x^2+3*b^2/c^2*x+(b^3-64)/c^3)*c^3)^(1/2)-91/53084160*((-b-2-2*I*3^(1/2))/c 
+(b-4)/c)*((x+(b-4)/c)/((-b-2-2*I*3^(1/2))/c+(b-4)/c))^(1/2)*((x-(-b-2+2*I 
*3^(1/2))/c)/(-(b-4)/c-(-b-2+2*I*3^(1/2))/c))^(1/2)*((x-(-b-2-2*I*3^(1/2)) 
/c)/(-(b-4)/c-(-b-2-2*I*3^(1/2))/c))^(1/2)/(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+ 
b^3-64)^(1/2)*EllipticF(((x+(b-4)/c)/((-b-2-2*I*3^(1/2))/c+(b-4)/c))^(1/2) 
,((-(b-4)/c-(-b-2-2*I*3^(1/2))/c)/(-(b-4)/c-(-b-2+2*I*3^(1/2))/c))^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 483 vs. \(2 (185) = 370\).

Time = 0.09 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.00 \[ \int \frac {1}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{7/2}} \, dx=-\frac {91 \, {\left (c^{9} x^{9} + 9 \, b c^{8} x^{8} + 36 \, b^{2} c^{7} x^{7} + 12 \, {\left (7 \, b^{3} - 16\right )} c^{6} x^{6} + 18 \, {\left (7 \, b^{4} - 64 \, b\right )} c^{5} x^{5} + b^{9} + 18 \, {\left (7 \, b^{5} - 160 \, b^{2}\right )} c^{4} x^{4} + 12 \, {\left (7 \, b^{6} - 320 \, b^{3} + 1024\right )} c^{3} x^{3} - 192 \, b^{6} + 36 \, {\left (b^{7} - 80 \, b^{4} + 1024 \, b\right )} c^{2} x^{2} + 12288 \, b^{3} + 9 \, {\left (b^{8} - 128 \, b^{5} + 4096 \, b^{2}\right )} c x - 262144\right )} \sqrt {c^{3}} {\rm weierstrassPInverse}\left (0, \frac {256}{c^{3}}, \frac {c x + b}{c}\right ) + {\left (91 \, c^{9} x^{7} + 637 \, b c^{8} x^{6} + 1911 \, b^{2} c^{7} x^{5} + 13 \, {\left (245 \, b^{3} - 1088\right )} c^{6} x^{4} + 13 \, {\left (245 \, b^{4} - 4352 \, b\right )} c^{5} x^{3} + 39 \, {\left (49 \, b^{5} - 2176 \, b^{2}\right )} c^{4} x^{2} + {\left (637 \, b^{6} - 56576 \, b^{3} + 643072\right )} c^{3} x + {\left (91 \, b^{7} - 14144 \, b^{4} + 643072 \, b\right )} c^{2}\right )} \sqrt {c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64}}{53084160 \, {\left (c^{12} x^{9} + 9 \, b c^{11} x^{8} + 36 \, b^{2} c^{10} x^{7} + 12 \, {\left (7 \, b^{3} - 16\right )} c^{9} x^{6} + 18 \, {\left (7 \, b^{4} - 64 \, b\right )} c^{8} x^{5} + 18 \, {\left (7 \, b^{5} - 160 \, b^{2}\right )} c^{7} x^{4} + 12 \, {\left (7 \, b^{6} - 320 \, b^{3} + 1024\right )} c^{6} x^{3} + 36 \, {\left (b^{7} - 80 \, b^{4} + 1024 \, b\right )} c^{5} x^{2} + 9 \, {\left (b^{8} - 128 \, b^{5} + 4096 \, b^{2}\right )} c^{4} x + {\left (b^{9} - 192 \, b^{6} + 12288 \, b^{3} - 262144\right )} c^{3}\right )}} \] Input:

integrate(1/(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(7/2),x, algorithm="fri 
cas")
 

Output:

-1/53084160*(91*(c^9*x^9 + 9*b*c^8*x^8 + 36*b^2*c^7*x^7 + 12*(7*b^3 - 16)* 
c^6*x^6 + 18*(7*b^4 - 64*b)*c^5*x^5 + b^9 + 18*(7*b^5 - 160*b^2)*c^4*x^4 + 
 12*(7*b^6 - 320*b^3 + 1024)*c^3*x^3 - 192*b^6 + 36*(b^7 - 80*b^4 + 1024*b 
)*c^2*x^2 + 12288*b^3 + 9*(b^8 - 128*b^5 + 4096*b^2)*c*x - 262144)*sqrt(c^ 
3)*weierstrassPInverse(0, 256/c^3, (c*x + b)/c) + (91*c^9*x^7 + 637*b*c^8* 
x^6 + 1911*b^2*c^7*x^5 + 13*(245*b^3 - 1088)*c^6*x^4 + 13*(245*b^4 - 4352* 
b)*c^5*x^3 + 39*(49*b^5 - 2176*b^2)*c^4*x^2 + (637*b^6 - 56576*b^3 + 64307 
2)*c^3*x + (91*b^7 - 14144*b^4 + 643072*b)*c^2)*sqrt(c^3*x^3 + 3*b*c^2*x^2 
 + 3*b^2*c*x + b^3 - 64))/(c^12*x^9 + 9*b*c^11*x^8 + 36*b^2*c^10*x^7 + 12* 
(7*b^3 - 16)*c^9*x^6 + 18*(7*b^4 - 64*b)*c^8*x^5 + 18*(7*b^5 - 160*b^2)*c^ 
7*x^4 + 12*(7*b^6 - 320*b^3 + 1024)*c^6*x^3 + 36*(b^7 - 80*b^4 + 1024*b)*c 
^5*x^2 + 9*(b^8 - 128*b^5 + 4096*b^2)*c^4*x + (b^9 - 192*b^6 + 12288*b^3 - 
 262144)*c^3)
 

Sympy [F]

\[ \int \frac {1}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{7/2}} \, dx=\int \frac {1}{\left (b^{3} + 3 b^{2} c x + 3 b c^{2} x^{2} + c^{3} x^{3} - 64\right )^{\frac {7}{2}}}\, dx \] Input:

integrate(1/(c**3*x**3+3*b*c**2*x**2+3*b**2*c*x+b**3-64)**(7/2),x)
 

Output:

Integral((b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)**(-7/2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{7/2}} \, dx=\int { \frac {1}{{\left (c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(7/2),x, algorithm="max 
ima")
 

Output:

integrate((c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x + b^3 - 64)^(-7/2), x)
 

Giac [F]

\[ \int \frac {1}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{7/2}} \, dx=\int { \frac {1}{{\left (c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate(1/(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(7/2),x, algorithm="gia 
c")
 

Output:

integrate((c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x + b^3 - 64)^(-7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{7/2}} \, dx=\int \frac {1}{{\left (b^3+3\,b^2\,c\,x+3\,b\,c^2\,x^2+c^3\,x^3-64\right )}^{7/2}} \,d x \] Input:

int(1/(b^3 + c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x - 64)^(7/2),x)
 

Output:

int(1/(b^3 + c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x - 64)^(7/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{7/2}} \, dx=\int \frac {\sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}}{c^{12} x^{12}+12 b \,c^{11} x^{11}+66 b^{2} c^{10} x^{10}+220 b^{3} c^{9} x^{9}+495 b^{4} c^{8} x^{8}+792 b^{5} c^{7} x^{7}+924 b^{6} c^{6} x^{6}-256 c^{9} x^{9}+792 b^{7} c^{5} x^{5}-2304 b \,c^{8} x^{8}+495 b^{8} c^{4} x^{4}-9216 b^{2} c^{7} x^{7}+220 b^{9} c^{3} x^{3}-21504 b^{3} c^{6} x^{6}+66 b^{10} c^{2} x^{2}-32256 b^{4} c^{5} x^{5}+12 b^{11} c x -32256 b^{5} c^{4} x^{4}+b^{12}-21504 b^{6} c^{3} x^{3}+24576 c^{6} x^{6}-9216 b^{7} c^{2} x^{2}+147456 b \,c^{5} x^{5}-2304 b^{8} c x +368640 b^{2} c^{4} x^{4}-256 b^{9}+491520 b^{3} c^{3} x^{3}+368640 b^{4} c^{2} x^{2}+147456 b^{5} c x +24576 b^{6}-1048576 c^{3} x^{3}-3145728 b \,c^{2} x^{2}-3145728 b^{2} c x -1048576 b^{3}[phone]}d x \] Input:

int(1/(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(7/2),x)
 

Output:

int(sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)/(b**12 + 12*b 
**11*c*x + 66*b**10*c**2*x**2 + 220*b**9*c**3*x**3 - 256*b**9 + 495*b**8*c 
**4*x**4 - 2304*b**8*c*x + 792*b**7*c**5*x**5 - 9216*b**7*c**2*x**2 + 924* 
b**6*c**6*x**6 - 21504*b**6*c**3*x**3 + 24576*b**6 + 792*b**5*c**7*x**7 - 
32256*b**5*c**4*x**4 + 147456*b**5*c*x + 495*b**4*c**8*x**8 - 32256*b**4*c 
**5*x**5 + 368640*b**4*c**2*x**2 + 220*b**3*c**9*x**9 - 21504*b**3*c**6*x* 
*6 + 491520*b**3*c**3*x**3 - 1048576*b**3 + 66*b**2*c**10*x**10 - 9216*b** 
2*c**7*x**7 + 368640*b**2*c**4*x**4 - 3145728*b**2*c*x + 12*b*c**11*x**11 
- 2304*b*c**8*x**8 + 147456*b*c**5*x**5 - 3145728*b*c**2*x**2 + c**12*x**1 
2 - 256*c**9*x**9 + 24576*c**6*x**6 - 1048576*c**3*x**3 + 16777216),x)