\(\int \frac {(c+d x^3)^2}{(a+b x^3)^{3/2}} \, dx\) [52]

Optimal result

Integrand size = 21, antiderivative size = 290 \[ \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {2 (b c-a d)^2 x}{3 a b^2 \sqrt {a+b x^3}}+\frac {2 d^2 x \sqrt {a+b x^3}}{5 b^2}+\frac {2 \sqrt {2+\sqrt {3}} \left (5 b^2 c^2+20 a b c d-16 a^2 d^2\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{15 \sqrt [4]{3} a b^{7/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \] Output:

2/3*(-a*d+b*c)^2*x/a/b^2/(b*x^3+a)^(1/2)+2/5*d^2*x*(b*x^3+a)^(1/2)/b^2+2/4 
5*(1/2*6^(1/2)+1/2*2^(1/2))*(-16*a^2*d^2+20*a*b*c*d+5*b^2*c^2)*(a^(1/3)+b^ 
(1/3)*x)*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/((1+3^(1/2))*a^(1/3)+b^( 
1/3)*x)^2)^(1/2)*EllipticF(((1-3^(1/2))*a^(1/3)+b^(1/3)*x)/((1+3^(1/2))*a^ 
(1/3)+b^(1/3)*x),I*3^(1/2)+2*I)*3^(3/4)/a/b^(7/3)/(a^(1/3)*(a^(1/3)+b^(1/3 
)*x)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x)^2)^(1/2)/(b*x^3+a)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 13.28 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.56 \[ \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {x \sqrt {1+\frac {b x^3}{a}} \left (40 a \left (14 c^2+7 c d x^3+2 d^2 x^6\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {3}{2},\frac {10}{3},-\frac {b x^3}{a}\right )-9 b x^3 \left (11 c^2+16 c d x^3+5 d^2 x^6\right ) \operatorname {Hypergeometric2F1}\left (\frac {4}{3},\frac {5}{2},\frac {13}{3},-\frac {b x^3}{a}\right )-27 b x^3 \left (c+d x^3\right )^2 \, _3F_2\left (\frac {4}{3},2,\frac {5}{2};1,\frac {13}{3};-\frac {b x^3}{a}\right )\right )}{560 a^2 \sqrt {a+b x^3}} \] Input:

Integrate[(c + d*x^3)^2/(a + b*x^3)^(3/2),x]
 

Output:

(x*Sqrt[1 + (b*x^3)/a]*(40*a*(14*c^2 + 7*c*d*x^3 + 2*d^2*x^6)*Hypergeometr 
ic2F1[1/3, 3/2, 10/3, -((b*x^3)/a)] - 9*b*x^3*(11*c^2 + 16*c*d*x^3 + 5*d^2 
*x^6)*Hypergeometric2F1[4/3, 5/2, 13/3, -((b*x^3)/a)] - 27*b*x^3*(c + d*x^ 
3)^2*HypergeometricPFQ[{4/3, 2, 5/2}, {1, 13/3}, -((b*x^3)/a)]))/(560*a^2* 
Sqrt[a + b*x^3])
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {930, 27, 913, 759}

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.

\(\displaystyle \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 930

\(\displaystyle \frac {2 \int \frac {c (b c+2 a d)-d (5 b c-8 a d) x^3}{2 \sqrt {b x^3+a}}dx}{3 a b}+\frac {2 x \left (c+d x^3\right ) (b c-a d)}{3 a b \sqrt {a+b x^3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {c (b c+2 a d)-d (5 b c-8 a d) x^3}{\sqrt {b x^3+a}}dx}{3 a b}+\frac {2 x \left (c+d x^3\right ) (b c-a d)}{3 a b \sqrt {a+b x^3}}\)

\(\Big \downarrow \) 913

\(\displaystyle \frac {\frac {\left (-16 a^2 d^2+20 a b c d+5 b^2 c^2\right ) \int \frac {1}{\sqrt {b x^3+a}}dx}{5 b}-\frac {2 d x \sqrt {a+b x^3} (5 b c-8 a d)}{5 b}}{3 a b}+\frac {2 x \left (c+d x^3\right ) (b c-a d)}{3 a b \sqrt {a+b x^3}}\)

\(\Big \downarrow \) 759

\(\displaystyle \frac {\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (-16 a^2 d^2+20 a b c d+5 b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt [4]{3} b^{4/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {2 d x \sqrt {a+b x^3} (5 b c-8 a d)}{5 b}}{3 a b}+\frac {2 x \left (c+d x^3\right ) (b c-a d)}{3 a b \sqrt {a+b x^3}}\)

Input:

Int[(c + d*x^3)^2/(a + b*x^3)^(3/2),x]
 

Output:

(2*(b*c - a*d)*x*(c + d*x^3))/(3*a*b*Sqrt[a + b*x^3]) + ((-2*d*(5*b*c - 8* 
a*d)*x*Sqrt[a + b*x^3])/(5*b) + (2*Sqrt[2 + Sqrt[3]]*(5*b^2*c^2 + 20*a*b*c 
*d - 16*a^2*d^2)*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + 
 b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticF[ArcSin[((1 
- Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 
 4*Sqrt[3]])/(5*3^(1/4)*b^(4/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + 
 Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3]))/(3*a*b)
 

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 759

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] & 
& PosQ[a]
 

rule 913

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si 
mp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(p + 1) + 1))), x] - Simp[(a*d - b*c*(n*( 
p + 1) + 1))/(b*(n*(p + 1) + 1))   Int[(a + b*x^n)^p, x], x] /; FreeQ[{a, b 
, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
 

rule 930

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[(a*d - c*b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*b*n*(p + 
1))), x] - Simp[1/(a*b*n*(p + 1))   Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q 
- 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*( 
p + q) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 
 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q, x]
 

Maple [A] (verified)

Time = 2.08 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.37

method	result	size
elliptic	\(\frac {2 x \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{3 b^{2} a \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}+\frac {2 d^{2} x \sqrt {b \,x^{3}+a}}{5 b^{2}}-\frac {2 i \left (-\frac {d \left (a d -2 b c \right )}{b^{2}}+\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{3 a \,b^{2}}-\frac {2 a \,d^{2}}{5 b^{2}}\right ) \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{3 b \sqrt {b \,x^{3}+a}}\)	\(398\)
default	\(\text {Expression too large to display}\)	\(940\)
risch	\(\text {Expression too large to display}\)	\(966\)

Input:

int((d*x^3+c)^2/(b*x^3+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

2/3/b^2*x/a*(a^2*d^2-2*a*b*c*d+b^2*c^2)/((x^3+a/b)*b)^(1/2)+2/5*d^2*x*(b*x 
^3+a)^(1/2)/b^2-2/3*I*(-d*(a*d-2*b*c)/b^2+1/3*(a^2*d^2-2*a*b*c*d+b^2*c^2)/ 
a/b^2-2/5*a/b^2*d^2)*3^(1/2)/b*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1 
/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a 
*b^2)^(1/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2) 
*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a 
*b^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b 
^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2), 
(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2 
)^(1/3)))^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.49 \[ \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {2 \, {\left ({\left (5 \, a b^{2} c^{2} + 20 \, a^{2} b c d - 16 \, a^{3} d^{2} + {\left (5 \, b^{3} c^{2} + 20 \, a b^{2} c d - 16 \, a^{2} b d^{2}\right )} x^{3}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + {\left (3 \, a b^{2} d^{2} x^{4} + {\left (5 \, b^{3} c^{2} - 10 \, a b^{2} c d + 8 \, a^{2} b d^{2}\right )} x\right )} \sqrt {b x^{3} + a}\right )}}{15 \, {\left (a b^{4} x^{3} + a^{2} b^{3}\right )}} \] Input:

integrate((d*x^3+c)^2/(b*x^3+a)^(3/2),x, algorithm="fricas")
 

Output:

2/15*((5*a*b^2*c^2 + 20*a^2*b*c*d - 16*a^3*d^2 + (5*b^3*c^2 + 20*a*b^2*c*d 
 - 16*a^2*b*d^2)*x^3)*sqrt(b)*weierstrassPInverse(0, -4*a/b, x) + (3*a*b^2 
*d^2*x^4 + (5*b^3*c^2 - 10*a*b^2*c*d + 8*a^2*b*d^2)*x)*sqrt(b*x^3 + a))/(a 
*b^4*x^3 + a^2*b^3)
 

Sympy [F]

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

integrate((d*x**3+c)**2/(b*x**3+a)**(3/2),x)
 

Output:

Integral((c + d*x**3)**2/(a + b*x**3)**(3/2), x)
 

Maxima [F]

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

integrate((d*x^3+c)^2/(b*x^3+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate((d*x^3 + c)^2/(b*x^3 + a)^(3/2), x)
 

Giac [F]

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

integrate((d*x^3+c)^2/(b*x^3+a)^(3/2),x, algorithm="giac")
 

Output:

integrate((d*x^3 + c)^2/(b*x^3 + a)^(3/2), x)
 

Mupad [F(-1)]

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

int((c + d*x^3)^2/(a + b*x^3)^(3/2),x)
 

Output:

int((c + d*x^3)^2/(a + b*x^3)^(3/2), x)
 

Reduce [F]

\[ \int \frac {\left (c+d x^3\right )^2}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {16 \sqrt {b \,x^{3}+a}\, a \,d^{2} x -20 \sqrt {b \,x^{3}+a}\, b c d x +2 \sqrt {b \,x^{3}+a}\, b \,d^{2} x^{4}-16 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a^{3} d^{2}+20 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a^{2} b c d -16 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a^{2} b \,d^{2} x^{3}+5 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a \,b^{2} c^{2}+20 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a \,b^{2} c d \,x^{3}+5 \left (\int \frac {\sqrt {b \,x^{3}+a}}{b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) b^{3} c^{2} x^{3}}{5 b^{2} \left (b \,x^{3}+a \right )} \] Input:

int((d*x^3+c)^2/(b*x^3+a)^(3/2),x)
                                                                                    
                                                                                    
 

Output:

(16*sqrt(a + b*x**3)*a*d**2*x - 20*sqrt(a + b*x**3)*b*c*d*x + 2*sqrt(a + b 
*x**3)*b*d**2*x**4 - 16*int(sqrt(a + b*x**3)/(a**2 + 2*a*b*x**3 + b**2*x** 
6),x)*a**3*d**2 + 20*int(sqrt(a + b*x**3)/(a**2 + 2*a*b*x**3 + b**2*x**6), 
x)*a**2*b*c*d - 16*int(sqrt(a + b*x**3)/(a**2 + 2*a*b*x**3 + b**2*x**6),x) 
*a**2*b*d**2*x**3 + 5*int(sqrt(a + b*x**3)/(a**2 + 2*a*b*x**3 + b**2*x**6) 
,x)*a*b**2*c**2 + 20*int(sqrt(a + b*x**3)/(a**2 + 2*a*b*x**3 + b**2*x**6), 
x)*a*b**2*c*d*x**3 + 5*int(sqrt(a + b*x**3)/(a**2 + 2*a*b*x**3 + b**2*x**6 
),x)*b**3*c**2*x**3)/(5*b**2*(a + b*x**3))