\(\int \frac {\sqrt [3]{c+d x}}{(a+b x)^{3/2}} \, dx\) [418]

Optimal result

Integrand size = 19, antiderivative size = 366 \[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{3/2}} \, dx=-\frac {2 \sqrt [3]{c+d x}}{b \sqrt {a+b x}}-\frac {4 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} b^{4/3} \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.19 \[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{3/2}} \, dx=-\frac {2 \sqrt [3]{c+d x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{3},\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )}{b \sqrt {a+b x} \sqrt [3]{\frac {b (c+d x)}{b c-a d}}} \] Input:

Integrate[(c + d*x)^(1/3)/(a + b*x)^(3/2),x]
 

Output:

(-2*(c + d*x)^(1/3)*Hypergeometric2F1[-1/2, -1/3, 1/2, (d*(a + b*x))/(-(b* 
c) + a*d)])/(b*Sqrt[a + b*x]*((b*(c + d*x))/(b*c - a*d))^(1/3))
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {57, 73, 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.

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

\(\Big \downarrow \) 57

\(\displaystyle \frac {2 d \int \frac {1}{\sqrt {a+b x} (c+d x)^{2/3}}dx}{3 b}-\frac {2 \sqrt [3]{c+d x}}{b \sqrt {a+b x}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {2 \int \frac {1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [3]{c+d x}}{b}-\frac {2 \sqrt [3]{c+d x}}{b \sqrt {a+b x}}\)

\(\Big \downarrow \) 760

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

Input:

Int[(c + d*x)^(1/3)/(a + b*x)^(3/2),x]
 

Output:

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

Defintions of rubi rules used

rule 57

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & 
& GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m 
+ n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c 
, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

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]
 

Maple [F]

\[\int \frac {\left (x d +c \right )^{\frac {1}{3}}}{\left (b x +a \right )^{\frac {3}{2}}}d x\]

Input:

int((d*x+c)^(1/3)/(b*x+a)^(3/2),x)
 

Output:

int((d*x+c)^(1/3)/(b*x+a)^(3/2),x)
 

Fricas [F]

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

integrate((d*x+c)^(1/3)/(b*x+a)^(3/2),x, algorithm="fricas")
 

Output:

integral(sqrt(b*x + a)*(d*x + c)^(1/3)/(b^2*x^2 + 2*a*b*x + a^2), x)
 

Sympy [F]

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

integrate((d*x+c)**(1/3)/(b*x+a)**(3/2),x)
 

Output:

Integral((c + d*x)**(1/3)/(a + b*x)**(3/2), x)
 

Maxima [F]

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

integrate((d*x+c)^(1/3)/(b*x+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate((d*x + c)^(1/3)/(b*x + a)^(3/2), x)
 

Giac [F]

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

integrate((d*x+c)^(1/3)/(b*x+a)^(3/2),x, algorithm="giac")
 

Output:

integrate((d*x + c)^(1/3)/(b*x + a)^(3/2), x)
 

Mupad [F(-1)]

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

int((c + d*x)^(1/3)/(a + b*x)^(3/2),x)
 

Output:

int((c + d*x)^(1/3)/(a + b*x)^(3/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt [3]{c+d x}}{(a+b x)^{3/2}} \, dx=\frac {6 \left (d x +c \right )^{\frac {1}{3}} \sqrt {b x +a}\, c +4 \left (\int \frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b x +a}\, x}{2 a \,b^{2} d^{2} x^{3}-3 b^{3} c d \,x^{3}+4 a^{2} b \,d^{2} x^{2}-4 a \,b^{2} c d \,x^{2}-3 b^{3} c^{2} x^{2}+2 a^{3} d^{2} x +a^{2} b c d x -6 a \,b^{2} c^{2} x +2 a^{3} c d -3 a^{2} b \,c^{2}}d x \right ) a^{3} d^{3}-10 \left (\int \frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b x +a}\, x}{2 a \,b^{2} d^{2} x^{3}-3 b^{3} c d \,x^{3}+4 a^{2} b \,d^{2} x^{2}-4 a \,b^{2} c d \,x^{2}-3 b^{3} c^{2} x^{2}+2 a^{3} d^{2} x +a^{2} b c d x -6 a \,b^{2} c^{2} x +2 a^{3} c d -3 a^{2} b \,c^{2}}d x \right ) a^{2} b c \,d^{2}+4 \left (\int \frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b x +a}\, x}{2 a \,b^{2} d^{2} x^{3}-3 b^{3} c d \,x^{3}+4 a^{2} b \,d^{2} x^{2}-4 a \,b^{2} c d \,x^{2}-3 b^{3} c^{2} x^{2}+2 a^{3} d^{2} x +a^{2} b c d x -6 a \,b^{2} c^{2} x +2 a^{3} c d -3 a^{2} b \,c^{2}}d x \right ) a^{2} b \,d^{3} x +6 \left (\int \frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b x +a}\, x}{2 a \,b^{2} d^{2} x^{3}-3 b^{3} c d \,x^{3}+4 a^{2} b \,d^{2} x^{2}-4 a \,b^{2} c d \,x^{2}-3 b^{3} c^{2} x^{2}+2 a^{3} d^{2} x +a^{2} b c d x -6 a \,b^{2} c^{2} x +2 a^{3} c d -3 a^{2} b \,c^{2}}d x \right ) a \,b^{2} c^{2} d -10 \left (\int \frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b x +a}\, x}{2 a \,b^{2} d^{2} x^{3}-3 b^{3} c d \,x^{3}+4 a^{2} b \,d^{2} x^{2}-4 a \,b^{2} c d \,x^{2}-3 b^{3} c^{2} x^{2}+2 a^{3} d^{2} x +a^{2} b c d x -6 a \,b^{2} c^{2} x +2 a^{3} c d -3 a^{2} b \,c^{2}}d x \right ) a \,b^{2} c \,d^{2} x +6 \left (\int \frac {\left (d x +c \right )^{\frac {1}{3}} \sqrt {b x +a}\, x}{2 a \,b^{2} d^{2} x^{3}-3 b^{3} c d \,x^{3}+4 a^{2} b \,d^{2} x^{2}-4 a \,b^{2} c d \,x^{2}-3 b^{3} c^{2} x^{2}+2 a^{3} d^{2} x +a^{2} b c d x -6 a \,b^{2} c^{2} x +2 a^{3} c d -3 a^{2} b \,c^{2}}d x \right ) b^{3} c^{2} d x}{2 a b d x -3 b^{2} c x +2 a^{2} d -3 a b c} \] Input:

int((d*x+c)^(1/3)/(b*x+a)^(3/2),x)
 

Output:

(2*(3*(c + d*x)**(1/3)*sqrt(a + b*x)*c + 2*int(((c + d*x)**(1/3)*sqrt(a + 
b*x)*x)/(2*a**3*c*d + 2*a**3*d**2*x - 3*a**2*b*c**2 + a**2*b*c*d*x + 4*a** 
2*b*d**2*x**2 - 6*a*b**2*c**2*x - 4*a*b**2*c*d*x**2 + 2*a*b**2*d**2*x**3 - 
 3*b**3*c**2*x**2 - 3*b**3*c*d*x**3),x)*a**3*d**3 - 5*int(((c + d*x)**(1/3 
)*sqrt(a + b*x)*x)/(2*a**3*c*d + 2*a**3*d**2*x - 3*a**2*b*c**2 + a**2*b*c* 
d*x + 4*a**2*b*d**2*x**2 - 6*a*b**2*c**2*x - 4*a*b**2*c*d*x**2 + 2*a*b**2* 
d**2*x**3 - 3*b**3*c**2*x**2 - 3*b**3*c*d*x**3),x)*a**2*b*c*d**2 + 2*int(( 
(c + d*x)**(1/3)*sqrt(a + b*x)*x)/(2*a**3*c*d + 2*a**3*d**2*x - 3*a**2*b*c 
**2 + a**2*b*c*d*x + 4*a**2*b*d**2*x**2 - 6*a*b**2*c**2*x - 4*a*b**2*c*d*x 
**2 + 2*a*b**2*d**2*x**3 - 3*b**3*c**2*x**2 - 3*b**3*c*d*x**3),x)*a**2*b*d 
**3*x + 3*int(((c + d*x)**(1/3)*sqrt(a + b*x)*x)/(2*a**3*c*d + 2*a**3*d**2 
*x - 3*a**2*b*c**2 + a**2*b*c*d*x + 4*a**2*b*d**2*x**2 - 6*a*b**2*c**2*x - 
 4*a*b**2*c*d*x**2 + 2*a*b**2*d**2*x**3 - 3*b**3*c**2*x**2 - 3*b**3*c*d*x* 
*3),x)*a*b**2*c**2*d - 5*int(((c + d*x)**(1/3)*sqrt(a + b*x)*x)/(2*a**3*c* 
d + 2*a**3*d**2*x - 3*a**2*b*c**2 + a**2*b*c*d*x + 4*a**2*b*d**2*x**2 - 6* 
a*b**2*c**2*x - 4*a*b**2*c*d*x**2 + 2*a*b**2*d**2*x**3 - 3*b**3*c**2*x**2 
- 3*b**3*c*d*x**3),x)*a*b**2*c*d**2*x + 3*int(((c + d*x)**(1/3)*sqrt(a + b 
*x)*x)/(2*a**3*c*d + 2*a**3*d**2*x - 3*a**2*b*c**2 + a**2*b*c*d*x + 4*a**2 
*b*d**2*x**2 - 6*a*b**2*c**2*x - 4*a*b**2*c*d*x**2 + 2*a*b**2*d**2*x**3 - 
3*b**3*c**2*x**2 - 3*b**3*c*d*x**3),x)*b**3*c**2*d*x))/(2*a**2*d - 3*a*...