\(\int \frac {\sqrt [4]{c+d x}}{(a+b x)^{3/2}} \, dx\) [434]

Optimal result

Integrand size = 19, antiderivative size = 104 \[ \int \frac {\sqrt [4]{c+d x}}{(a+b x)^{3/2}} \, dx=-\frac {2 \sqrt [4]{c+d x}}{b \sqrt {a+b x}}+\frac {2 \sqrt [4]{b c-a d} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{5/4} \sqrt {a+b x}} \] Output:

-2*(d*x+c)^(1/4)/b/(b*x+a)^(1/2)+2*(-a*d+b*c)^(1/4)*(-d*(b*x+a)/(-a*d+b*c) 
)^(1/2)*EllipticF(b^(1/4)*(d*x+c)^(1/4)/(-a*d+b*c)^(1/4),I)/b^(5/4)/(b*x+a 
)^(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 = 0.68 \[ \int \frac {\sqrt [4]{c+d x}}{(a+b x)^{3/2}} \, dx=-\frac {2 \sqrt [4]{c+d x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{4},\frac {1}{2},\frac {d (a+b x)}{-b c+a d}\right )}{b \sqrt {a+b x} \sqrt [4]{\frac {b (c+d x)}{b c-a d}}} \] Input:

Integrate[(c + d*x)^(1/4)/(a + b*x)^(3/2),x]
 

Output:

(-2*(c + d*x)^(1/4)*Hypergeometric2F1[-1/2, -1/4, 1/2, (d*(a + b*x))/(-(b* 
c) + a*d)])/(b*Sqrt[a + b*x]*((b*(c + d*x))/(b*c - a*d))^(1/4))
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.15, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {57, 73, 765, 762}

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.

\(\displaystyle \int \frac {\sqrt [4]{c+d x}}{(a+b x)^{3/2}} \, dx\)

\(\Big \downarrow \) 57

\(\displaystyle \frac {d \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/4}}dx}{2 b}-\frac {2 \sqrt [4]{c+d x}}{b \sqrt {a+b x}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {2 \int \frac {1}{\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}d\sqrt [4]{c+d x}}{b}-\frac {2 \sqrt [4]{c+d x}}{b \sqrt {a+b x}}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {2 \sqrt {1-\frac {b (c+d x)}{b c-a d}} \int \frac {1}{\sqrt {1-\frac {b (c+d x)}{b c-a d}}}d\sqrt [4]{c+d x}}{b \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}-\frac {2 \sqrt [4]{c+d x}}{b \sqrt {a+b x}}\)

\(\Big \downarrow \) 762

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

Input:

Int[(c + d*x)^(1/4)/(a + b*x)^(3/2),x]
 

Output:

(-2*(c + d*x)^(1/4))/(b*Sqrt[a + b*x]) + (2*(b*c - a*d)^(1/4)*Sqrt[1 - (b* 
(c + d*x))/(b*c - a*d)]*EllipticF[ArcSin[(b^(1/4)*(c + d*x)^(1/4))/(b*c - 
a*d)^(1/4)], -1])/(b^(5/4)*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 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 0]
 

Maple [F]

\[\int \frac {\left (x d +c \right )^{\frac {1}{4}}}{\left (b x +a \right )^{\frac {3}{2}}}d x\]

Input:

int((d*x+c)^(1/4)/(b*x+a)^(3/2),x)
 

Output:

int((d*x+c)^(1/4)/(b*x+a)^(3/2),x)
 

Fricas [F]

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

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

Output:

integral(sqrt(b*x + a)*(d*x + c)^(1/4)/(b^2*x^2 + 2*a*b*x + a^2), x)
 

Sympy [F]

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

integrate((d*x+c)**(1/4)/(b*x+a)**(3/2),x)
 

Output:

Integral((c + d*x)**(1/4)/(a + b*x)**(3/2), x)
 

Maxima [F]

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

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

Output:

integrate((d*x + c)^(1/4)/(b*x + a)^(3/2), x)
 

Giac [F]

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

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

Output:

integrate((d*x + c)^(1/4)/(b*x + a)^(3/2), x)
 

Mupad [F(-1)]

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

int((c + d*x)^(1/4)/(a + b*x)^(3/2),x)
 

Output:

int((c + d*x)^(1/4)/(a + b*x)^(3/2), x)
 

Reduce [F]

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

int((d*x+c)^(1/4)/(b*x+a)^(3/2),x)
 

Output:

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