\(\int \frac {\sqrt [4]{c+d x}}{\sqrt {a+b x}} \, dx\) [433]

Optimal result

Integrand size = 19, antiderivative size = 111 \[ \int \frac {\sqrt [4]{c+d x}}{\sqrt {a+b x}} \, dx=\frac {4 \sqrt {a+b x} \sqrt [4]{c+d x}}{3 b}+\frac {4 (b c-a d)^{5/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{3 b^{5/4} d \sqrt {a+b x}} \] Output:

4/3*(b*x+a)^(1/2)*(d*x+c)^(1/4)/b+4/3*(-a*d+b*c)^(5/4)*(-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)/d/( 
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 = 0.64 \[ \int \frac {\sqrt [4]{c+d x}}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {a+b x} \sqrt [4]{c+d x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{2},\frac {d (a+b x)}{-b c+a d}\right )}{b \sqrt [4]{\frac {b (c+d x)}{b c-a d}}} \] Input:

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

Output:

(2*Sqrt[a + b*x]*(c + d*x)^(1/4)*Hypergeometric2F1[-1/4, 1/2, 3/2, (d*(a + 
 b*x))/(-(b*c) + a*d)])/(b*((b*(c + d*x))/(b*c - a*d))^(1/4))
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {60, 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.

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

Input:

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

Output:

(4*Sqrt[a + b*x]*(c + d*x)^(1/4))/(3*b) + (4*(b*c - a*d)^(5/4)*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])/(3*b^(5/4)*d*Sqrt[a - (b*c)/d + (b*(c + d*x))/d])
 

Defintions of rubi rules used

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \frac {\sqrt [4]{c+d x}}{\sqrt {a+b x}} \, dx=\int \frac {\sqrt [4]{c + d x}}{\sqrt {a + b x}}\, dx \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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