\(\int \frac {\sqrt [3]{c+d x}}{\sqrt {a+b x} (3 b c-4 a d-b d x)} \, dx\) [1619]

Optimal result

Integrand size = 35, antiderivative size = 105 \[ \int \frac {\sqrt [3]{c+d x}}{\sqrt {a+b x} (3 b c-4 a d-b d x)} \, dx=\frac {3 \sqrt {-\frac {d (a+b x)}{b c-a d}} (c+d x)^{4/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},1,\frac {7}{3},\frac {b (c+d x)}{b c-a d},\frac {b (c+d x)}{4 (b c-a d)}\right )}{16 d (b c-a d) \sqrt {a+b x}} \] Output:

3/16*(-d*(b*x+a)/(-a*d+b*c))^(1/2)*(d*x+c)^(4/3)*AppellF1(4/3,1,1/2,7/3,b* 
(d*x+c)/(-4*a*d+4*b*c),b*(d*x+c)/(-a*d+b*c))/d/(-a*d+b*c)/(b*x+a)^(1/2)
 

Mathematica [A] (verified)

Time = 10.08 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt [3]{c+d x}}{\sqrt {a+b x} (3 b c-4 a d-b d x)} \, dx=\frac {2 \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{3},1,\frac {3}{2},\frac {d (a+b x)}{-b c+a d},\frac {d (a+b x)}{3 b c-3 a d}\right )}{3 b^2 (c+d x)^{2/3}} \] Input:

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

Output:

(2*Sqrt[a + b*x]*((b*(c + d*x))/(b*c - a*d))^(2/3)*AppellF1[1/2, -1/3, 1, 
3/2, (d*(a + b*x))/(-(b*c) + a*d), (d*(a + b*x))/(3*b*c - 3*a*d)])/(3*b^2* 
(c + d*x)^(2/3))
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.15, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {149, 1013, 1012}

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.

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

\(\Big \downarrow \) 149

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1012

\(\displaystyle \frac {3 (c+d x)^{4/3} \sqrt {1-\frac {b (c+d x)}{b c-a d}} \operatorname {AppellF1}\left (\frac {4}{3},1,\frac {1}{2},\frac {7}{3},\frac {b (c+d x)}{4 (b c-a d)},\frac {b (c+d x)}{b c-a d}\right )}{16 d (b c-a d) \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}\)

Input:

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

Output:

(3*(c + d*x)^(4/3)*Sqrt[1 - (b*(c + d*x))/(b*c - a*d)]*AppellF1[4/3, 1, 1/ 
2, 7/3, (b*(c + d*x))/(4*(b*c - a*d)), (b*(c + d*x))/(b*c - a*d)])/(16*d*( 
b*c - a*d)*Sqrt[a - (b*c)/d + (b*(c + d*x))/d])
 

Defintions of rubi rules used

rule 149

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) 
)^(p_.), x_] :> With[{k = Denominator[m]}, Simp[k/b   Subst[Int[x^(k*(m + 1 
) - 1)*(c - a*(d/b) + d*(x^k/b))^n*(e - a*(f/b) + f*(x^k/b))^p, x], x, (a + 
 b*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && FractionQ[m] && 
IntegerQ[2*n] && IntegerQ[p]
 

rule 1012

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m 
+ 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, 
 b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n 
 - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 1013

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^ 
n/a))^FracPart[p])   Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; 
 FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & 
& NeQ[m, n - 1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{c+d x}}{\sqrt {a+b x} (3 b c-4 a d-b d x)} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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