\(\int \frac {1}{\sqrt {a+b x} (c+d x)^{2/3}} \, dx\) [428]

Optimal result

Integrand size = 19, antiderivative size = 345 \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{2/3}} \, dx=-\frac {2\ 3^{3/4} \sqrt {2-\sqrt {3}} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right ) \sqrt {\frac {(b c-a d)^{2/3}+\sqrt [3]{b} \sqrt [3]{b c-a d} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}{\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [3]{b} d \sqrt {a+b x} \sqrt {-\frac {\sqrt [3]{b c-a d} \left (\sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b c-a d}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )^2}}} \] Output:

-2*3^(3/4)*(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)*E 
llipticF(((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))/b^(1/3)/d/(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 = 0.21 \[ \int \frac {1}{\sqrt {a+b x} (c+d x)^{2/3}} \, dx=\frac {2 \sqrt {a+b x} \left (\frac {b (c+d x)}{b c-a d}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {3}{2},\frac {d (a+b x)}{-b c+a d}\right )}{b (c+d x)^{2/3}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {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.

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 760

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

Input:

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

Output:

(-2*3^(3/4)*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]])/(b^(1/3)*d*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 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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