3.31.25 \(\int \frac {1}{\sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}} \, dx\) [3025]

3.31.25.1 Optimal result

Integrand size = 26, antiderivative size = 1283 \[ \int \frac {1}{\sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}} \, dx=\frac {3 \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\left (d (3 b c+a d)+4 b d^2 x\right )^2}}{2 b^{2/3} d^3 \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (b c-a d)^{2/3} \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {\left (d (3 b c+a d)+4 b d^2 x\right )^2} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2 \sqrt [3]{b} (b c-a d)^{2/3} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}+4 b^{2/3} ((c+d x) (a d+b (c+2 d x)))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}\right )|-7-4 \sqrt {3}\right )}{4 b^{2/3} d \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}}}+\frac {3^{3/4} (b c-a d)^{2/3} \sqrt [3]{(c+d x) (b c+a d+2 b d x)} \sqrt {\left (d (3 b c+a d)+4 b d^2 x\right )^2} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2 \sqrt [3]{b} (b c-a d)^{2/3} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}+4 b^{2/3} ((c+d x) (a d+b (c+2 d x)))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}}\right ),-7-4 \sqrt {3}\right )}{\sqrt {2} b^{2/3} d \sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x} (3 b c+a d+4 b d x) \sqrt {d^2 (3 b c+a d+4 b d x)^2} \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2 \sqrt [3]{b} \sqrt [3]{(c+d x) (a d+b (c+2 d x))}\right )^2}}} \]

output

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

3.31.25.2 Mathematica [C] (verified)

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

Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.07 \[ \int \frac {1}{\sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}} \, dx=\frac {3 (c+d x)^{2/3} \sqrt [3]{\frac {a d+b (c+2 d x)}{-b c+a d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {2 b (c+d x)}{b c-a d}\right )}{2 d \sqrt [3]{a d+b (c+2 d x)}} \]

input

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

output

(3*(c + d*x)^(2/3)*((a*d + b*(c + 2*d*x))/(-(b*c) + a*d))^(1/3)*Hypergeome 
tric2F1[1/3, 2/3, 5/3, (2*b*(c + d*x))/(b*c - a*d)])/(2*d*(a*d + b*(c + 2* 
d*x))^(1/3))
 

3.31.25.3 Rubi [C] (verified)

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

Time = 0.17 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.07, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {80, 79}

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.

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

\(\displaystyle \frac {3 (c+d x)^{2/3} \sqrt [3]{-\frac {a d+b c+2 b d x}{b c-a d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {5}{3},\frac {2 b (c+d x)}{b c-a d}\right )}{2 d \sqrt [3]{a d+b c+2 b d x}}\)

input

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

output

(3*(c + d*x)^(2/3)*(-((b*c + a*d + 2*b*d*x)/(b*c - a*d)))^(1/3)*Hypergeome 
tric2F1[1/3, 2/3, 5/3, (2*b*(c + d*x))/(b*c - a*d)])/(2*d*(b*c + a*d + 2*b 
*d*x)^(1/3))
 

3.31.25.3.1 Defintions of rubi rules used

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 80

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c 
 + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) 
^FracPart[n])   Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) 
), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integ 
erQ[n] && (RationalQ[m] ||  !SimplerQ[n + 1, m + 1])
 

3.31.25.4 Maple [F]

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

input

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

output

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

3.31.25.5 Fricas [F]

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

input

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

output

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

3.31.25.6 Sympy [F]

\[ \int \frac {1}{\sqrt [3]{c+d x} \sqrt [3]{b c+a d+2 b d x}} \, dx=\int \frac {1}{\sqrt [3]{c + d x} \sqrt [3]{a d + b c + 2 b d x}}\, dx \]

input

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

output

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

3.31.25.7 Maxima [F]

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

input

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

output

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

3.31.25.8 Giac [F]

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

input

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

output

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

3.31.25.9 Mupad [F(-1)]

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

input

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

output

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