3.16.96 \(\int \frac {(a+b x)^{2/3}}{\sqrt [3]{c+d x}} \, dx\) [1596]

3.16.96.1 Optimal result

Integrand size = 19, antiderivative size = 1293 \[ \int \frac {(a+b x)^{2/3}}{\sqrt [3]{c+d x}} \, dx=\frac {3 (a+b x)^{2/3} (c+d x)^{2/3}}{4 d}-\frac {3 (b c-a d) \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \sqrt {(a d+b (c+2 d x))^2}}{2 \sqrt [3]{2} b^{2/3} d^{5/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (b c-a d)^{5/3} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right )|-7-4 \sqrt {3}\right )}{4 \sqrt [3]{2} b^{2/3} d^{5/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \sqrt {(a d+b (c+2 d x))^2}}-\frac {3^{3/4} (b c-a d)^{5/3} \sqrt [3]{(a+b x) (c+d x)} \sqrt {(b c+a d+2 b d x)^2} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right ) \sqrt {\frac {(b c-a d)^{4/3}-2^{2/3} \sqrt [3]{b} \sqrt [3]{d} (b c-a d)^{2/3} \sqrt [3]{(a+b x) (c+d x)}+2 \sqrt [3]{2} b^{2/3} d^{2/3} ((a+b x) (c+d x))^{2/3}}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}{\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}}\right ),-7-4 \sqrt {3}\right )}{2^{5/6} b^{2/3} d^{5/3} \sqrt [3]{a+b x} \sqrt [3]{c+d x} (b c+a d+2 b d x) \sqrt {\frac {(b c-a d)^{2/3} \left ((b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )}{\left (\left (1+\sqrt {3}\right ) (b c-a d)^{2/3}+2^{2/3} \sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{(a+b x) (c+d x)}\right )^2}} \sqrt {(a d+b (c+2 d x))^2}} \]

output

3/4*(b*x+a)^(2/3)*(d*x+c)^(2/3)/d-3/4*(-a*d+b*c)*((b*x+a)*(d*x+c))^(1/3)*( 
(2*b*d*x+a*d+b*c)^2)^(1/2)*((a*d+b*(2*d*x+c))^2)^(1/2)*2^(2/3)/b^(2/3)/d^( 
5/3)/(b*x+a)^(1/3)/(d*x+c)^(1/3)/(2*b*d*x+a*d+b*c)/(2^(2/3)*b^(1/3)*d^(1/3 
)*((b*x+a)*(d*x+c))^(1/3)+(-a*d+b*c)^(2/3)*(1+3^(1/2)))-1/2*3^(3/4)*(-a*d+ 
b*c)^(5/3)*((b*x+a)*(d*x+c))^(1/3)*((-a*d+b*c)^(2/3)+2^(2/3)*b^(1/3)*d^(1/ 
3)*((b*x+a)*(d*x+c))^(1/3))*EllipticF((2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)*(d 
*x+c))^(1/3)+(-a*d+b*c)^(2/3)*(1-3^(1/2)))/(2^(2/3)*b^(1/3)*d^(1/3)*((b*x+ 
a)*(d*x+c))^(1/3)+(-a*d+b*c)^(2/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*((2*b*d*x+a 
*d+b*c)^2)^(1/2)*(((-a*d+b*c)^(4/3)-2^(2/3)*b^(1/3)*d^(1/3)*(-a*d+b*c)^(2/ 
3)*((b*x+a)*(d*x+c))^(1/3)+2*2^(1/3)*b^(2/3)*d^(2/3)*((b*x+a)*(d*x+c))^(2/ 
3))/(2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)*(d*x+c))^(1/3)+(-a*d+b*c)^(2/3)*(1+3 
^(1/2)))^2)^(1/2)*2^(1/6)/b^(2/3)/d^(5/3)/(b*x+a)^(1/3)/(d*x+c)^(1/3)/(2*b 
*d*x+a*d+b*c)/((a*d+b*(2*d*x+c))^2)^(1/2)/((-a*d+b*c)^(2/3)*((-a*d+b*c)^(2 
/3)+2^(2/3)*b^(1/3)*d^(1/3)*((b*x+a)*(d*x+c))^(1/3))/(2^(2/3)*b^(1/3)*d^(1 
/3)*((b*x+a)*(d*x+c))^(1/3)+(-a*d+b*c)^(2/3)*(1+3^(1/2)))^2)^(1/2)+3/8*3^( 
1/4)*(-a*d+b*c)^(5/3)*((b*x+a)*(d*x+c))^(1/3)*((-a*d+b*c)^(2/3)+2^(2/3)*b^ 
(1/3)*d^(1/3)*((b*x+a)*(d*x+c))^(1/3))*EllipticE((2^(2/3)*b^(1/3)*d^(1/3)* 
((b*x+a)*(d*x+c))^(1/3)+(-a*d+b*c)^(2/3)*(1-3^(1/2)))/(2^(2/3)*b^(1/3)*d^( 
1/3)*((b*x+a)*(d*x+c))^(1/3)+(-a*d+b*c)^(2/3)*(1+3^(1/2))),I*3^(1/2)+2*I)* 
((2*b*d*x+a*d+b*c)^2)^(1/2)*(1/2*6^(1/2)-1/2*2^(1/2))*(((-a*d+b*c)^(4/3...
 

3.16.96.2 Mathematica [C] (verified)

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

Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.06 \[ \int \frac {(a+b x)^{2/3}}{\sqrt [3]{c+d x}} \, dx=\frac {3 (a+b x)^{5/3} \sqrt [3]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {5}{3},\frac {8}{3},\frac {d (a+b x)}{-b c+a d}\right )}{5 b \sqrt [3]{c+d x}} \]

input

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

output

(3*(a + b*x)^(5/3)*((b*(c + d*x))/(b*c - a*d))^(1/3)*Hypergeometric2F1[1/3 
, 5/3, 8/3, (d*(a + b*x))/(-(b*c) + a*d)])/(5*b*(c + d*x)^(1/3))
 

3.16.96.3 Rubi [C] (verified)

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

Time = 0.16 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.06, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, 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.

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

\(\displaystyle \frac {3 (a+b x)^{5/3} \sqrt [3]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {5}{3},\frac {8}{3},-\frac {d (a+b x)}{b c-a d}\right )}{5 b \sqrt [3]{c+d x}}\)

input

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

output

(3*(a + b*x)^(5/3)*((b*(c + d*x))/(b*c - a*d))^(1/3)*Hypergeometric2F1[1/3 
, 5/3, 8/3, -((d*(a + b*x))/(b*c - a*d))])/(5*b*(c + d*x)^(1/3))
 

3.16.96.3.1 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])
 

3.16.96.4 Maple [F]

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

input

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

output

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

3.16.96.5 Fricas [F]

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

input

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

output

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

3.16.96.6 Sympy [F]

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

input

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

output

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

3.16.96.7 Maxima [F]

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

input

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

output

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

3.16.96.8 Giac [F]

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

input

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

output

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

3.16.96.9 Mupad [F(-1)]

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

input

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

output

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