\(\int \sqrt [3]{a+b x} \sqrt [3]{c+d x} \, dx\) [525]

Optimal result

Integrand size = 19, antiderivative size = 58 \[ \int \sqrt [3]{a+b x} \sqrt [3]{c+d x} \, dx=\frac {3 (a+b x)^{4/3} (c+d x)^{4/3} \operatorname {Hypergeometric2F1}\left (1,\frac {8}{3},\frac {7}{3},-\frac {d (a+b x)}{b c-a d}\right )}{4 (b c-a d)} \] Output:

3*(b*x+a)^(4/3)*(d*x+c)^(4/3)*hypergeom([1, 8/3],[7/3],-d*(b*x+a)/(-a*d+b* 
c))/(-4*a*d+4*b*c)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.26 \[ \int \sqrt [3]{a+b x} \sqrt [3]{c+d x} \, dx=\frac {3 (a+b x)^{4/3} \sqrt [3]{c+d x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {4}{3},\frac {7}{3},\frac {d (a+b x)}{-b c+a d}\right )}{4 b \sqrt [3]{\frac {b (c+d x)}{b c-a d}}} \] Input:

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

Output:

(3*(a + b*x)^(4/3)*(c + d*x)^(1/3)*Hypergeometric2F1[-1/3, 4/3, 7/3, (d*(a 
 + b*x))/(-(b*c) + a*d)])/(4*b*((b*(c + d*x))/(b*c - a*d))^(1/3))
 

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.28, 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.

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

\(\displaystyle \frac {3 (a+b x)^{4/3} \sqrt [3]{c+d x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {4}{3},\frac {7}{3},-\frac {d (a+b x)}{b c-a d}\right )}{4 b \sqrt [3]{\frac {b (c+d x)}{b c-a d}}}\)

Input:

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

Output:

(3*(a + b*x)^(4/3)*(c + d*x)^(1/3)*Hypergeometric2F1[-1/3, 4/3, 7/3, -((d* 
(a + b*x))/(b*c - a*d))])/(4*b*((b*(c + d*x))/(b*c - a*d))^(1/3))
 

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])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

\[ \int \sqrt [3]{a+b x} \sqrt [3]{c+d x} \, dx=\int \sqrt [3]{a + b x} \sqrt [3]{c + d x}\, dx \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \sqrt [3]{a+b x} \sqrt [3]{c+d x} \, dx=\frac {6 \left (d x +c \right )^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}} a c +3 \left (d x +c \right )^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}} a d x +3 \left (d x +c \right )^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}} b c x +\left (\int \frac {\left (d x +c \right )^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}} x}{a b \,d^{2} x^{2}+b^{2} c d \,x^{2}+a^{2} d^{2} x +2 a b c d x +b^{2} c^{2} x +a^{2} c d +a b \,c^{2}}d x \right ) a^{3} d^{3}-\left (\int \frac {\left (d x +c \right )^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}} x}{a b \,d^{2} x^{2}+b^{2} c d \,x^{2}+a^{2} d^{2} x +2 a b c d x +b^{2} c^{2} x +a^{2} c d +a b \,c^{2}}d x \right ) a^{2} b c \,d^{2}-\left (\int \frac {\left (d x +c \right )^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}} x}{a b \,d^{2} x^{2}+b^{2} c d \,x^{2}+a^{2} d^{2} x +2 a b c d x +b^{2} c^{2} x +a^{2} c d +a b \,c^{2}}d x \right ) a \,b^{2} c^{2} d +\left (\int \frac {\left (d x +c \right )^{\frac {1}{3}} \left (b x +a \right )^{\frac {1}{3}} x}{a b \,d^{2} x^{2}+b^{2} c d \,x^{2}+a^{2} d^{2} x +2 a b c d x +b^{2} c^{2} x +a^{2} c d +a b \,c^{2}}d x \right ) b^{3} c^{3}}{5 a d +5 b c} \] Input:

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

Output:

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