\(\int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{4/3}} \, dx\) [563]

Optimal result

Integrand size = 19, antiderivative size = 58 \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{4/3}} \, dx=\frac {3 (a+b x)^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {5}{3},-\frac {d (a+b x)}{b c-a d}\right )}{2 (b c-a d) \sqrt [3]{c+d x}} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{4/3}} \, dx=\frac {3 (a+b x)^{2/3} \left (\frac {b (c+d x)}{b c-a d}\right )^{4/3} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {4}{3},\frac {5}{3},\frac {d (a+b x)}{-b c+a d}\right )}{2 b (c+d x)^{4/3}} \] Input:

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

Output:

(3*(a + b*x)^(2/3)*((b*(c + d*x))/(b*c - a*d))^(4/3)*Hypergeometric2F1[2/3 
, 4/3, 5/3, (d*(a + b*x))/(-(b*c) + a*d)])/(2*b*(c + d*x)^(4/3))
 

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.40, 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.

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

\(\displaystyle \frac {3 (a+b x)^{2/3} \sqrt [3]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {4}{3},\frac {5}{3},-\frac {d (a+b x)}{b c-a d}\right )}{2 \sqrt [3]{c+d x} (b c-a d)}\)

Input:

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

Output:

(3*(a + b*x)^(2/3)*((b*(c + d*x))/(b*c - a*d))^(1/3)*Hypergeometric2F1[2/3 
, 4/3, 5/3, -((d*(a + b*x))/(b*c - a*d))])/(2*(b*c - a*d)*(c + d*x)^(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]

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

Input:

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

Output:

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

Fricas [F]

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

integrate(1/(b*x+a)^(1/3)/(d*x+c)^(4/3),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(1/(b*x+a)^(1/3)/(d*x+c)^(4/3),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(1/(b*x+a)^(1/3)/(d*x+c)^(4/3),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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