\(\int \frac {(a+b x)^{3/2}}{\sqrt [5]{c+d x}} \, dx\) [482]

Optimal result

Integrand size = 19, antiderivative size = 58 \[ \int \frac {(a+b x)^{3/2}}{\sqrt [5]{c+d x}} \, dx=\frac {2 (a+b x)^{5/2} (c+d x)^{4/5} \operatorname {Hypergeometric2F1}\left (1,\frac {33}{10},\frac {7}{2},-\frac {d (a+b x)}{b c-a d}\right )}{5 (b c-a d)} \] Output:

2*(b*x+a)^(5/2)*(d*x+c)^(4/5)*hypergeom([1, 33/10],[7/2],-d*(b*x+a)/(-a*d+ 
b*c))/(-5*a*d+5*b*c)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.26 \[ \int \frac {(a+b x)^{3/2}}{\sqrt [5]{c+d x}} \, dx=\frac {2 (a+b x)^{5/2} \sqrt [5]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{5},\frac {5}{2},\frac {7}{2},\frac {d (a+b x)}{-b c+a d}\right )}{5 b \sqrt [5]{c+d x}} \] Input:

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

Output:

(2*(a + b*x)^(5/2)*((b*(c + d*x))/(b*c - a*d))^(1/5)*Hypergeometric2F1[1/5 
, 5/2, 7/2, (d*(a + b*x))/(-(b*c) + a*d)])/(5*b*(c + d*x)^(1/5))
 

Rubi [A] (verified)

Time = 0.17 (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.

\(\displaystyle \int \frac {(a+b x)^{3/2}}{\sqrt [5]{c+d x}} \, dx\)

\(\Big \downarrow \) 80

\(\displaystyle \frac {\sqrt [5]{\frac {b (c+d x)}{b c-a d}} \int \frac {(a+b x)^{3/2}}{\sqrt [5]{\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}}dx}{\sqrt [5]{c+d x}}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {2 (a+b x)^{5/2} \sqrt [5]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{5},\frac {5}{2},\frac {7}{2},-\frac {d (a+b x)}{b c-a d}\right )}{5 b \sqrt [5]{c+d x}}\)

Input:

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

Output:

(2*(a + b*x)^(5/2)*((b*(c + d*x))/(b*c - a*d))^(1/5)*Hypergeometric2F1[1/5 
, 5/2, 7/2, -((d*(a + b*x))/(b*c - a*d))])/(5*b*(c + d*x)^(1/5))
 

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]

\[\int \frac {\left (b x +a \right )^{\frac {3}{2}}}{\left (x d +c \right )^{\frac {1}{5}}}d x\]

Input:

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

Output:

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

Fricas [F]

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

integrate((b*x+a)^(3/2)/(d*x+c)^(1/5),x, algorithm="fricas")
 

Output:

integral((b*x + a)^(3/2)/(d*x + c)^(1/5), x)
 

Sympy [F]

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

integrate((b*x+a)**(3/2)/(d*x+c)**(1/5),x)
 

Output:

Integral((a + b*x)**(3/2)/(c + d*x)**(1/5), x)
 

Maxima [F]

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

integrate((b*x+a)^(3/2)/(d*x+c)^(1/5),x, algorithm="maxima")
 

Output:

integrate((b*x + a)^(3/2)/(d*x + c)^(1/5), x)
 

Giac [F]

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

integrate((b*x+a)^(3/2)/(d*x+c)^(1/5),x, algorithm="giac")
 

Output:

integrate((b*x + a)^(3/2)/(d*x + c)^(1/5), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(a + b*x)/(c + d*x)**(1/5),x)*a + int((sqrt(a + b*x)*x)/(c + d*x)* 
*(1/5),x)*b