\(\int (a+b x)^{7/6} (c+d x)^{13/6} \, dx\) [702]

Optimal result

Integrand size = 19, antiderivative size = 58 \[ \int (a+b x)^{7/6} (c+d x)^{13/6} \, dx=\frac {6 (a+b x)^{13/6} (c+d x)^{19/6} \operatorname {Hypergeometric2F1}\left (1,\frac {16}{3},\frac {19}{6},-\frac {d (a+b x)}{b c-a d}\right )}{13 (b c-a d)} \] Output:

6*(b*x+a)^(13/6)*(d*x+c)^(19/6)*hypergeom([1, 16/3],[19/6],-d*(b*x+a)/(-a* 
d+b*c))/(-13*a*d+13*b*c)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.26 \[ \int (a+b x)^{7/6} (c+d x)^{13/6} \, dx=\frac {6 (a+b x)^{13/6} (c+d x)^{13/6} \operatorname {Hypergeometric2F1}\left (-\frac {13}{6},\frac {13}{6},\frac {19}{6},\frac {d (a+b x)}{-b c+a d}\right )}{13 b \left (\frac {b (c+d x)}{b c-a d}\right )^{13/6}} \] Input:

Integrate[(a + b*x)^(7/6)*(c + d*x)^(13/6),x]
 

Output:

(6*(a + b*x)^(13/6)*(c + d*x)^(13/6)*Hypergeometric2F1[-13/6, 13/6, 19/6, 
(d*(a + b*x))/(-(b*c) + a*d)])/(13*b*((b*(c + d*x))/(b*c - a*d))^(13/6))
 

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.45, 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.

\(\displaystyle \int (a+b x)^{7/6} (c+d x)^{13/6} \, dx\)

\(\Big \downarrow \) 80

\(\displaystyle \frac {\sqrt [6]{c+d x} (b c-a d)^2 \int (a+b x)^{7/6} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{13/6}dx}{b^2 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {6 (a+b x)^{13/6} \sqrt [6]{c+d x} (b c-a d)^2 \operatorname {Hypergeometric2F1}\left (-\frac {13}{6},\frac {13}{6},\frac {19}{6},-\frac {d (a+b x)}{b c-a d}\right )}{13 b^3 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}}\)

Input:

Int[(a + b*x)^(7/6)*(c + d*x)^(13/6),x]
 

Output:

(6*(b*c - a*d)^2*(a + b*x)^(13/6)*(c + d*x)^(1/6)*Hypergeometric2F1[-13/6, 
 13/6, 19/6, -((d*(a + b*x))/(b*c - a*d))])/(13*b^3*((b*(c + d*x))/(b*c - 
a*d))^(1/6))
 

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]

\[\int \left (b x +a \right )^{\frac {7}{6}} \left (x d +c \right )^{\frac {13}{6}}d x\]

Input:

int((b*x+a)^(7/6)*(d*x+c)^(13/6),x)
 

Output:

int((b*x+a)^(7/6)*(d*x+c)^(13/6),x)
 

Fricas [F]

\[ \int (a+b x)^{7/6} (c+d x)^{13/6} \, dx=\int { {\left (b x + a\right )}^{\frac {7}{6}} {\left (d x + c\right )}^{\frac {13}{6}} \,d x } \] Input:

integrate((b*x+a)^(7/6)*(d*x+c)^(13/6),x, algorithm="fricas")
 

Output:

integral((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) 
*(b*x + a)^(1/6)*(d*x + c)^(1/6), x)
 

Sympy [F(-1)]

Timed out. \[ \int (a+b x)^{7/6} (c+d x)^{13/6} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**(7/6)*(d*x+c)**(13/6),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (a+b x)^{7/6} (c+d x)^{13/6} \, dx=\int { {\left (b x + a\right )}^{\frac {7}{6}} {\left (d x + c\right )}^{\frac {13}{6}} \,d x } \] Input:

integrate((b*x+a)^(7/6)*(d*x+c)^(13/6),x, algorithm="maxima")
 

Output:

integrate((b*x + a)^(7/6)*(d*x + c)^(13/6), x)
 

Giac [F]

\[ \int (a+b x)^{7/6} (c+d x)^{13/6} \, dx=\int { {\left (b x + a\right )}^{\frac {7}{6}} {\left (d x + c\right )}^{\frac {13}{6}} \,d x } \] Input:

integrate((b*x+a)^(7/6)*(d*x+c)^(13/6),x, algorithm="giac")
 

Output:

integrate((b*x + a)^(7/6)*(d*x + c)^(13/6), x)
 

Mupad [F(-1)]

Timed out. \[ \int (a+b x)^{7/6} (c+d x)^{13/6} \, dx=\int {\left (a+b\,x\right )}^{7/6}\,{\left (c+d\,x\right )}^{13/6} \,d x \] Input:

int((a + b*x)^(7/6)*(c + d*x)^(13/6),x)
 

Output:

int((a + b*x)^(7/6)*(c + d*x)^(13/6), x)
 

Reduce [F]

\[ \int (a+b x)^{7/6} (c+d x)^{13/6} \, dx=\left (\int \left (d x +c \right )^{\frac {1}{6}} \left (b x +a \right )^{\frac {1}{6}} x^{3}d x \right ) b \,d^{2}+\left (\int \left (d x +c \right )^{\frac {1}{6}} \left (b x +a \right )^{\frac {1}{6}} x^{2}d x \right ) a \,d^{2}+2 \left (\int \left (d x +c \right )^{\frac {1}{6}} \left (b x +a \right )^{\frac {1}{6}} x^{2}d x \right ) b c d +2 \left (\int \left (d x +c \right )^{\frac {1}{6}} \left (b x +a \right )^{\frac {1}{6}} x d x \right ) a c d +\left (\int \left (d x +c \right )^{\frac {1}{6}} \left (b x +a \right )^{\frac {1}{6}} x d x \right ) b \,c^{2}+\left (\int \left (d x +c \right )^{\frac {1}{6}} \left (b x +a \right )^{\frac {1}{6}}d x \right ) a \,c^{2} \] Input:

int((b*x+a)^(7/6)*(d*x+c)^(13/6),x)
 

Output:

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