Integrand size = 19, antiderivative size = 58 \[ \int \sqrt [6]{a+b x} (c+d x)^{13/6} \, dx=\frac {6 (a+b x)^{7/6} (c+d x)^{19/6} \operatorname {Hypergeometric2F1}\left (1,\frac {13}{3},\frac {13}{6},-\frac {d (a+b x)}{b c-a d}\right )}{7 (b c-a d)} \] Output:
6*(b*x+a)^(7/6)*(d*x+c)^(19/6)*hypergeom([1, 13/3],[13/6],-d*(b*x+a)/(-a*d +b*c))/(-7*a*d+7*b*c)
Time = 0.06 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.26 \[ \int \sqrt [6]{a+b x} (c+d x)^{13/6} \, dx=\frac {6 (a+b x)^{7/6} (c+d x)^{13/6} \operatorname {Hypergeometric2F1}\left (-\frac {13}{6},\frac {7}{6},\frac {13}{6},\frac {d (a+b x)}{-b c+a d}\right )}{7 b \left (\frac {b (c+d x)}{b c-a d}\right )^{13/6}} \] Input:
Integrate[(a + b*x)^(1/6)*(c + d*x)^(13/6),x]
Output:
(6*(a + b*x)^(7/6)*(c + d*x)^(13/6)*Hypergeometric2F1[-13/6, 7/6, 13/6, (d *(a + b*x))/(-(b*c) + a*d)])/(7*b*((b*(c + d*x))/(b*c - a*d))^(13/6))
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.
\(\displaystyle \int \sqrt [6]{a+b x} (c+d x)^{13/6} \, dx\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {\sqrt [6]{c+d x} (b c-a d)^2 \int \sqrt [6]{a+b x} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{13/6}dx}{b^2 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {6 (a+b x)^{7/6} \sqrt [6]{c+d x} (b c-a d)^2 \operatorname {Hypergeometric2F1}\left (-\frac {13}{6},\frac {7}{6},\frac {13}{6},-\frac {d (a+b x)}{b c-a d}\right )}{7 b^3 \sqrt [6]{\frac {b (c+d x)}{b c-a d}}}\) |
Input:
Int[(a + b*x)^(1/6)*(c + d*x)^(13/6),x]
Output:
(6*(b*c - a*d)^2*(a + b*x)^(7/6)*(c + d*x)^(1/6)*Hypergeometric2F1[-13/6, 7/6, 13/6, -((d*(a + b*x))/(b*c - a*d))])/(7*b^3*((b*(c + d*x))/(b*c - a*d ))^(1/6))
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]))
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])
\[\int \left (b x +a \right )^{\frac {1}{6}} \left (x d +c \right )^{\frac {13}{6}}d x\]
Input:
int((b*x+a)^(1/6)*(d*x+c)^(13/6),x)
Output:
int((b*x+a)^(1/6)*(d*x+c)^(13/6),x)
\[ \int \sqrt [6]{a+b x} (c+d x)^{13/6} \, dx=\int { {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {13}{6}} \,d x } \] Input:
integrate((b*x+a)^(1/6)*(d*x+c)^(13/6),x, algorithm="fricas")
Output:
integral((d^2*x^2 + 2*c*d*x + c^2)*(b*x + a)^(1/6)*(d*x + c)^(1/6), x)
Timed out. \[ \int \sqrt [6]{a+b x} (c+d x)^{13/6} \, dx=\text {Timed out} \] Input:
integrate((b*x+a)**(1/6)*(d*x+c)**(13/6),x)
Output:
Timed out
\[ \int \sqrt [6]{a+b x} (c+d x)^{13/6} \, dx=\int { {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {13}{6}} \,d x } \] Input:
integrate((b*x+a)^(1/6)*(d*x+c)^(13/6),x, algorithm="maxima")
Output:
integrate((b*x + a)^(1/6)*(d*x + c)^(13/6), x)
\[ \int \sqrt [6]{a+b x} (c+d x)^{13/6} \, dx=\int { {\left (b x + a\right )}^{\frac {1}{6}} {\left (d x + c\right )}^{\frac {13}{6}} \,d x } \] Input:
integrate((b*x+a)^(1/6)*(d*x+c)^(13/6),x, algorithm="giac")
Output:
integrate((b*x + a)^(1/6)*(d*x + c)^(13/6), x)
Timed out. \[ \int \sqrt [6]{a+b x} (c+d x)^{13/6} \, dx=\int {\left (a+b\,x\right )}^{1/6}\,{\left (c+d\,x\right )}^{13/6} \,d x \] Input:
int((a + b*x)^(1/6)*(c + d*x)^(13/6),x)
Output:
int((a + b*x)^(1/6)*(c + d*x)^(13/6), x)
\[ \int \sqrt [6]{a+b x} (c+d x)^{13/6} \, dx=\left (\int \left (d x +c \right )^{\frac {1}{6}} \left (b x +a \right )^{\frac {1}{6}} x^{2}d x \right ) d^{2}+2 \left (\int \left (d x +c \right )^{\frac {1}{6}} \left (b x +a \right )^{\frac {1}{6}} x d x \right ) c d +\left (\int \left (d x +c \right )^{\frac {1}{6}} \left (b x +a \right )^{\frac {1}{6}}d x \right ) c^{2} \] Input:
int((b*x+a)^(1/6)*(d*x+c)^(13/6),x)
Output:
int((c + d*x)**(1/6)*(a + b*x)**(1/6)*x**2,x)*d**2 + 2*int((c + d*x)**(1/6 )*(a + b*x)**(1/6)*x,x)*c*d + int((c + d*x)**(1/6)*(a + b*x)**(1/6),x)*c** 2