Integrand size = 20, antiderivative size = 66 \[ \int \sqrt [5]{a-b x} (a+b x)^{6/5} \, dx=\frac {5 \sqrt [5]{2} \sqrt [5]{a-b x} (a+b x)^{11/5} \operatorname {Hypergeometric2F1}\left (-\frac {1}{5},\frac {11}{5},\frac {16}{5},\frac {a+b x}{2 a}\right )}{11 b \sqrt [5]{1-\frac {b x}{a}}} \] Output:
5/11*2^(1/5)*(-b*x+a)^(1/5)*(b*x+a)^(11/5)*hypergeom([-1/5, 11/5],[16/5],1 /2*(b*x+a)/a)/b/(1-b*x/a)^(1/5)
Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.03 \[ \int \sqrt [5]{a-b x} (a+b x)^{6/5} \, dx=-\frac {5 \sqrt [5]{2} a (a-b x)^{6/5} \sqrt [5]{a+b x} \operatorname {Hypergeometric2F1}\left (-\frac {6}{5},\frac {6}{5},\frac {11}{5},\frac {a-b x}{2 a}\right )}{3 b \sqrt [5]{\frac {a+b x}{a}}} \] Input:
Integrate[(a - b*x)^(1/5)*(a + b*x)^(6/5),x]
Output:
(-5*2^(1/5)*a*(a - b*x)^(6/5)*(a + b*x)^(1/5)*Hypergeometric2F1[-6/5, 6/5, 11/5, (a - b*x)/(2*a)])/(3*b*((a + b*x)/a)^(1/5))
Time = 0.16 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {80, 27, 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.
\(\displaystyle \int \sqrt [5]{a-b x} (a+b x)^{6/5} \, dx\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {2 \sqrt [5]{2} a \sqrt [5]{a+b x} \int \frac {\sqrt [5]{a-b x} \left (\frac {b x}{a}+1\right )^{6/5}}{2 \sqrt [5]{2}}dx}{\sqrt [5]{\frac {a+b x}{a}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a \sqrt [5]{a+b x} \int \sqrt [5]{a-b x} \left (\frac {b x}{a}+1\right )^{6/5}dx}{\sqrt [5]{\frac {a+b x}{a}}}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle -\frac {5 \sqrt [5]{2} a (a-b x)^{6/5} \sqrt [5]{a+b x} \operatorname {Hypergeometric2F1}\left (-\frac {6}{5},\frac {6}{5},\frac {11}{5},\frac {a-b x}{2 a}\right )}{3 b \sqrt [5]{\frac {a+b x}{a}}}\) |
Input:
Int[(a - b*x)^(1/5)*(a + b*x)^(6/5),x]
Output:
(-5*2^(1/5)*a*(a - b*x)^(6/5)*(a + b*x)^(1/5)*Hypergeometric2F1[-6/5, 6/5, 11/5, (a - b*x)/(2*a)])/(3*b*((a + b*x)/a)^(1/5))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
\[\int \left (-b x +a \right )^{\frac {1}{5}} \left (b x +a \right )^{\frac {6}{5}}d x\]
Input:
int((-b*x+a)^(1/5)*(b*x+a)^(6/5),x)
Output:
int((-b*x+a)^(1/5)*(b*x+a)^(6/5),x)
\[ \int \sqrt [5]{a-b x} (a+b x)^{6/5} \, dx=\int { {\left (b x + a\right )}^{\frac {6}{5}} {\left (-b x + a\right )}^{\frac {1}{5}} \,d x } \] Input:
integrate((-b*x+a)^(1/5)*(b*x+a)^(6/5),x, algorithm="fricas")
Output:
integral((b*x + a)^(6/5)*(-b*x + a)^(1/5), x)
\[ \int \sqrt [5]{a-b x} (a+b x)^{6/5} \, dx=\int \sqrt [5]{a - b x} \left (a + b x\right )^{\frac {6}{5}}\, dx \] Input:
integrate((-b*x+a)**(1/5)*(b*x+a)**(6/5),x)
Output:
Integral((a - b*x)**(1/5)*(a + b*x)**(6/5), x)
\[ \int \sqrt [5]{a-b x} (a+b x)^{6/5} \, dx=\int { {\left (b x + a\right )}^{\frac {6}{5}} {\left (-b x + a\right )}^{\frac {1}{5}} \,d x } \] Input:
integrate((-b*x+a)^(1/5)*(b*x+a)^(6/5),x, algorithm="maxima")
Output:
integrate((b*x + a)^(6/5)*(-b*x + a)^(1/5), x)
\[ \int \sqrt [5]{a-b x} (a+b x)^{6/5} \, dx=\int { {\left (b x + a\right )}^{\frac {6}{5}} {\left (-b x + a\right )}^{\frac {1}{5}} \,d x } \] Input:
integrate((-b*x+a)^(1/5)*(b*x+a)^(6/5),x, algorithm="giac")
Output:
integrate((b*x + a)^(6/5)*(-b*x + a)^(1/5), x)
Timed out. \[ \int \sqrt [5]{a-b x} (a+b x)^{6/5} \, dx=\int {\left (a+b\,x\right )}^{6/5}\,{\left (a-b\,x\right )}^{1/5} \,d x \] Input:
int((a + b*x)^(6/5)*(a - b*x)^(1/5),x)
Output:
int((a + b*x)^(6/5)*(a - b*x)^(1/5), x)
\[ \int \sqrt [5]{a-b x} (a+b x)^{6/5} \, dx=\left (\int \left (b x +a \right )^{\frac {1}{5}} \left (-b x +a \right )^{\frac {1}{5}} x d x \right ) b +\left (\int \left (b x +a \right )^{\frac {1}{5}} \left (-b x +a \right )^{\frac {1}{5}}d x \right ) a \] Input:
int((-b*x+a)^(1/5)*(b*x+a)^(6/5),x)
Output:
int((a + b*x)**(1/5)*(a - b*x)**(1/5)*x,x)*b + int((a + b*x)**(1/5)*(a - b *x)**(1/5),x)*a