Integrand size = 20, antiderivative size = 66 \[ \int \sqrt [5]{a-b x} \sqrt [5]{a+b x} \, dx=-\frac {5 (a-b x)^{6/5} \sqrt [5]{a+b x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{5},\frac {6}{5},\frac {11}{5},\frac {a-b x}{2 a}\right )}{3\ 2^{4/5} b \sqrt [5]{1+\frac {b x}{a}}} \] Output:
-5/6*(-b*x+a)^(6/5)*(b*x+a)^(1/5)*hypergeom([-1/5, 6/5],[11/5],1/2*(-b*x+a )/a)*2^(1/5)/b/(1+b*x/a)^(1/5)
Time = 0.02 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.02 \[ \int \sqrt [5]{a-b x} \sqrt [5]{a+b x} \, dx=-\frac {5 (a-b x)^{6/5} \sqrt [5]{a+b x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{5},\frac {6}{5},\frac {11}{5},\frac {a-b x}{2 a}\right )}{3\ 2^{4/5} b \sqrt [5]{\frac {a+b x}{a}}} \] Input:
Integrate[(a - b*x)^(1/5)*(a + b*x)^(1/5),x]
Output:
(-5*(a - b*x)^(6/5)*(a + b*x)^(1/5)*Hypergeometric2F1[-1/5, 6/5, 11/5, (a - b*x)/(2*a)])/(3*2^(4/5)*b*((a + b*x)/a)^(1/5))
Time = 0.16 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.88, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {46, 238, 237}
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.
\(\displaystyle \int \sqrt [5]{a-b x} \sqrt [5]{a+b x} \, dx\) |
\(\Big \downarrow \) 46 |
\(\displaystyle \frac {\sqrt [5]{a-b x} \sqrt [5]{a+b x} \int \sqrt [5]{a^2-b^2 x^2}dx}{\sqrt [5]{a^2-b^2 x^2}}\) |
\(\Big \downarrow \) 238 |
\(\displaystyle \frac {\sqrt [5]{a-b x} \sqrt [5]{a+b x} \int \sqrt [5]{1-\frac {b^2 x^2}{a^2}}dx}{\sqrt [5]{1-\frac {b^2 x^2}{a^2}}}\) |
\(\Big \downarrow \) 237 |
\(\displaystyle \frac {x \sqrt [5]{a-b x} \sqrt [5]{a+b x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{5},\frac {1}{2},\frac {3}{2},\frac {b^2 x^2}{a^2}\right )}{\sqrt [5]{1-\frac {b^2 x^2}{a^2}}}\) |
Input:
Int[(a - b*x)^(1/5)*(a + b*x)^(1/5),x]
Output:
(x*(a - b*x)^(1/5)*(a + b*x)^(1/5)*Hypergeometric2F1[-1/5, 1/2, 3/2, (b^2* x^2)/a^2])/(1 - (b^2*x^2)/a^2)^(1/5)
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^FracPart[m]*((c + d*x)^FracPart[m]/(a*c + b*d*x^2)^FracPart[m]) I nt[(a*c + b*d*x^2)^m, x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && !IntegerQ[2*m]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[- p, 1/2, 1/2 + 1, (-b)*(x^2/a)], x] /; FreeQ[{a, b, p}, x] && !IntegerQ[2*p ] && GtQ[a, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2) ^FracPart[p]/(1 + b*(x^2/a))^FracPart[p]) Int[(1 + b*(x^2/a))^p, x], x] / ; FreeQ[{a, b, p}, x] && !IntegerQ[2*p] && !GtQ[a, 0]
\[\int \left (-b x +a \right )^{\frac {1}{5}} \left (b x +a \right )^{\frac {1}{5}}d x\]
Input:
int((-b*x+a)^(1/5)*(b*x+a)^(1/5),x)
Output:
int((-b*x+a)^(1/5)*(b*x+a)^(1/5),x)
\[ \int \sqrt [5]{a-b x} \sqrt [5]{a+b x} \, dx=\int { {\left (b x + a\right )}^{\frac {1}{5}} {\left (-b x + a\right )}^{\frac {1}{5}} \,d x } \] Input:
integrate((-b*x+a)^(1/5)*(b*x+a)^(1/5),x, algorithm="fricas")
Output:
integral((b*x + a)^(1/5)*(-b*x + a)^(1/5), x)
\[ \int \sqrt [5]{a-b x} \sqrt [5]{a+b x} \, dx=\int \sqrt [5]{a - b x} \sqrt [5]{a + b x}\, dx \] Input:
integrate((-b*x+a)**(1/5)*(b*x+a)**(1/5),x)
Output:
Integral((a - b*x)**(1/5)*(a + b*x)**(1/5), x)
\[ \int \sqrt [5]{a-b x} \sqrt [5]{a+b x} \, dx=\int { {\left (b x + a\right )}^{\frac {1}{5}} {\left (-b x + a\right )}^{\frac {1}{5}} \,d x } \] Input:
integrate((-b*x+a)^(1/5)*(b*x+a)^(1/5),x, algorithm="maxima")
Output:
integrate((b*x + a)^(1/5)*(-b*x + a)^(1/5), x)
\[ \int \sqrt [5]{a-b x} \sqrt [5]{a+b x} \, dx=\int { {\left (b x + a\right )}^{\frac {1}{5}} {\left (-b x + a\right )}^{\frac {1}{5}} \,d x } \] Input:
integrate((-b*x+a)^(1/5)*(b*x+a)^(1/5),x, algorithm="giac")
Output:
integrate((b*x + a)^(1/5)*(-b*x + a)^(1/5), x)
Timed out. \[ \int \sqrt [5]{a-b x} \sqrt [5]{a+b x} \, dx=\int {\left (a+b\,x\right )}^{1/5}\,{\left (a-b\,x\right )}^{1/5} \,d x \] Input:
int((a + b*x)^(1/5)*(a - b*x)^(1/5),x)
Output:
int((a + b*x)^(1/5)*(a - b*x)^(1/5), x)
\[ \int \sqrt [5]{a-b x} \sqrt [5]{a+b x} \, dx=\int \left (b x +a \right )^{\frac {1}{5}} \left (-b x +a \right )^{\frac {1}{5}}d x \] Input:
int((-b*x+a)^(1/5)*(b*x+a)^(1/5),x)
Output:
int((a + b*x)**(1/5)*(a - b*x)**(1/5),x)