Integrand size = 27, antiderivative size = 99 \[ \int (c x)^{-1-\frac {j}{2}} \sqrt {a x^j+b x^n} \, dx=-\frac {2 (c x)^{-j/2} \sqrt {a x^j+b x^n}}{c (j-n)}+\frac {2 \sqrt {a} x^{j/2} (c x)^{-j/2} \text {arctanh}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{c (j-n)} \] Output:
-2*(a*x^j+b*x^n)^(1/2)/c/(j-n)/((c*x)^(1/2*j))+2*a^(1/2)*x^(1/2*j)*arctanh (a^(1/2)*x^(1/2*j)/(a*x^j+b*x^n)^(1/2))/c/(j-n)/((c*x)^(1/2*j))
Time = 0.24 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.10 \[ \int (c x)^{-1-\frac {j}{2}} \sqrt {a x^j+b x^n} \, dx=-\frac {2 (c x)^{-j/2} \left (a x^j+b x^n-\sqrt {a} \sqrt {b} x^{\frac {j+n}{2}} \sqrt {1+\frac {a x^{j-n}}{b}} \text {arcsinh}\left (\frac {\sqrt {a} x^{\frac {j-n}{2}}}{\sqrt {b}}\right )\right )}{c (j-n) \sqrt {a x^j+b x^n}} \] Input:
Integrate[(c*x)^(-1 - j/2)*Sqrt[a*x^j + b*x^n],x]
Output:
(-2*(a*x^j + b*x^n - Sqrt[a]*Sqrt[b]*x^((j + n)/2)*Sqrt[1 + (a*x^(j - n))/ b]*ArcSinh[(Sqrt[a]*x^((j - n)/2))/Sqrt[b]]))/(c*(j - n)*(c*x)^(j/2)*Sqrt[ a*x^j + b*x^n])
Time = 0.43 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1937, 1934, 1935, 219}
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.
\(\displaystyle \int (c x)^{-\frac {j}{2}-1} \sqrt {a x^j+b x^n} \, dx\) |
\(\Big \downarrow \) 1937 |
\(\displaystyle \frac {x^{j/2} (c x)^{-j/2} \int x^{-\frac {j}{2}-1} \sqrt {a x^j+b x^n}dx}{c}\) |
\(\Big \downarrow \) 1934 |
\(\displaystyle \frac {x^{j/2} (c x)^{-j/2} \left (a \int \frac {x^{\frac {j-2}{2}}}{\sqrt {a x^j+b x^n}}dx-\frac {2 x^{-j/2} \sqrt {a x^j+b x^n}}{j-n}\right )}{c}\) |
\(\Big \downarrow \) 1935 |
\(\displaystyle \frac {x^{j/2} (c x)^{-j/2} \left (\frac {2 a \int \frac {1}{1-\frac {a x^j}{a x^j+b x^n}}d\frac {x^{j/2}}{\sqrt {a x^j+b x^n}}}{j-n}-\frac {2 x^{-j/2} \sqrt {a x^j+b x^n}}{j-n}\right )}{c}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {x^{j/2} (c x)^{-j/2} \left (\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{j-n}-\frac {2 x^{-j/2} \sqrt {a x^j+b x^n}}{j-n}\right )}{c}\) |
Input:
Int[(c*x)^(-1 - j/2)*Sqrt[a*x^j + b*x^n],x]
Output:
(x^(j/2)*((-2*Sqrt[a*x^j + b*x^n])/((j - n)*x^(j/2)) + (2*Sqrt[a]*ArcTanh[ (Sqrt[a]*x^(j/2))/Sqrt[a*x^j + b*x^n]])/(j - n)))/(c*(c*x)^(j/2))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[(c*x)^(m + 1)*((a*x^j + b*x^n)^p/(c*p*(n - j))), x] + Simp[a/c^j Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, j, m , n}, x] && IGtQ[p + 1/2, 0] && NeQ[n, j] && EqQ[Simplify[m + j*p + 1], 0] && (IntegerQ[j] || GtQ[c, 0])
Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp [-2/(n - j) Subst[Int[1/(1 - a*x^2), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]
Int[((c_)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^IntPart[m]*((c*x)^FracPart[m]/x^FracPart[m]) Int[x^m*(a*x^j + b *x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && IntegerQ[p + 1/2] && NeQ[n, j] && EqQ[Simplify[m + j*p + 1], 0]
\[\int \left (c x \right )^{-1-\frac {j}{2}} \sqrt {a \,x^{j}+b \,x^{n}}d x\]
Input:
int((c*x)^(-1-1/2*j)*(a*x^j+b*x^n)^(1/2),x)
Output:
int((c*x)^(-1-1/2*j)*(a*x^j+b*x^n)^(1/2),x)
Exception generated. \[ \int (c x)^{-1-\frac {j}{2}} \sqrt {a x^j+b x^n} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c*x)^(-1-1/2*j)*(a*x^j+b*x^n)^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
\[ \int (c x)^{-1-\frac {j}{2}} \sqrt {a x^j+b x^n} \, dx=\int \left (c x\right )^{- \frac {j}{2} - 1} \sqrt {a x^{j} + b x^{n}}\, dx \] Input:
integrate((c*x)**(-1-1/2*j)*(a*x**j+b*x**n)**(1/2),x)
Output:
Integral((c*x)**(-j/2 - 1)*sqrt(a*x**j + b*x**n), x)
\[ \int (c x)^{-1-\frac {j}{2}} \sqrt {a x^j+b x^n} \, dx=\int { \sqrt {a x^{j} + b x^{n}} \left (c x\right )^{-\frac {1}{2} \, j - 1} \,d x } \] Input:
integrate((c*x)^(-1-1/2*j)*(a*x^j+b*x^n)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(a*x^j + b*x^n)*(c*x)^(-1/2*j - 1), x)
\[ \int (c x)^{-1-\frac {j}{2}} \sqrt {a x^j+b x^n} \, dx=\int { \sqrt {a x^{j} + b x^{n}} \left (c x\right )^{-\frac {1}{2} \, j - 1} \,d x } \] Input:
integrate((c*x)^(-1-1/2*j)*(a*x^j+b*x^n)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(a*x^j + b*x^n)*(c*x)^(-1/2*j - 1), x)
Timed out. \[ \int (c x)^{-1-\frac {j}{2}} \sqrt {a x^j+b x^n} \, dx=\int \frac {\sqrt {a\,x^j+b\,x^n}}{{\left (c\,x\right )}^{\frac {j}{2}+1}} \,d x \] Input:
int((a*x^j + b*x^n)^(1/2)/(c*x)^(j/2 + 1),x)
Output:
int((a*x^j + b*x^n)^(1/2)/(c*x)^(j/2 + 1), x)
\[ \int (c x)^{-1-\frac {j}{2}} \sqrt {a x^j+b x^n} \, dx=\frac {-2 \sqrt {x^{j} a +x^{n} b}+x^{\frac {j}{2}} \left (\int \frac {x^{j} \sqrt {x^{j} a +x^{n} b}}{x^{\frac {3 j}{2}} a x +x^{\frac {j}{2}+n} b x}d x \right ) a j -x^{\frac {j}{2}} \left (\int \frac {x^{j} \sqrt {x^{j} a +x^{n} b}}{x^{\frac {3 j}{2}} a x +x^{\frac {j}{2}+n} b x}d x \right ) a n}{x^{\frac {j}{2}} c^{\frac {j}{2}} c \left (j -n \right )} \] Input:
int((c*x)^(-1-1/2*j)*(a*x^j+b*x^n)^(1/2),x)
Output:
( - 2*sqrt(x**j*a + x**n*b) + x**(j/2)*int((x**j*sqrt(x**j*a + x**n*b))/(x **((3*j)/2)*a*x + x**((j + 2*n)/2)*b*x),x)*a*j - x**(j/2)*int((x**j*sqrt(x **j*a + x**n*b))/(x**((3*j)/2)*a*x + x**((j + 2*n)/2)*b*x),x)*a*n)/(x**(j/ 2)*c**(j/2)*c*(j - n))