\(\int (c x)^{-1-\frac {3 j}{2}} (a x^j+b x^n)^{3/2} \, dx\) [352]

Optimal result

Integrand size = 27, antiderivative size = 141 \[ \int (c x)^{-1-\frac {3 j}{2}} \left (a x^j+b x^n\right )^{3/2} \, dx=-\frac {2 a x^j (c x)^{-3 j/2} \sqrt {a x^j+b x^n}}{c (j-n)}-\frac {2 (c x)^{-3 j/2} \left (a x^j+b x^n\right )^{3/2}}{3 c (j-n)}+\frac {2 a^{3/2} x^{3 j/2} (c x)^{-3 j/2} \text {arctanh}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{c (j-n)} \] Output:

-2*a*x^j*(a*x^j+b*x^n)^(1/2)/c/(j-n)/((c*x)^(3/2*j))-2/3*(a*x^j+b*x^n)^(3/ 
2)/c/(j-n)/((c*x)^(3/2*j))+2*a^(3/2)*x^(3/2*j)*arctanh(a^(1/2)*x^(1/2*j)/( 
a*x^j+b*x^n)^(1/2))/c/(j-n)/((c*x)^(3/2*j))
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.93 \[ \int (c x)^{-1-\frac {3 j}{2}} \left (a x^j+b x^n\right )^{3/2} \, dx=-\frac {2 (c x)^{-3 j/2} \left (4 a^2 x^{2 j}+b^2 x^{2 n}+5 a b x^{j+n}-3 a^{3/2} \sqrt {b} x^{\frac {1}{2} (3 j+n)} \sqrt {1+\frac {a x^{j-n}}{b}} \text {arcsinh}\left (\frac {\sqrt {a} x^{\frac {j-n}{2}}}{\sqrt {b}}\right )\right )}{3 c (j-n) \sqrt {a x^j+b x^n}} \] Input:

Integrate[(c*x)^(-1 - (3*j)/2)*(a*x^j + b*x^n)^(3/2),x]
 

Output:

(-2*(4*a^2*x^(2*j) + b^2*x^(2*n) + 5*a*b*x^(j + n) - 3*a^(3/2)*Sqrt[b]*x^( 
(3*j + n)/2)*Sqrt[1 + (a*x^(j - n))/b]*ArcSinh[(Sqrt[a]*x^((j - n)/2))/Sqr 
t[b]]))/(3*c*(j - n)*(c*x)^((3*j)/2)*Sqrt[a*x^j + b*x^n])
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1937, 1934, 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.

\(\displaystyle \int (c x)^{-\frac {3 j}{2}-1} \left (a x^j+b x^n\right )^{3/2} \, dx\)

\(\Big \downarrow \) 1937

\(\displaystyle \frac {x^{3 j/2} (c x)^{-3 j/2} \int x^{-\frac {3 j}{2}-1} \left (a x^j+b x^n\right )^{3/2}dx}{c}\)

\(\Big \downarrow \) 1934

\(\displaystyle \frac {x^{3 j/2} (c x)^{-3 j/2} \left (a \int x^{-\frac {j}{2}-1} \sqrt {a x^j+b x^n}dx-\frac {2 x^{-3 j/2} \left (a x^j+b x^n\right )^{3/2}}{3 (j-n)}\right )}{c}\)

\(\Big \downarrow \) 1934

\(\displaystyle \frac {x^{3 j/2} (c x)^{-3 j/2} \left (a \left (a \int \frac {x^{\frac {j-2}{2}}}{\sqrt {a x^j+b x^n}}dx-\frac {2 x^{-j/2} \sqrt {a x^j+b x^n}}{j-n}\right )-\frac {2 x^{-3 j/2} \left (a x^j+b x^n\right )^{3/2}}{3 (j-n)}\right )}{c}\)

\(\Big \downarrow \) 1935

\(\displaystyle \frac {x^{3 j/2} (c x)^{-3 j/2} \left (a \left (\frac {2 a \int \frac {1}{1-\frac {a x^j}{a x^j+b x^n}}d\frac {x^{j/2}}{\sqrt {a x^j+b x^n}}}{j-n}-\frac {2 x^{-j/2} \sqrt {a x^j+b x^n}}{j-n}\right )-\frac {2 x^{-3 j/2} \left (a x^j+b x^n\right )^{3/2}}{3 (j-n)}\right )}{c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {x^{3 j/2} (c x)^{-3 j/2} \left (a \left (\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{j-n}-\frac {2 x^{-j/2} \sqrt {a x^j+b x^n}}{j-n}\right )-\frac {2 x^{-3 j/2} \left (a x^j+b x^n\right )^{3/2}}{3 (j-n)}\right )}{c}\)

Input:

Int[(c*x)^(-1 - (3*j)/2)*(a*x^j + b*x^n)^(3/2),x]
 

Output:

(x^((3*j)/2)*((-2*(a*x^j + b*x^n)^(3/2))/(3*(j - n)*x^((3*j)/2)) + a*((-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)^((3*j)/2))
 

Defintions of rubi rules used

rule 219

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])
 

rule 1934

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])
 

rule 1935

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]
 

rule 1937

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]
 

Maple [F]

\[\int \left (c x \right )^{-1-\frac {3 j}{2}} \left (a \,x^{j}+b \,x^{n}\right )^{\frac {3}{2}}d x\]

Input:

int((c*x)^(-1-3/2*j)*(a*x^j+b*x^n)^(3/2),x)
 

Output:

int((c*x)^(-1-3/2*j)*(a*x^j+b*x^n)^(3/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int (c x)^{-1-\frac {3 j}{2}} \left (a x^j+b x^n\right )^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((c*x)^(-1-3/2*j)*(a*x^j+b*x^n)^(3/2),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

Sympy [F]

\[ \int (c x)^{-1-\frac {3 j}{2}} \left (a x^j+b x^n\right )^{3/2} \, dx=\int \left (c x\right )^{- \frac {3 j}{2} - 1} \left (a x^{j} + b x^{n}\right )^{\frac {3}{2}}\, dx \] Input:

integrate((c*x)**(-1-3/2*j)*(a*x**j+b*x**n)**(3/2),x)
 

Output:

Integral((c*x)**(-3*j/2 - 1)*(a*x**j + b*x**n)**(3/2), x)
 

Maxima [F]

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

integrate((c*x)^(-1-3/2*j)*(a*x^j+b*x^n)^(3/2),x, algorithm="maxima")
 

Output:

integrate((a*x^j + b*x^n)^(3/2)*(c*x)^(-3/2*j - 1), x)
 

Giac [F]

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

integrate((c*x)^(-1-3/2*j)*(a*x^j+b*x^n)^(3/2),x, algorithm="giac")
 

Output:

integrate((a*x^j + b*x^n)^(3/2)*(c*x)^(-3/2*j - 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int (c x)^{-1-\frac {3 j}{2}} \left (a x^j+b x^n\right )^{3/2} \, dx=\int \frac {{\left (a\,x^j+b\,x^n\right )}^{3/2}}{{\left (c\,x\right )}^{\frac {3\,j}{2}+1}} \,d x \] Input:

int((a*x^j + b*x^n)^(3/2)/(c*x)^((3*j)/2 + 1),x)
 

Output:

int((a*x^j + b*x^n)^(3/2)/(c*x)^((3*j)/2 + 1), x)
 

Reduce [F]

\[ \int (c x)^{-1-\frac {3 j}{2}} \left (a x^j+b x^n\right )^{3/2} \, dx=\frac {-8 x^{j} \sqrt {x^{j} a +x^{n} b}\, a -2 x^{n} \sqrt {x^{j} a +x^{n} b}\, b +3 x^{\frac {3 j}{2}} \left (\int \frac {x^{2 j} \sqrt {x^{j} a +x^{n} b}}{x^{\frac {5 j}{2}} a x +x^{\frac {3 j}{2}+n} b x}d x \right ) a^{2} j -3 x^{\frac {3 j}{2}} \left (\int \frac {x^{2 j} \sqrt {x^{j} a +x^{n} b}}{x^{\frac {5 j}{2}} a x +x^{\frac {3 j}{2}+n} b x}d x \right ) a^{2} n}{3 x^{\frac {3 j}{2}} c^{\frac {3 j}{2}} c \left (j -n \right )} \] Input:

int((c*x)^(-1-3/2*j)*(a*x^j+b*x^n)^(3/2),x)
 

Output:

( - 8*x**j*sqrt(x**j*a + x**n*b)*a - 2*x**n*sqrt(x**j*a + x**n*b)*b + 3*x* 
*((3*j)/2)*int((x**(2*j)*sqrt(x**j*a + x**n*b))/(x**((5*j)/2)*a*x + x**((3 
*j + 2*n)/2)*b*x),x)*a**2*j - 3*x**((3*j)/2)*int((x**(2*j)*sqrt(x**j*a + x 
**n*b))/(x**((5*j)/2)*a*x + x**((3*j + 2*n)/2)*b*x),x)*a**2*n)/(3*x**((3*j 
)/2)*c**((3*j)/2)*c*(j - n))