Integrand size = 23, antiderivative size = 122 \[ \int (c x)^{7/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2} \, dx=\frac {2 a c^2 (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}{3+n}+\frac {2 (c x)^{9/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2}}{3 c (3+n)}-\frac {2 a^{3/2} c^4 \sqrt {x} \text {arctanh}\left (\frac {\sqrt {a}}{x^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}\right )}{(3+n) \sqrt {c x}} \] Output:
2*a*c^2*(c*x)^(3/2)*(a/x^3+b*x^n)^(1/2)/(3+n)+2/3*(c*x)^(9/2)*(a/x^3+b*x^n )^(3/2)/c/(3+n)-2*a^(3/2)*c^4*x^(1/2)*arctanh(a^(1/2)/x^(3/2)/(a/x^3+b*x^n )^(1/2))/(3+n)/(c*x)^(1/2)
Time = 0.10 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.82 \[ \int (c x)^{7/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2} \, dx=\frac {2 c^2 (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n} \left (\sqrt {a+b x^{3+n}} \left (4 a+b x^{3+n}\right )-3 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b x^{3+n}}}{\sqrt {a}}\right )\right )}{3 (3+n) \sqrt {a+b x^{3+n}}} \] Input:
Integrate[(c*x)^(7/2)*(a/x^3 + b*x^n)^(3/2),x]
Output:
(2*c^2*(c*x)^(3/2)*Sqrt[a/x^3 + b*x^n]*(Sqrt[a + b*x^(3 + n)]*(4*a + b*x^( 3 + n)) - 3*a^(3/2)*ArcTanh[Sqrt[a + b*x^(3 + n)]/Sqrt[a]]))/(3*(3 + n)*Sq rt[a + b*x^(3 + n)])
Time = 0.51 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1934, 1934, 1937, 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.
\(\displaystyle \int (c x)^{7/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 1934 |
\(\displaystyle a c^3 \int \sqrt {c x} \sqrt {b x^n+\frac {a}{x^3}}dx+\frac {2 (c x)^{9/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2}}{3 c (n+3)}\) |
\(\Big \downarrow \) 1934 |
\(\displaystyle a c^3 \left (a c^3 \int \frac {1}{(c x)^{5/2} \sqrt {b x^n+\frac {a}{x^3}}}dx+\frac {2 (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}{c (n+3)}\right )+\frac {2 (c x)^{9/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2}}{3 c (n+3)}\) |
\(\Big \downarrow \) 1937 |
\(\displaystyle a c^3 \left (\frac {a c \sqrt {x} \int \frac {1}{x^{5/2} \sqrt {b x^n+\frac {a}{x^3}}}dx}{\sqrt {c x}}+\frac {2 (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}{c (n+3)}\right )+\frac {2 (c x)^{9/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2}}{3 c (n+3)}\) |
\(\Big \downarrow \) 1935 |
\(\displaystyle a c^3 \left (\frac {2 (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}{c (n+3)}-\frac {2 a c \sqrt {x} \int \frac {1}{1-\frac {a}{x^3 \left (b x^n+\frac {a}{x^3}\right )}}d\frac {1}{x^{3/2} \sqrt {b x^n+\frac {a}{x^3}}}}{(n+3) \sqrt {c x}}\right )+\frac {2 (c x)^{9/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2}}{3 c (n+3)}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle a c^3 \left (\frac {2 (c x)^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}{c (n+3)}-\frac {2 \sqrt {a} c \sqrt {x} \text {arctanh}\left (\frac {\sqrt {a}}{x^{3/2} \sqrt {\frac {a}{x^3}+b x^n}}\right )}{(n+3) \sqrt {c x}}\right )+\frac {2 (c x)^{9/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2}}{3 c (n+3)}\) |
Input:
Int[(c*x)^(7/2)*(a/x^3 + b*x^n)^(3/2),x]
Output:
(2*(c*x)^(9/2)*(a/x^3 + b*x^n)^(3/2))/(3*c*(3 + n)) + a*c^3*((2*(c*x)^(3/2 )*Sqrt[a/x^3 + b*x^n])/(c*(3 + n)) - (2*Sqrt[a]*c*Sqrt[x]*ArcTanh[Sqrt[a]/ (x^(3/2)*Sqrt[a/x^3 + b*x^n])])/((3 + n)*Sqrt[c*x]))
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]
\[\int \left (c x \right )^{\frac {7}{2}} \left (\frac {a}{x^{3}}+b \,x^{n}\right )^{\frac {3}{2}}d x\]
Input:
int((c*x)^(7/2)*(a/x^3+b*x^n)^(3/2),x)
Output:
int((c*x)^(7/2)*(a/x^3+b*x^n)^(3/2),x)
Exception generated. \[ \int (c x)^{7/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c*x)^(7/2)*(a/x^3+b*x^n)^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
Timed out. \[ \int (c x)^{7/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2} \, dx=\text {Timed out} \] Input:
integrate((c*x)**(7/2)*(a/x**3+b*x**n)**(3/2),x)
Output:
Timed out
\[ \int (c x)^{7/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2} \, dx=\int { {\left (b x^{n} + \frac {a}{x^{3}}\right )}^{\frac {3}{2}} \left (c x\right )^{\frac {7}{2}} \,d x } \] Input:
integrate((c*x)^(7/2)*(a/x^3+b*x^n)^(3/2),x, algorithm="maxima")
Output:
integrate((b*x^n + a/x^3)^(3/2)*(c*x)^(7/2), x)
\[ \int (c x)^{7/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2} \, dx=\int { {\left (b x^{n} + \frac {a}{x^{3}}\right )}^{\frac {3}{2}} \left (c x\right )^{\frac {7}{2}} \,d x } \] Input:
integrate((c*x)^(7/2)*(a/x^3+b*x^n)^(3/2),x, algorithm="giac")
Output:
integrate((b*x^n + a/x^3)^(3/2)*(c*x)^(7/2), x)
Timed out. \[ \int (c x)^{7/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2} \, dx=\int {\left (c\,x\right )}^{7/2}\,{\left (b\,x^n+\frac {a}{x^3}\right )}^{3/2} \,d x \] Input:
int((c*x)^(7/2)*(b*x^n + a/x^3)^(3/2),x)
Output:
int((c*x)^(7/2)*(b*x^n + a/x^3)^(3/2), x)
\[ \int (c x)^{7/2} \left (\frac {a}{x^3}+b x^n\right )^{3/2} \, dx=\sqrt {c}\, c^{3} \left (\left (\int \frac {\sqrt {x^{n} b \,x^{3}+a}}{x}d x \right ) a +\left (\int x^{n} \sqrt {x^{n} b \,x^{3}+a}\, x^{2}d x \right ) b \right ) \] Input:
int((c*x)^(7/2)*(a/x^3+b*x^n)^(3/2),x)
Output:
sqrt(c)*c**3*(int(sqrt(x**n*b*x**3 + a)/x,x)*a + int(x**n*sqrt(x**n*b*x**3 + a)*x**2,x)*b)