Integrand size = 23, antiderivative size = 128 \[ \int \frac {\left (a x^3+b x^n\right )^{3/2}}{(c x)^{11/2}} \, dx=-\frac {2 a \sqrt {a x^3+b x^n}}{c^4 (3-n) (c x)^{3/2}}-\frac {2 \left (a x^3+b x^n\right )^{3/2}}{3 c (3-n) (c x)^{9/2}}+\frac {2 a^{3/2} \sqrt {c x} \text {arctanh}\left (\frac {\sqrt {a} x^{3/2}}{\sqrt {a x^3+b x^n}}\right )}{c^6 (3-n) \sqrt {x}} \] Output:
-2*a*(a*x^3+b*x^n)^(1/2)/c^4/(3-n)/(c*x)^(3/2)-2/3*(a*x^3+b*x^n)^(3/2)/c/( 3-n)/(c*x)^(9/2)+2*a^(3/2)*(c*x)^(1/2)*arctanh(a^(1/2)*x^(3/2)/(a*x^3+b*x^ n)^(1/2))/c^6/(3-n)/x^(1/2)
Time = 1.61 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a x^3+b x^n\right )^{3/2}}{(c x)^{11/2}} \, dx=\frac {2 \sqrt {c x} \left (4 a^2 x^6+b^2 x^{2 n}+5 a b x^{3+n}-3 a^{3/2} \sqrt {b} x^{\frac {9+n}{2}} \sqrt {1+\frac {a x^{3-n}}{b}} \text {arcsinh}\left (\frac {\sqrt {a} x^{\frac {3}{2}-\frac {n}{2}}}{\sqrt {b}}\right )\right )}{3 c^6 (-3+n) x^5 \sqrt {a x^3+b x^n}} \] Input:
Integrate[(a*x^3 + b*x^n)^(3/2)/(c*x)^(11/2),x]
Output:
(2*Sqrt[c*x]*(4*a^2*x^6 + b^2*x^(2*n) + 5*a*b*x^(3 + n) - 3*a^(3/2)*Sqrt[b ]*x^((9 + n)/2)*Sqrt[1 + (a*x^(3 - n))/b]*ArcSinh[(Sqrt[a]*x^(3/2 - n/2))/ Sqrt[b]]))/(3*c^6*(-3 + n)*x^5*Sqrt[a*x^3 + b*x^n])
Time = 0.54 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.04, 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.
\(\displaystyle \int \frac {\left (a x^3+b x^n\right )^{3/2}}{(c x)^{11/2}} \, dx\) |
\(\Big \downarrow \) 1934 |
\(\displaystyle \frac {a \int \frac {\sqrt {b x^n+a x^3}}{(c x)^{5/2}}dx}{c^3}-\frac {2 \left (a x^3+b x^n\right )^{3/2}}{3 c (3-n) (c x)^{9/2}}\) |
\(\Big \downarrow \) 1934 |
\(\displaystyle \frac {a \left (\frac {a \int \frac {\sqrt {c x}}{\sqrt {b x^n+a x^3}}dx}{c^3}-\frac {2 \sqrt {a x^3+b x^n}}{c (3-n) (c x)^{3/2}}\right )}{c^3}-\frac {2 \left (a x^3+b x^n\right )^{3/2}}{3 c (3-n) (c x)^{9/2}}\) |
\(\Big \downarrow \) 1937 |
\(\displaystyle \frac {a \left (\frac {a \sqrt {c x} \int \frac {\sqrt {x}}{\sqrt {b x^n+a x^3}}dx}{c^3 \sqrt {x}}-\frac {2 \sqrt {a x^3+b x^n}}{c (3-n) (c x)^{3/2}}\right )}{c^3}-\frac {2 \left (a x^3+b x^n\right )^{3/2}}{3 c (3-n) (c x)^{9/2}}\) |
\(\Big \downarrow \) 1935 |
\(\displaystyle \frac {a \left (\frac {2 a \sqrt {c x} \int \frac {1}{1-\frac {a x^3}{b x^n+a x^3}}d\frac {x^{3/2}}{\sqrt {b x^n+a x^3}}}{c^3 (3-n) \sqrt {x}}-\frac {2 \sqrt {a x^3+b x^n}}{c (3-n) (c x)^{3/2}}\right )}{c^3}-\frac {2 \left (a x^3+b x^n\right )^{3/2}}{3 c (3-n) (c x)^{9/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {a \left (\frac {2 \sqrt {a} \sqrt {c x} \text {arctanh}\left (\frac {\sqrt {a} x^{3/2}}{\sqrt {a x^3+b x^n}}\right )}{c^3 (3-n) \sqrt {x}}-\frac {2 \sqrt {a x^3+b x^n}}{c (3-n) (c x)^{3/2}}\right )}{c^3}-\frac {2 \left (a x^3+b x^n\right )^{3/2}}{3 c (3-n) (c x)^{9/2}}\) |
Input:
Int[(a*x^3 + b*x^n)^(3/2)/(c*x)^(11/2),x]
Output:
(-2*(a*x^3 + b*x^n)^(3/2))/(3*c*(3 - n)*(c*x)^(9/2)) + (a*((-2*Sqrt[a*x^3 + b*x^n])/(c*(3 - n)*(c*x)^(3/2)) + (2*Sqrt[a]*Sqrt[c*x]*ArcTanh[(Sqrt[a]* x^(3/2))/Sqrt[a*x^3 + b*x^n]])/(c^3*(3 - n)*Sqrt[x])))/c^3
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]
\[\int \frac {\left (a \,x^{3}+b \,x^{n}\right )^{\frac {3}{2}}}{\left (c x \right )^{\frac {11}{2}}}d x\]
Input:
int((a*x^3+b*x^n)^(3/2)/(c*x)^(11/2),x)
Output:
int((a*x^3+b*x^n)^(3/2)/(c*x)^(11/2),x)
Exception generated. \[ \int \frac {\left (a x^3+b x^n\right )^{3/2}}{(c x)^{11/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x^3+b*x^n)^(3/2)/(c*x)^(11/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (has polynomial part)
Timed out. \[ \int \frac {\left (a x^3+b x^n\right )^{3/2}}{(c x)^{11/2}} \, dx=\text {Timed out} \] Input:
integrate((a*x**3+b*x**n)**(3/2)/(c*x)**(11/2),x)
Output:
Timed out
\[ \int \frac {\left (a x^3+b x^n\right )^{3/2}}{(c x)^{11/2}} \, dx=\int { \frac {{\left (a x^{3} + b x^{n}\right )}^{\frac {3}{2}}}{\left (c x\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((a*x^3+b*x^n)^(3/2)/(c*x)^(11/2),x, algorithm="maxima")
Output:
integrate((a*x^3 + b*x^n)^(3/2)/(c*x)^(11/2), x)
\[ \int \frac {\left (a x^3+b x^n\right )^{3/2}}{(c x)^{11/2}} \, dx=\int { \frac {{\left (a x^{3} + b x^{n}\right )}^{\frac {3}{2}}}{\left (c x\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((a*x^3+b*x^n)^(3/2)/(c*x)^(11/2),x, algorithm="giac")
Output:
integrate((a*x^3 + b*x^n)^(3/2)/(c*x)^(11/2), x)
Timed out. \[ \int \frac {\left (a x^3+b x^n\right )^{3/2}}{(c x)^{11/2}} \, dx=\int \frac {{\left (b\,x^n+a\,x^3\right )}^{3/2}}{{\left (c\,x\right )}^{11/2}} \,d x \] Input:
int((b*x^n + a*x^3)^(3/2)/(c*x)^(11/2),x)
Output:
int((b*x^n + a*x^3)^(3/2)/(c*x)^(11/2), x)
\[ \int \frac {\left (a x^3+b x^n\right )^{3/2}}{(c x)^{11/2}} \, dx=\frac {\left (\int \frac {\sqrt {x^{n} b +a \,x^{3}}}{\sqrt {x}\, x^{2}}d x \right ) a +\left (\int \frac {x^{n} \sqrt {x^{n} b +a \,x^{3}}}{\sqrt {x}\, x^{5}}d x \right ) b}{\sqrt {c}\, c^{5}} \] Input:
int((a*x^3+b*x^n)^(3/2)/(c*x)^(11/2),x)
Output:
(int(sqrt(x**n*b + a*x**3)/(sqrt(x)*x**2),x)*a + int((x**n*sqrt(x**n*b + a *x**3))/(sqrt(x)*x**5),x)*b)/(sqrt(c)*c**5)