Integrand size = 20, antiderivative size = 123 \[ \int \frac {x^4 (a+b x)^n}{\sqrt {c x^2}} \, dx=-\frac {a^3 x (a+b x)^{1+n}}{b^4 (1+n) \sqrt {c x^2}}+\frac {3 a^2 x (a+b x)^{2+n}}{b^4 (2+n) \sqrt {c x^2}}-\frac {3 a x (a+b x)^{3+n}}{b^4 (3+n) \sqrt {c x^2}}+\frac {x (a+b x)^{4+n}}{b^4 (4+n) \sqrt {c x^2}} \]
-a^3*x*(b*x+a)^(1+n)/b^4/(1+n)/(c*x^2)^(1/2)+3*a^2*x*(b*x+a)^(2+n)/b^4/(2+ n)/(c*x^2)^(1/2)-3*a*x*(b*x+a)^(3+n)/b^4/(3+n)/(c*x^2)^(1/2)+x*(b*x+a)^(4+ n)/b^4/(4+n)/(c*x^2)^(1/2)
Time = 0.03 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.78 \[ \int \frac {x^4 (a+b x)^n}{\sqrt {c x^2}} \, dx=\frac {x (a+b x)^{1+n} \left (-6 a^3+6 a^2 b (1+n) x-3 a b^2 \left (2+3 n+n^2\right ) x^2+b^3 \left (6+11 n+6 n^2+n^3\right ) x^3\right )}{b^4 (1+n) (2+n) (3+n) (4+n) \sqrt {c x^2}} \]
(x*(a + b*x)^(1 + n)*(-6*a^3 + 6*a^2*b*(1 + n)*x - 3*a*b^2*(2 + 3*n + n^2) *x^2 + b^3*(6 + 11*n + 6*n^2 + n^3)*x^3))/(b^4*(1 + n)*(2 + n)*(3 + n)*(4 + n)*Sqrt[c*x^2])
Time = 0.20 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.76, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {30, 53, 2009}
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.
| \(\displaystyle \int \frac {x^4 (a+b x)^n}{\sqrt {c x^2}} \, dx\) |
| \(\Big \downarrow \) 30 |
| \(\displaystyle \frac {x \int x^3 (a+b x)^ndx}{\sqrt {c x^2}}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {x \int \left (-\frac {a^3 (a+b x)^n}{b^3}+\frac {3 a^2 (a+b x)^{n+1}}{b^3}-\frac {3 a (a+b x)^{n+2}}{b^3}+\frac {(a+b x)^{n+3}}{b^3}\right )dx}{\sqrt {c x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x \left (-\frac {a^3 (a+b x)^{n+1}}{b^4 (n+1)}+\frac {3 a^2 (a+b x)^{n+2}}{b^4 (n+2)}-\frac {3 a (a+b x)^{n+3}}{b^4 (n+3)}+\frac {(a+b x)^{n+4}}{b^4 (n+4)}\right )}{\sqrt {c x^2}}\) |
(x*(-((a^3*(a + b*x)^(1 + n))/(b^4*(1 + n))) + (3*a^2*(a + b*x)^(2 + n))/( b^4*(2 + n)) - (3*a*(a + b*x)^(3 + n))/(b^4*(3 + n)) + (a + b*x)^(4 + n)/( b^4*(4 + n))))/Sqrt[c*x^2]
3.10.46.3.1 Defintions of rubi rules used
Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & & !IntegerQ[p]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Time = 0.14 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.09
| method | result | size |
| gosper | \(-\frac {x \left (b x +a \right )^{1+n} \left (-b^{3} n^{3} x^{3}-6 b^{3} n^{2} x^{3}+3 a \,b^{2} n^{2} x^{2}-11 b^{3} n \,x^{3}+9 a \,b^{2} n \,x^{2}-6 b^{3} x^{3}-6 a^{2} b n x +6 a \,b^{2} x^{2}-6 a^{2} b x +6 a^{3}\right )}{b^{4} \sqrt {c \,x^{2}}\, \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}\) | \(134\) |
| risch | \(-\frac {x \left (-b^{4} n^{3} x^{4}-a \,b^{3} n^{3} x^{3}-6 b^{4} n^{2} x^{4}-3 a \,b^{3} n^{2} x^{3}-11 b^{4} n \,x^{4}+3 a^{2} b^{2} n^{2} x^{2}-2 x^{3} a n \,b^{3}-6 b^{4} x^{4}+3 a^{2} n \,x^{2} b^{2}-6 x \,a^{3} n b +6 a^{4}\right ) \left (b x +a \right )^{n}}{\sqrt {c \,x^{2}}\, \left (3+n \right ) \left (4+n \right ) \left (2+n \right ) \left (1+n \right ) b^{4}}\) | \(154\) |
-x/b^4/(c*x^2)^(1/2)*(b*x+a)^(1+n)/(n^4+10*n^3+35*n^2+50*n+24)*(-b^3*n^3*x ^3-6*b^3*n^2*x^3+3*a*b^2*n^2*x^2-11*b^3*n*x^3+9*a*b^2*n*x^2-6*b^3*x^3-6*a^ 2*b*n*x+6*a*b^2*x^2-6*a^2*b*x+6*a^3)
Time = 0.23 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.28 \[ \int \frac {x^4 (a+b x)^n}{\sqrt {c x^2}} \, dx=\frac {{\left (6 \, a^{3} b n x + {\left (b^{4} n^{3} + 6 \, b^{4} n^{2} + 11 \, b^{4} n + 6 \, b^{4}\right )} x^{4} - 6 \, a^{4} + {\left (a b^{3} n^{3} + 3 \, a b^{3} n^{2} + 2 \, a b^{3} n\right )} x^{3} - 3 \, {\left (a^{2} b^{2} n^{2} + a^{2} b^{2} n\right )} x^{2}\right )} \sqrt {c x^{2}} {\left (b x + a\right )}^{n}}{{\left (b^{4} c n^{4} + 10 \, b^{4} c n^{3} + 35 \, b^{4} c n^{2} + 50 \, b^{4} c n + 24 \, b^{4} c\right )} x} \]
(6*a^3*b*n*x + (b^4*n^3 + 6*b^4*n^2 + 11*b^4*n + 6*b^4)*x^4 - 6*a^4 + (a*b ^3*n^3 + 3*a*b^3*n^2 + 2*a*b^3*n)*x^3 - 3*(a^2*b^2*n^2 + a^2*b^2*n)*x^2)*s qrt(c*x^2)*(b*x + a)^n/((b^4*c*n^4 + 10*b^4*c*n^3 + 35*b^4*c*n^2 + 50*b^4* c*n + 24*b^4*c)*x)
\[ \int \frac {x^4 (a+b x)^n}{\sqrt {c x^2}} \, dx=\begin {cases} \frac {a^{n} x^{5}}{4 \sqrt {c x^{2}}} & \text {for}\: b = 0 \\\int \frac {x^{4}}{\sqrt {c x^{2}} \left (a + b x\right )^{4}}\, dx & \text {for}\: n = -4 \\\int \frac {x^{4}}{\sqrt {c x^{2}} \left (a + b x\right )^{3}}\, dx & \text {for}\: n = -3 \\\int \frac {x^{4}}{\sqrt {c x^{2}} \left (a + b x\right )^{2}}\, dx & \text {for}\: n = -2 \\\int \frac {x^{4}}{\sqrt {c x^{2}} \left (a + b x\right )}\, dx & \text {for}\: n = -1 \\- \frac {6 a^{4} x \left (a + b x\right )^{n}}{b^{4} n^{4} \sqrt {c x^{2}} + 10 b^{4} n^{3} \sqrt {c x^{2}} + 35 b^{4} n^{2} \sqrt {c x^{2}} + 50 b^{4} n \sqrt {c x^{2}} + 24 b^{4} \sqrt {c x^{2}}} + \frac {6 a^{3} b n x^{2} \left (a + b x\right )^{n}}{b^{4} n^{4} \sqrt {c x^{2}} + 10 b^{4} n^{3} \sqrt {c x^{2}} + 35 b^{4} n^{2} \sqrt {c x^{2}} + 50 b^{4} n \sqrt {c x^{2}} + 24 b^{4} \sqrt {c x^{2}}} - \frac {3 a^{2} b^{2} n^{2} x^{3} \left (a + b x\right )^{n}}{b^{4} n^{4} \sqrt {c x^{2}} + 10 b^{4} n^{3} \sqrt {c x^{2}} + 35 b^{4} n^{2} \sqrt {c x^{2}} + 50 b^{4} n \sqrt {c x^{2}} + 24 b^{4} \sqrt {c x^{2}}} - \frac {3 a^{2} b^{2} n x^{3} \left (a + b x\right )^{n}}{b^{4} n^{4} \sqrt {c x^{2}} + 10 b^{4} n^{3} \sqrt {c x^{2}} + 35 b^{4} n^{2} \sqrt {c x^{2}} + 50 b^{4} n \sqrt {c x^{2}} + 24 b^{4} \sqrt {c x^{2}}} + \frac {a b^{3} n^{3} x^{4} \left (a + b x\right )^{n}}{b^{4} n^{4} \sqrt {c x^{2}} + 10 b^{4} n^{3} \sqrt {c x^{2}} + 35 b^{4} n^{2} \sqrt {c x^{2}} + 50 b^{4} n \sqrt {c x^{2}} + 24 b^{4} \sqrt {c x^{2}}} + \frac {3 a b^{3} n^{2} x^{4} \left (a + b x\right )^{n}}{b^{4} n^{4} \sqrt {c x^{2}} + 10 b^{4} n^{3} \sqrt {c x^{2}} + 35 b^{4} n^{2} \sqrt {c x^{2}} + 50 b^{4} n \sqrt {c x^{2}} + 24 b^{4} \sqrt {c x^{2}}} + \frac {2 a b^{3} n x^{4} \left (a + b x\right )^{n}}{b^{4} n^{4} \sqrt {c x^{2}} + 10 b^{4} n^{3} \sqrt {c x^{2}} + 35 b^{4} n^{2} \sqrt {c x^{2}} + 50 b^{4} n \sqrt {c x^{2}} + 24 b^{4} \sqrt {c x^{2}}} + \frac {b^{4} n^{3} x^{5} \left (a + b x\right )^{n}}{b^{4} n^{4} \sqrt {c x^{2}} + 10 b^{4} n^{3} \sqrt {c x^{2}} + 35 b^{4} n^{2} \sqrt {c x^{2}} + 50 b^{4} n \sqrt {c x^{2}} + 24 b^{4} \sqrt {c x^{2}}} + \frac {6 b^{4} n^{2} x^{5} \left (a + b x\right )^{n}}{b^{4} n^{4} \sqrt {c x^{2}} + 10 b^{4} n^{3} \sqrt {c x^{2}} + 35 b^{4} n^{2} \sqrt {c x^{2}} + 50 b^{4} n \sqrt {c x^{2}} + 24 b^{4} \sqrt {c x^{2}}} + \frac {11 b^{4} n x^{5} \left (a + b x\right )^{n}}{b^{4} n^{4} \sqrt {c x^{2}} + 10 b^{4} n^{3} \sqrt {c x^{2}} + 35 b^{4} n^{2} \sqrt {c x^{2}} + 50 b^{4} n \sqrt {c x^{2}} + 24 b^{4} \sqrt {c x^{2}}} + \frac {6 b^{4} x^{5} \left (a + b x\right )^{n}}{b^{4} n^{4} \sqrt {c x^{2}} + 10 b^{4} n^{3} \sqrt {c x^{2}} + 35 b^{4} n^{2} \sqrt {c x^{2}} + 50 b^{4} n \sqrt {c x^{2}} + 24 b^{4} \sqrt {c x^{2}}} & \text {otherwise} \end {cases} \]
Piecewise((a**n*x**5/(4*sqrt(c*x**2)), Eq(b, 0)), (Integral(x**4/(sqrt(c*x **2)*(a + b*x)**4), x), Eq(n, -4)), (Integral(x**4/(sqrt(c*x**2)*(a + b*x) **3), x), Eq(n, -3)), (Integral(x**4/(sqrt(c*x**2)*(a + b*x)**2), x), Eq(n , -2)), (Integral(x**4/(sqrt(c*x**2)*(a + b*x)), x), Eq(n, -1)), (-6*a**4* x*(a + b*x)**n/(b**4*n**4*sqrt(c*x**2) + 10*b**4*n**3*sqrt(c*x**2) + 35*b* *4*n**2*sqrt(c*x**2) + 50*b**4*n*sqrt(c*x**2) + 24*b**4*sqrt(c*x**2)) + 6* a**3*b*n*x**2*(a + b*x)**n/(b**4*n**4*sqrt(c*x**2) + 10*b**4*n**3*sqrt(c*x **2) + 35*b**4*n**2*sqrt(c*x**2) + 50*b**4*n*sqrt(c*x**2) + 24*b**4*sqrt(c *x**2)) - 3*a**2*b**2*n**2*x**3*(a + b*x)**n/(b**4*n**4*sqrt(c*x**2) + 10* b**4*n**3*sqrt(c*x**2) + 35*b**4*n**2*sqrt(c*x**2) + 50*b**4*n*sqrt(c*x**2 ) + 24*b**4*sqrt(c*x**2)) - 3*a**2*b**2*n*x**3*(a + b*x)**n/(b**4*n**4*sqr t(c*x**2) + 10*b**4*n**3*sqrt(c*x**2) + 35*b**4*n**2*sqrt(c*x**2) + 50*b** 4*n*sqrt(c*x**2) + 24*b**4*sqrt(c*x**2)) + a*b**3*n**3*x**4*(a + b*x)**n/( b**4*n**4*sqrt(c*x**2) + 10*b**4*n**3*sqrt(c*x**2) + 35*b**4*n**2*sqrt(c*x **2) + 50*b**4*n*sqrt(c*x**2) + 24*b**4*sqrt(c*x**2)) + 3*a*b**3*n**2*x**4 *(a + b*x)**n/(b**4*n**4*sqrt(c*x**2) + 10*b**4*n**3*sqrt(c*x**2) + 35*b** 4*n**2*sqrt(c*x**2) + 50*b**4*n*sqrt(c*x**2) + 24*b**4*sqrt(c*x**2)) + 2*a *b**3*n*x**4*(a + b*x)**n/(b**4*n**4*sqrt(c*x**2) + 10*b**4*n**3*sqrt(c*x* *2) + 35*b**4*n**2*sqrt(c*x**2) + 50*b**4*n*sqrt(c*x**2) + 24*b**4*sqrt(c* x**2)) + b**4*n**3*x**5*(a + b*x)**n/(b**4*n**4*sqrt(c*x**2) + 10*b**4*...
Time = 0.22 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.85 \[ \int \frac {x^4 (a+b x)^n}{\sqrt {c x^2}} \, dx=\frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4} \sqrt {c}} \]
((n^3 + 6*n^2 + 11*n + 6)*b^4*x^4 + (n^3 + 3*n^2 + 2*n)*a*b^3*x^3 - 3*(n^2 + n)*a^2*b^2*x^2 + 6*a^3*b*n*x - 6*a^4)*(b*x + a)^n/((n^4 + 10*n^3 + 35*n ^2 + 50*n + 24)*b^4*sqrt(c))
Exception generated. \[ \int \frac {x^4 (a+b x)^n}{\sqrt {c x^2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{1,[0,3,1,0,0]%%%} / %%%{1,[0,0,0,1,1]%%%} Error: Bad Argum ent Value
Time = 0.59 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.51 \[ \int \frac {x^4 (a+b x)^n}{\sqrt {c x^2}} \, dx=\frac {{\left (a+b\,x\right )}^n\,\left (\frac {x^5\,\left (n^3+6\,n^2+11\,n+6\right )}{n^4+10\,n^3+35\,n^2+50\,n+24}-\frac {6\,a^4\,x}{b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {6\,a^3\,n\,x^2}{b^3\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {a\,n\,x^4\,\left (n^2+3\,n+2\right )}{b\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}-\frac {3\,a^2\,n\,x^3\,\left (n+1\right )}{b^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}\right )}{\sqrt {c\,x^2}} \]
((a + b*x)^n*((x^5*(11*n + 6*n^2 + n^3 + 6))/(50*n + 35*n^2 + 10*n^3 + n^4 + 24) - (6*a^4*x)/(b^4*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (6*a^3*n*x^ 2)/(b^3*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) + (a*n*x^4*(3*n + n^2 + 2))/( b*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)) - (3*a^2*n*x^3*(n + 1))/(b^2*(50*n + 35*n^2 + 10*n^3 + n^4 + 24))))/(c*x^2)^(1/2)