\(\int \frac {(a+b x)^{9/2}}{x^5} \, dx\) [379]

Optimal result

Integrand size = 13, antiderivative size = 124 \[ \int \frac {(a+b x)^{9/2}}{x^5} \, dx=2 b^4 \sqrt {a+b x}-\frac {a^4 \sqrt {a+b x}}{4 x^4}-\frac {11 a^3 b \sqrt {a+b x}}{8 x^3}-\frac {105 a^2 b^2 \sqrt {a+b x}}{32 x^2}-\frac {325 a b^3 \sqrt {a+b x}}{64 x}-\frac {315}{64} \sqrt {a} b^4 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \] Output:

2*b^4*(b*x+a)^(1/2)-1/4*a^4*(b*x+a)^(1/2)/x^4-11/8*a^3*b*(b*x+a)^(1/2)/x^3 
-105/32*a^2*b^2*(b*x+a)^(1/2)/x^2-325/64*a*b^3*(b*x+a)^(1/2)/x-315/64*a^(1 
/2)*b^4*arctanh((b*x+a)^(1/2)/a^(1/2))
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.69 \[ \int \frac {(a+b x)^{9/2}}{x^5} \, dx=\frac {1}{64} \left (-\frac {\sqrt {a+b x} \left (16 a^4+88 a^3 b x+210 a^2 b^2 x^2+325 a b^3 x^3-128 b^4 x^4\right )}{x^4}-315 \sqrt {a} b^4 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right ) \] Input:

Integrate[(a + b*x)^(9/2)/x^5,x]
 

Output:

(-((Sqrt[a + b*x]*(16*a^4 + 88*a^3*b*x + 210*a^2*b^2*x^2 + 325*a*b^3*x^3 - 
 128*b^4*x^4))/x^4) - 315*Sqrt[a]*b^4*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/64
 

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {51, 51, 51, 51, 60, 73, 221}

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.

\(\displaystyle \int \frac {(a+b x)^{9/2}}{x^5} \, dx\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {9}{8} b \int \frac {(a+b x)^{7/2}}{x^4}dx-\frac {(a+b x)^{9/2}}{4 x^4}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {9}{8} b \left (\frac {7}{6} b \int \frac {(a+b x)^{5/2}}{x^3}dx-\frac {(a+b x)^{7/2}}{3 x^3}\right )-\frac {(a+b x)^{9/2}}{4 x^4}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {9}{8} b \left (\frac {7}{6} b \left (\frac {5}{4} b \int \frac {(a+b x)^{3/2}}{x^2}dx-\frac {(a+b x)^{5/2}}{2 x^2}\right )-\frac {(a+b x)^{7/2}}{3 x^3}\right )-\frac {(a+b x)^{9/2}}{4 x^4}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {9}{8} b \left (\frac {7}{6} b \left (\frac {5}{4} b \left (\frac {3}{2} b \int \frac {\sqrt {a+b x}}{x}dx-\frac {(a+b x)^{3/2}}{x}\right )-\frac {(a+b x)^{5/2}}{2 x^2}\right )-\frac {(a+b x)^{7/2}}{3 x^3}\right )-\frac {(a+b x)^{9/2}}{4 x^4}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {9}{8} b \left (\frac {7}{6} b \left (\frac {5}{4} b \left (\frac {3}{2} b \left (a \int \frac {1}{x \sqrt {a+b x}}dx+2 \sqrt {a+b x}\right )-\frac {(a+b x)^{3/2}}{x}\right )-\frac {(a+b x)^{5/2}}{2 x^2}\right )-\frac {(a+b x)^{7/2}}{3 x^3}\right )-\frac {(a+b x)^{9/2}}{4 x^4}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {9}{8} b \left (\frac {7}{6} b \left (\frac {5}{4} b \left (\frac {3}{2} b \left (\frac {2 a \int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}}{b}+2 \sqrt {a+b x}\right )-\frac {(a+b x)^{3/2}}{x}\right )-\frac {(a+b x)^{5/2}}{2 x^2}\right )-\frac {(a+b x)^{7/2}}{3 x^3}\right )-\frac {(a+b x)^{9/2}}{4 x^4}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {9}{8} b \left (\frac {7}{6} b \left (\frac {5}{4} b \left (\frac {3}{2} b \left (2 \sqrt {a+b x}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )-\frac {(a+b x)^{3/2}}{x}\right )-\frac {(a+b x)^{5/2}}{2 x^2}\right )-\frac {(a+b x)^{7/2}}{3 x^3}\right )-\frac {(a+b x)^{9/2}}{4 x^4}\)

Input:

Int[(a + b*x)^(9/2)/x^5,x]
 

Output:

-1/4*(a + b*x)^(9/2)/x^4 + (9*b*(-1/3*(a + b*x)^(7/2)/x^3 + (7*b*(-1/2*(a 
+ b*x)^(5/2)/x^2 + (5*b*(-((a + b*x)^(3/2)/x) + (3*b*(2*Sqrt[a + b*x] - 2* 
Sqrt[a]*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]))/2))/4))/6))/8
 

Defintions of rubi rules used

rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.62

method	result	size
risch	\(-\frac {a \sqrt {b x +a}\, \left (325 b^{3} x^{3}+210 a \,b^{2} x^{2}+88 a^{2} b x +16 a^{3}\right )}{64 x^{4}}+\frac {b^{4} \left (256 \sqrt {b x +a}-630 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )\right )}{128}\)	\(77\)
derivativedivides	\(2 b^{4} \left (\sqrt {b x +a}-a \left (\frac {\frac {325 \left (b x +a \right )^{\frac {7}{2}}}{128}-\frac {765 a \left (b x +a \right )^{\frac {5}{2}}}{128}+\frac {643 a^{2} \left (b x +a \right )^{\frac {3}{2}}}{128}-\frac {187 a^{3} \sqrt {b x +a}}{128}}{b^{4} x^{4}}+\frac {315 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 \sqrt {a}}\right )\right )\)	\(86\)
default	\(2 b^{4} \left (\sqrt {b x +a}-a \left (\frac {\frac {325 \left (b x +a \right )^{\frac {7}{2}}}{128}-\frac {765 a \left (b x +a \right )^{\frac {5}{2}}}{128}+\frac {643 a^{2} \left (b x +a \right )^{\frac {3}{2}}}{128}-\frac {187 a^{3} \sqrt {b x +a}}{128}}{b^{4} x^{4}}+\frac {315 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 \sqrt {a}}\right )\right )\)	\(86\)
pseudoelliptic	\(\frac {-315 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) a \,b^{4} x^{4}+128 b^{4} x^{4} \sqrt {a}\, \sqrt {b x +a}-325 a^{\frac {3}{2}} b^{3} x^{3} \sqrt {b x +a}-210 a^{\frac {5}{2}} b^{2} x^{2} \sqrt {b x +a}-88 a^{\frac {7}{2}} b x \sqrt {b x +a}-16 a^{\frac {9}{2}} \sqrt {b x +a}}{64 x^{4} \sqrt {a}}\)	\(111\)

Input:

int((b*x+a)^(9/2)/x^5,x,method=_RETURNVERBOSE)
 

Output:

-1/64*a*(b*x+a)^(1/2)*(325*b^3*x^3+210*a*b^2*x^2+88*a^2*b*x+16*a^3)/x^4+1/ 
128*b^4*(256*(b*x+a)^(1/2)-630*a^(1/2)*arctanh((b*x+a)^(1/2)/a^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.40 \[ \int \frac {(a+b x)^{9/2}}{x^5} \, dx=\left [\frac {315 \, \sqrt {a} b^{4} x^{4} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (128 \, b^{4} x^{4} - 325 \, a b^{3} x^{3} - 210 \, a^{2} b^{2} x^{2} - 88 \, a^{3} b x - 16 \, a^{4}\right )} \sqrt {b x + a}}{128 \, x^{4}}, \frac {315 \, \sqrt {-a} b^{4} x^{4} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x + a}}\right ) + {\left (128 \, b^{4} x^{4} - 325 \, a b^{3} x^{3} - 210 \, a^{2} b^{2} x^{2} - 88 \, a^{3} b x - 16 \, a^{4}\right )} \sqrt {b x + a}}{64 \, x^{4}}\right ] \] Input:

integrate((b*x+a)^(9/2)/x^5,x, algorithm="fricas")
 

Output:

[1/128*(315*sqrt(a)*b^4*x^4*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 
 2*(128*b^4*x^4 - 325*a*b^3*x^3 - 210*a^2*b^2*x^2 - 88*a^3*b*x - 16*a^4)*s 
qrt(b*x + a))/x^4, 1/64*(315*sqrt(-a)*b^4*x^4*arctan(sqrt(-a)/sqrt(b*x + a 
)) + (128*b^4*x^4 - 325*a*b^3*x^3 - 210*a^2*b^2*x^2 - 88*a^3*b*x - 16*a^4) 
*sqrt(b*x + a))/x^4]
 

Sympy [A] (verification not implemented)

Time = 8.37 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.47 \[ \int \frac {(a+b x)^{9/2}}{x^5} \, dx=- \frac {315 \sqrt {a} b^{4} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{64} - \frac {a^{5}}{4 \sqrt {b} x^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {13 a^{4} \sqrt {b}}{8 x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {149 a^{3} b^{\frac {3}{2}}}{32 x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {535 a^{2} b^{\frac {5}{2}}}{64 x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {197 a b^{\frac {7}{2}}}{64 \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {2 b^{\frac {9}{2}} \sqrt {x}}{\sqrt {\frac {a}{b x} + 1}} \] Input:

integrate((b*x+a)**(9/2)/x**5,x)
 

Output:

-315*sqrt(a)*b**4*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/64 - a**5/(4*sqrt(b)*x* 
*(9/2)*sqrt(a/(b*x) + 1)) - 13*a**4*sqrt(b)/(8*x**(7/2)*sqrt(a/(b*x) + 1)) 
 - 149*a**3*b**(3/2)/(32*x**(5/2)*sqrt(a/(b*x) + 1)) - 535*a**2*b**(5/2)/( 
64*x**(3/2)*sqrt(a/(b*x) + 1)) - 197*a*b**(7/2)/(64*sqrt(x)*sqrt(a/(b*x) + 
 1)) + 2*b**(9/2)*sqrt(x)/sqrt(a/(b*x) + 1)
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.25 \[ \int \frac {(a+b x)^{9/2}}{x^5} \, dx=\frac {315}{128} \, \sqrt {a} b^{4} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right ) + 2 \, \sqrt {b x + a} b^{4} - \frac {325 \, {\left (b x + a\right )}^{\frac {7}{2}} a b^{4} - 765 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} b^{4} + 643 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b^{4} - 187 \, \sqrt {b x + a} a^{4} b^{4}}{64 \, {\left ({\left (b x + a\right )}^{4} - 4 \, {\left (b x + a\right )}^{3} a + 6 \, {\left (b x + a\right )}^{2} a^{2} - 4 \, {\left (b x + a\right )} a^{3} + a^{4}\right )}} \] Input:

integrate((b*x+a)^(9/2)/x^5,x, algorithm="maxima")
 

Output:

315/128*sqrt(a)*b^4*log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a) 
)) + 2*sqrt(b*x + a)*b^4 - 1/64*(325*(b*x + a)^(7/2)*a*b^4 - 765*(b*x + a) 
^(5/2)*a^2*b^4 + 643*(b*x + a)^(3/2)*a^3*b^4 - 187*sqrt(b*x + a)*a^4*b^4)/ 
((b*x + a)^4 - 4*(b*x + a)^3*a + 6*(b*x + a)^2*a^2 - 4*(b*x + a)*a^3 + a^4 
)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^{9/2}}{x^5} \, dx=\frac {\frac {315 \, a b^{5} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + 128 \, \sqrt {b x + a} b^{5} - \frac {325 \, {\left (b x + a\right )}^{\frac {7}{2}} a b^{5} - 765 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} b^{5} + 643 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b^{5} - 187 \, \sqrt {b x + a} a^{4} b^{5}}{b^{4} x^{4}}}{64 \, b} \] Input:

integrate((b*x+a)^(9/2)/x^5,x, algorithm="giac")
 

Output:

1/64*(315*a*b^5*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) + 128*sqrt(b*x + a 
)*b^5 - (325*(b*x + a)^(7/2)*a*b^5 - 765*(b*x + a)^(5/2)*a^2*b^5 + 643*(b* 
x + a)^(3/2)*a^3*b^5 - 187*sqrt(b*x + a)*a^4*b^5)/(b^4*x^4))/b
 

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.76 \[ \int \frac {(a+b x)^{9/2}}{x^5} \, dx=2\,b^4\,\sqrt {a+b\,x}+\frac {187\,a^4\,\sqrt {a+b\,x}}{64\,x^4}-\frac {643\,a^3\,{\left (a+b\,x\right )}^{3/2}}{64\,x^4}+\frac {765\,a^2\,{\left (a+b\,x\right )}^{5/2}}{64\,x^4}-\frac {325\,a\,{\left (a+b\,x\right )}^{7/2}}{64\,x^4}+\frac {\sqrt {a}\,b^4\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,315{}\mathrm {i}}{64} \] Input:

int((a + b*x)^(9/2)/x^5,x)
 

Output:

2*b^4*(a + b*x)^(1/2) + (187*a^4*(a + b*x)^(1/2))/(64*x^4) - (643*a^3*(a + 
 b*x)^(3/2))/(64*x^4) + (765*a^2*(a + b*x)^(5/2))/(64*x^4) + (a^(1/2)*b^4* 
atan(((a + b*x)^(1/2)*1i)/a^(1/2))*315i)/64 - (325*a*(a + b*x)^(7/2))/(64* 
x^4)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^{9/2}}{x^5} \, dx=\frac {-32 \sqrt {b x +a}\, a^{4}-176 \sqrt {b x +a}\, a^{3} b x -420 \sqrt {b x +a}\, a^{2} b^{2} x^{2}-650 \sqrt {b x +a}\, a \,b^{3} x^{3}+256 \sqrt {b x +a}\, b^{4} x^{4}+315 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) b^{4} x^{4}-315 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) b^{4} x^{4}}{128 x^{4}} \] Input:

int((b*x+a)^(9/2)/x^5,x)
 

Output:

( - 32*sqrt(a + b*x)*a**4 - 176*sqrt(a + b*x)*a**3*b*x - 420*sqrt(a + b*x) 
*a**2*b**2*x**2 - 650*sqrt(a + b*x)*a*b**3*x**3 + 256*sqrt(a + b*x)*b**4*x 
**4 + 315*sqrt(a)*log(sqrt(a + b*x) - sqrt(a))*b**4*x**4 - 315*sqrt(a)*log 
(sqrt(a + b*x) + sqrt(a))*b**4*x**4)/(128*x**4)