\(\int \frac {(a+b x)^{9/2}}{x^4} \, dx\) [378]

Optimal result

Integrand size = 13, antiderivative size = 121 \[ \int \frac {(a+b x)^{9/2}}{x^4} \, dx=8 a b^3 \sqrt {a+b x}-\frac {a^4 \sqrt {a+b x}}{3 x^3}-\frac {25 a^3 b \sqrt {a+b x}}{12 x^2}-\frac {55 a^2 b^2 \sqrt {a+b x}}{8 x}+\frac {2}{3} b^3 (a+b x)^{3/2}-\frac {105}{8} a^{3/2} b^3 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \] Output:

8*a*b^3*(b*x+a)^(1/2)-1/3*a^4*(b*x+a)^(1/2)/x^3-25/12*a^3*b*(b*x+a)^(1/2)/ 
x^2-55/8*a^2*b^2*(b*x+a)^(1/2)/x+2/3*b^3*(b*x+a)^(3/2)-105/8*a^(3/2)*b^3*a 
rctanh((b*x+a)^(1/2)/a^(1/2))
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.70 \[ \int \frac {(a+b x)^{9/2}}{x^4} \, dx=\frac {1}{24} \left (\frac {\sqrt {a+b x} \left (-8 a^4-50 a^3 b x-165 a^2 b^2 x^2+208 a b^3 x^3+16 b^4 x^4\right )}{x^3}-315 a^{3/2} b^3 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right ) \] Input:

Integrate[(a + b*x)^(9/2)/x^4,x]
 

Output:

((Sqrt[a + b*x]*(-8*a^4 - 50*a^3*b*x - 165*a^2*b^2*x^2 + 208*a*b^3*x^3 + 1 
6*b^4*x^4))/x^3 - 315*a^(3/2)*b^3*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/24
 

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.95, 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, 60, 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.

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

\(\Big \downarrow \) 51

\(\displaystyle \frac {3}{2} b \int \frac {(a+b x)^{7/2}}{x^3}dx-\frac {(a+b x)^{9/2}}{3 x^3}\)

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[(a + b*x)^(9/2)/x^4,x]
 

Output:

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

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 = 0.19 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.64

method	result	size
risch	\(-\frac {a^{2} \sqrt {b x +a}\, \left (165 b^{2} x^{2}+50 a b x +8 a^{2}\right )}{24 x^{3}}+\frac {b^{3} \left (\frac {32 \left (b x +a \right )^{\frac {3}{2}}}{3}+128 a \sqrt {b x +a}-210 a^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )\right )}{16}\)	\(78\)
pseudoelliptic	\(-\frac {105 \left (\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) a^{2} b^{3} x^{3}-\frac {16 \sqrt {b x +a}\, \left (\sqrt {a}\, b^{4} x^{4}+13 a^{\frac {3}{2}} b^{3} x^{3}-\frac {165 a^{\frac {5}{2}} b^{2} x^{2}}{16}-\frac {25 a^{\frac {7}{2}} b x}{8}-\frac {a^{\frac {9}{2}}}{2}\right )}{315}\right )}{8 \sqrt {a}\, x^{3}}\)	\(86\)
derivativedivides	\(2 b^{3} \left (\frac {\left (b x +a \right )^{\frac {3}{2}}}{3}+4 a \sqrt {b x +a}-a^{2} \left (-\frac {-\frac {55 \left (b x +a \right )^{\frac {5}{2}}}{16}+\frac {35 a \left (b x +a \right )^{\frac {3}{2}}}{6}-\frac {41 a^{2} \sqrt {b x +a}}{16}}{b^{3} x^{3}}+\frac {105 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{16 \sqrt {a}}\right )\right )\)	\(89\)
default	\(2 b^{3} \left (\frac {\left (b x +a \right )^{\frac {3}{2}}}{3}+4 a \sqrt {b x +a}-a^{2} \left (-\frac {-\frac {55 \left (b x +a \right )^{\frac {5}{2}}}{16}+\frac {35 a \left (b x +a \right )^{\frac {3}{2}}}{6}-\frac {41 a^{2} \sqrt {b x +a}}{16}}{b^{3} x^{3}}+\frac {105 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{16 \sqrt {a}}\right )\right )\)	\(89\)

Input:

int((b*x+a)^(9/2)/x^4,x,method=_RETURNVERBOSE)
 

Output:

-1/24*a^2*(b*x+a)^(1/2)*(165*b^2*x^2+50*a*b*x+8*a^2)/x^3+1/16*b^3*(32/3*(b 
*x+a)^(3/2)+128*a*(b*x+a)^(1/2)-210*a^(3/2)*arctanh((b*x+a)^(1/2)/a^(1/2)) 
)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.45 \[ \int \frac {(a+b x)^{9/2}}{x^4} \, dx=\left [\frac {315 \, a^{\frac {3}{2}} b^{3} x^{3} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (16 \, b^{4} x^{4} + 208 \, a b^{3} x^{3} - 165 \, a^{2} b^{2} x^{2} - 50 \, a^{3} b x - 8 \, a^{4}\right )} \sqrt {b x + a}}{48 \, x^{3}}, \frac {315 \, \sqrt {-a} a b^{3} x^{3} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x + a}}\right ) + {\left (16 \, b^{4} x^{4} + 208 \, a b^{3} x^{3} - 165 \, a^{2} b^{2} x^{2} - 50 \, a^{3} b x - 8 \, a^{4}\right )} \sqrt {b x + a}}{24 \, x^{3}}\right ] \] Input:

integrate((b*x+a)^(9/2)/x^4,x, algorithm="fricas")
 

Output:

[1/48*(315*a^(3/2)*b^3*x^3*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 
2*(16*b^4*x^4 + 208*a*b^3*x^3 - 165*a^2*b^2*x^2 - 50*a^3*b*x - 8*a^4)*sqrt 
(b*x + a))/x^3, 1/24*(315*sqrt(-a)*a*b^3*x^3*arctan(sqrt(-a)/sqrt(b*x + a) 
) + (16*b^4*x^4 + 208*a*b^3*x^3 - 165*a^2*b^2*x^2 - 50*a^3*b*x - 8*a^4)*sq 
rt(b*x + a))/x^3]
 

Sympy [A] (verification not implemented)

Time = 8.13 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.52 \[ \int \frac {(a+b x)^{9/2}}{x^4} \, dx=- \frac {105 a^{\frac {3}{2}} b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{8} - \frac {a^{5}}{3 \sqrt {b} x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {29 a^{4} \sqrt {b}}{12 x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {215 a^{3} b^{\frac {3}{2}}}{24 x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {43 a^{2} b^{\frac {5}{2}}}{24 \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {28 a b^{\frac {7}{2}} \sqrt {x}}{3 \sqrt {\frac {a}{b x} + 1}} + \frac {2 b^{\frac {9}{2}} x^{\frac {3}{2}}}{3 \sqrt {\frac {a}{b x} + 1}} \] Input:

integrate((b*x+a)**(9/2)/x**4,x)
 

Output:

-105*a**(3/2)*b**3*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/8 - a**5/(3*sqrt(b)*x* 
*(7/2)*sqrt(a/(b*x) + 1)) - 29*a**4*sqrt(b)/(12*x**(5/2)*sqrt(a/(b*x) + 1) 
) - 215*a**3*b**(3/2)/(24*x**(3/2)*sqrt(a/(b*x) + 1)) + 43*a**2*b**(5/2)/( 
24*sqrt(x)*sqrt(a/(b*x) + 1)) + 28*a*b**(7/2)*sqrt(x)/(3*sqrt(a/(b*x) + 1) 
) + 2*b**(9/2)*x**(3/2)/(3*sqrt(a/(b*x) + 1))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.20 \[ \int \frac {(a+b x)^{9/2}}{x^4} \, dx=\frac {105}{16} \, a^{\frac {3}{2}} b^{3} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right ) + \frac {2}{3} \, {\left (b x + a\right )}^{\frac {3}{2}} b^{3} + 8 \, \sqrt {b x + a} a b^{3} - \frac {165 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} b^{3} - 280 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b^{3} + 123 \, \sqrt {b x + a} a^{4} b^{3}}{24 \, {\left ({\left (b x + a\right )}^{3} - 3 \, {\left (b x + a\right )}^{2} a + 3 \, {\left (b x + a\right )} a^{2} - a^{3}\right )}} \] Input:

integrate((b*x+a)^(9/2)/x^4,x, algorithm="maxima")
 

Output:

105/16*a^(3/2)*b^3*log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a)) 
) + 2/3*(b*x + a)^(3/2)*b^3 + 8*sqrt(b*x + a)*a*b^3 - 1/24*(165*(b*x + a)^ 
(5/2)*a^2*b^3 - 280*(b*x + a)^(3/2)*a^3*b^3 + 123*sqrt(b*x + a)*a^4*b^3)/( 
(b*x + a)^3 - 3*(b*x + a)^2*a + 3*(b*x + a)*a^2 - a^3)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.78 \[ \int \frac {(a+b x)^{9/2}}{x^4} \, dx=\frac {1}{24} \, {\left (\frac {315 \, a^{2} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + 16 \, {\left (b x + a\right )}^{\frac {3}{2}} + 192 \, \sqrt {b x + a} a - \frac {165 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 280 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 123 \, \sqrt {b x + a} a^{4}}{b^{3} x^{3}}\right )} b^{3} \] Input:

integrate((b*x+a)^(9/2)/x^4,x, algorithm="giac")
 

Output:

1/24*(315*a^2*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) + 16*(b*x + a)^(3/2) 
 + 192*sqrt(b*x + a)*a - (165*(b*x + a)^(5/2)*a^2 - 280*(b*x + a)^(3/2)*a^ 
3 + 123*sqrt(b*x + a)*a^4)/(b^3*x^3))*b^3
 

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b x)^{9/2}}{x^4} \, dx=\frac {2\,b^3\,{\left (a+b\,x\right )}^{3/2}}{3}+\frac {\frac {41\,a^4\,b^3\,\sqrt {a+b\,x}}{8}-\frac {35\,a^3\,b^3\,{\left (a+b\,x\right )}^{3/2}}{3}+\frac {55\,a^2\,b^3\,{\left (a+b\,x\right )}^{5/2}}{8}}{3\,a\,{\left (a+b\,x\right )}^2-3\,a^2\,\left (a+b\,x\right )-{\left (a+b\,x\right )}^3+a^3}+8\,a\,b^3\,\sqrt {a+b\,x}+\frac {a^{3/2}\,b^3\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,105{}\mathrm {i}}{8} \] Input:

int((a + b*x)^(9/2)/x^4,x)
 

Output:

(2*b^3*(a + b*x)^(3/2))/3 + ((41*a^4*b^3*(a + b*x)^(1/2))/8 - (35*a^3*b^3* 
(a + b*x)^(3/2))/3 + (55*a^2*b^3*(a + b*x)^(5/2))/8)/(3*a*(a + b*x)^2 - 3* 
a^2*(a + b*x) - (a + b*x)^3 + a^3) + (a^(3/2)*b^3*atan(((a + b*x)^(1/2)*1i 
)/a^(1/2))*105i)/8 + 8*a*b^3*(a + b*x)^(1/2)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x)^{9/2}}{x^4} \, dx=\frac {-16 \sqrt {b x +a}\, a^{4}-100 \sqrt {b x +a}\, a^{3} b x -330 \sqrt {b x +a}\, a^{2} b^{2} x^{2}+416 \sqrt {b x +a}\, a \,b^{3} x^{3}+32 \sqrt {b x +a}\, b^{4} x^{4}+315 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a \,b^{3} x^{3}-315 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a \,b^{3} x^{3}}{48 x^{3}} \] Input:

int((b*x+a)^(9/2)/x^4,x)
 

Output:

( - 16*sqrt(a + b*x)*a**4 - 100*sqrt(a + b*x)*a**3*b*x - 330*sqrt(a + b*x) 
*a**2*b**2*x**2 + 416*sqrt(a + b*x)*a*b**3*x**3 + 32*sqrt(a + b*x)*b**4*x* 
*4 + 315*sqrt(a)*log(sqrt(a + b*x) - sqrt(a))*a*b**3*x**3 - 315*sqrt(a)*lo 
g(sqrt(a + b*x) + sqrt(a))*a*b**3*x**3)/(48*x**3)