\(\int x^{3/2} (a+b x)^{5/2} \, dx\) [446]

Optimal result

Integrand size = 15, antiderivative size = 144 \[ \int x^{3/2} (a+b x)^{5/2} \, dx=-\frac {3 a^4 \sqrt {x} \sqrt {a+b x}}{128 b^2}+\frac {a^3 x^{3/2} \sqrt {a+b x}}{64 b}+\frac {31}{80} a^2 x^{5/2} \sqrt {a+b x}+\frac {21}{40} a b x^{7/2} \sqrt {a+b x}+\frac {1}{5} b^2 x^{9/2} \sqrt {a+b x}+\frac {3 a^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{5/2}} \] Output:

-3/128*a^4*x^(1/2)*(b*x+a)^(1/2)/b^2+1/64*a^3*x^(3/2)*(b*x+a)^(1/2)/b+31/8 
0*a^2*x^(5/2)*(b*x+a)^(1/2)+21/40*a*b*x^(7/2)*(b*x+a)^(1/2)+1/5*b^2*x^(9/2 
)*(b*x+a)^(1/2)+3/128*a^5*arctanh(b^(1/2)*x^(1/2)/(b*x+a)^(1/2))/b^(5/2)
 

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.74 \[ \int x^{3/2} (a+b x)^{5/2} \, dx=\frac {\sqrt {b} \sqrt {x} \sqrt {a+b x} \left (-15 a^4+10 a^3 b x+248 a^2 b^2 x^2+336 a b^3 x^3+128 b^4 x^4\right )+30 a^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )}{640 b^{5/2}} \] Input:

Integrate[x^(3/2)*(a + b*x)^(5/2),x]
 

Output:

(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(-15*a^4 + 10*a^3*b*x + 248*a^2*b^2*x^2 + 3 
36*a*b^3*x^3 + 128*b^4*x^4) + 30*a^5*ArcTanh[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + 
 Sqrt[a + b*x])])/(640*b^(5/2))
 

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {60, 60, 60, 60, 60, 65, 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.

\(\displaystyle \int x^{3/2} (a+b x)^{5/2} \, dx\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {1}{2} a \int x^{3/2} (a+b x)^{3/2}dx+\frac {1}{5} x^{5/2} (a+b x)^{5/2}\)

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 65

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

\(\Big \downarrow \) 219

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

Input:

Int[x^(3/2)*(a + b*x)^(5/2),x]
 

Output:

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

Defintions of rubi rules used

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 65

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2   Sub 
st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d 
}, x] &&  !GtQ[c, 0]
 

rule 219

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])
 

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.76

method	result	size
risch	\(-\frac {\left (-128 b^{4} x^{4}-336 a \,x^{3} b^{3}-248 a^{2} b^{2} x^{2}-10 a^{3} b x +15 a^{4}\right ) \sqrt {x}\, \sqrt {b x +a}}{640 b^{2}}+\frac {3 a^{5} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{256 b^{\frac {5}{2}} \sqrt {x}\, \sqrt {b x +a}}\)	\(109\)
default	\(\frac {x^{\frac {3}{2}} \left (b x +a \right )^{\frac {7}{2}}}{5 b}-\frac {3 a \left (\frac {\sqrt {x}\, \left (b x +a \right )^{\frac {7}{2}}}{4 b}-\frac {a \left (\frac {\sqrt {x}\, \left (b x +a \right )^{\frac {5}{2}}}{3}+\frac {5 a \left (\frac {\sqrt {x}\, \left (b x +a \right )^{\frac {3}{2}}}{2}+\frac {3 a \left (\sqrt {x}\, \sqrt {b x +a}+\frac {a \sqrt {x \left (b x +a \right )}\, \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 \sqrt {b x +a}\, \sqrt {x}\, \sqrt {b}}\right )}{4}\right )}{6}\right )}{8 b}\right )}{10 b}\)	\(138\)

Input:

int(x^(3/2)*(b*x+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/640*(-128*b^4*x^4-336*a*b^3*x^3-248*a^2*b^2*x^2-10*a^3*b*x+15*a^4)*x^(1 
/2)*(b*x+a)^(1/2)/b^2+3/256*a^5/b^(5/2)*ln((1/2*a+b*x)/b^(1/2)+(b*x^2+a*x) 
^(1/2))*(x*(b*x+a))^(1/2)/x^(1/2)/(b*x+a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.26 \[ \int x^{3/2} (a+b x)^{5/2} \, dx=\left [\frac {15 \, a^{5} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (128 \, b^{5} x^{4} + 336 \, a b^{4} x^{3} + 248 \, a^{2} b^{3} x^{2} + 10 \, a^{3} b^{2} x - 15 \, a^{4} b\right )} \sqrt {b x + a} \sqrt {x}}{1280 \, b^{3}}, -\frac {15 \, a^{5} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {b x + a}}\right ) - {\left (128 \, b^{5} x^{4} + 336 \, a b^{4} x^{3} + 248 \, a^{2} b^{3} x^{2} + 10 \, a^{3} b^{2} x - 15 \, a^{4} b\right )} \sqrt {b x + a} \sqrt {x}}{640 \, b^{3}}\right ] \] Input:

integrate(x^(3/2)*(b*x+a)^(5/2),x, algorithm="fricas")
 

Output:

[1/1280*(15*a^5*sqrt(b)*log(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) + 
 2*(128*b^5*x^4 + 336*a*b^4*x^3 + 248*a^2*b^3*x^2 + 10*a^3*b^2*x - 15*a^4* 
b)*sqrt(b*x + a)*sqrt(x))/b^3, -1/640*(15*a^5*sqrt(-b)*arctan(sqrt(-b)*sqr 
t(x)/sqrt(b*x + a)) - (128*b^5*x^4 + 336*a*b^4*x^3 + 248*a^2*b^3*x^2 + 10* 
a^3*b^2*x - 15*a^4*b)*sqrt(b*x + a)*sqrt(x))/b^3]
 

Sympy [A] (verification not implemented)

Time = 28.03 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.25 \[ \int x^{3/2} (a+b x)^{5/2} \, dx=- \frac {3 a^{\frac {9}{2}} \sqrt {x}}{128 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {a^{\frac {7}{2}} x^{\frac {3}{2}}}{128 b \sqrt {1 + \frac {b x}{a}}} + \frac {129 a^{\frac {5}{2}} x^{\frac {5}{2}}}{320 \sqrt {1 + \frac {b x}{a}}} + \frac {73 a^{\frac {3}{2}} b x^{\frac {7}{2}}}{80 \sqrt {1 + \frac {b x}{a}}} + \frac {29 \sqrt {a} b^{2} x^{\frac {9}{2}}}{40 \sqrt {1 + \frac {b x}{a}}} + \frac {3 a^{5} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{128 b^{\frac {5}{2}}} + \frac {b^{3} x^{\frac {11}{2}}}{5 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \] Input:

integrate(x**(3/2)*(b*x+a)**(5/2),x)
 

Output:

-3*a**(9/2)*sqrt(x)/(128*b**2*sqrt(1 + b*x/a)) - a**(7/2)*x**(3/2)/(128*b* 
sqrt(1 + b*x/a)) + 129*a**(5/2)*x**(5/2)/(320*sqrt(1 + b*x/a)) + 73*a**(3/ 
2)*b*x**(7/2)/(80*sqrt(1 + b*x/a)) + 29*sqrt(a)*b**2*x**(9/2)/(40*sqrt(1 + 
 b*x/a)) + 3*a**5*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(128*b**(5/2)) + b**3*x** 
(11/2)/(5*sqrt(a)*sqrt(1 + b*x/a))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (104) = 208\).

Time = 0.12 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.47 \[ \int x^{3/2} (a+b x)^{5/2} \, dx=-\frac {3 \, a^{5} \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x + a}}{\sqrt {x}}}{\sqrt {b} + \frac {\sqrt {b x + a}}{\sqrt {x}}}\right )}{256 \, b^{\frac {5}{2}}} - \frac {\frac {15 \, \sqrt {b x + a} a^{5} b^{4}}{\sqrt {x}} - \frac {70 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} b^{3}}{x^{\frac {3}{2}}} + \frac {128 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{5} b^{2}}{x^{\frac {5}{2}}} + \frac {70 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{5} b}{x^{\frac {7}{2}}} - \frac {15 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{5}}{x^{\frac {9}{2}}}}{640 \, {\left (b^{7} - \frac {5 \, {\left (b x + a\right )} b^{6}}{x} + \frac {10 \, {\left (b x + a\right )}^{2} b^{5}}{x^{2}} - \frac {10 \, {\left (b x + a\right )}^{3} b^{4}}{x^{3}} + \frac {5 \, {\left (b x + a\right )}^{4} b^{3}}{x^{4}} - \frac {{\left (b x + a\right )}^{5} b^{2}}{x^{5}}\right )}} \] Input:

integrate(x^(3/2)*(b*x+a)^(5/2),x, algorithm="maxima")
 

Output:

-3/256*a^5*log(-(sqrt(b) - sqrt(b*x + a)/sqrt(x))/(sqrt(b) + sqrt(b*x + a) 
/sqrt(x)))/b^(5/2) - 1/640*(15*sqrt(b*x + a)*a^5*b^4/sqrt(x) - 70*(b*x + a 
)^(3/2)*a^5*b^3/x^(3/2) + 128*(b*x + a)^(5/2)*a^5*b^2/x^(5/2) + 70*(b*x + 
a)^(7/2)*a^5*b/x^(7/2) - 15*(b*x + a)^(9/2)*a^5/x^(9/2))/(b^7 - 5*(b*x + a 
)*b^6/x + 10*(b*x + a)^2*b^5/x^2 - 10*(b*x + a)^3*b^4/x^3 + 5*(b*x + a)^4* 
b^3/x^4 - (b*x + a)^5*b^2/x^5)
 

Giac [F(-1)]

Timed out. \[ \int x^{3/2} (a+b x)^{5/2} \, dx=\text {Timed out} \] Input:

integrate(x^(3/2)*(b*x+a)^(5/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int x^{3/2} (a+b x)^{5/2} \, dx=\int x^{3/2}\,{\left (a+b\,x\right )}^{5/2} \,d x \] Input:

int(x^(3/2)*(a + b*x)^(5/2),x)
 

Output:

int(x^(3/2)*(a + b*x)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.79 \[ \int x^{3/2} (a+b x)^{5/2} \, dx=\frac {-15 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b +10 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{2} x +248 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{3} x^{2}+336 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{4} x^{3}+128 \sqrt {x}\, \sqrt {b x +a}\, b^{5} x^{4}+15 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{5}}{640 b^{3}} \] Input:

int(x^(3/2)*(b*x+a)^(5/2),x)
 

Output:

( - 15*sqrt(x)*sqrt(a + b*x)*a**4*b + 10*sqrt(x)*sqrt(a + b*x)*a**3*b**2*x 
 + 248*sqrt(x)*sqrt(a + b*x)*a**2*b**3*x**2 + 336*sqrt(x)*sqrt(a + b*x)*a* 
b**4*x**3 + 128*sqrt(x)*sqrt(a + b*x)*b**5*x**4 + 15*sqrt(b)*log((sqrt(a + 
 b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**5)/(640*b**3)