3.2.73 \(\int \frac {\sqrt {b x^{2/3}+a x}}{x^3} \, dx\) [173]

3.2.73.1 Optimal result

Integrand size = 19, antiderivative size = 178 \[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^3} \, dx=-\frac {3 \sqrt {b x^{2/3}+a x}}{5 x^2}-\frac {3 a \sqrt {b x^{2/3}+a x}}{40 b x^{5/3}}+\frac {7 a^2 \sqrt {b x^{2/3}+a x}}{80 b^2 x^{4/3}}-\frac {7 a^3 \sqrt {b x^{2/3}+a x}}{64 b^3 x}+\frac {21 a^4 \sqrt {b x^{2/3}+a x}}{128 b^4 x^{2/3}}-\frac {21 a^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{128 b^{9/2}} \]

output

-21/128*a^5*arctanh(x^(1/3)*b^(1/2)/(b*x^(2/3)+a*x)^(1/2))/b^(9/2)-3/5*(b* 
x^(2/3)+a*x)^(1/2)/x^2-3/40*a*(b*x^(2/3)+a*x)^(1/2)/b/x^(5/3)+7/80*a^2*(b* 
x^(2/3)+a*x)^(1/2)/b^2/x^(4/3)-7/64*a^3*(b*x^(2/3)+a*x)^(1/2)/b^3/x+21/128 
*a^4*(b*x^(2/3)+a*x)^(1/2)/b^4/x^(2/3)
 

3.2.73.2 Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^3} \, dx=\frac {\sqrt {b x^{2/3}+a x} \left (-384 b^4-48 a b^3 \sqrt [3]{x}+56 a^2 b^2 x^{2/3}-70 a^3 b x+105 a^4 x^{4/3}\right )}{640 b^4 x^2}-\frac {21 a^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{128 b^{9/2}} \]

input

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

output

(Sqrt[b*x^(2/3) + a*x]*(-384*b^4 - 48*a*b^3*x^(1/3) + 56*a^2*b^2*x^(2/3) - 
 70*a^3*b*x + 105*a^4*x^(4/3)))/(640*b^4*x^2) - (21*a^5*ArcTanh[(Sqrt[b]*x 
^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(128*b^(9/2))
 

3.2.73.3 Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {1926, 1931, 1931, 1931, 1931, 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.

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

\(\Big \downarrow \) 1926

\(\displaystyle \frac {1}{10} a \int \frac {1}{x^2 \sqrt {x^{2/3} b+a x}}dx-\frac {3 \sqrt {a x+b x^{2/3}}}{5 x^2}\)

\(\Big \downarrow \) 1931

\(\displaystyle \frac {1}{10} a \left (-\frac {7 a \int \frac {1}{x^{5/3} \sqrt {x^{2/3} b+a x}}dx}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{5 x^2}\)

\(\Big \downarrow \) 1931

\(\displaystyle \frac {1}{10} a \left (-\frac {7 a \left (-\frac {5 a \int \frac {1}{x^{4/3} \sqrt {x^{2/3} b+a x}}dx}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{5 x^2}\)

\(\Big \downarrow \) 1931

\(\displaystyle \frac {1}{10} a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \int \frac {1}{x \sqrt {x^{2/3} b+a x}}dx}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{5 x^2}\)

\(\Big \downarrow \) 1931

\(\displaystyle \frac {1}{10} a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (-\frac {a \int \frac {1}{x^{2/3} \sqrt {x^{2/3} b+a x}}dx}{2 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{5 x^2}\)

\(\Big \downarrow \) 1935

\(\displaystyle \frac {1}{10} a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (\frac {3 a \int \frac {1}{1-\frac {b x^{2/3}}{x^{2/3} b+a x}}d\frac {\sqrt [3]{x}}{\sqrt {x^{2/3} b+a x}}}{b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{5 x^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{10} a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (\frac {3 a \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{b^{3/2}}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{5 x^2}\)

input

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

output

(-3*Sqrt[b*x^(2/3) + a*x])/(5*x^2) + (a*((-3*Sqrt[b*x^(2/3) + a*x])/(4*b*x 
^(5/3)) - (7*a*(-(Sqrt[b*x^(2/3) + a*x]/(b*x^(4/3))) - (5*a*((-3*Sqrt[b*x^ 
(2/3) + a*x])/(2*b*x) - (3*a*((-3*Sqrt[b*x^(2/3) + a*x])/(b*x^(2/3)) + (3* 
a*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/b^(3/2)))/(4*b)))/(6*b 
)))/(8*b)))/10
 

3.2.73.3.1 Defintions of rubi rules used

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])
 

rule 1926

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*(m + j*p + 1))), x] - Simp[b*p 
*((n - j)/(c^n*(m + j*p + 1)))   Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p - 1), 
 x], x] /; FreeQ[{a, b, c}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (Integer 
sQ[j, n] || GtQ[c, 0]) && GtQ[p, 0] && LtQ[m + j*p + 1, 0]
 

rule 1931

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[c^(j - 1)*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(m + j*p 
+ 1))), x] - Simp[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1)))   I 
nt[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] 
 &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[ 
m + j*p + 1, 0]
 

rule 1935

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]
 

3.2.73.4 Maple [A] (verified)

Time = 2.03 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.70

method	result	size
derivativedivides	\(-\frac {\sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (-105 b^{\frac {9}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {9}{2}}+490 b^{\frac {11}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {7}{2}}-896 b^{\frac {13}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {5}{2}}+105 \,\operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) b^{4} a^{5} x^{\frac {5}{3}}+790 b^{\frac {15}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}}+105 b^{\frac {17}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\right )}{640 x^{2} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {17}{2}}}\)	\(125\)
default	\(-\frac {\sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (-105 b^{\frac {9}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {9}{2}}+490 b^{\frac {11}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {7}{2}}-896 b^{\frac {13}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {5}{2}}+105 \,\operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) b^{4} a^{5} x^{\frac {5}{3}}+790 b^{\frac {15}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}}+105 b^{\frac {17}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\right )}{640 x^{2} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {17}{2}}}\)	\(125\)

input

int((b*x^(2/3)+a*x)^(1/2)/x^3,x,method=_RETURNVERBOSE)
 

output

-1/640*(b*x^(2/3)+a*x)^(1/2)*(-105*b^(9/2)*(b+a*x^(1/3))^(9/2)+490*b^(11/2 
)*(b+a*x^(1/3))^(7/2)-896*b^(13/2)*(b+a*x^(1/3))^(5/2)+105*arctanh((b+a*x^ 
(1/3))^(1/2)/b^(1/2))*b^4*a^5*x^(5/3)+790*b^(15/2)*(b+a*x^(1/3))^(3/2)+105 
*b^(17/2)*(b+a*x^(1/3))^(1/2))/x^2/(b+a*x^(1/3))^(1/2)/b^(17/2)
 

3.2.73.5 Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^3} \, dx=\text {Timed out} \]

input

integrate((b*x^(2/3)+a*x)^(1/2)/x^3,x, algorithm="fricas")
 

output

Timed out
 

3.2.73.6 Sympy [F]

\[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^3} \, dx=\int \frac {\sqrt {a x + b x^{\frac {2}{3}}}}{x^{3}}\, dx \]

input

integrate((b*x**(2/3)+a*x)**(1/2)/x**3,x)
 

output

Integral(sqrt(a*x + b*x**(2/3))/x**3, x)
 

3.2.73.7 Maxima [F]

\[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^3} \, dx=\int { \frac {\sqrt {a x + b x^{\frac {2}{3}}}}{x^{3}} \,d x } \]

input

integrate((b*x^(2/3)+a*x)^(1/2)/x^3,x, algorithm="maxima")
 

output

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

3.2.73.8 Giac [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^3} \, dx=\frac {\frac {105 \, a^{6} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{4}} + \frac {105 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{6} - 490 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{6} b + 896 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{6} b^{2} - 790 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{6} b^{3} - 105 \, \sqrt {a x^{\frac {1}{3}} + b} a^{6} b^{4}}{a^{5} b^{4} x^{\frac {5}{3}}}}{640 \, a} \]

input

integrate((b*x^(2/3)+a*x)^(1/2)/x^3,x, algorithm="giac")
 

output

1/640*(105*a^6*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^4) + (105* 
(a*x^(1/3) + b)^(9/2)*a^6 - 490*(a*x^(1/3) + b)^(7/2)*a^6*b + 896*(a*x^(1/ 
3) + b)^(5/2)*a^6*b^2 - 790*(a*x^(1/3) + b)^(3/2)*a^6*b^3 - 105*sqrt(a*x^( 
1/3) + b)*a^6*b^4)/(a^5*b^4*x^(5/3)))/a
 

3.2.73.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^3} \, dx=\int \frac {\sqrt {a\,x+b\,x^{2/3}}}{x^3} \,d x \]

input

int((a*x + b*x^(2/3))^(1/2)/x^3,x)
 

output

int((a*x + b*x^(2/3))^(1/2)/x^3, x)