\(\int \frac {x}{\sqrt {b x^{2/3}+a x}} \, dx\) [167]

Optimal result

Integrand size = 17, antiderivative size = 137 \[ \int \frac {x}{\sqrt {b x^{2/3}+a x}} \, dx=-\frac {128 b^3 \sqrt {b x^{2/3}+a x}}{105 a^4}+\frac {256 b^4 \sqrt {b x^{2/3}+a x}}{105 a^5 \sqrt [3]{x}}+\frac {32 b^2 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{35 a^3}-\frac {16 b x^{2/3} \sqrt {b x^{2/3}+a x}}{21 a^2}+\frac {2 x \sqrt {b x^{2/3}+a x}}{3 a} \] Output:

-128/105*b^3*(b*x^(2/3)+a*x)^(1/2)/a^4+256/105*b^4*(b*x^(2/3)+a*x)^(1/2)/a 
^5/x^(1/3)+32/35*b^2*x^(1/3)*(b*x^(2/3)+a*x)^(1/2)/a^3-16/21*b*x^(2/3)*(b* 
x^(2/3)+a*x)^(1/2)/a^2+2/3*x*(b*x^(2/3)+a*x)^(1/2)/a
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.54 \[ \int \frac {x}{\sqrt {b x^{2/3}+a x}} \, dx=\frac {2 \sqrt {b x^{2/3}+a x} \left (128 b^4-64 a b^3 \sqrt [3]{x}+48 a^2 b^2 x^{2/3}-40 a^3 b x+35 a^4 x^{4/3}\right )}{105 a^5 \sqrt [3]{x}} \] Input:

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

Output:

(2*Sqrt[b*x^(2/3) + a*x]*(128*b^4 - 64*a*b^3*x^(1/3) + 48*a^2*b^2*x^(2/3) 
- 40*a^3*b*x + 35*a^4*x^(4/3)))/(105*a^5*x^(1/3))
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.10, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {1922, 1922, 1922, 1908, 1920}

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.

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

\(\Big \downarrow \) 1922

\(\displaystyle \frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \int \frac {x^{2/3}}{\sqrt {x^{2/3} b+a x}}dx}{9 a}\)

\(\Big \downarrow \) 1922

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

\(\Big \downarrow \) 1922

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

\(\Big \downarrow \) 1908

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

\(\Big \downarrow \) 1920

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 1908

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + 
b*x^n)^(p + 1)/(a*(j*p + 1)*x^(j - 1)), x] - Simp[b*((n*p + n - j + 1)/(a*( 
j*p + 1)))   Int[x^(n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, j, n, 
p}, x] &&  !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(n*p + n - j + 1)/(n 
- j)], 0] && NeQ[j*p + 1, 0]
 

rule 1920

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*(n - j 
)*(p + 1))), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[ 
n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])
 

rule 1922

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, j, m, n, 
p}, x] &&  !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(m + n*p + n - j + 1) 
/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c, 0])
 

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.50

method	result	size
derivativedivides	\(\frac {2 x^{\frac {1}{3}} \left (x^{\frac {1}{3}} a +b \right ) \left (35 a^{4} x^{\frac {4}{3}}-40 a^{3} b x +48 a^{2} b^{2} x^{\frac {2}{3}}-64 a \,b^{3} x^{\frac {1}{3}}+128 b^{4}\right )}{105 \sqrt {b \,x^{\frac {2}{3}}+a x}\, a^{5}}\)	\(68\)
default	\(\frac {2 x^{\frac {1}{3}} \left (x^{\frac {1}{3}} a +b \right ) \left (35 a^{4} x^{\frac {4}{3}}-40 a^{3} b x +48 a^{2} b^{2} x^{\frac {2}{3}}-64 a \,b^{3} x^{\frac {1}{3}}+128 b^{4}\right )}{105 \sqrt {b \,x^{\frac {2}{3}}+a x}\, a^{5}}\)	\(68\)

Input:

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

Output:

2/105*x^(1/3)*(x^(1/3)*a+b)*(35*a^4*x^(4/3)-40*a^3*b*x+48*a^2*b^2*x^(2/3)- 
64*a*b^3*x^(1/3)+128*b^4)/(b*x^(2/3)+a*x)^(1/2)/a^5
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (101) = 202\).

Time = 125.32 (sec) , antiderivative size = 502, normalized size of antiderivative = 3.66 \[ \int \frac {x}{\sqrt {b x^{2/3}+a x}} \, dx=-\frac {2 \, {\left (2 \, {\left (805306368 \, b^{10} + 167772160 \, b^{9} + 786432 \, {\left (64 \, a^{3} - 3\right )} b^{7} - 15728640 \, b^{8} - 4096 \, {\left (11264 \, a^{3} - 53\right )} b^{6} - 101920 \, a^{6} - 1536 \, {\left (5504 \, a^{3} + 1\right )} b^{5} - 48 \, {\left (1966080 \, a^{6} + 114688 \, a^{3} + 3\right )} b^{4} - 1792 \, {\left (36864 \, a^{6} - 103 \, a^{3}\right )} b^{3} - 192 \, {\left (65280 \, a^{6} + 23 \, a^{3}\right )} b^{2} + 15 \, {\left (188416 \, a^{6} + 3 \, a^{3}\right )} b\right )} x - {\left (35 \, {\left (16777216 \, a^{4} b^{6} + 6291456 \, a^{4} b^{5} + 196608 \, a^{4} b^{4} - 262144 \, a^{7} - 114688 \, a^{4} b^{3} - 2304 \, a^{4} b^{2} + 864 \, a^{4} b - 27 \, a^{4}\right )} x^{2} + 48 \, {\left (16777216 \, a^{2} b^{8} + 6291456 \, a^{2} b^{7} + 196608 \, a^{2} b^{6} - 114688 \, a^{2} b^{5} - 2304 \, a^{2} b^{4} + 864 \, a^{2} b^{3} - {\left (262144 \, a^{5} + 27 \, a^{2}\right )} b^{2}\right )} x^{\frac {4}{3}} - 64 \, {\left (16777216 \, a b^{9} + 6291456 \, a b^{8} + 196608 \, a b^{7} - 114688 \, a b^{6} - 2304 \, a b^{5} + 864 \, a b^{4} - {\left (262144 \, a^{4} + 27 \, a\right )} b^{3}\right )} x + 8 \, {\left (268435456 \, b^{10} + 100663296 \, b^{9} + 3145728 \, b^{8} - 1835008 \, b^{7} - 36864 \, b^{6} - 16 \, {\left (262144 \, a^{3} + 27\right )} b^{4} + 13824 \, b^{5} - 5 \, {\left (16777216 \, a^{3} b^{7} + 6291456 \, a^{3} b^{6} + 196608 \, a^{3} b^{5} - 114688 \, a^{3} b^{4} - 2304 \, a^{3} b^{3} + 864 \, a^{3} b^{2} - {\left (262144 \, a^{6} + 27 \, a^{3}\right )} b\right )} x\right )} x^{\frac {2}{3}}\right )} \sqrt {a x + b x^{\frac {2}{3}}}\right )}}{105 \, {\left (16777216 \, a^{5} b^{6} + 6291456 \, a^{5} b^{5} + 196608 \, a^{5} b^{4} - 262144 \, a^{8} - 114688 \, a^{5} b^{3} - 2304 \, a^{5} b^{2} + 864 \, a^{5} b - 27 \, a^{5}\right )} x} \] Input:

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

Output:

-2/105*(2*(805306368*b^10 + 167772160*b^9 + 786432*(64*a^3 - 3)*b^7 - 1572 
8640*b^8 - 4096*(11264*a^3 - 53)*b^6 - 101920*a^6 - 1536*(5504*a^3 + 1)*b^ 
5 - 48*(1966080*a^6 + 114688*a^3 + 3)*b^4 - 1792*(36864*a^6 - 103*a^3)*b^3 
 - 192*(65280*a^6 + 23*a^3)*b^2 + 15*(188416*a^6 + 3*a^3)*b)*x - (35*(1677 
7216*a^4*b^6 + 6291456*a^4*b^5 + 196608*a^4*b^4 - 262144*a^7 - 114688*a^4* 
b^3 - 2304*a^4*b^2 + 864*a^4*b - 27*a^4)*x^2 + 48*(16777216*a^2*b^8 + 6291 
456*a^2*b^7 + 196608*a^2*b^6 - 114688*a^2*b^5 - 2304*a^2*b^4 + 864*a^2*b^3 
 - (262144*a^5 + 27*a^2)*b^2)*x^(4/3) - 64*(16777216*a*b^9 + 6291456*a*b^8 
 + 196608*a*b^7 - 114688*a*b^6 - 2304*a*b^5 + 864*a*b^4 - (262144*a^4 + 27 
*a)*b^3)*x + 8*(268435456*b^10 + 100663296*b^9 + 3145728*b^8 - 1835008*b^7 
 - 36864*b^6 - 16*(262144*a^3 + 27)*b^4 + 13824*b^5 - 5*(16777216*a^3*b^7 
+ 6291456*a^3*b^6 + 196608*a^3*b^5 - 114688*a^3*b^4 - 2304*a^3*b^3 + 864*a 
^3*b^2 - (262144*a^6 + 27*a^3)*b)*x)*x^(2/3))*sqrt(a*x + b*x^(2/3)))/((167 
77216*a^5*b^6 + 6291456*a^5*b^5 + 196608*a^5*b^4 - 262144*a^8 - 114688*a^5 
*b^3 - 2304*a^5*b^2 + 864*a^5*b - 27*a^5)*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.58 \[ \int \frac {x}{\sqrt {b x^{2/3}+a x}} \, dx=-\frac {256 \, b^{\frac {9}{2}}}{105 \, a^{5}} + \frac {2 \, {\left (35 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} - 180 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b + 378 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{2} - 420 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{3} + 315 \, \sqrt {a x^{\frac {1}{3}} + b} b^{4}\right )}}{105 \, a^{5}} \] Input:

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

Output:

-256/105*b^(9/2)/a^5 + 2/105*(35*(a*x^(1/3) + b)^(9/2) - 180*(a*x^(1/3) + 
b)^(7/2)*b + 378*(a*x^(1/3) + b)^(5/2)*b^2 - 420*(a*x^(1/3) + b)^(3/2)*b^3 
 + 315*sqrt(a*x^(1/3) + b)*b^4)/a^5
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.39 \[ \int \frac {x}{\sqrt {b x^{2/3}+a x}} \, dx=\frac {2 \sqrt {x^{\frac {1}{3}} a +b}\, \left (48 x^{\frac {2}{3}} a^{2} b^{2}+35 x^{\frac {4}{3}} a^{4}-64 x^{\frac {1}{3}} a \,b^{3}-40 a^{3} b x +128 b^{4}\right )}{105 a^{5}} \] Input:

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

Output:

(2*sqrt(x**(1/3)*a + b)*(48*x**(2/3)*a**2*b**2 + 35*x**(1/3)*a**4*x - 64*x 
**(1/3)*a*b**3 - 40*a**3*b*x + 128*b**4))/(105*a**5)