Integrand size = 19, antiderivative size = 225 \[ \int \frac {x^2}{\sqrt {b x^{2/3}+a x}} \, dx=\frac {2048 b^6 \sqrt {b x^{2/3}+a x}}{2145 a^7}-\frac {4096 b^7 \sqrt {b x^{2/3}+a x}}{2145 a^8 \sqrt [3]{x}}-\frac {512 b^5 \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{715 a^6}+\frac {256 b^4 x^{2/3} \sqrt {b x^{2/3}+a x}}{429 a^5}-\frac {224 b^3 x \sqrt {b x^{2/3}+a x}}{429 a^4}+\frac {336 b^2 x^{4/3} \sqrt {b x^{2/3}+a x}}{715 a^3}-\frac {28 b x^{5/3} \sqrt {b x^{2/3}+a x}}{65 a^2}+\frac {2 x^2 \sqrt {b x^{2/3}+a x}}{5 a} \] Output:
2048/2145*b^6*(b*x^(2/3)+a*x)^(1/2)/a^7-4096/2145*b^7*(b*x^(2/3)+a*x)^(1/2 )/a^8/x^(1/3)-512/715*b^5*x^(1/3)*(b*x^(2/3)+a*x)^(1/2)/a^6+256/429*b^4*x^ (2/3)*(b*x^(2/3)+a*x)^(1/2)/a^5-224/429*b^3*x*(b*x^(2/3)+a*x)^(1/2)/a^4+33 6/715*b^2*x^(4/3)*(b*x^(2/3)+a*x)^(1/2)/a^3-28/65*b*x^(5/3)*(b*x^(2/3)+a*x )^(1/2)/a^2+2/5*x^2*(b*x^(2/3)+a*x)^(1/2)/a
Time = 0.09 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.49 \[ \int \frac {x^2}{\sqrt {b x^{2/3}+a x}} \, dx=\frac {2 \sqrt {b x^{2/3}+a x} \left (-2048 b^7+1024 a b^6 \sqrt [3]{x}-768 a^2 b^5 x^{2/3}+640 a^3 b^4 x-560 a^4 b^3 x^{4/3}+504 a^5 b^2 x^{5/3}-462 a^6 b x^2+429 a^7 x^{7/3}\right )}{2145 a^8 \sqrt [3]{x}} \] Input:
Integrate[x^2/Sqrt[b*x^(2/3) + a*x],x]
Output:
(2*Sqrt[b*x^(2/3) + a*x]*(-2048*b^7 + 1024*a*b^6*x^(1/3) - 768*a^2*b^5*x^( 2/3) + 640*a^3*b^4*x - 560*a^4*b^3*x^(4/3) + 504*a^5*b^2*x^(5/3) - 462*a^6 *b*x^2 + 429*a^7*x^(7/3)))/(2145*a^8*x^(1/3))
Time = 0.83 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {1922, 1922, 1922, 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.
| \(\displaystyle \int \frac {x^2}{\sqrt {a x+b x^{2/3}}} \, dx\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \int \frac {x^{5/3}}{\sqrt {x^{2/3} b+a x}}dx}{15 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \int \frac {x^{4/3}}{\sqrt {x^{2/3} b+a x}}dx}{13 a}\right )}{15 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \int \frac {x}{\sqrt {x^{2/3} b+a x}}dx}{11 a}\right )}{13 a}\right )}{15 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \int \frac {x^{2/3}}{\sqrt {x^{2/3} b+a x}}dx}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \left (\frac {6 x^{2/3} \sqrt {a x+b x^{2/3}}}{7 a}-\frac {6 b \int \frac {\sqrt [3]{x}}{\sqrt {x^{2/3} b+a x}}dx}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \left (\frac {6 x^{2/3} \sqrt {a x+b x^{2/3}}}{7 a}-\frac {6 b \left (\frac {6 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{5 a}-\frac {4 b \int \frac {1}{\sqrt {x^{2/3} b+a x}}dx}{5 a}\right )}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\) |
| \(\Big \downarrow \) 1908 |
| \(\displaystyle \frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \left (\frac {6 x^{2/3} \sqrt {a x+b x^{2/3}}}{7 a}-\frac {6 b \left (\frac {6 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{5 a}-\frac {4 b \left (\frac {2 \sqrt {a x+b x^{2/3}}}{a}-\frac {2 b \int \frac {1}{\sqrt [3]{x} \sqrt {x^{2/3} b+a x}}dx}{3 a}\right )}{5 a}\right )}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\) |
| \(\Big \downarrow \) 1920 |
| \(\displaystyle \frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \left (\frac {6 x^{2/3} \sqrt {a x+b x^{2/3}}}{7 a}-\frac {6 b \left (\frac {6 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{5 a}-\frac {4 b \left (\frac {2 \sqrt {a x+b x^{2/3}}}{a}-\frac {4 b \sqrt {a x+b x^{2/3}}}{a^2 \sqrt [3]{x}}\right )}{5 a}\right )}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\) |
Input:
Int[x^2/Sqrt[b*x^(2/3) + a*x],x]
Output:
(2*x^2*Sqrt[b*x^(2/3) + a*x])/(5*a) - (14*b*((6*x^(5/3)*Sqrt[b*x^(2/3) + a *x])/(13*a) - (12*b*((6*x^(4/3)*Sqrt[b*x^(2/3) + a*x])/(11*a) - (10*b*((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)))/(11*a)))/(13*a)))/(15*a)
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]
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])
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])
Time = 0.35 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.45
| method | result | size |
| derivativedivides | \(\frac {2 x^{\frac {1}{3}} \left (x^{\frac {1}{3}} a +b \right ) \left (429 a^{7} x^{\frac {7}{3}}-462 a^{6} b \,x^{2}+504 a^{5} b^{2} x^{\frac {5}{3}}-560 a^{4} b^{3} x^{\frac {4}{3}}+640 a^{3} b^{4} x -768 a^{2} b^{5} x^{\frac {2}{3}}+1024 a \,b^{6} x^{\frac {1}{3}}-2048 b^{7}\right )}{2145 \sqrt {b \,x^{\frac {2}{3}}+a x}\, a^{8}}\) | \(101\) |
| default | \(\frac {2 x^{\frac {1}{3}} \left (x^{\frac {1}{3}} a +b \right ) \left (429 a^{7} x^{\frac {7}{3}}-462 a^{6} b \,x^{2}+504 a^{5} b^{2} x^{\frac {5}{3}}-560 a^{4} b^{3} x^{\frac {4}{3}}+640 a^{3} b^{4} x -768 a^{2} b^{5} x^{\frac {2}{3}}+1024 a \,b^{6} x^{\frac {1}{3}}-2048 b^{7}\right )}{2145 \sqrt {b \,x^{\frac {2}{3}}+a x}\, a^{8}}\) | \(101\) |
Input:
int(x^2/(b*x^(2/3)+a*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/2145*x^(1/3)*(x^(1/3)*a+b)*(429*a^7*x^(7/3)-462*a^6*b*x^2+504*a^5*b^2*x^ (5/3)-560*a^4*b^3*x^(4/3)+640*a^3*b^4*x-768*a^2*b^5*x^(2/3)+1024*a*b^6*x^( 1/3)-2048*b^7)/(b*x^(2/3)+a*x)^(1/2)/a^8
Leaf count of result is larger than twice the leaf count of optimal. 768 vs. \(2 (167) = 334\).
Time = 136.13 (sec) , antiderivative size = 768, normalized size of antiderivative = 3.41 \[ \int \frac {x^2}{\sqrt {b x^{2/3}+a x}} \, dx =\text {Too large to display} \] Input:
integrate(x^2/(b*x^(2/3)+a*x)^(1/2),x, algorithm="fricas")
Output:
1/2145*((51539607552*b^13 + 10737418240*b^12 + 50331648*(64*a^3 - 3)*b^10 - 1006632960*b^11 - 262144*(11264*a^3 - 53)*b^9 + 4996992*a^9 - 98304*(550 4*a^3 + 1)*b^8 + 3072*(3194880*a^6 - 114688*a^3 - 3)*b^7 + 114688*(18816*a ^6 + 103*a^3)*b^6 - 12288*(48816*a^6 + 23*a^3)*b^5 + 192*(21626880*a^9 + 4 95872*a^6 + 15*a^3)*b^4 + 256*(10690560*a^9 - 24073*a^6)*b^3 + 3744*(13312 0*a^9 + 49*a^6)*b^2 - 297*(450560*a^9 + 7*a^6)*b)*x + 2*(429*(16777216*a^7 *b^6 + 6291456*a^7*b^5 + 196608*a^7*b^4 - 262144*a^10 - 114688*a^7*b^3 - 2 304*a^7*b^2 + 864*a^7*b - 27*a^7)*x^3 - 560*(16777216*a^4*b^9 + 6291456*a^ 4*b^8 + 196608*a^4*b^7 - 114688*a^4*b^6 - 2304*a^4*b^5 + 864*a^4*b^4 - (26 2144*a^7 + 27*a^4)*b^3)*x^2 + 1024*(16777216*a*b^12 + 6291456*a*b^11 + 196 608*a*b^10 - 114688*a*b^9 - 2304*a*b^8 + 864*a*b^7 - (262144*a^4 + 27*a)*b ^6)*x - 2*(17179869184*b^13 + 6442450944*b^12 + 201326592*b^11 - 117440512 *b^10 - 2359296*b^9 - 1024*(262144*a^3 + 27)*b^7 + 884736*b^8 + 231*(16777 216*a^6*b^7 + 6291456*a^6*b^6 + 196608*a^6*b^5 - 114688*a^6*b^4 - 2304*a^6 *b^3 + 864*a^6*b^2 - (262144*a^9 + 27*a^6)*b)*x^2 - 320*(16777216*a^3*b^10 + 6291456*a^3*b^9 + 196608*a^3*b^8 - 114688*a^3*b^7 - 2304*a^3*b^6 + 864* a^3*b^5 - (262144*a^6 + 27*a^3)*b^4)*x)*x^(2/3) + 24*(21*(16777216*a^5*b^8 + 6291456*a^5*b^7 + 196608*a^5*b^6 - 114688*a^5*b^5 - 2304*a^5*b^4 + 864* a^5*b^3 - (262144*a^8 + 27*a^5)*b^2)*x^2 - 32*(16777216*a^2*b^11 + 6291456 *a^2*b^10 + 196608*a^2*b^9 - 114688*a^2*b^8 - 2304*a^2*b^7 + 864*a^2*b^...
\[ \int \frac {x^2}{\sqrt {b x^{2/3}+a x}} \, dx=\int \frac {x^{2}}{\sqrt {a x + b x^{\frac {2}{3}}}}\, dx \] Input:
integrate(x**2/(b*x**(2/3)+a*x)**(1/2),x)
Output:
Integral(x**2/sqrt(a*x + b*x**(2/3)), x)
\[ \int \frac {x^2}{\sqrt {b x^{2/3}+a x}} \, dx=\int { \frac {x^{2}}{\sqrt {a x + b x^{\frac {2}{3}}}} \,d x } \] Input:
integrate(x^2/(b*x^(2/3)+a*x)^(1/2),x, algorithm="maxima")
Output:
integrate(x^2/sqrt(a*x + b*x^(2/3)), x)
Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.54 \[ \int \frac {x^2}{\sqrt {b x^{2/3}+a x}} \, dx=\frac {4096 \, b^{\frac {15}{2}}}{2145 \, a^{8}} + \frac {2 \, {\left (429 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} - 3465 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b + 12285 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{2} - 25025 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{3} + 32175 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{4} - 27027 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{5} + 15015 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{6} - 6435 \, \sqrt {a x^{\frac {1}{3}} + b} b^{7}\right )}}{2145 \, a^{8}} \] Input:
integrate(x^2/(b*x^(2/3)+a*x)^(1/2),x, algorithm="giac")
Output:
4096/2145*b^(15/2)/a^8 + 2/2145*(429*(a*x^(1/3) + b)^(15/2) - 3465*(a*x^(1 /3) + b)^(13/2)*b + 12285*(a*x^(1/3) + b)^(11/2)*b^2 - 25025*(a*x^(1/3) + b)^(9/2)*b^3 + 32175*(a*x^(1/3) + b)^(7/2)*b^4 - 27027*(a*x^(1/3) + b)^(5/ 2)*b^5 + 15015*(a*x^(1/3) + b)^(3/2)*b^6 - 6435*sqrt(a*x^(1/3) + b)*b^7)/a ^8
Timed out. \[ \int \frac {x^2}{\sqrt {b x^{2/3}+a x}} \, dx=\int \frac {x^2}{\sqrt {a\,x+b\,x^{2/3}}} \,d x \] Input:
int(x^2/(a*x + b*x^(2/3))^(1/2),x)
Output:
int(x^2/(a*x + b*x^(2/3))^(1/2), x)
Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.39 \[ \int \frac {x^2}{\sqrt {b x^{2/3}+a x}} \, dx=\frac {2 \sqrt {x^{\frac {1}{3}} a +b}\, \left (504 x^{\frac {5}{3}} a^{5} b^{2}-768 x^{\frac {2}{3}} a^{2} b^{5}+429 x^{\frac {7}{3}} a^{7}-560 x^{\frac {4}{3}} a^{4} b^{3}+1024 x^{\frac {1}{3}} a \,b^{6}-462 a^{6} b \,x^{2}+640 a^{3} b^{4} x -2048 b^{7}\right )}{2145 a^{8}} \] Input:
int(x^2/(b*x^(2/3)+a*x)^(1/2),x)
Output:
(2*sqrt(x**(1/3)*a + b)*(504*x**(2/3)*a**5*b**2*x - 768*x**(2/3)*a**2*b**5 + 429*x**(1/3)*a**7*x**2 - 560*x**(1/3)*a**4*b**3*x + 1024*x**(1/3)*a*b** 6 - 462*a**6*b*x**2 + 640*a**3*b**4*x - 2048*b**7))/(2145*a**8)