Integrand size = 15, antiderivative size = 109 \[ \int \sqrt {b x^{2/3}+a x} \, dx=\frac {2 \left (b x^{2/3}+a x\right )^{3/2}}{3 a}-\frac {32 b^3 \left (b x^{2/3}+a x\right )^{3/2}}{105 a^4 x}+\frac {16 b^2 \left (b x^{2/3}+a x\right )^{3/2}}{35 a^3 x^{2/3}}-\frac {4 b \left (b x^{2/3}+a x\right )^{3/2}}{7 a^2 \sqrt [3]{x}} \] Output:
2/3*(b*x^(2/3)+a*x)^(3/2)/a-32/105*b^3*(b*x^(2/3)+a*x)^(3/2)/a^4/x+16/35*b ^2*(b*x^(2/3)+a*x)^(3/2)/a^3/x^(2/3)-4/7*b*(b*x^(2/3)+a*x)^(3/2)/a^2/x^(1/ 3)
Time = 0.05 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.68 \[ \int \sqrt {b x^{2/3}+a x} \, dx=\frac {2 \sqrt {b x^{2/3}+a x} \left (-16 b^4+8 a b^3 \sqrt [3]{x}-6 a^2 b^2 x^{2/3}+5 a^3 b x+35 a^4 x^{4/3}\right )}{105 a^4 \sqrt [3]{x}} \] Input:
Integrate[Sqrt[b*x^(2/3) + a*x],x]
Output:
(2*Sqrt[b*x^(2/3) + a*x]*(-16*b^4 + 8*a*b^3*x^(1/3) - 6*a^2*b^2*x^(2/3) + 5*a^3*b*x + 35*a^4*x^(4/3)))/(105*a^4*x^(1/3))
Time = 0.44 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1908, 1922, 1922, 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.
\(\displaystyle \int \sqrt {a x+b x^{2/3}} \, dx\) |
\(\Big \downarrow \) 1908 |
\(\displaystyle \frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{3 a}-\frac {2 b \int \frac {\sqrt {x^{2/3} b+a x}}{\sqrt [3]{x}}dx}{3 a}\) |
\(\Big \downarrow \) 1922 |
\(\displaystyle \frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{3 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{3/2}}{7 a \sqrt [3]{x}}-\frac {4 b \int \frac {\sqrt {x^{2/3} b+a x}}{x^{2/3}}dx}{7 a}\right )}{3 a}\) |
\(\Big \downarrow \) 1922 |
\(\displaystyle \frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{3 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{3/2}}{7 a \sqrt [3]{x}}-\frac {4 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{3/2}}{5 a x^{2/3}}-\frac {2 b \int \frac {\sqrt {x^{2/3} b+a x}}{x}dx}{5 a}\right )}{7 a}\right )}{3 a}\) |
\(\Big \downarrow \) 1920 |
\(\displaystyle \frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{3 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{3/2}}{7 a \sqrt [3]{x}}-\frac {4 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{3/2}}{5 a x^{2/3}}-\frac {4 b \left (a x+b x^{2/3}\right )^{3/2}}{5 a^2 x}\right )}{7 a}\right )}{3 a}\) |
Input:
Int[Sqrt[b*x^(2/3) + a*x],x]
Output:
(2*(b*x^(2/3) + a*x)^(3/2))/(3*a) - (2*b*((6*(b*x^(2/3) + a*x)^(3/2))/(7*a *x^(1/3)) - (4*b*((-4*b*(b*x^(2/3) + a*x)^(3/2))/(5*a^2*x) + (6*(b*x^(2/3) + a*x)^(3/2))/(5*a*x^(2/3))))/(7*a)))/(3*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 = 0.38 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.52
method | result | size |
derivativedivides | \(\frac {2 \sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (x^{\frac {1}{3}} a +b \right ) \left (35 a^{3} x -30 a^{2} b \,x^{\frac {2}{3}}+24 a \,b^{2} x^{\frac {1}{3}}-16 b^{3}\right )}{105 x^{\frac {1}{3}} a^{4}}\) | \(57\) |
default | \(-\frac {2 \sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (x^{\frac {1}{3}} a +b \right ) \left (30 a^{2} b \,x^{\frac {2}{3}}-24 a \,b^{2} x^{\frac {1}{3}}-35 a^{3} x +16 b^{3}\right )}{105 x^{\frac {1}{3}} a^{4}}\) | \(57\) |
Input:
int((b*x^(2/3)+a*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/105*(b*x^(2/3)+a*x)^(1/2)*(x^(1/3)*a+b)*(35*a^3*x-30*a^2*b*x^(2/3)+24*a* b^2*x^(1/3)-16*b^3)/x^(1/3)/a^4
Leaf count of result is larger than twice the leaf count of optimal. 501 vs. \(2 (81) = 162\).
Time = 120.60 (sec) , antiderivative size = 501, normalized size of antiderivative = 4.60 \[ \int \sqrt {b x^{2/3}+a x} \, dx=\frac {{\left (805306368 \, b^{10} + 167772160 \, b^{9} + 786432 \, {\left (64 \, a^{3} - 3\right )} b^{7} - 15728640 \, b^{8} - 4096 \, {\left (11264 \, a^{3} - 53\right )} b^{6} + 815360 \, a^{6} - 1536 \, {\left (5504 \, a^{3} + 1\right )} b^{5} - 48 \, {\left (15728640 \, a^{6} + 114688 \, a^{3} + 3\right )} b^{4} - 1792 \, {\left (221184 \, a^{6} - 103 \, a^{3}\right )} b^{3} - 192 \, {\left (307200 \, a^{6} + 23 \, a^{3}\right )} b^{2} - 15 \, {\left (499712 \, a^{6} - 3 \, a^{3}\right )} b\right )} x + 4 \, {\left (35 \, {\left (16777216 \, a^{4} b^{6} + 6291456 \, a^{4} b^{5} + 196608 \, a^{4} b^{4} - 262144 \, a^{7} - 114688 \, a^{4} b^{3} - 2304 \, a^{4} b^{2} + 864 \, a^{4} b - 27 \, a^{4}\right )} x^{2} - 6 \, {\left (16777216 \, a^{2} b^{8} + 6291456 \, a^{2} b^{7} + 196608 \, a^{2} b^{6} - 114688 \, a^{2} b^{5} - 2304 \, a^{2} b^{4} + 864 \, a^{2} b^{3} - {\left (262144 \, a^{5} + 27 \, a^{2}\right )} b^{2}\right )} x^{\frac {4}{3}} + 8 \, {\left (16777216 \, a b^{9} + 6291456 \, a b^{8} + 196608 \, a b^{7} - 114688 \, a b^{6} - 2304 \, a b^{5} + 864 \, a b^{4} - {\left (262144 \, a^{4} + 27 \, a\right )} b^{3}\right )} x - {\left (268435456 \, b^{10} + 100663296 \, b^{9} + 3145728 \, b^{8} - 1835008 \, b^{7} - 36864 \, b^{6} - 16 \, {\left (262144 \, a^{3} + 27\right )} b^{4} + 13824 \, b^{5} - 5 \, {\left (16777216 \, a^{3} b^{7} + 6291456 \, a^{3} b^{6} + 196608 \, a^{3} b^{5} - 114688 \, a^{3} b^{4} - 2304 \, a^{3} b^{3} + 864 \, a^{3} b^{2} - {\left (262144 \, a^{6} + 27 \, a^{3}\right )} b\right )} x\right )} x^{\frac {2}{3}}\right )} \sqrt {a x + b x^{\frac {2}{3}}}}{210 \, {\left (16777216 \, a^{4} b^{6} + 6291456 \, a^{4} b^{5} + 196608 \, a^{4} b^{4} - 262144 \, a^{7} - 114688 \, a^{4} b^{3} - 2304 \, a^{4} b^{2} + 864 \, a^{4} b - 27 \, a^{4}\right )} x} \] Input:
integrate((b*x^(2/3)+a*x)^(1/2),x, algorithm="fricas")
Output:
1/210*((805306368*b^10 + 167772160*b^9 + 786432*(64*a^3 - 3)*b^7 - 1572864 0*b^8 - 4096*(11264*a^3 - 53)*b^6 + 815360*a^6 - 1536*(5504*a^3 + 1)*b^5 - 48*(15728640*a^6 + 114688*a^3 + 3)*b^4 - 1792*(221184*a^6 - 103*a^3)*b^3 - 192*(307200*a^6 + 23*a^3)*b^2 - 15*(499712*a^6 - 3*a^3)*b)*x + 4*(35*(16 777216*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 - 6*(16777216*a^2*b^8 + 629 1456*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) + 8*(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 - (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)))/((16777 216*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)
\[ \int \sqrt {b x^{2/3}+a x} \, dx=\int \sqrt {a x + b x^{\frac {2}{3}}}\, dx \] Input:
integrate((b*x**(2/3)+a*x)**(1/2),x)
Output:
Integral(sqrt(a*x + b*x**(2/3)), x)
\[ \int \sqrt {b x^{2/3}+a x} \, dx=\int { \sqrt {a x + b x^{\frac {2}{3}}} \,d x } \] Input:
integrate((b*x^(2/3)+a*x)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(a*x + b*x^(2/3)), x)
Time = 0.30 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.31 \[ \int \sqrt {b x^{2/3}+a x} \, dx=\frac {32 \, b^{\frac {9}{2}}}{105 \, a^{4}} + \frac {2 \, {\left (\frac {9 \, {\left (5 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} - 21 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b + 35 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{2} - 35 \, \sqrt {a x^{\frac {1}{3}} + b} b^{3}\right )} b}{a^{3}} + \frac {35 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} - 180 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b + 378 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{2} - 420 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{3} + 315 \, \sqrt {a x^{\frac {1}{3}} + b} b^{4}}{a^{3}}\right )}}{105 \, a} \] Input:
integrate((b*x^(2/3)+a*x)^(1/2),x, algorithm="giac")
Output:
32/105*b^(9/2)/a^4 + 2/105*(9*(5*(a*x^(1/3) + b)^(7/2) - 21*(a*x^(1/3) + b )^(5/2)*b + 35*(a*x^(1/3) + b)^(3/2)*b^2 - 35*sqrt(a*x^(1/3) + b)*b^3)*b/a ^3 + (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^3)/a
Time = 8.74 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.37 \[ \int \sqrt {b x^{2/3}+a x} \, dx=\frac {3\,x\,\sqrt {a\,x+b\,x^{2/3}}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},4;\ 5;\ -\frac {a\,x^{1/3}}{b}\right )}{4\,\sqrt {\frac {a\,x^{1/3}}{b}+1}} \] Input:
int((a*x + b*x^(2/3))^(1/2),x)
Output:
(3*x*(a*x + b*x^(2/3))^(1/2)*hypergeom([-1/2, 4], 5, -(a*x^(1/3))/b))/(4*( (a*x^(1/3))/b + 1)^(1/2))
Time = 0.23 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.50 \[ \int \sqrt {b x^{2/3}+a x} \, dx=\frac {2 \sqrt {x^{\frac {1}{3}} a +b}\, \left (-6 x^{\frac {2}{3}} a^{2} b^{2}+35 x^{\frac {4}{3}} a^{4}+8 x^{\frac {1}{3}} a \,b^{3}+5 a^{3} b x -16 b^{4}\right )}{105 a^{4}} \] Input:
int((b*x^(2/3)+a*x)^(1/2),x)
Output:
(2*sqrt(x**(1/3)*a + b)*( - 6*x**(2/3)*a**2*b**2 + 35*x**(1/3)*a**4*x + 8* x**(1/3)*a*b**3 + 5*a**3*b*x - 16*b**4))/(105*a**4)