Integrand size = 15, antiderivative size = 169 \[ \int \left (b x^{2/3}+a x\right )^{3/2} \, dx=\frac {2 \left (b x^{2/3}+a x\right )^{5/2}}{5 a}-\frac {512 b^5 \left (b x^{2/3}+a x\right )^{5/2}}{15015 a^6 x^{5/3}}+\frac {256 b^4 \left (b x^{2/3}+a x\right )^{5/2}}{3003 a^5 x^{4/3}}-\frac {64 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{429 a^4 x}+\frac {32 b^2 \left (b x^{2/3}+a x\right )^{5/2}}{143 a^3 x^{2/3}}-\frac {4 b \left (b x^{2/3}+a x\right )^{5/2}}{13 a^2 \sqrt [3]{x}} \]
2/5*(b*x^(2/3)+a*x)^(5/2)/a-512/15015*b^5*(b*x^(2/3)+a*x)^(5/2)/a^6/x^(5/3 )+256/3003*b^4*(b*x^(2/3)+a*x)^(5/2)/a^5/x^(4/3)-64/429*b^3*(b*x^(2/3)+a*x )^(5/2)/a^4/x+32/143*b^2*(b*x^(2/3)+a*x)^(5/2)/a^3/x^(2/3)-4/13*b*(b*x^(2/ 3)+a*x)^(5/2)/a^2/x^(1/3)
Time = 6.21 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.56 \[ \int \left (b x^{2/3}+a x\right )^{3/2} \, dx=\frac {2 \left (b+a \sqrt [3]{x}\right ) \left (b x^{2/3}+a x\right )^{3/2} \left (-256 b^5+640 a b^4 \sqrt [3]{x}-1120 a^2 b^3 x^{2/3}+1680 a^3 b^2 x-2310 a^4 b x^{4/3}+3003 a^5 x^{5/3}\right )}{15015 a^6 x} \]
(2*(b + a*x^(1/3))*(b*x^(2/3) + a*x)^(3/2)*(-256*b^5 + 640*a*b^4*x^(1/3) - 1120*a^2*b^3*x^(2/3) + 1680*a^3*b^2*x - 2310*a^4*b*x^(4/3) + 3003*a^5*x^( 5/3)))/(15015*a^6*x)
Time = 0.37 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1908, 1922, 1922, 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.
| \(\displaystyle \int \left (a x+b x^{2/3}\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1908 |
| \(\displaystyle \frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{5 a}-\frac {2 b \int \frac {\left (x^{2/3} b+a x\right )^{3/2}}{\sqrt [3]{x}}dx}{3 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{5 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{13 a \sqrt [3]{x}}-\frac {8 b \int \frac {\left (x^{2/3} b+a x\right )^{3/2}}{x^{2/3}}dx}{13 a}\right )}{3 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{5 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{13 a \sqrt [3]{x}}-\frac {8 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{11 a x^{2/3}}-\frac {6 b \int \frac {\left (x^{2/3} b+a x\right )^{3/2}}{x}dx}{11 a}\right )}{13 a}\right )}{3 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{5 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{13 a \sqrt [3]{x}}-\frac {8 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{11 a x^{2/3}}-\frac {6 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{3 a x}-\frac {4 b \int \frac {\left (x^{2/3} b+a x\right )^{3/2}}{x^{4/3}}dx}{9 a}\right )}{11 a}\right )}{13 a}\right )}{3 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{5 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{13 a \sqrt [3]{x}}-\frac {8 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{11 a x^{2/3}}-\frac {6 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{3 a x}-\frac {4 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{7 a x^{4/3}}-\frac {2 b \int \frac {\left (x^{2/3} b+a x\right )^{3/2}}{x^{5/3}}dx}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{3 a}\) |
| \(\Big \downarrow \) 1920 |
| \(\displaystyle \frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{5 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{13 a \sqrt [3]{x}}-\frac {8 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{11 a x^{2/3}}-\frac {6 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{5/2}}{3 a x}-\frac {4 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{5/2}}{7 a x^{4/3}}-\frac {12 b \left (a x+b x^{2/3}\right )^{5/2}}{35 a^2 x^{5/3}}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{3 a}\) |
(2*(b*x^(2/3) + a*x)^(5/2))/(5*a) - (2*b*((6*(b*x^(2/3) + a*x)^(5/2))/(13* a*x^(1/3)) - (8*b*((6*(b*x^(2/3) + a*x)^(5/2))/(11*a*x^(2/3)) - (6*b*((2*( b*x^(2/3) + a*x)^(5/2))/(3*a*x) - (4*b*((-12*b*(b*x^(2/3) + a*x)^(5/2))/(3 5*a^2*x^(5/3)) + (6*(b*x^(2/3) + a*x)^(5/2))/(7*a*x^(4/3))))/(9*a)))/(11*a )))/(13*a)))/(3*a)
3.2.78.3.1 Defintions of rubi rules used
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 = 2.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.47
| method | result | size |
| derivativedivides | \(\frac {2 \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (b +a \,x^{\frac {1}{3}}\right ) \left (3003 a^{5} x^{\frac {5}{3}}-2310 a^{4} b \,x^{\frac {4}{3}}+1680 a^{3} b^{2} x -1120 a^{2} b^{3} x^{\frac {2}{3}}+640 a \,b^{4} x^{\frac {1}{3}}-256 b^{5}\right )}{15015 x \,a^{6}}\) | \(79\) |
| default | \(\frac {2 \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (b +a \,x^{\frac {1}{3}}\right ) \left (3003 a^{5} x^{\frac {5}{3}}-2310 a^{4} b \,x^{\frac {4}{3}}+1680 a^{3} b^{2} x -1120 a^{2} b^{3} x^{\frac {2}{3}}+640 a \,b^{4} x^{\frac {1}{3}}-256 b^{5}\right )}{15015 x \,a^{6}}\) | \(79\) |
2/15015*(b*x^(2/3)+a*x)^(3/2)*(b+a*x^(1/3))*(3003*a^5*x^(5/3)-2310*a^4*b*x ^(4/3)+1680*a^3*b^2*x-1120*a^2*b^3*x^(2/3)+640*a*b^4*x^(1/3)-256*b^5)/x/a^ 6
Leaf count of result is larger than twice the leaf count of optimal. 768 vs. \(2 (125) = 250\).
Time = 153.11 (sec) , antiderivative size = 768, normalized size of antiderivative = 4.54 \[ \int \left (b x^{2/3}+a x\right )^{3/2} \, dx=\frac {2 \, {\left (4 \, {\left (805306368 \, b^{13} + 167772160 \, b^{12} + 786432 \, {\left (64 \, a^{3} - 3\right )} b^{10} - 15728640 \, b^{11} - 4096 \, {\left (11264 \, a^{3} - 53\right )} b^{9} + 4372368 \, a^{9} - 1536 \, {\left (5504 \, a^{3} + 1\right )} b^{8} - 48 \, {\left (242810880 \, a^{6} + 114688 \, a^{3} + 3\right )} b^{7} - 1792 \, {\left (1353984 \, a^{6} - 103 \, a^{3}\right )} b^{6} + 192 \, {\left (1152384 \, a^{6} - 23 \, a^{3}\right )} b^{5} - 3 \, {\left (3633315840 \, a^{9} - 12027392 \, a^{6} - 15 \, a^{3}\right )} b^{4} - 112 \, {\left (35389440 \, a^{9} + 29281 \, a^{6}\right )} b^{3} - 819 \, {\left (368640 \, a^{9} - 31 \, a^{6}\right )} b^{2} + 693 \, {\left (40960 \, a^{9} + 3 \, a^{6}\right )} b\right )} x + {\left (3003 \, {\left (16777216 \, a^{7} b^{6} + 6291456 \, a^{7} b^{5} + 196608 \, a^{7} b^{4} - 262144 \, a^{10} - 114688 \, a^{7} b^{3} - 2304 \, a^{7} b^{2} + 864 \, a^{7} b - 27 \, a^{7}\right )} x^{3} - 70 \, {\left (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} - {\left (262144 \, a^{7} + 27 \, a^{4}\right )} b^{3}\right )} x^{2} + 128 \, {\left (16777216 \, a b^{12} + 6291456 \, a b^{11} + 196608 \, a b^{10} - 114688 \, a b^{9} - 2304 \, a b^{8} + 864 \, a b^{7} - {\left (262144 \, a^{4} + 27 \, a\right )} b^{6}\right )} x - 16 \, {\left (268435456 \, b^{13} + 100663296 \, b^{12} + 3145728 \, b^{11} - 1835008 \, b^{10} - 36864 \, b^{9} - 16 \, {\left (262144 \, a^{3} + 27\right )} b^{7} + 13824 \, b^{8} - 231 \, {\left (16777216 \, 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} - {\left (262144 \, a^{9} + 27 \, a^{6}\right )} b\right )} x^{2} - 5 \, {\left (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} - {\left (262144 \, a^{6} + 27 \, a^{3}\right )} b^{4}\right )} x\right )} x^{\frac {2}{3}} + 3 \, {\left (21 \, {\left (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} - {\left (262144 \, a^{8} + 27 \, a^{5}\right )} b^{2}\right )} x^{2} - 32 \, {\left (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^{6} - {\left (262144 \, a^{5} + 27 \, a^{2}\right )} b^{5}\right )} x\right )} x^{\frac {1}{3}}\right )} \sqrt {a x + b x^{\frac {2}{3}}}\right )}}{15015 \, {\left (16777216 \, a^{6} b^{6} + 6291456 \, a^{6} b^{5} + 196608 \, a^{6} b^{4} - 262144 \, a^{9} - 114688 \, a^{6} b^{3} - 2304 \, a^{6} b^{2} + 864 \, a^{6} b - 27 \, a^{6}\right )} x} \]
2/15015*(4*(805306368*b^13 + 167772160*b^12 + 786432*(64*a^3 - 3)*b^10 - 1 5728640*b^11 - 4096*(11264*a^3 - 53)*b^9 + 4372368*a^9 - 1536*(5504*a^3 + 1)*b^8 - 48*(242810880*a^6 + 114688*a^3 + 3)*b^7 - 1792*(1353984*a^6 - 103 *a^3)*b^6 + 192*(1152384*a^6 - 23*a^3)*b^5 - 3*(3633315840*a^9 - 12027392* a^6 - 15*a^3)*b^4 - 112*(35389440*a^9 + 29281*a^6)*b^3 - 819*(368640*a^9 - 31*a^6)*b^2 + 693*(40960*a^9 + 3*a^6)*b)*x + (3003*(16777216*a^7*b^6 + 62 91456*a^7*b^5 + 196608*a^7*b^4 - 262144*a^10 - 114688*a^7*b^3 - 2304*a^7*b ^2 + 864*a^7*b - 27*a^7)*x^3 - 70*(16777216*a^4*b^9 + 6291456*a^4*b^8 + 19 6608*a^4*b^7 - 114688*a^4*b^6 - 2304*a^4*b^5 + 864*a^4*b^4 - (262144*a^7 + 27*a^4)*b^3)*x^2 + 128*(16777216*a*b^12 + 6291456*a*b^11 + 196608*a*b^10 - 114688*a*b^9 - 2304*a*b^8 + 864*a*b^7 - (262144*a^4 + 27*a)*b^6)*x - 16* (268435456*b^13 + 100663296*b^12 + 3145728*b^11 - 1835008*b^10 - 36864*b^9 - 16*(262144*a^3 + 27)*b^7 + 13824*b^8 - 231*(16777216*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 - 5*(16777216*a^3*b^10 + 6291456*a^3*b^9 + 196 608*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) + 3*(21*(16777216*a^5*b^8 + 6291456*a^5*b^7 + 1966 08*a^5*b^6 - 114688*a^5*b^5 - 2304*a^5*b^4 + 864*a^5*b^3 - (262144*a^8 + 2 7*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^6 - (262144*a^5 + 27*a^2)...
\[ \int \left (b x^{2/3}+a x\right )^{3/2} \, dx=\int \left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}\, dx \]
\[ \int \left (b x^{2/3}+a x\right )^{3/2} \, dx=\int { {\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (125) = 250\).
Time = 0.31 (sec) , antiderivative size = 434, normalized size of antiderivative = 2.57 \[ \int \left (b x^{2/3}+a x\right )^{3/2} \, dx=\frac {2}{3003} \, b {\left (\frac {256 \, b^{\frac {13}{2}}}{a^{6}} + \frac {\frac {13 \, {\left (63 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} - 385 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b + 990 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{2} - 1386 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{3} + 1155 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{4} - 693 \, \sqrt {a x^{\frac {1}{3}} + b} b^{5}\right )} b}{a^{5}} + \frac {3 \, {\left (231 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} - 1638 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b + 5005 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{2} - 8580 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{3} + 9009 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{4} - 6006 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{5} + 3003 \, \sqrt {a x^{\frac {1}{3}} + b} b^{6}\right )}}{a^{5}}}{a}\right )} - \frac {2}{15015} \, a {\left (\frac {1024 \, b^{\frac {15}{2}}}{a^{7}} - \frac {\frac {15 \, {\left (231 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} - 1638 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b + 5005 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{2} - 8580 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{3} + 9009 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{4} - 6006 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{5} + 3003 \, \sqrt {a x^{\frac {1}{3}} + b} b^{6}\right )} b}{a^{6}} + \frac {7 \, {\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 )}}{a^{6}}}{a}\right )} \]
2/3003*b*(256*b^(13/2)/a^6 + (13*(63*(a*x^(1/3) + b)^(11/2) - 385*(a*x^(1/ 3) + b)^(9/2)*b + 990*(a*x^(1/3) + b)^(7/2)*b^2 - 1386*(a*x^(1/3) + b)^(5/ 2)*b^3 + 1155*(a*x^(1/3) + b)^(3/2)*b^4 - 693*sqrt(a*x^(1/3) + b)*b^5)*b/a ^5 + 3*(231*(a*x^(1/3) + b)^(13/2) - 1638*(a*x^(1/3) + b)^(11/2)*b + 5005* (a*x^(1/3) + b)^(9/2)*b^2 - 8580*(a*x^(1/3) + b)^(7/2)*b^3 + 9009*(a*x^(1/ 3) + b)^(5/2)*b^4 - 6006*(a*x^(1/3) + b)^(3/2)*b^5 + 3003*sqrt(a*x^(1/3) + b)*b^6)/a^5)/a) - 2/15015*a*(1024*b^(15/2)/a^7 - (15*(231*(a*x^(1/3) + b) ^(13/2) - 1638*(a*x^(1/3) + b)^(11/2)*b + 5005*(a*x^(1/3) + b)^(9/2)*b^2 - 8580*(a*x^(1/3) + b)^(7/2)*b^3 + 9009*(a*x^(1/3) + b)^(5/2)*b^4 - 6006*(a *x^(1/3) + b)^(3/2)*b^5 + 3003*sqrt(a*x^(1/3) + b)*b^6)*b/a^6 + 7*(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^6)/a)
Time = 10.89 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.24 \[ \int \left (b x^{2/3}+a x\right )^{3/2} \, dx=\frac {x\,{\left (a\,x+b\,x^{2/3}\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},6;\ 7;\ -\frac {a\,x^{1/3}}{b}\right )}{2\,{\left (\frac {a\,x^{1/3}}{b}+1\right )}^{3/2}} \]