Integrand size = 17, antiderivative size = 195 \[ \int x \sqrt {b x^{2/3}+a x} \, dx=-\frac {128 b^3 \left (b x^{2/3}+a x\right )^{3/2}}{429 a^4}+\frac {2048 b^6 \left (b x^{2/3}+a x\right )^{3/2}}{15015 a^7 x}-\frac {1024 b^5 \left (b x^{2/3}+a x\right )^{3/2}}{5005 a^6 x^{2/3}}+\frac {256 b^4 \left (b x^{2/3}+a x\right )^{3/2}}{1001 a^5 \sqrt [3]{x}}+\frac {48 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{143 a^3}-\frac {24 b x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{65 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{3/2}}{5 a} \]
-128/429*b^3*(b*x^(2/3)+a*x)^(3/2)/a^4+2048/15015*b^6*(b*x^(2/3)+a*x)^(3/2 )/a^7/x-1024/5005*b^5*(b*x^(2/3)+a*x)^(3/2)/a^6/x^(2/3)+256/1001*b^4*(b*x^ (2/3)+a*x)^(3/2)/a^5/x^(1/3)+48/143*b^2*x^(1/3)*(b*x^(2/3)+a*x)^(3/2)/a^3- 24/65*b*x^(2/3)*(b*x^(2/3)+a*x)^(3/2)/a^2+2/5*x*(b*x^(2/3)+a*x)^(3/2)/a
Time = 0.11 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.49 \[ \int x \sqrt {b x^{2/3}+a x} \, dx=\frac {2 \left (b x^{2/3}+a x\right )^{3/2} \left (1024 b^6-1536 a b^5 \sqrt [3]{x}+1920 a^2 b^4 x^{2/3}-2240 a^3 b^3 x+2520 a^4 b^2 x^{4/3}-2772 a^5 b x^{5/3}+3003 a^6 x^2\right )}{15015 a^7 x} \]
(2*(b*x^(2/3) + a*x)^(3/2)*(1024*b^6 - 1536*a*b^5*x^(1/3) + 1920*a^2*b^4*x ^(2/3) - 2240*a^3*b^3*x + 2520*a^4*b^2*x^(4/3) - 2772*a^5*b*x^(5/3) + 3003 *a^6*x^2))/(15015*a^7*x)
Time = 0.43 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {1922, 1922, 1922, 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.
| \(\displaystyle \int x \sqrt {a x+b x^{2/3}} \, dx\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x \left (a x+b x^{2/3}\right )^{3/2}}{5 a}-\frac {4 b \int x^{2/3} \sqrt {x^{2/3} b+a x}dx}{5 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x \left (a x+b x^{2/3}\right )^{3/2}}{5 a}-\frac {4 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{13 a}-\frac {10 b \int \sqrt [3]{x} \sqrt {x^{2/3} b+a x}dx}{13 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x \left (a x+b x^{2/3}\right )^{3/2}}{5 a}-\frac {4 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{13 a}-\frac {10 b \left (\frac {6 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{11 a}-\frac {8 b \int \sqrt {x^{2/3} b+a x}dx}{11 a}\right )}{13 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 1908 |
| \(\displaystyle \frac {2 x \left (a x+b x^{2/3}\right )^{3/2}}{5 a}-\frac {4 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{13 a}-\frac {10 b \left (\frac {6 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{11 a}-\frac {8 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{3 a}-\frac {2 b \int \frac {\sqrt {x^{2/3} b+a x}}{\sqrt [3]{x}}dx}{3 a}\right )}{11 a}\right )}{13 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x \left (a x+b x^{2/3}\right )^{3/2}}{5 a}-\frac {4 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{13 a}-\frac {10 b \left (\frac {6 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{11 a}-\frac {8 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{3 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{3/2}}{7 a \sqrt [3]{x}}-\frac {4 b \int \frac {\sqrt {x^{2/3} b+a x}}{x^{2/3}}dx}{7 a}\right )}{3 a}\right )}{11 a}\right )}{13 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x \left (a x+b x^{2/3}\right )^{3/2}}{5 a}-\frac {4 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{13 a}-\frac {10 b \left (\frac {6 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{11 a}-\frac {8 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{3 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{3/2}}{7 a \sqrt [3]{x}}-\frac {4 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{3/2}}{5 a x^{2/3}}-\frac {2 b \int \frac {\sqrt {x^{2/3} b+a x}}{x}dx}{5 a}\right )}{7 a}\right )}{3 a}\right )}{11 a}\right )}{13 a}\right )}{5 a}\) |
| \(\Big \downarrow \) 1920 |
| \(\displaystyle \frac {2 x \left (a x+b x^{2/3}\right )^{3/2}}{5 a}-\frac {4 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{13 a}-\frac {10 b \left (\frac {6 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{11 a}-\frac {8 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{3 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{3/2}}{7 a \sqrt [3]{x}}-\frac {4 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{3/2}}{5 a x^{2/3}}-\frac {4 b \left (a x+b x^{2/3}\right )^{3/2}}{5 a^2 x}\right )}{7 a}\right )}{3 a}\right )}{11 a}\right )}{13 a}\right )}{5 a}\) |
(2*x*(b*x^(2/3) + a*x)^(3/2))/(5*a) - (4*b*((6*x^(2/3)*(b*x^(2/3) + a*x)^( 3/2))/(13*a) - (10*b*((6*x^(1/3)*(b*x^(2/3) + a*x)^(3/2))/(11*a) - (8*b*(( 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)))/(11*a)))/(13*a)))/(5*a)
3.2.69.3.1 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 = 2.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.46
| method | result | size |
| derivativedivides | \(\frac {2 \sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (b +a \,x^{\frac {1}{3}}\right ) \left (3003 a^{6} x^{2}-2772 a^{5} b \,x^{\frac {5}{3}}+2520 a^{4} b^{2} x^{\frac {4}{3}}-2240 a^{3} b^{3} x +1920 a^{2} x^{\frac {2}{3}} b^{4}-1536 a \,b^{5} x^{\frac {1}{3}}+1024 b^{6}\right )}{15015 x^{\frac {1}{3}} a^{7}}\) | \(90\) |
| default | \(-\frac {2 \sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (b +a \,x^{\frac {1}{3}}\right ) \left (2772 a^{5} b \,x^{\frac {5}{3}}-2520 a^{4} b^{2} x^{\frac {4}{3}}-1920 a^{2} x^{\frac {2}{3}} b^{4}-3003 a^{6} x^{2}+1536 a \,b^{5} x^{\frac {1}{3}}+2240 a^{3} b^{3} x -1024 b^{6}\right )}{15015 x^{\frac {1}{3}} a^{7}}\) | \(90\) |
2/15015*(b*x^(2/3)+a*x)^(1/2)*(b+a*x^(1/3))*(3003*a^6*x^2-2772*a^5*b*x^(5/ 3)+2520*a^4*b^2*x^(4/3)-2240*a^3*b^3*x+1920*a^2*x^(2/3)*b^4-1536*a*b^5*x^( 1/3)+1024*b^6)/x^(1/3)/a^7
Leaf count of result is larger than twice the leaf count of optimal. 767 vs. \(2 (145) = 290\).
Time = 147.95 (sec) , antiderivative size = 767, normalized size of antiderivative = 3.93 \[ \int x \sqrt {b x^{2/3}+a x} \, dx=-\frac {{\left (51539607552 \, b^{13} + 10737418240 \, b^{12} + 50331648 \, {\left (64 \, a^{3} - 3\right )} b^{10} - 1006632960 \, b^{11} - 262144 \, {\left (11264 \, a^{3} - 53\right )} b^{9} - 69957888 \, a^{9} - 98304 \, {\left (5504 \, a^{3} + 1\right )} b^{8} + 3072 \, {\left (3194880 \, a^{6} - 114688 \, a^{3} - 3\right )} b^{7} + 114688 \, {\left (18816 \, a^{6} + 103 \, a^{3}\right )} b^{6} - 12288 \, {\left (48816 \, a^{6} + 23 \, a^{3}\right )} b^{5} + 192 \, {\left (302776320 \, a^{9} + 495872 \, a^{6} + 15 \, a^{3}\right )} b^{4} + 1792 \, {\left (16588800 \, a^{9} - 3439 \, a^{6}\right )} b^{3} + 26208 \, {\left (163840 \, a^{9} + 7 \, a^{6}\right )} b^{2} + 693 \, {\left (1024000 \, a^{9} - 3 \, a^{6}\right )} b\right )} x - 4 \, {\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} + 280 \, {\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} - 512 \, {\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 + {\left (17179869184 \, b^{13} + 6442450944 \, b^{12} + 201326592 \, b^{11} - 117440512 \, b^{10} - 2359296 \, b^{9} - 1024 \, {\left (262144 \, a^{3} + 27\right )} b^{7} + 884736 \, 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} - 320 \, {\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}} - 12 \, {\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}}}}{30030 \, {\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} \]
-1/30030*((51539607552*b^13 + 10737418240*b^12 + 50331648*(64*a^3 - 3)*b^1 0 - 1006632960*b^11 - 262144*(11264*a^3 - 53)*b^9 - 69957888*a^9 - 98304*( 5504*a^3 + 1)*b^8 + 3072*(3194880*a^6 - 114688*a^3 - 3)*b^7 + 114688*(1881 6*a^6 + 103*a^3)*b^6 - 12288*(48816*a^6 + 23*a^3)*b^5 + 192*(302776320*a^9 + 495872*a^6 + 15*a^3)*b^4 + 1792*(16588800*a^9 - 3439*a^6)*b^3 + 26208*( 163840*a^9 + 7*a^6)*b^2 + 693*(1024000*a^9 - 3*a^6)*b)*x - 4*(3003*(167772 16*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)*x^3 + 280*(16777216*a^4*b^9 + 6291 456*a^4*b^8 + 196608*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 - 512*(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 + (17179869184*b^13 + 6442450944*b^12 + 201326592*b^11 - 117440 512*b^10 - 2359296*b^9 - 1024*(262144*a^3 + 27)*b^7 + 884736*b^8 + 231*(16 777216*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 + 8 64*a^3*b^5 - (262144*a^6 + 27*a^3)*b^4)*x)*x^(2/3) - 12*(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 + 8 64*a^5*b^3 - (262144*a^8 + 27*a^5)*b^2)*x^2 - 32*(16777216*a^2*b^11 + 6291 456*a^2*b^10 + 196608*a^2*b^9 - 114688*a^2*b^8 - 2304*a^2*b^7 + 864*a^2...
\[ \int x \sqrt {b x^{2/3}+a x} \, dx=\int x \sqrt {a x + b x^{\frac {2}{3}}}\, dx \]
\[ \int x \sqrt {b x^{2/3}+a x} \, dx=\int { \sqrt {a x + b x^{\frac {2}{3}}} x \,d x } \]
Time = 0.30 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.17 \[ \int x \sqrt {b x^{2/3}+a x} \, dx=-\frac {2048 \, b^{\frac {15}{2}}}{15015 \, a^{7}} + \frac {2 \, {\left (\frac {15 \, {\left (231 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} - 1638 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b + 5005 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{2} - 8580 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{3} + 9009 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{4} - 6006 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{5} + 3003 \, \sqrt {a x^{\frac {1}{3}} + b} b^{6}\right )} b}{a^{6}} + \frac {7 \, {\left (429 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} - 3465 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b + 12285 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{2} - 25025 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{3} + 32175 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{4} - 27027 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{5} + 15015 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{6} - 6435 \, \sqrt {a x^{\frac {1}{3}} + b} b^{7}\right )}}{a^{6}}\right )}}{15015 \, a} \]
-2048/15015*b^(15/2)/a^7 + 2/15015*(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)^(1 5/2) - 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
Timed out. \[ \int x \sqrt {b x^{2/3}+a x} \, dx=\int x\,\sqrt {a\,x+b\,x^{2/3}} \,d x \]