Integrand size = 19, antiderivative size = 371 \[ \int x^3 \sqrt {b x^{2/3}+a x} \, dx=-\frac {524288 b^9 \left (b x^{2/3}+a x\right )^{3/2}}{4345965 a^{10}}+\frac {8388608 b^{12} \left (b x^{2/3}+a x\right )^{3/2}}{152108775 a^{13} x}-\frac {4194304 b^{11} \left (b x^{2/3}+a x\right )^{3/2}}{50702925 a^{12} x^{2/3}}+\frac {1048576 b^{10} \left (b x^{2/3}+a x\right )^{3/2}}{10140585 a^{11} \sqrt [3]{x}}+\frac {65536 b^8 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2}}{482885 a^9}-\frac {360448 b^7 x^{2/3} \left (b x^{2/3}+a x\right )^{3/2}}{2414425 a^8}+\frac {90112 b^6 x \left (b x^{2/3}+a x\right )^{3/2}}{557175 a^7}-\frac {45056 b^5 x^{4/3} \left (b x^{2/3}+a x\right )^{3/2}}{260015 a^6}+\frac {2816 b^4 x^{5/3} \left (b x^{2/3}+a x\right )^{3/2}}{15295 a^5}-\frac {1408 b^3 x^2 \left (b x^{2/3}+a x\right )^{3/2}}{7245 a^4}+\frac {352 b^2 x^{7/3} \left (b x^{2/3}+a x\right )^{3/2}}{1725 a^3}-\frac {16 b x^{8/3} \left (b x^{2/3}+a x\right )^{3/2}}{75 a^2}+\frac {2 x^3 \left (b x^{2/3}+a x\right )^{3/2}}{9 a} \]
-524288/4345965*b^9*(b*x^(2/3)+a*x)^(3/2)/a^10+8388608/152108775*b^12*(b*x ^(2/3)+a*x)^(3/2)/a^13/x-4194304/50702925*b^11*(b*x^(2/3)+a*x)^(3/2)/a^12/ x^(2/3)+1048576/10140585*b^10*(b*x^(2/3)+a*x)^(3/2)/a^11/x^(1/3)+65536/482 885*b^8*x^(1/3)*(b*x^(2/3)+a*x)^(3/2)/a^9-360448/2414425*b^7*x^(2/3)*(b*x^ (2/3)+a*x)^(3/2)/a^8+90112/557175*b^6*x*(b*x^(2/3)+a*x)^(3/2)/a^7-45056/26 0015*b^5*x^(4/3)*(b*x^(2/3)+a*x)^(3/2)/a^6+2816/15295*b^4*x^(5/3)*(b*x^(2/ 3)+a*x)^(3/2)/a^5-1408/7245*b^3*x^2*(b*x^(2/3)+a*x)^(3/2)/a^4+352/1725*b^2 *x^(7/3)*(b*x^(2/3)+a*x)^(3/2)/a^3-16/75*b*x^(8/3)*(b*x^(2/3)+a*x)^(3/2)/a ^2+2/9*x^3*(b*x^(2/3)+a*x)^(3/2)/a
Time = 0.18 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.50 \[ \int x^3 \sqrt {b x^{2/3}+a x} \, dx=\frac {2 \sqrt {b x^{2/3}+a x} \left (4194304 b^{13}-2097152 a b^{12} \sqrt [3]{x}+1572864 a^2 b^{11} x^{2/3}-1310720 a^3 b^{10} x+1146880 a^4 b^9 x^{4/3}-1032192 a^5 b^8 x^{5/3}+946176 a^6 b^7 x^2-878592 a^7 b^6 x^{7/3}+823680 a^8 b^5 x^{8/3}-777920 a^9 b^4 x^3+739024 a^{10} b^3 x^{10/3}-705432 a^{11} b^2 x^{11/3}+676039 a^{12} b x^4+16900975 a^{13} x^{13/3}\right )}{152108775 a^{13} \sqrt [3]{x}} \]
(2*Sqrt[b*x^(2/3) + a*x]*(4194304*b^13 - 2097152*a*b^12*x^(1/3) + 1572864* a^2*b^11*x^(2/3) - 1310720*a^3*b^10*x + 1146880*a^4*b^9*x^(4/3) - 1032192* a^5*b^8*x^(5/3) + 946176*a^6*b^7*x^2 - 878592*a^7*b^6*x^(7/3) + 823680*a^8 *b^5*x^(8/3) - 777920*a^9*b^4*x^3 + 739024*a^10*b^3*x^(10/3) - 705432*a^11 *b^2*x^(11/3) + 676039*a^12*b*x^4 + 16900975*a^13*x^(13/3)))/(152108775*a^ 13*x^(1/3))
Time = 0.88 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.18, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {1922, 1922, 1922, 1922, 1922, 1922, 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.
| \(\displaystyle \int x^3 \sqrt {a x+b x^{2/3}} \, dx\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^3 \left (a x+b x^{2/3}\right )^{3/2}}{9 a}-\frac {8 b \int x^{8/3} \sqrt {x^{2/3} b+a x}dx}{9 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^3 \left (a x+b x^{2/3}\right )^{3/2}}{9 a}-\frac {8 b \left (\frac {6 x^{8/3} \left (a x+b x^{2/3}\right )^{3/2}}{25 a}-\frac {22 b \int x^{7/3} \sqrt {x^{2/3} b+a x}dx}{25 a}\right )}{9 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^3 \left (a x+b x^{2/3}\right )^{3/2}}{9 a}-\frac {8 b \left (\frac {6 x^{8/3} \left (a x+b x^{2/3}\right )^{3/2}}{25 a}-\frac {22 b \left (\frac {6 x^{7/3} \left (a x+b x^{2/3}\right )^{3/2}}{23 a}-\frac {20 b \int x^2 \sqrt {x^{2/3} b+a x}dx}{23 a}\right )}{25 a}\right )}{9 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^3 \left (a x+b x^{2/3}\right )^{3/2}}{9 a}-\frac {8 b \left (\frac {6 x^{8/3} \left (a x+b x^{2/3}\right )^{3/2}}{25 a}-\frac {22 b \left (\frac {6 x^{7/3} \left (a x+b x^{2/3}\right )^{3/2}}{23 a}-\frac {20 b \left (\frac {2 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7 a}-\frac {6 b \int x^{5/3} \sqrt {x^{2/3} b+a x}dx}{7 a}\right )}{23 a}\right )}{25 a}\right )}{9 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^3 \left (a x+b x^{2/3}\right )^{3/2}}{9 a}-\frac {8 b \left (\frac {6 x^{8/3} \left (a x+b x^{2/3}\right )^{3/2}}{25 a}-\frac {22 b \left (\frac {6 x^{7/3} \left (a x+b x^{2/3}\right )^{3/2}}{23 a}-\frac {20 b \left (\frac {2 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7 a}-\frac {6 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{19 a}-\frac {16 b \int x^{4/3} \sqrt {x^{2/3} b+a x}dx}{19 a}\right )}{7 a}\right )}{23 a}\right )}{25 a}\right )}{9 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^3 \left (a x+b x^{2/3}\right )^{3/2}}{9 a}-\frac {8 b \left (\frac {6 x^{8/3} \left (a x+b x^{2/3}\right )^{3/2}}{25 a}-\frac {22 b \left (\frac {6 x^{7/3} \left (a x+b x^{2/3}\right )^{3/2}}{23 a}-\frac {20 b \left (\frac {2 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7 a}-\frac {6 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{19 a}-\frac {16 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{17 a}-\frac {14 b \int x \sqrt {x^{2/3} b+a x}dx}{17 a}\right )}{19 a}\right )}{7 a}\right )}{23 a}\right )}{25 a}\right )}{9 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^3 \left (a x+b x^{2/3}\right )^{3/2}}{9 a}-\frac {8 b \left (\frac {6 x^{8/3} \left (a x+b x^{2/3}\right )^{3/2}}{25 a}-\frac {22 b \left (\frac {6 x^{7/3} \left (a x+b x^{2/3}\right )^{3/2}}{23 a}-\frac {20 b \left (\frac {2 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7 a}-\frac {6 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{19 a}-\frac {16 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{17 a}-\frac {14 b \left (\frac {2 x \left (a x+b x^{2/3}\right )^{3/2}}{5 a}-\frac {4 b \int x^{2/3} \sqrt {x^{2/3} b+a x}dx}{5 a}\right )}{17 a}\right )}{19 a}\right )}{7 a}\right )}{23 a}\right )}{25 a}\right )}{9 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^3 \left (a x+b x^{2/3}\right )^{3/2}}{9 a}-\frac {8 b \left (\frac {6 x^{8/3} \left (a x+b x^{2/3}\right )^{3/2}}{25 a}-\frac {22 b \left (\frac {6 x^{7/3} \left (a x+b x^{2/3}\right )^{3/2}}{23 a}-\frac {20 b \left (\frac {2 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7 a}-\frac {6 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{19 a}-\frac {16 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{17 a}-\frac {14 b \left (\frac {2 x \left (a x+b x^{2/3}\right )^{3/2}}{5 a}-\frac {4 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{13 a}-\frac {10 b \int \sqrt [3]{x} \sqrt {x^{2/3} b+a x}dx}{13 a}\right )}{5 a}\right )}{17 a}\right )}{19 a}\right )}{7 a}\right )}{23 a}\right )}{25 a}\right )}{9 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^3 \left (a x+b x^{2/3}\right )^{3/2}}{9 a}-\frac {8 b \left (\frac {6 x^{8/3} \left (a x+b x^{2/3}\right )^{3/2}}{25 a}-\frac {22 b \left (\frac {6 x^{7/3} \left (a x+b x^{2/3}\right )^{3/2}}{23 a}-\frac {20 b \left (\frac {2 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7 a}-\frac {6 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{19 a}-\frac {16 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{17 a}-\frac {14 b \left (\frac {2 x \left (a x+b x^{2/3}\right )^{3/2}}{5 a}-\frac {4 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{13 a}-\frac {10 b \left (\frac {6 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{11 a}-\frac {8 b \int \sqrt {x^{2/3} b+a x}dx}{11 a}\right )}{13 a}\right )}{5 a}\right )}{17 a}\right )}{19 a}\right )}{7 a}\right )}{23 a}\right )}{25 a}\right )}{9 a}\) |
| \(\Big \downarrow \) 1908 |
| \(\displaystyle \frac {2 x^3 \left (a x+b x^{2/3}\right )^{3/2}}{9 a}-\frac {8 b \left (\frac {6 x^{8/3} \left (a x+b x^{2/3}\right )^{3/2}}{25 a}-\frac {22 b \left (\frac {6 x^{7/3} \left (a x+b x^{2/3}\right )^{3/2}}{23 a}-\frac {20 b \left (\frac {2 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7 a}-\frac {6 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{19 a}-\frac {16 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{17 a}-\frac {14 b \left (\frac {2 x \left (a x+b x^{2/3}\right )^{3/2}}{5 a}-\frac {4 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{13 a}-\frac {10 b \left (\frac {6 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{11 a}-\frac {8 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{3 a}-\frac {2 b \int \frac {\sqrt {x^{2/3} b+a x}}{\sqrt [3]{x}}dx}{3 a}\right )}{11 a}\right )}{13 a}\right )}{5 a}\right )}{17 a}\right )}{19 a}\right )}{7 a}\right )}{23 a}\right )}{25 a}\right )}{9 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^3 \left (a x+b x^{2/3}\right )^{3/2}}{9 a}-\frac {8 b \left (\frac {6 x^{8/3} \left (a x+b x^{2/3}\right )^{3/2}}{25 a}-\frac {22 b \left (\frac {6 x^{7/3} \left (a x+b x^{2/3}\right )^{3/2}}{23 a}-\frac {20 b \left (\frac {2 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7 a}-\frac {6 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{19 a}-\frac {16 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{17 a}-\frac {14 b \left (\frac {2 x \left (a x+b x^{2/3}\right )^{3/2}}{5 a}-\frac {4 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{13 a}-\frac {10 b \left (\frac {6 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{11 a}-\frac {8 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{3 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{3/2}}{7 a \sqrt [3]{x}}-\frac {4 b \int \frac {\sqrt {x^{2/3} b+a x}}{x^{2/3}}dx}{7 a}\right )}{3 a}\right )}{11 a}\right )}{13 a}\right )}{5 a}\right )}{17 a}\right )}{19 a}\right )}{7 a}\right )}{23 a}\right )}{25 a}\right )}{9 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^3 \left (a x+b x^{2/3}\right )^{3/2}}{9 a}-\frac {8 b \left (\frac {6 x^{8/3} \left (a x+b x^{2/3}\right )^{3/2}}{25 a}-\frac {22 b \left (\frac {6 x^{7/3} \left (a x+b x^{2/3}\right )^{3/2}}{23 a}-\frac {20 b \left (\frac {2 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7 a}-\frac {6 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{19 a}-\frac {16 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{17 a}-\frac {14 b \left (\frac {2 x \left (a x+b x^{2/3}\right )^{3/2}}{5 a}-\frac {4 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{13 a}-\frac {10 b \left (\frac {6 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{11 a}-\frac {8 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{3 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{3/2}}{7 a \sqrt [3]{x}}-\frac {4 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{3/2}}{5 a x^{2/3}}-\frac {2 b \int \frac {\sqrt {x^{2/3} b+a x}}{x}dx}{5 a}\right )}{7 a}\right )}{3 a}\right )}{11 a}\right )}{13 a}\right )}{5 a}\right )}{17 a}\right )}{19 a}\right )}{7 a}\right )}{23 a}\right )}{25 a}\right )}{9 a}\) |
| \(\Big \downarrow \) 1920 |
| \(\displaystyle \frac {2 x^3 \left (a x+b x^{2/3}\right )^{3/2}}{9 a}-\frac {8 b \left (\frac {6 x^{8/3} \left (a x+b x^{2/3}\right )^{3/2}}{25 a}-\frac {22 b \left (\frac {6 x^{7/3} \left (a x+b x^{2/3}\right )^{3/2}}{23 a}-\frac {20 b \left (\frac {2 x^2 \left (a x+b x^{2/3}\right )^{3/2}}{7 a}-\frac {6 b \left (\frac {6 x^{5/3} \left (a x+b x^{2/3}\right )^{3/2}}{19 a}-\frac {16 b \left (\frac {6 x^{4/3} \left (a x+b x^{2/3}\right )^{3/2}}{17 a}-\frac {14 b \left (\frac {2 x \left (a x+b x^{2/3}\right )^{3/2}}{5 a}-\frac {4 b \left (\frac {6 x^{2/3} \left (a x+b x^{2/3}\right )^{3/2}}{13 a}-\frac {10 b \left (\frac {6 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{3/2}}{11 a}-\frac {8 b \left (\frac {2 \left (a x+b x^{2/3}\right )^{3/2}}{3 a}-\frac {2 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{3/2}}{7 a \sqrt [3]{x}}-\frac {4 b \left (\frac {6 \left (a x+b x^{2/3}\right )^{3/2}}{5 a x^{2/3}}-\frac {4 b \left (a x+b x^{2/3}\right )^{3/2}}{5 a^2 x}\right )}{7 a}\right )}{3 a}\right )}{11 a}\right )}{13 a}\right )}{5 a}\right )}{17 a}\right )}{19 a}\right )}{7 a}\right )}{23 a}\right )}{25 a}\right )}{9 a}\) |
(2*x^3*(b*x^(2/3) + a*x)^(3/2))/(9*a) - (8*b*((6*x^(8/3)*(b*x^(2/3) + a*x) ^(3/2))/(25*a) - (22*b*((6*x^(7/3)*(b*x^(2/3) + a*x)^(3/2))/(23*a) - (20*b *((2*x^2*(b*x^(2/3) + a*x)^(3/2))/(7*a) - (6*b*((6*x^(5/3)*(b*x^(2/3) + a* x)^(3/2))/(19*a) - (16*b*((6*x^(4/3)*(b*x^(2/3) + a*x)^(3/2))/(17*a) - (14 *b*((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)) )/(17*a)))/(19*a)))/(7*a)))/(23*a)))/(25*a)))/(9*a)
3.2.67.3.1 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 = 2.08 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.42
| method | result | size |
| derivativedivides | \(\frac {2 \sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (b +a \,x^{\frac {1}{3}}\right ) \left (16900975 a^{12} x^{4}-16224936 a^{11} b \,x^{\frac {11}{3}}+15519504 a^{10} b^{2} x^{\frac {10}{3}}-14780480 a^{9} x^{3} b^{3}+14002560 a^{8} b^{4} x^{\frac {8}{3}}-13178880 a^{7} b^{5} x^{\frac {7}{3}}+12300288 a^{6} b^{6} x^{2}-11354112 a^{5} b^{7} x^{\frac {5}{3}}+10321920 a^{4} b^{8} x^{\frac {4}{3}}-9175040 a^{3} b^{9} x +7864320 a^{2} b^{10} x^{\frac {2}{3}}-6291456 a \,b^{11} x^{\frac {1}{3}}+4194304 b^{12}\right )}{152108775 x^{\frac {1}{3}} a^{13}}\) | \(156\) |
| default | \(-\frac {2 \sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (b +a \,x^{\frac {1}{3}}\right ) \left (16224936 a^{11} b \,x^{\frac {11}{3}}-15519504 a^{10} b^{2} x^{\frac {10}{3}}-14002560 a^{8} b^{4} x^{\frac {8}{3}}+13178880 a^{7} b^{5} x^{\frac {7}{3}}+11354112 a^{5} b^{7} x^{\frac {5}{3}}-10321920 a^{4} b^{8} x^{\frac {4}{3}}-16900975 a^{12} x^{4}+14780480 a^{9} x^{3} b^{3}-7864320 a^{2} b^{10} x^{\frac {2}{3}}-12300288 a^{6} b^{6} x^{2}+6291456 a \,b^{11} x^{\frac {1}{3}}+9175040 a^{3} b^{9} x -4194304 b^{12}\right )}{152108775 x^{\frac {1}{3}} a^{13}}\) | \(156\) |
2/152108775*(b*x^(2/3)+a*x)^(1/2)*(b+a*x^(1/3))*(16900975*a^12*x^4-1622493 6*a^11*b*x^(11/3)+15519504*a^10*b^2*x^(10/3)-14780480*a^9*x^3*b^3+14002560 *a^8*b^4*x^(8/3)-13178880*a^7*b^5*x^(7/3)+12300288*a^6*b^6*x^2-11354112*a^ 5*b^7*x^(5/3)+10321920*a^4*b^8*x^(4/3)-9175040*a^3*b^9*x+7864320*a^2*b^10* x^(2/3)-6291456*a*b^11*x^(1/3)+4194304*b^12)/x^(1/3)/a^13
Leaf count of result is larger than twice the leaf count of optimal. 1293 vs. \(2 (277) = 554\).
Time = 179.80 (sec) , antiderivative size = 1293, normalized size of antiderivative = 3.49 \[ \int x^3 \sqrt {b x^{2/3}+a x} \, dx=\text {Too large to display} \]
-1/304217550*((211106232532992*b^19 + 43980465111040*b^18 + 206158430208*( 64*a^3 - 3)*b^16 - 4123168604160*b^17 - 1073741824*(11264*a^3 - 53)*b^15 - 393725113600*a^15 - 402653184*(5504*a^3 + 1)*b^14 + 12582912*(3194880*a^6 - 114688*a^3 - 3)*b^13 + 469762048*(18816*a^6 + 103*a^3)*b^12 - 50331648* (48816*a^6 + 23*a^3)*b^11 - 786432*(45731840*a^9 - 495872*a^6 - 15*a^3)*b^ 10 - 7340032*(1349120*a^9 + 3439*a^6)*b^9 + 250478592*(5600*a^9 + 3*a^6)*b ^8 + 12288*(2616979456*a^12 - 21542400*a^9 - 693*a^6)*b^7 + 212992*(437436 16*a^12 + 89111*a^9)*b^6 - 638976*(1652476*a^12 + 935*a^9)*b^5 + 42432*(72 17086464*a^15 + 4969216*a^12 + 165*a^9)*b^4 + 7524608*(20570112*a^15 - 210 1*a^12)*b^3 + 2821728*(7815168*a^15 + 181*a^12)*b^2 + 2028117*(2072576*a^1 5 - 3*a^12)*b)*x - 4*(16900975*(16777216*a^13*b^6 + 6291456*a^13*b^5 + 196 608*a^13*b^4 - 262144*a^16 - 114688*a^13*b^3 - 2304*a^13*b^2 + 864*a^13*b - 27*a^13)*x^5 + 739024*(16777216*a^10*b^9 + 6291456*a^10*b^8 + 196608*a^1 0*b^7 - 114688*a^10*b^6 - 2304*a^10*b^5 + 864*a^10*b^4 - (262144*a^13 + 27 *a^10)*b^3)*x^4 - 878592*(16777216*a^7*b^12 + 6291456*a^7*b^11 + 196608*a^ 7*b^10 - 114688*a^7*b^9 - 2304*a^7*b^8 + 864*a^7*b^7 - (262144*a^10 + 27*a ^7)*b^6)*x^3 + 1146880*(16777216*a^4*b^15 + 6291456*a^4*b^14 + 196608*a^4* b^13 - 114688*a^4*b^12 - 2304*a^4*b^11 + 864*a^4*b^10 - (262144*a^7 + 27*a ^4)*b^9)*x^2 - 2097152*(16777216*a*b^18 + 6291456*a*b^17 + 196608*a*b^16 - 114688*a*b^15 - 2304*a*b^14 + 864*a*b^13 - (262144*a^4 + 27*a)*b^12)*x...
\[ \int x^3 \sqrt {b x^{2/3}+a x} \, dx=\int x^{3} \sqrt {a x + b x^{\frac {2}{3}}}\, dx \]
\[ \int x^3 \sqrt {b x^{2/3}+a x} \, dx=\int { \sqrt {a x + b x^{\frac {2}{3}}} x^{3} \,d x } \]
Time = 0.30 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.07 \[ \int x^3 \sqrt {b x^{2/3}+a x} \, dx=-\frac {8388608 \, b^{\frac {27}{2}}}{152108775 \, a^{13}} + \frac {2 \, {\left (\frac {27 \, {\left (676039 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {25}{2}} - 8817900 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {23}{2}} b + 53117350 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} b^{2} - 195695500 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} b^{3} + 492116625 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b^{4} - 892371480 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{5} + 1201269300 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{6} - 1216870200 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{7} + 929553625 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{8} - 531173500 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{9} + 223092870 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{10} - 67603900 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{11} + 16900975 \, \sqrt {a x^{\frac {1}{3}} + b} b^{12}\right )} b}{a^{12}} + \frac {13 \, {\left (1300075 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {27}{2}} - 18253053 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {25}{2}} b + 119041650 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {23}{2}} b^{2} - 478056150 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} b^{3} + 1320944625 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} b^{4} - 2657429775 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b^{5} + 4015671660 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{6} - 4633467300 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{7} + 4106936925 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{8} - 2788660875 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{9} + 1434168450 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{10} - 547591590 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{11} + 152108775 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{12} - 35102025 \, \sqrt {a x^{\frac {1}{3}} + b} b^{13}\right )}}{a^{12}}\right )}}{152108775 \, a} \]
-8388608/152108775*b^(27/2)/a^13 + 2/152108775*(27*(676039*(a*x^(1/3) + b) ^(25/2) - 8817900*(a*x^(1/3) + b)^(23/2)*b + 53117350*(a*x^(1/3) + b)^(21/ 2)*b^2 - 195695500*(a*x^(1/3) + b)^(19/2)*b^3 + 492116625*(a*x^(1/3) + b)^ (17/2)*b^4 - 892371480*(a*x^(1/3) + b)^(15/2)*b^5 + 1201269300*(a*x^(1/3) + b)^(13/2)*b^6 - 1216870200*(a*x^(1/3) + b)^(11/2)*b^7 + 929553625*(a*x^( 1/3) + b)^(9/2)*b^8 - 531173500*(a*x^(1/3) + b)^(7/2)*b^9 + 223092870*(a*x ^(1/3) + b)^(5/2)*b^10 - 67603900*(a*x^(1/3) + b)^(3/2)*b^11 + 16900975*sq rt(a*x^(1/3) + b)*b^12)*b/a^12 + 13*(1300075*(a*x^(1/3) + b)^(27/2) - 1825 3053*(a*x^(1/3) + b)^(25/2)*b + 119041650*(a*x^(1/3) + b)^(23/2)*b^2 - 478 056150*(a*x^(1/3) + b)^(21/2)*b^3 + 1320944625*(a*x^(1/3) + b)^(19/2)*b^4 - 2657429775*(a*x^(1/3) + b)^(17/2)*b^5 + 4015671660*(a*x^(1/3) + b)^(15/2 )*b^6 - 4633467300*(a*x^(1/3) + b)^(13/2)*b^7 + 4106936925*(a*x^(1/3) + b) ^(11/2)*b^8 - 2788660875*(a*x^(1/3) + b)^(9/2)*b^9 + 1434168450*(a*x^(1/3) + b)^(7/2)*b^10 - 547591590*(a*x^(1/3) + b)^(5/2)*b^11 + 152108775*(a*x^( 1/3) + b)^(3/2)*b^12 - 35102025*sqrt(a*x^(1/3) + b)*b^13)/a^12)/a
Timed out. \[ \int x^3 \sqrt {b x^{2/3}+a x} \, dx=\int x^3\,\sqrt {a\,x+b\,x^{2/3}} \,d x \]