Integrand size = 19, antiderivative size = 401 \[ \int \frac {x^4}{\sqrt {b x^{2/3}+a x}} \, dx=\frac {8388608 b^{12} \sqrt {b x^{2/3}+a x}}{11700675 a^{13}}-\frac {16777216 b^{13} \sqrt {b x^{2/3}+a x}}{11700675 a^{14} \sqrt [3]{x}}-\frac {2097152 b^{11} \sqrt [3]{x} \sqrt {b x^{2/3}+a x}}{3900225 a^{12}}+\frac {1048576 b^{10} x^{2/3} \sqrt {b x^{2/3}+a x}}{2340135 a^{11}}-\frac {131072 b^9 x \sqrt {b x^{2/3}+a x}}{334305 a^{10}}+\frac {65536 b^8 x^{4/3} \sqrt {b x^{2/3}+a x}}{185725 a^9}-\frac {180224 b^7 x^{5/3} \sqrt {b x^{2/3}+a x}}{557175 a^8}+\frac {1171456 b^6 x^2 \sqrt {b x^{2/3}+a x}}{3900225 a^7}-\frac {73216 b^5 x^{7/3} \sqrt {b x^{2/3}+a x}}{260015 a^6}+\frac {36608 b^4 x^{8/3} \sqrt {b x^{2/3}+a x}}{137655 a^5}-\frac {9152 b^3 x^3 \sqrt {b x^{2/3}+a x}}{36225 a^4}+\frac {416 b^2 x^{10/3} \sqrt {b x^{2/3}+a x}}{1725 a^3}-\frac {52 b x^{11/3} \sqrt {b x^{2/3}+a x}}{225 a^2}+\frac {2 x^4 \sqrt {b x^{2/3}+a x}}{9 a} \]
8388608/11700675*b^12*(b*x^(2/3)+a*x)^(1/2)/a^13-16777216/11700675*b^13*(b *x^(2/3)+a*x)^(1/2)/a^14/x^(1/3)-2097152/3900225*b^11*x^(1/3)*(b*x^(2/3)+a *x)^(1/2)/a^12+1048576/2340135*b^10*x^(2/3)*(b*x^(2/3)+a*x)^(1/2)/a^11-131 072/334305*b^9*x*(b*x^(2/3)+a*x)^(1/2)/a^10+65536/185725*b^8*x^(4/3)*(b*x^ (2/3)+a*x)^(1/2)/a^9-180224/557175*b^7*x^(5/3)*(b*x^(2/3)+a*x)^(1/2)/a^8+1 171456/3900225*b^6*x^2*(b*x^(2/3)+a*x)^(1/2)/a^7-73216/260015*b^5*x^(7/3)* (b*x^(2/3)+a*x)^(1/2)/a^6+36608/137655*b^4*x^(8/3)*(b*x^(2/3)+a*x)^(1/2)/a ^5-9152/36225*b^3*x^3*(b*x^(2/3)+a*x)^(1/2)/a^4+416/1725*b^2*x^(10/3)*(b*x ^(2/3)+a*x)^(1/2)/a^3-52/225*b*x^(11/3)*(b*x^(2/3)+a*x)^(1/2)/a^2+2/9*x^4* (b*x^(2/3)+a*x)^(1/2)/a
Time = 0.19 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.46 \[ \int \frac {x^4}{\sqrt {b x^{2/3}+a x}} \, dx=\frac {2 \sqrt {b x^{2/3}+a x} \left (-8388608 b^{13}+4194304 a b^{12} \sqrt [3]{x}-3145728 a^2 b^{11} x^{2/3}+2621440 a^3 b^{10} x-2293760 a^4 b^9 x^{4/3}+2064384 a^5 b^8 x^{5/3}-1892352 a^6 b^7 x^2+1757184 a^7 b^6 x^{7/3}-1647360 a^8 b^5 x^{8/3}+1555840 a^9 b^4 x^3-1478048 a^{10} b^3 x^{10/3}+1410864 a^{11} b^2 x^{11/3}-1352078 a^{12} b x^4+1300075 a^{13} x^{13/3}\right )}{11700675 a^{14} \sqrt [3]{x}} \]
(2*Sqrt[b*x^(2/3) + a*x]*(-8388608*b^13 + 4194304*a*b^12*x^(1/3) - 3145728 *a^2*b^11*x^(2/3) + 2621440*a^3*b^10*x - 2293760*a^4*b^9*x^(4/3) + 2064384 *a^5*b^8*x^(5/3) - 1892352*a^6*b^7*x^2 + 1757184*a^7*b^6*x^(7/3) - 1647360 *a^8*b^5*x^(8/3) + 1555840*a^9*b^4*x^3 - 1478048*a^10*b^3*x^(10/3) + 14108 64*a^11*b^2*x^(11/3) - 1352078*a^12*b*x^4 + 1300075*a^13*x^(13/3)))/(11700 675*a^14*x^(1/3))
Time = 0.92 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.17, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {1922, 1922, 1922, 1922, 1922, 1922, 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.
| \(\displaystyle \int \frac {x^4}{\sqrt {a x+b x^{2/3}}} \, dx\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^4 \sqrt {a x+b x^{2/3}}}{9 a}-\frac {26 b \int \frac {x^{11/3}}{\sqrt {x^{2/3} b+a x}}dx}{27 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^4 \sqrt {a x+b x^{2/3}}}{9 a}-\frac {26 b \left (\frac {6 x^{11/3} \sqrt {a x+b x^{2/3}}}{25 a}-\frac {24 b \int \frac {x^{10/3}}{\sqrt {x^{2/3} b+a x}}dx}{25 a}\right )}{27 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^4 \sqrt {a x+b x^{2/3}}}{9 a}-\frac {26 b \left (\frac {6 x^{11/3} \sqrt {a x+b x^{2/3}}}{25 a}-\frac {24 b \left (\frac {6 x^{10/3} \sqrt {a x+b x^{2/3}}}{23 a}-\frac {22 b \int \frac {x^3}{\sqrt {x^{2/3} b+a x}}dx}{23 a}\right )}{25 a}\right )}{27 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^4 \sqrt {a x+b x^{2/3}}}{9 a}-\frac {26 b \left (\frac {6 x^{11/3} \sqrt {a x+b x^{2/3}}}{25 a}-\frac {24 b \left (\frac {6 x^{10/3} \sqrt {a x+b x^{2/3}}}{23 a}-\frac {22 b \left (\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \int \frac {x^{8/3}}{\sqrt {x^{2/3} b+a x}}dx}{21 a}\right )}{23 a}\right )}{25 a}\right )}{27 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^4 \sqrt {a x+b x^{2/3}}}{9 a}-\frac {26 b \left (\frac {6 x^{11/3} \sqrt {a x+b x^{2/3}}}{25 a}-\frac {24 b \left (\frac {6 x^{10/3} \sqrt {a x+b x^{2/3}}}{23 a}-\frac {22 b \left (\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \int \frac {x^{7/3}}{\sqrt {x^{2/3} b+a x}}dx}{19 a}\right )}{21 a}\right )}{23 a}\right )}{25 a}\right )}{27 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^4 \sqrt {a x+b x^{2/3}}}{9 a}-\frac {26 b \left (\frac {6 x^{11/3} \sqrt {a x+b x^{2/3}}}{25 a}-\frac {24 b \left (\frac {6 x^{10/3} \sqrt {a x+b x^{2/3}}}{23 a}-\frac {22 b \left (\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \int \frac {x^2}{\sqrt {x^{2/3} b+a x}}dx}{17 a}\right )}{19 a}\right )}{21 a}\right )}{23 a}\right )}{25 a}\right )}{27 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^4 \sqrt {a x+b x^{2/3}}}{9 a}-\frac {26 b \left (\frac {6 x^{11/3} \sqrt {a x+b x^{2/3}}}{25 a}-\frac {24 b \left (\frac {6 x^{10/3} \sqrt {a x+b x^{2/3}}}{23 a}-\frac {22 b \left (\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \int \frac {x^{5/3}}{\sqrt {x^{2/3} b+a x}}dx}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{23 a}\right )}{25 a}\right )}{27 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^4 \sqrt {a x+b x^{2/3}}}{9 a}-\frac {26 b \left (\frac {6 x^{11/3} \sqrt {a x+b x^{2/3}}}{25 a}-\frac {24 b \left (\frac {6 x^{10/3} \sqrt {a x+b x^{2/3}}}{23 a}-\frac {22 b \left (\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \int \frac {x^{4/3}}{\sqrt {x^{2/3} b+a x}}dx}{13 a}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{23 a}\right )}{25 a}\right )}{27 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^4 \sqrt {a x+b x^{2/3}}}{9 a}-\frac {26 b \left (\frac {6 x^{11/3} \sqrt {a x+b x^{2/3}}}{25 a}-\frac {24 b \left (\frac {6 x^{10/3} \sqrt {a x+b x^{2/3}}}{23 a}-\frac {22 b \left (\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \int \frac {x}{\sqrt {x^{2/3} b+a x}}dx}{11 a}\right )}{13 a}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{23 a}\right )}{25 a}\right )}{27 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^4 \sqrt {a x+b x^{2/3}}}{9 a}-\frac {26 b \left (\frac {6 x^{11/3} \sqrt {a x+b x^{2/3}}}{25 a}-\frac {24 b \left (\frac {6 x^{10/3} \sqrt {a x+b x^{2/3}}}{23 a}-\frac {22 b \left (\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \int \frac {x^{2/3}}{\sqrt {x^{2/3} b+a x}}dx}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{23 a}\right )}{25 a}\right )}{27 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^4 \sqrt {a x+b x^{2/3}}}{9 a}-\frac {26 b \left (\frac {6 x^{11/3} \sqrt {a x+b x^{2/3}}}{25 a}-\frac {24 b \left (\frac {6 x^{10/3} \sqrt {a x+b x^{2/3}}}{23 a}-\frac {22 b \left (\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \left (\frac {6 x^{2/3} \sqrt {a x+b x^{2/3}}}{7 a}-\frac {6 b \int \frac {\sqrt [3]{x}}{\sqrt {x^{2/3} b+a x}}dx}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{23 a}\right )}{25 a}\right )}{27 a}\) |
| \(\Big \downarrow \) 1922 |
| \(\displaystyle \frac {2 x^4 \sqrt {a x+b x^{2/3}}}{9 a}-\frac {26 b \left (\frac {6 x^{11/3} \sqrt {a x+b x^{2/3}}}{25 a}-\frac {24 b \left (\frac {6 x^{10/3} \sqrt {a x+b x^{2/3}}}{23 a}-\frac {22 b \left (\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \left (\frac {6 x^{2/3} \sqrt {a x+b x^{2/3}}}{7 a}-\frac {6 b \left (\frac {6 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{5 a}-\frac {4 b \int \frac {1}{\sqrt {x^{2/3} b+a x}}dx}{5 a}\right )}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{23 a}\right )}{25 a}\right )}{27 a}\) |
| \(\Big \downarrow \) 1908 |
| \(\displaystyle \frac {2 x^4 \sqrt {a x+b x^{2/3}}}{9 a}-\frac {26 b \left (\frac {6 x^{11/3} \sqrt {a x+b x^{2/3}}}{25 a}-\frac {24 b \left (\frac {6 x^{10/3} \sqrt {a x+b x^{2/3}}}{23 a}-\frac {22 b \left (\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \left (\frac {6 x^{2/3} \sqrt {a x+b x^{2/3}}}{7 a}-\frac {6 b \left (\frac {6 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{5 a}-\frac {4 b \left (\frac {2 \sqrt {a x+b x^{2/3}}}{a}-\frac {2 b \int \frac {1}{\sqrt [3]{x} \sqrt {x^{2/3} b+a x}}dx}{3 a}\right )}{5 a}\right )}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{23 a}\right )}{25 a}\right )}{27 a}\) |
| \(\Big \downarrow \) 1920 |
| \(\displaystyle \frac {2 x^4 \sqrt {a x+b x^{2/3}}}{9 a}-\frac {26 b \left (\frac {6 x^{11/3} \sqrt {a x+b x^{2/3}}}{25 a}-\frac {24 b \left (\frac {6 x^{10/3} \sqrt {a x+b x^{2/3}}}{23 a}-\frac {22 b \left (\frac {2 x^3 \sqrt {a x+b x^{2/3}}}{7 a}-\frac {20 b \left (\frac {6 x^{8/3} \sqrt {a x+b x^{2/3}}}{19 a}-\frac {18 b \left (\frac {6 x^{7/3} \sqrt {a x+b x^{2/3}}}{17 a}-\frac {16 b \left (\frac {2 x^2 \sqrt {a x+b x^{2/3}}}{5 a}-\frac {14 b \left (\frac {6 x^{5/3} \sqrt {a x+b x^{2/3}}}{13 a}-\frac {12 b \left (\frac {6 x^{4/3} \sqrt {a x+b x^{2/3}}}{11 a}-\frac {10 b \left (\frac {2 x \sqrt {a x+b x^{2/3}}}{3 a}-\frac {8 b \left (\frac {6 x^{2/3} \sqrt {a x+b x^{2/3}}}{7 a}-\frac {6 b \left (\frac {6 \sqrt [3]{x} \sqrt {a x+b x^{2/3}}}{5 a}-\frac {4 b \left (\frac {2 \sqrt {a x+b x^{2/3}}}{a}-\frac {4 b \sqrt {a x+b x^{2/3}}}{a^2 \sqrt [3]{x}}\right )}{5 a}\right )}{7 a}\right )}{9 a}\right )}{11 a}\right )}{13 a}\right )}{15 a}\right )}{17 a}\right )}{19 a}\right )}{21 a}\right )}{23 a}\right )}{25 a}\right )}{27 a}\) |
(2*x^4*Sqrt[b*x^(2/3) + a*x])/(9*a) - (26*b*((6*x^(11/3)*Sqrt[b*x^(2/3) + a*x])/(25*a) - (24*b*((6*x^(10/3)*Sqrt[b*x^(2/3) + a*x])/(23*a) - (22*b*(( 2*x^3*Sqrt[b*x^(2/3) + a*x])/(7*a) - (20*b*((6*x^(8/3)*Sqrt[b*x^(2/3) + a* x])/(19*a) - (18*b*((6*x^(7/3)*Sqrt[b*x^(2/3) + a*x])/(17*a) - (16*b*((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*Sq rt[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)))/(17*a)))/(19*a)))/(21*a)))/(23*a)))/(25* a)))/(27*a)
3.2.85.3.1 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 = 2.07 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.42
| method | result | size |
| derivativedivides | \(\frac {2 x^{\frac {1}{3}} \left (b +a \,x^{\frac {1}{3}}\right ) \left (1300075 a^{13} x^{\frac {13}{3}}-1352078 a^{12} b \,x^{4}+1410864 a^{11} b^{2} x^{\frac {11}{3}}-1478048 a^{10} b^{3} x^{\frac {10}{3}}+1555840 a^{9} b^{4} x^{3}-1647360 a^{8} b^{5} x^{\frac {8}{3}}+1757184 a^{7} b^{6} x^{\frac {7}{3}}-1892352 a^{6} b^{7} x^{2}+2064384 a^{5} b^{8} x^{\frac {5}{3}}-2293760 a^{4} b^{9} x^{\frac {4}{3}}+2621440 a^{3} b^{10} x -3145728 a^{2} b^{11} x^{\frac {2}{3}}+4194304 a \,b^{12} x^{\frac {1}{3}}-8388608 b^{13}\right )}{11700675 \sqrt {b \,x^{\frac {2}{3}}+a x}\, a^{14}}\) | \(167\) |
| default | \(\frac {2 x^{\frac {1}{3}} \left (b +a \,x^{\frac {1}{3}}\right ) \left (1300075 a^{13} x^{\frac {13}{3}}-1352078 a^{12} b \,x^{4}+1410864 a^{11} b^{2} x^{\frac {11}{3}}-1478048 a^{10} b^{3} x^{\frac {10}{3}}+1555840 a^{9} b^{4} x^{3}-1647360 a^{8} b^{5} x^{\frac {8}{3}}+1757184 a^{7} b^{6} x^{\frac {7}{3}}-1892352 a^{6} b^{7} x^{2}+2064384 a^{5} b^{8} x^{\frac {5}{3}}-2293760 a^{4} b^{9} x^{\frac {4}{3}}+2621440 a^{3} b^{10} x -3145728 a^{2} b^{11} x^{\frac {2}{3}}+4194304 a \,b^{12} x^{\frac {1}{3}}-8388608 b^{13}\right )}{11700675 \sqrt {b \,x^{\frac {2}{3}}+a x}\, a^{14}}\) | \(167\) |
2/11700675*x^(1/3)*(b+a*x^(1/3))*(1300075*a^13*x^(13/3)-1352078*a^12*b*x^4 +1410864*a^11*b^2*x^(11/3)-1478048*a^10*b^3*x^(10/3)+1555840*a^9*b^4*x^3-1 647360*a^8*b^5*x^(8/3)+1757184*a^7*b^6*x^(7/3)-1892352*a^6*b^7*x^2+2064384 *a^5*b^8*x^(5/3)-2293760*a^4*b^9*x^(4/3)+2621440*a^3*b^10*x-3145728*a^2*b^ 11*x^(2/3)+4194304*a*b^12*x^(1/3)-8388608*b^13)/(b*x^(2/3)+a*x)^(1/2)/a^14
Leaf count of result is larger than twice the leaf count of optimal. 1294 vs. \(2 (299) = 598\).
Time = 148.43 (sec) , antiderivative size = 1294, normalized size of antiderivative = 3.23 \[ \int \frac {x^4}{\sqrt {b x^{2/3}+a x}} \, dx=\text {Too large to display} \]
1/11700675*((211106232532992*b^19 + 43980465111040*b^18 + 206158430208*(64 *a^3 - 3)*b^16 - 4123168604160*b^17 - 1073741824*(11264*a^3 - 53)*b^15 + 1 5143273600*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*(48 816*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*(43743616* a^12 + 89111*a^9)*b^6 - 638976*(1652476*a^12 + 935*a^9)*b^5 + 3264*(360854 3232*a^15 + 64599808*a^12 + 2145*a^9)*b^4 + 578816*(13049856*a^15 - 27313* a^12)*b^3 + 217056*(6211584*a^15 + 2353*a^12)*b^2 - 156009*(2547712*a^15 + 39*a^12)*b)*x + 2*(1300075*(16777216*a^13*b^6 + 6291456*a^13*b^5 + 196608 *a^13*b^4 - 262144*a^16 - 114688*a^13*b^3 - 2304*a^13*b^2 + 864*a^13*b - 2 7*a^13)*x^5 - 1478048*(16777216*a^10*b^9 + 6291456*a^10*b^8 + 196608*a^10* 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 + 1757184*(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 - 2293760*(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 + 4194304*(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 ...
\[ \int \frac {x^4}{\sqrt {b x^{2/3}+a x}} \, dx=\int \frac {x^{4}}{\sqrt {a x + b x^{\frac {2}{3}}}}\, dx \]
\[ \int \frac {x^4}{\sqrt {b x^{2/3}+a x}} \, dx=\int { \frac {x^{4}}{\sqrt {a x + b x^{\frac {2}{3}}}} \,d x } \]
Time = 0.65 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.51 \[ \int \frac {x^4}{\sqrt {b x^{2/3}+a x}} \, dx=\frac {16777216 \, b^{\frac {27}{2}}}{11700675 \, a^{14}} + \frac {2 \, {\left (1300075 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {27}{2}} - 18253053 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {25}{2}} b + 119041650 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {23}{2}} b^{2} - 478056150 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} b^{3} + 1320944625 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} b^{4} - 2657429775 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b^{5} + 4015671660 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{6} - 4633467300 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{7} + 4106936925 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{8} - 2788660875 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{9} + 1434168450 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{10} - 547591590 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{11} + 152108775 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{12} - 35102025 \, \sqrt {a x^{\frac {1}{3}} + b} b^{13}\right )}}{11700675 \, a^{14}} \]
16777216/11700675*b^(27/2)/a^14 + 2/11700675*(1300075*(a*x^(1/3) + b)^(27/ 2) - 18253053*(a*x^(1/3) + b)^(25/2)*b + 119041650*(a*x^(1/3) + b)^(23/2)* b^2 - 478056150*(a*x^(1/3) + b)^(21/2)*b^3 + 1320944625*(a*x^(1/3) + b)^(1 9/2)*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 + 1521087 75*(a*x^(1/3) + b)^(3/2)*b^12 - 35102025*sqrt(a*x^(1/3) + b)*b^13)/a^14
Timed out. \[ \int \frac {x^4}{\sqrt {b x^{2/3}+a x}} \, dx=\int \frac {x^4}{\sqrt {a\,x+b\,x^{2/3}}} \,d x \]