Integrand size = 20, antiderivative size = 667 \[ \int \frac {c+d x}{x^2 \left (a+b x^2\right )^{4/3}} \, dx=\frac {3 (a d-b c x)}{2 a^2 \sqrt [3]{a+b x^2}}-\frac {c \left (a+b x^2\right )^{2/3}}{a^2 x}-\frac {5 b c x}{2 a^2 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}+\frac {\sqrt {3} d \arctan \left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^2}}{\sqrt {3} \sqrt [3]{a}}\right )}{2 a^{4/3}}+\frac {5 \sqrt [4]{3} \sqrt {2+\sqrt {3}} c \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )|-7+4 \sqrt {3}\right )}{4 a^{5/3} x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {5 c \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt {2} \sqrt [4]{3} a^{5/3} x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {d \log (x)}{2 a^{4/3}}+\frac {3 d \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{4 a^{4/3}} \] Output:
3/2*(-b*c*x+a*d)/a^2/(b*x^2+a)^(1/3)-c*(b*x^2+a)^(2/3)/a^2/x-5/2*b*c*x/a^2 /((1-3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3))+1/2*3^(1/2)*d*arctan(1/3*(a^(1/3)+2 *(b*x^2+a)^(1/3))*3^(1/2)/a^(1/3))/a^(4/3)+5/4*3^(1/4)*(1/2*6^(1/2)+1/2*2^ (1/2))*c*(a^(1/3)-(b*x^2+a)^(1/3))*((a^(2/3)+a^(1/3)*(b*x^2+a)^(1/3)+(b*x^ 2+a)^(2/3))/((1-3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3))^2)^(1/2)*EllipticE(((1+3 ^(1/2))*a^(1/3)-(b*x^2+a)^(1/3))/((1-3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3)),2*I -I*3^(1/2))/a^(5/3)/x/(-a^(1/3)*(a^(1/3)-(b*x^2+a)^(1/3))/((1-3^(1/2))*a^( 1/3)-(b*x^2+a)^(1/3))^2)^(1/2)-5/6*c*(a^(1/3)-(b*x^2+a)^(1/3))*((a^(2/3)+a ^(1/3)*(b*x^2+a)^(1/3)+(b*x^2+a)^(2/3))/((1-3^(1/2))*a^(1/3)-(b*x^2+a)^(1/ 3))^2)^(1/2)*EllipticF(((1+3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3))/((1-3^(1/2))* a^(1/3)-(b*x^2+a)^(1/3)),2*I-I*3^(1/2))*2^(1/2)*3^(3/4)/a^(5/3)/x/(-a^(1/3 )*(a^(1/3)-(b*x^2+a)^(1/3))/((1-3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3))^2)^(1/2) -1/2*d*ln(x)/a^(4/3)+3/4*d*ln(a^(1/3)-(b*x^2+a)^(1/3))/a^(4/3)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.12 \[ \int \frac {c+d x}{x^2 \left (a+b x^2\right )^{4/3}} \, dx=\frac {-2 c \sqrt [3]{1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {4}{3},\frac {1}{2},-\frac {b x^2}{a}\right )+3 d x \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},1,\frac {2}{3},1+\frac {b x^2}{a}\right )}{2 a x \sqrt [3]{a+b x^2}} \] Input:
Integrate[(c + d*x)/(x^2*(a + b*x^2)^(4/3)),x]
Output:
(-2*c*(1 + (b*x^2)/a)^(1/3)*Hypergeometric2F1[-1/2, 4/3, 1/2, -((b*x^2)/a) ] + 3*d*x*Hypergeometric2F1[-1/3, 1, 2/3, 1 + (b*x^2)/a])/(2*a*x*(a + b*x^ 2)^(1/3))
Time = 0.72 (sec) , antiderivative size = 733, normalized size of antiderivative = 1.10, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {542, 243, 61, 67, 16, 253, 264, 233, 833, 760, 1082, 217, 2418}
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.
| \(\displaystyle \int \frac {c+d x}{x^2 \left (a+b x^2\right )^{4/3}} \, dx\) |
| \(\Big \downarrow \) 542 |
| \(\displaystyle c \int \frac {1}{x^2 \left (b x^2+a\right )^{4/3}}dx+d \int \frac {1}{x \left (b x^2+a\right )^{4/3}}dx\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle c \int \frac {1}{x^2 \left (b x^2+a\right )^{4/3}}dx+\frac {1}{2} d \int \frac {1}{x^2 \left (b x^2+a\right )^{4/3}}dx^2\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle c \int \frac {1}{x^2 \left (b x^2+a\right )^{4/3}}dx+\frac {1}{2} d \left (\frac {\int \frac {1}{x^2 \sqrt [3]{b x^2+a}}dx^2}{a}+\frac {3}{a \sqrt [3]{a+b x^2}}\right )\) |
| \(\Big \downarrow \) 67 |
| \(\displaystyle \frac {1}{2} d \left (\frac {\frac {3}{2} \int \frac {1}{x^4+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}-\frac {3 \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x^2}}\right )+c \int \frac {1}{x^2 \left (b x^2+a\right )^{4/3}}dx\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{2} d \left (\frac {\frac {3}{2} \int \frac {1}{x^4+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x^2}}\right )+c \int \frac {1}{x^2 \left (b x^2+a\right )^{4/3}}dx\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {1}{2} d \left (\frac {\frac {3}{2} \int \frac {1}{x^4+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x^2}}\right )+c \left (\frac {5 \int \frac {1}{x^2 \sqrt [3]{b x^2+a}}dx}{2 a}+\frac {3}{2 a x \sqrt [3]{a+b x^2}}\right )\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {1}{2} d \left (\frac {\frac {3}{2} \int \frac {1}{x^4+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x^2}}\right )+c \left (\frac {5 \left (\frac {b \int \frac {1}{\sqrt [3]{b x^2+a}}dx}{3 a}-\frac {\left (a+b x^2\right )^{2/3}}{a x}\right )}{2 a}+\frac {3}{2 a x \sqrt [3]{a+b x^2}}\right )\) |
| \(\Big \downarrow \) 233 |
| \(\displaystyle \frac {1}{2} d \left (\frac {\frac {3}{2} \int \frac {1}{x^4+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x^2}}\right )+c \left (\frac {5 \left (\frac {\sqrt {b x^2} \int \frac {\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}}{2 a x}-\frac {\left (a+b x^2\right )^{2/3}}{a x}\right )}{2 a}+\frac {3}{2 a x \sqrt [3]{a+b x^2}}\right )\) |
| \(\Big \downarrow \) 833 |
| \(\displaystyle \frac {1}{2} d \left (\frac {\frac {3}{2} \int \frac {1}{x^4+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x^2}}\right )+c \left (\frac {5 \left (\frac {\sqrt {b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}\right )}{2 a x}-\frac {\left (a+b x^2\right )^{2/3}}{a x}\right )}{2 a}+\frac {3}{2 a x \sqrt [3]{a+b x^2}}\right )\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle c \left (\frac {5 \left (\frac {\sqrt {b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}\right )}{2 a x}-\frac {\left (a+b x^2\right )^{2/3}}{a x}\right )}{2 a}+\frac {3}{2 a x \sqrt [3]{a+b x^2}}\right )+\frac {1}{2} d \left (\frac {\frac {3}{2} \int \frac {1}{x^4+a^{2/3}+\sqrt [3]{a} \sqrt [3]{b x^2+a}}d\sqrt [3]{b x^2+a}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x^2}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle c \left (\frac {5 \left (\frac {\sqrt {b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}\right )}{2 a x}-\frac {\left (a+b x^2\right )^{2/3}}{a x}\right )}{2 a}+\frac {3}{2 a x \sqrt [3]{a+b x^2}}\right )+\frac {1}{2} d \left (\frac {-\frac {3 \int \frac {1}{-x^4-3}d\left (\frac {2 \sqrt [3]{b x^2+a}}{\sqrt [3]{a}}+1\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x^2}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle c \left (\frac {5 \left (\frac {\sqrt {b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}\right )}{2 a x}-\frac {\left (a+b x^2\right )^{2/3}}{a x}\right )}{2 a}+\frac {3}{2 a x \sqrt [3]{a+b x^2}}\right )+\frac {1}{2} d \left (\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x^2}}\right )\) |
| \(\Big \downarrow \) 2418 |
| \(\displaystyle c \left (\frac {5 \left (\frac {\sqrt {b x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {2 \sqrt {b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )}{2 a x}-\frac {\left (a+b x^2\right )^{2/3}}{a x}\right )}{2 a}+\frac {3}{2 a x \sqrt [3]{a+b x^2}}\right )+\frac {1}{2} d \left (\frac {\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}-\frac {\log \left (x^2\right )}{2 \sqrt [3]{a}}}{a}+\frac {3}{a \sqrt [3]{a+b x^2}}\right )\) |
Input:
Int[(c + d*x)/(x^2*(a + b*x^2)^(4/3)),x]
Output:
c*(3/(2*a*x*(a + b*x^2)^(1/3)) + (5*(-((a + b*x^2)^(2/3)/(a*x)) + (Sqrt[b* x^2]*((-2*Sqrt[b*x^2])/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3)) + (3^(1 /4)*Sqrt[2 + Sqrt[3]]*a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a + b*x^2 )^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(S qrt[b*x^2]*Sqrt[-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - Sqrt[3])*a ^(1/3) - (a + b*x^2)^(1/3))^2)]) - (2*Sqrt[2 - Sqrt[3]]*(1 + Sqrt[3])*a^(1 /3)*(a^(1/3) - (a + b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a + b*x^2)^(1/3 ) + (a + b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))^2]*Elli pticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))/((1 - Sqrt[3])*a^ (1/3) - (a + b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[b*x^2]*Sqrt[-( (a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^ 2)^(1/3))^2)])))/(2*a*x)))/(2*a)) + (d*(3/(a*(a + b*x^2)^(1/3)) + ((Sqrt[3 ]*ArcTan[(1 + (2*(a + b*x^2)^(1/3))/a^(1/3))/Sqrt[3]])/a^(1/3) - Log[x^2]/ (2*a^(1/3)) + (3*Log[a^(1/3) - (a + b*x^2)^(1/3)])/(2*a^(1/3)))/a))/2
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / ; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[3*(Sqrt[b*x^2]/(2*b*x)) Subst[Int[x/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b }, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c Int[x^m*(a + b*x^2)^p, x], x] + Simp[d Int[x^(m + 1)*(a + b*x^2 )^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && IntegerQ[m] && !IntegerQ[2*p]
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) *s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x ] && NegQ[a]
Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] ], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r) Int[1/Sqrt[a + b*x ^3], x], x] + Simp[1/r Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x ]] /; FreeQ[{a, b}, x] && NegQ[a]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Simplify[(1 + Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 + Sqrt[3])*(d/c) ]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x] + S imp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( (1 - Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
\[\int \frac {d x +c}{x^{2} \left (b \,x^{2}+a \right )^{\frac {4}{3}}}d x\]
Input:
int((d*x+c)/x^2/(b*x^2+a)^(4/3),x)
Output:
int((d*x+c)/x^2/(b*x^2+a)^(4/3),x)
Timed out. \[ \int \frac {c+d x}{x^2 \left (a+b x^2\right )^{4/3}} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)/x^2/(b*x^2+a)^(4/3),x, algorithm="fricas")
Output:
Timed out
Time = 5.49 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.11 \[ \int \frac {c+d x}{x^2 \left (a+b x^2\right )^{4/3}} \, dx=- \frac {d \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {4}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{2 b^{\frac {4}{3}} x^{\frac {8}{3}} \Gamma \left (\frac {7}{3}\right )} - \frac {c {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {4}{3} \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{a^{\frac {4}{3}} x} \] Input:
integrate((d*x+c)/x**2/(b*x**2+a)**(4/3),x)
Output:
-d*gamma(4/3)*hyper((4/3, 4/3), (7/3,), a*exp_polar(I*pi)/(b*x**2))/(2*b** (4/3)*x**(8/3)*gamma(7/3)) - c*hyper((-1/2, 4/3), (1/2,), b*x**2*exp_polar (I*pi)/a)/(a**(4/3)*x)
\[ \int \frac {c+d x}{x^2 \left (a+b x^2\right )^{4/3}} \, dx=\int { \frac {d x + c}{{\left (b x^{2} + a\right )}^{\frac {4}{3}} x^{2}} \,d x } \] Input:
integrate((d*x+c)/x^2/(b*x^2+a)^(4/3),x, algorithm="maxima")
Output:
integrate((d*x + c)/((b*x^2 + a)^(4/3)*x^2), x)
\[ \int \frac {c+d x}{x^2 \left (a+b x^2\right )^{4/3}} \, dx=\int { \frac {d x + c}{{\left (b x^{2} + a\right )}^{\frac {4}{3}} x^{2}} \,d x } \] Input:
integrate((d*x+c)/x^2/(b*x^2+a)^(4/3),x, algorithm="giac")
Output:
integrate((d*x + c)/((b*x^2 + a)^(4/3)*x^2), x)
Time = 9.89 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.27 \[ \int \frac {c+d x}{x^2 \left (a+b x^2\right )^{4/3}} \, dx=\frac {3\,d}{2\,a\,{\left (b\,x^2+a\right )}^{1/3}}+\frac {d\,\ln \left (18\,a\,d^2\,{\left (b\,x^2+a\right )}^{1/3}-18\,a^{4/3}\,d^2\right )}{2\,a^{4/3}}-\frac {\ln \left (18\,a\,d^2\,{\left (b\,x^2+a\right )}^{1/3}-\frac {9\,a^{4/3}\,{\left (d-\sqrt {3}\,d\,1{}\mathrm {i}\right )}^2}{2}\right )\,\left (d-\sqrt {3}\,d\,1{}\mathrm {i}\right )}{4\,a^{4/3}}-\frac {\ln \left (18\,a\,d^2\,{\left (b\,x^2+a\right )}^{1/3}-\frac {9\,a^{4/3}\,{\left (d+\sqrt {3}\,d\,1{}\mathrm {i}\right )}^2}{2}\right )\,\left (d+\sqrt {3}\,d\,1{}\mathrm {i}\right )}{4\,a^{4/3}}-\frac {3\,c\,{\left (\frac {a}{b\,x^2}+1\right )}^{4/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {4}{3},\frac {11}{6};\ \frac {17}{6};\ -\frac {a}{b\,x^2}\right )}{11\,x\,{\left (b\,x^2+a\right )}^{4/3}} \] Input:
int((c + d*x)/(x^2*(a + b*x^2)^(4/3)),x)
Output:
(3*d)/(2*a*(a + b*x^2)^(1/3)) + (d*log(18*a*d^2*(a + b*x^2)^(1/3) - 18*a^( 4/3)*d^2))/(2*a^(4/3)) - (log(18*a*d^2*(a + b*x^2)^(1/3) - (9*a^(4/3)*(d - 3^(1/2)*d*1i)^2)/2)*(d - 3^(1/2)*d*1i))/(4*a^(4/3)) - (log(18*a*d^2*(a + b*x^2)^(1/3) - (9*a^(4/3)*(d + 3^(1/2)*d*1i)^2)/2)*(d + 3^(1/2)*d*1i))/(4* a^(4/3)) - (3*c*(a/(b*x^2) + 1)^(4/3)*hypergeom([4/3, 11/6], 17/6, -a/(b*x ^2)))/(11*x*(a + b*x^2)^(4/3))
\[ \int \frac {c+d x}{x^2 \left (a+b x^2\right )^{4/3}} \, dx=\left (\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{3}} a \,x^{2}+\left (b \,x^{2}+a \right )^{\frac {1}{3}} b \,x^{4}}d x \right ) c +\left (\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{3}} a x +\left (b \,x^{2}+a \right )^{\frac {1}{3}} b \,x^{3}}d x \right ) d \] Input:
int((d*x+c)/x^2/(b*x^2+a)^(4/3),x)
Output:
int(1/((a + b*x**2)**(1/3)*a*x**2 + (a + b*x**2)**(1/3)*b*x**4),x)*c + int (1/((a + b*x**2)**(1/3)*a*x + (a + b*x**2)**(1/3)*b*x**3),x)*d