3.25.92 \(\int \frac {1}{(a+b x+c x^2)^{7/3}} \, dx\) [2492]

3.25.92.1 Optimal result

Integrand size = 14, antiderivative size = 993 \[ \int \frac {1}{\left (a+b x+c x^2\right )^{7/3}} \, dx=-\frac {3 (b+2 c x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}+\frac {15 c (b+2 c x)}{2 \left (b^2-4 a c\right )^2 \sqrt [3]{a+b x+c x^2}}-\frac {15 c^{4/3} (b+2 c x)}{\sqrt [3]{2} \left (b^2-4 a c\right )^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}+\frac {15 \sqrt [4]{3} \sqrt {2-\sqrt {3}} c^{4/3} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt [3]{2} \left (b^2-4 a c\right )^{5/3} (b+2 c x) \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}}}-\frac {5 \sqrt [6]{2} 3^{3/4} c^{4/3} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}\right ),-7-4 \sqrt {3}\right )}{\left (b^2-4 a c\right )^{5/3} (b+2 c x) \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}}} \]

output

-3/4*(2*c*x+b)/(-4*a*c+b^2)/(c*x^2+b*x+a)^(4/3)+15/2*c*(2*c*x+b)/(-4*a*c+b 
^2)^2/(c*x^2+b*x+a)^(1/3)-15/2*c^(4/3)*(2*c*x+b)*2^(2/3)/(-4*a*c+b^2)^2/(2 
^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3)+(-4*a*c+b^2)^(1/3)*(1+3^(1/2)))-5*2^(1/ 
6)*3^(3/4)*c^(4/3)*((-4*a*c+b^2)^(1/3)+2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3) 
)*EllipticF((2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3)+(-4*a*c+b^2)^(1/3)*(1-3^( 
1/2)))/(2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3)+(-4*a*c+b^2)^(1/3)*(1+3^(1/2)) 
),I*3^(1/2)+2*I)*(((-4*a*c+b^2)^(2/3)-2^(2/3)*c^(1/3)*(-4*a*c+b^2)^(1/3)*( 
c*x^2+b*x+a)^(1/3)+2*2^(1/3)*c^(2/3)*(c*x^2+b*x+a)^(2/3))/(2^(2/3)*c^(1/3) 
*(c*x^2+b*x+a)^(1/3)+(-4*a*c+b^2)^(1/3)*(1+3^(1/2)))^2)^(1/2)/(-4*a*c+b^2) 
^(5/3)/(2*c*x+b)/((-4*a*c+b^2)^(1/3)*((-4*a*c+b^2)^(1/3)+2^(2/3)*c^(1/3)*( 
c*x^2+b*x+a)^(1/3))/(2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3)+(-4*a*c+b^2)^(1/3 
)*(1+3^(1/2)))^2)^(1/2)+15/4*3^(1/4)*c^(4/3)*((-4*a*c+b^2)^(1/3)+2^(2/3)*c 
^(1/3)*(c*x^2+b*x+a)^(1/3))*EllipticE((2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3) 
+(-4*a*c+b^2)^(1/3)*(1-3^(1/2)))/(2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3)+(-4* 
a*c+b^2)^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(1/2*6^(1/2)-1/2*2^(1/2))*(((-4 
*a*c+b^2)^(2/3)-2^(2/3)*c^(1/3)*(-4*a*c+b^2)^(1/3)*(c*x^2+b*x+a)^(1/3)+2*2 
^(1/3)*c^(2/3)*(c*x^2+b*x+a)^(2/3))/(2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3)+( 
-4*a*c+b^2)^(1/3)*(1+3^(1/2)))^2)^(1/2)*2^(2/3)/(-4*a*c+b^2)^(5/3)/(2*c*x+ 
b)/((-4*a*c+b^2)^(1/3)*((-4*a*c+b^2)^(1/3)+2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^( 
1/3))/(2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3)+(-4*a*c+b^2)^(1/3)*(1+3^(1/2...
 

3.25.92.2 Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.14 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.14 \[ \int \frac {1}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\frac {(b+2 c x) \left (3 \sqrt [3]{2} \left (-b^2+10 b c x+2 c \left (7 a+5 c x^2\right )\right )-20 c (a+x (b+c x)) \sqrt [3]{\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )\right )}{4 \sqrt [3]{2} \left (b^2-4 a c\right )^2 (a+x (b+c x))^{4/3}} \]

input

Integrate[(a + b*x + c*x^2)^(-7/3),x]
 

output

((b + 2*c*x)*(3*2^(1/3)*(-b^2 + 10*b*c*x + 2*c*(7*a + 5*c*x^2)) - 20*c*(a 
+ x*(b + c*x))*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/3)*Hypergeometric 
2F1[1/3, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(4*2^(1/3)*(b^2 - 4*a*c) 
^2*(a + x*(b + c*x))^(4/3))
 

3.25.92.3 Rubi [A] (warning: unable to verify)

Time = 0.77 (sec) , antiderivative size = 1112, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1089, 1089, 1095, 832, 759, 2416}

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.

\(\displaystyle \int \frac {1}{\left (a+b x+c x^2\right )^{7/3}} \, dx\)

\(\Big \downarrow \) 1089

\(\displaystyle -\frac {5 c \int \frac {1}{\left (c x^2+b x+a\right )^{4/3}}dx}{2 \left (b^2-4 a c\right )}-\frac {3 (b+2 c x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}\)

\(\Big \downarrow \) 1089

\(\displaystyle -\frac {5 c \left (\frac {2 c \int \frac {1}{\sqrt [3]{c x^2+b x+a}}dx}{b^2-4 a c}-\frac {3 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2}}\right )}{2 \left (b^2-4 a c\right )}-\frac {3 (b+2 c x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}\)

\(\Big \downarrow \) 1095

\(\displaystyle -\frac {5 c \left (\frac {6 c \sqrt {(b+2 c x)^2} \int \frac {\sqrt [3]{c x^2+b x+a}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [3]{c x^2+b x+a}}{\left (b^2-4 a c\right ) (b+2 c x)}-\frac {3 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2}}\right )}{2 \left (b^2-4 a c\right )}-\frac {3 (b+2 c x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}\)

\(\Big \downarrow \) 832

\(\displaystyle -\frac {5 c \left (\frac {6 c \sqrt {(b+2 c x)^2} \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [3]{c x^2+b x+a}}{2^{2/3} \sqrt [3]{c}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c} \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [3]{c x^2+b x+a}}{2^{2/3} \sqrt [3]{c}}\right )}{\left (b^2-4 a c\right ) (b+2 c x)}-\frac {3 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2}}\right )}{2 \left (b^2-4 a c\right )}-\frac {3 (b+2 c x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}\)

\(\Big \downarrow \) 759

\(\displaystyle -\frac {5 c \left (\frac {6 c \sqrt {(b+2 c x)^2} \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [3]{c x^2+b x+a}}{2^{2/3} \sqrt [3]{c}}-\frac {\left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt {\frac {-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+\left (b^2-4 a c\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [3]{2} \sqrt [4]{3} c^{2/3} \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}\right )}{\left (b^2-4 a c\right ) (b+2 c x)}-\frac {3 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2}}\right )}{2 \left (b^2-4 a c\right )}-\frac {3 (b+2 c x)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}\)

\(\Big \downarrow \) 2416

\(\displaystyle -\frac {3 (b+2 c x)}{4 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{4/3}}-\frac {5 c \left (\frac {6 c \sqrt {(b+2 c x)^2} \left (\frac {\frac {\sqrt [3]{2} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}{\sqrt [3]{c} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a} \sqrt [3]{b^2-4 a c}+2 \sqrt [3]{2} c^{2/3} \left (c x^2+b x+a\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right )|-7-4 \sqrt {3}\right )}{2^{2/3} \sqrt [3]{c} \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}}{2^{2/3} \sqrt [3]{c}}-\frac {\left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right ) \sqrt {\frac {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a} \sqrt [3]{b^2-4 a c}+2 \sqrt [3]{2} c^{2/3} \left (c x^2+b x+a\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [3]{2} \sqrt [4]{3} c^{2/3} \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}\right )^2}} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}\right )}{\left (b^2-4 a c\right ) (b+2 c x)}-\frac {3 (b+2 c x)}{\left (b^2-4 a c\right ) \sqrt [3]{c x^2+b x+a}}\right )}{2 \left (b^2-4 a c\right )}\)

input

Int[(a + b*x + c*x^2)^(-7/3),x]
 

output

(-3*(b + 2*c*x))/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(4/3)) - (5*c*((-3*(b 
+ 2*c*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)^(1/3)) + (6*c*Sqrt[(b + 2*c*x)^ 
2]*(((2^(1/3)*Sqrt[b^2 - 4*a*c + 4*c*(a + b*x + c*x^2)])/(c^(1/3)*((1 + Sq 
rt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))) - ( 
3^(1/4)*Sqrt[2 - Sqrt[3]]*(b^2 - 4*a*c)^(1/3)*((b^2 - 4*a*c)^(1/3) + 2^(2/ 
3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))*Sqrt[((b^2 - 4*a*c)^(2/3) - 2^(2/3)*c^ 
(1/3)*(b^2 - 4*a*c)^(1/3)*(a + b*x + c*x^2)^(1/3) + 2*2^(1/3)*c^(2/3)*(a + 
 b*x + c*x^2)^(2/3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)* 
(a + b*x + c*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 - Sqrt[3])*(b^2 - 4*a*c)^ 
(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))/((1 + Sqrt[3])*(b^2 - 4*a 
*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))], -7 - 4*Sqrt[3]])/(2 
^(2/3)*c^(1/3)*Sqrt[((b^2 - 4*a*c)^(1/3)*((b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^ 
(1/3)*(a + b*x + c*x^2)^(1/3)))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/ 
3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))^2]*Sqrt[b^2 - 4*a*c + 4*c*(a + b*x + c 
*x^2)]))/(2^(2/3)*c^(1/3)) - ((1 - Sqrt[3])*Sqrt[2 + Sqrt[3]]*(b^2 - 4*a*c 
)^(1/3)*((b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))*Sq 
rt[((b^2 - 4*a*c)^(2/3) - 2^(2/3)*c^(1/3)*(b^2 - 4*a*c)^(1/3)*(a + b*x + c 
*x^2)^(1/3) + 2*2^(1/3)*c^(2/3)*(a + b*x + c*x^2)^(2/3))/((1 + Sqrt[3])*(b 
^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))^2]*EllipticF[ 
ArcSin[((1 - Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + ...
 

3.25.92.3.1 Defintions of rubi rules used

rule 759

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] & 
& PosQ[a]
 

rule 832

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] && PosQ[a]
 

rule 1089

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 
 3)/((p + 1)*(b^2 - 4*a*c)))   Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre 
eQ[{a, b, c}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1095

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[3*(Sqrt[(b 
+ 2*c*x)^2]/(b + 2*c*x))   Subst[Int[x^(3*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4 
*c*x^3], x], x, (a + b*x + c*x^2)^(1/3)], x] /; FreeQ[{a, b, c}, x] && Inte 
gerQ[3*p]
 

rule 2416

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 + Sqrt 
[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] && PosQ[a] && Eq 
Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
 

3.25.92.4 Maple [F]

\[\int \frac {1}{\left (c \,x^{2}+b x +a \right )^{\frac {7}{3}}}d x\]

input

int(1/(c*x^2+b*x+a)^(7/3),x)
 

output

int(1/(c*x^2+b*x+a)^(7/3),x)
 

3.25.92.5 Fricas [F]

\[ \int \frac {1}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}}} \,d x } \]

input

integrate(1/(c*x^2+b*x+a)^(7/3),x, algorithm="fricas")
 

output

integral((c*x^2 + b*x + a)^(2/3)/(c^3*x^6 + 3*b*c^2*x^5 + 3*(b^2*c + a*c^2 
)*x^4 + 3*a^2*b*x + (b^3 + 6*a*b*c)*x^3 + a^3 + 3*(a*b^2 + a^2*c)*x^2), x)
 

3.25.92.6 Sympy [F]

\[ \int \frac {1}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\int \frac {1}{\left (a + b x + c x^{2}\right )^{\frac {7}{3}}}\, dx \]

input

integrate(1/(c*x**2+b*x+a)**(7/3),x)
 

output

Integral((a + b*x + c*x**2)**(-7/3), x)
 

3.25.92.7 Maxima [F]

\[ \int \frac {1}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}}} \,d x } \]

input

integrate(1/(c*x^2+b*x+a)^(7/3),x, algorithm="maxima")
 

output

integrate((c*x^2 + b*x + a)^(-7/3), x)
 

3.25.92.8 Giac [F]

\[ \int \frac {1}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}}} \,d x } \]

input

integrate(1/(c*x^2+b*x+a)^(7/3),x, algorithm="giac")
 

output

integrate((c*x^2 + b*x + a)^(-7/3), x)
 

3.25.92.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\int \frac {1}{{\left (c\,x^2+b\,x+a\right )}^{7/3}} \,d x \]

input

int(1/(a + b*x + c*x^2)^(7/3),x)
 

output

int(1/(a + b*x + c*x^2)^(7/3), x)