\(\int \frac {(e x)^{4/3} (c+d x)}{\sqrt {a+b x^2}} \, dx\) [1600]

Optimal result

Integrand size = 24, antiderivative size = 1049 \[ \int \frac {(e x)^{4/3} (c+d x)}{\sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Output:

3/4*c*e*(e*x)^(1/3)*(b*x^2+a)^(1/2)/b+3/7*d*(e*x)^(4/3)*(b*x^2+a)^(1/2)/b- 
12/7*a*d*e^2*(b*x^2+a)^(1/2)/b^(5/3)/((1+3^(1/2))*a^(1/3)*e^(2/3)+b^(1/3)* 
(e*x)^(2/3))+6/7*3^(1/4)*(1/2*6^(1/2)-1/2*2^(1/2))*a^(4/3)*d*e^(2/3)*(a^(1 
/3)*e^(2/3)+b^(1/3)*(e*x)^(2/3))*((a^(2/3)*e^(4/3)-a^(1/3)*b^(1/3)*e^(2/3) 
*(e*x)^(2/3)+b^(2/3)*(e*x)^(4/3))/((1+3^(1/2))*a^(1/3)*e^(2/3)+b^(1/3)*(e* 
x)^(2/3))^2)^(1/2)*EllipticE(((1-3^(1/2))*a^(1/3)*e^(2/3)+b^(1/3)*(e*x)^(2 
/3))/((1+3^(1/2))*a^(1/3)*e^(2/3)+b^(1/3)*(e*x)^(2/3)),I*3^(1/2)+2*I)/b^(5 
/3)/(b*x^2+a)^(1/2)/(a^(1/3)*e^(2/3)*(a^(1/3)*e^(2/3)+b^(1/3)*(e*x)^(2/3)) 
/((1+3^(1/2))*a^(1/3)*e^(2/3)+b^(1/3)*(e*x)^(2/3))^2)^(1/2)-1/8*3^(3/4)*a^ 
(2/3)*c*e^(1/3)*(e*x)^(1/3)*(a^(1/3)*e^(2/3)+b^(1/3)*(e*x)^(2/3))*((a^(2/3 
)*e^(4/3)-a^(1/3)*b^(1/3)*e^(2/3)*(e*x)^(2/3)+b^(2/3)*(e*x)^(4/3))/(a^(1/3 
)*e^(2/3)+(1+3^(1/2))*b^(1/3)*(e*x)^(2/3))^2)^(1/2)*InverseJacobiAM(arccos 
((a^(1/3)*e^(2/3)+(1-3^(1/2))*b^(1/3)*(e*x)^(2/3))/(a^(1/3)*e^(2/3)+(1+3^( 
1/2))*b^(1/3)*(e*x)^(2/3))),1/4*6^(1/2)+1/4*2^(1/2))/b/(b*x^2+a)^(1/2)/(b^ 
(1/3)*(e*x)^(2/3)*(a^(1/3)*e^(2/3)+b^(1/3)*(e*x)^(2/3))/(a^(1/3)*e^(2/3)+( 
1+3^(1/2))*b^(1/3)*(e*x)^(2/3))^2)^(1/2)-4/7*2^(1/2)*3^(3/4)*a^(4/3)*d*e^( 
2/3)*(a^(1/3)*e^(2/3)+b^(1/3)*(e*x)^(2/3))*((a^(2/3)*e^(4/3)-a^(1/3)*b^(1/ 
3)*e^(2/3)*(e*x)^(2/3)+b^(2/3)*(e*x)^(4/3))/((1+3^(1/2))*a^(1/3)*e^(2/3)+b 
^(1/3)*(e*x)^(2/3))^2)^(1/2)*EllipticF(((1-3^(1/2))*a^(1/3)*e^(2/3)+b^(1/3 
)*(e*x)^(2/3))/((1+3^(1/2))*a^(1/3)*e^(2/3)+b^(1/3)*(e*x)^(2/3)),I*3^(1...
 

Mathematica [C] (verified)

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

Time = 10.06 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.11 \[ \int \frac {(e x)^{4/3} (c+d x)}{\sqrt {a+b x^2}} \, dx=\frac {3 e \sqrt [3]{e x} \left ((7 c+4 d x) \left (a+b x^2\right )-7 a c \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {b x^2}{a}\right )-4 a d x \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},-\frac {b x^2}{a}\right )\right )}{28 b \sqrt {a+b x^2}} \] Input:

Integrate[((e*x)^(4/3)*(c + d*x))/Sqrt[a + b*x^2],x]
 

Output:

(3*e*(e*x)^(1/3)*((7*c + 4*d*x)*(a + b*x^2) - 7*a*c*Sqrt[1 + (b*x^2)/a]*Hy 
pergeometric2F1[1/6, 1/2, 7/6, -((b*x^2)/a)] - 4*a*d*x*Sqrt[1 + (b*x^2)/a] 
*Hypergeometric2F1[1/2, 2/3, 5/3, -((b*x^2)/a)]))/(28*b*Sqrt[a + b*x^2])
 

Rubi [A] (warning: unable to verify)

Time = 0.97 (sec) , antiderivative size = 1081, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {557, 262, 266, 766, 807, 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.

\(\displaystyle \int \frac {(e x)^{4/3} (c+d x)}{\sqrt {a+b x^2}} \, dx\)

\(\Big \downarrow \) 557

\(\displaystyle c \int \frac {(e x)^{4/3}}{\sqrt {b x^2+a}}dx+\frac {d \int \frac {(e x)^{7/3}}{\sqrt {b x^2+a}}dx}{e}\)

\(\Big \downarrow \) 262

\(\displaystyle c \left (\frac {3 e \sqrt [3]{e x} \sqrt {a+b x^2}}{4 b}-\frac {a e^2 \int \frac {1}{(e x)^{2/3} \sqrt {b x^2+a}}dx}{4 b}\right )+\frac {d \left (\frac {3 e (e x)^{4/3} \sqrt {a+b x^2}}{7 b}-\frac {4 a e^2 \int \frac {\sqrt [3]{e x}}{\sqrt {b x^2+a}}dx}{7 b}\right )}{e}\)

\(\Big \downarrow \) 266

\(\displaystyle c \left (\frac {3 e \sqrt [3]{e x} \sqrt {a+b x^2}}{4 b}-\frac {3 a e \int \frac {1}{\sqrt {b x^2+a}}d\sqrt [3]{e x}}{4 b}\right )+\frac {d \left (\frac {3 e (e x)^{4/3} \sqrt {a+b x^2}}{7 b}-\frac {12 a e \int \frac {e x}{\sqrt {b x^2+a}}d\sqrt [3]{e x}}{7 b}\right )}{e}\)

\(\Big \downarrow \) 766

\(\displaystyle \frac {d \left (\frac {3 e (e x)^{4/3} \sqrt {a+b x^2}}{7 b}-\frac {12 a e \int \frac {e x}{\sqrt {b x^2+a}}d\sqrt [3]{e x}}{7 b}\right )}{e}+c \left (\frac {3 e \sqrt [3]{e x} \sqrt {a+b x^2}}{4 b}-\frac {3^{3/4} a^{2/3} \sqrt [3]{e} \sqrt [3]{e x} \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right ) \sqrt {\frac {a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} e^{2/3} (e x)^{2/3}+b^{2/3} (e x)^{4/3}}{\left (\sqrt [3]{a} e^{2/3}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} (e x)^{2/3}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a} e^{2/3}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} (e x)^{2/3}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{8 b \sqrt {a+b x^2} \sqrt {\frac {\sqrt [3]{b} (e x)^{2/3} \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{\left (\sqrt [3]{a} e^{2/3}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} (e x)^{2/3}\right )^2}}}\right )\)

\(\Big \downarrow \) 807

\(\displaystyle \frac {d \left (\frac {3 e (e x)^{4/3} \sqrt {a+b x^2}}{7 b}-\frac {6 a e \int \frac {(e x)^{2/3}}{\sqrt {a+\frac {b x}{e}}}d(e x)^{2/3}}{7 b}\right )}{e}+c \left (\frac {3 e \sqrt [3]{e x} \sqrt {a+b x^2}}{4 b}-\frac {3^{3/4} a^{2/3} \sqrt [3]{e} \sqrt [3]{e x} \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right ) \sqrt {\frac {a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} e^{2/3} (e x)^{2/3}+b^{2/3} (e x)^{4/3}}{\left (\sqrt [3]{a} e^{2/3}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} (e x)^{2/3}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a} e^{2/3}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} (e x)^{2/3}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{8 b \sqrt {a+b x^2} \sqrt {\frac {\sqrt [3]{b} (e x)^{2/3} \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{\left (\sqrt [3]{a} e^{2/3}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} (e x)^{2/3}\right )^2}}}\right )\)

\(\Big \downarrow \) 832

\(\displaystyle \frac {d \left (\frac {3 e (e x)^{4/3} \sqrt {a+b x^2}}{7 b}-\frac {6 a e \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}{\sqrt {a+\frac {b x}{e}}}d(e x)^{2/3}}{\sqrt [3]{b}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} e^{2/3} \int \frac {1}{\sqrt {a+\frac {b x}{e}}}d(e x)^{2/3}}{\sqrt [3]{b}}\right )}{7 b}\right )}{e}+c \left (\frac {3 e \sqrt [3]{e x} \sqrt {a+b x^2}}{4 b}-\frac {3^{3/4} a^{2/3} \sqrt [3]{e} \sqrt [3]{e x} \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right ) \sqrt {\frac {a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} e^{2/3} (e x)^{2/3}+b^{2/3} (e x)^{4/3}}{\left (\sqrt [3]{a} e^{2/3}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} (e x)^{2/3}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a} e^{2/3}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} (e x)^{2/3}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{8 b \sqrt {a+b x^2} \sqrt {\frac {\sqrt [3]{b} (e x)^{2/3} \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{\left (\sqrt [3]{a} e^{2/3}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} (e x)^{2/3}\right )^2}}}\right )\)

\(\Big \downarrow \) 759

\(\displaystyle \frac {d \left (\frac {3 e (e x)^{4/3} \sqrt {a+b x^2}}{7 b}-\frac {6 a e \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}{\sqrt {a+\frac {b x}{e}}}d(e x)^{2/3}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} e^{2/3} \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right ) \sqrt {\frac {a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} e^{2/3} (e x)^{2/3}+b^{2/3} (e x)^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {a+\frac {b x}{e}} \sqrt {\frac {\sqrt [3]{a} e^{2/3} \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )^2}}}\right )}{7 b}\right )}{e}+c \left (\frac {3 e \sqrt [3]{e x} \sqrt {a+b x^2}}{4 b}-\frac {3^{3/4} a^{2/3} \sqrt [3]{e} \sqrt [3]{e x} \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right ) \sqrt {\frac {a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} e^{2/3} (e x)^{2/3}+b^{2/3} (e x)^{4/3}}{\left (\sqrt [3]{a} e^{2/3}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} (e x)^{2/3}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a} e^{2/3}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} (e x)^{2/3}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{8 b \sqrt {a+b x^2} \sqrt {\frac {\sqrt [3]{b} (e x)^{2/3} \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{\left (\sqrt [3]{a} e^{2/3}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} (e x)^{2/3}\right )^2}}}\right )\)

\(\Big \downarrow \) 2416

\(\displaystyle c \left (\frac {3 e \sqrt [3]{e x} \sqrt {b x^2+a}}{4 b}-\frac {3^{3/4} a^{2/3} \sqrt [3]{e} \sqrt [3]{e x} \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right ) \sqrt {\frac {a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{4/3}}{\left (\sqrt [3]{a} e^{2/3}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} (e x)^{2/3}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{a} e^{2/3}+\left (1-\sqrt {3}\right ) \sqrt [3]{b} (e x)^{2/3}}{\sqrt [3]{a} e^{2/3}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} (e x)^{2/3}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{8 b \sqrt {b x^2+a} \sqrt {\frac {\sqrt [3]{b} (e x)^{2/3} \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{\left (\sqrt [3]{a} e^{2/3}+\left (1+\sqrt {3}\right ) \sqrt [3]{b} (e x)^{2/3}\right )^2}}}\right )+\frac {d \left (\frac {3 e (e x)^{4/3} \sqrt {b x^2+a}}{7 b}-\frac {6 a e \left (\frac {\frac {2 e^2 \sqrt {a+\frac {b x}{e}}}{\sqrt [3]{b} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} e^{2/3} \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right ) \sqrt {\frac {a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt {a+\frac {b x}{e}} \sqrt {\frac {\sqrt [3]{a} e^{2/3} \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )^2}}}}{\sqrt [3]{b}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} e^{2/3} \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right ) \sqrt {\frac {a^{2/3} e^{4/3}-\sqrt [3]{a} \sqrt [3]{b} (e x)^{2/3} e^{2/3}+b^{2/3} (e x)^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {a+\frac {b x}{e}} \sqrt {\frac {\sqrt [3]{a} e^{2/3} \left (\sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} e^{2/3}+\sqrt [3]{b} (e x)^{2/3}\right )^2}}}\right )}{7 b}\right )}{e}\)

Input:

Int[((e*x)^(4/3)*(c + d*x))/Sqrt[a + b*x^2],x]
 

Output:

c*((3*e*(e*x)^(1/3)*Sqrt[a + b*x^2])/(4*b) - (3^(3/4)*a^(2/3)*e^(1/3)*(e*x 
)^(1/3)*(a^(1/3)*e^(2/3) + b^(1/3)*(e*x)^(2/3))*Sqrt[(a^(2/3)*e^(4/3) - a^ 
(1/3)*b^(1/3)*e^(2/3)*(e*x)^(2/3) + b^(2/3)*(e*x)^(4/3))/(a^(1/3)*e^(2/3) 
+ (1 + Sqrt[3])*b^(1/3)*(e*x)^(2/3))^2]*EllipticF[ArcCos[(a^(1/3)*e^(2/3) 
+ (1 - Sqrt[3])*b^(1/3)*(e*x)^(2/3))/(a^(1/3)*e^(2/3) + (1 + Sqrt[3])*b^(1 
/3)*(e*x)^(2/3))], (2 + Sqrt[3])/4])/(8*b*Sqrt[a + b*x^2]*Sqrt[(b^(1/3)*(e 
*x)^(2/3)*(a^(1/3)*e^(2/3) + b^(1/3)*(e*x)^(2/3)))/(a^(1/3)*e^(2/3) + (1 + 
 Sqrt[3])*b^(1/3)*(e*x)^(2/3))^2])) + (d*((3*e*(e*x)^(4/3)*Sqrt[a + b*x^2] 
)/(7*b) - (6*a*e*(((2*e^2*Sqrt[a + (b*x)/e])/(b^(1/3)*((1 + Sqrt[3])*a^(1/ 
3)*e^(2/3) + b^(1/3)*(e*x)^(2/3))) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(1/3)*e^ 
(2/3)*(a^(1/3)*e^(2/3) + b^(1/3)*(e*x)^(2/3))*Sqrt[(a^(2/3)*e^(4/3) + b^(2 
/3)*(e*x)^(2/3) - a^(1/3)*b^(1/3)*e^(2/3)*(e*x)^(2/3))/((1 + Sqrt[3])*a^(1 
/3)*e^(2/3) + b^(1/3)*(e*x)^(2/3))^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1 
/3)*e^(2/3) + b^(1/3)*(e*x)^(2/3))/((1 + Sqrt[3])*a^(1/3)*e^(2/3) + b^(1/3 
)*(e*x)^(2/3))], -7 - 4*Sqrt[3]])/(b^(1/3)*Sqrt[a + (b*x)/e]*Sqrt[(a^(1/3) 
*e^(2/3)*(a^(1/3)*e^(2/3) + b^(1/3)*(e*x)^(2/3)))/((1 + Sqrt[3])*a^(1/3)*e 
^(2/3) + b^(1/3)*(e*x)^(2/3))^2]))/b^(1/3) - (2*(1 - Sqrt[3])*Sqrt[2 + Sqr 
t[3]]*a^(1/3)*e^(2/3)*(a^(1/3)*e^(2/3) + b^(1/3)*(e*x)^(2/3))*Sqrt[(a^(2/3 
)*e^(4/3) + b^(2/3)*(e*x)^(2/3) - a^(1/3)*b^(1/3)*e^(2/3)*(e*x)^(2/3))/((1 
 + Sqrt[3])*a^(1/3)*e^(2/3) + b^(1/3)*(e*x)^(2/3))^2]*EllipticF[ArcSin[...
 

Defintions of rubi rules used

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 557

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[c   Int[(e*x)^m*(a + b*x^2)^p, x], x] + Simp[d/e   Int[(e*x)^( 
m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x]
 

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 766

Int[1/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[x*(s + r*x^2)*(Sqrt[(s^2 - r*s*x^2 + r^2*x^4)/ 
(s + (1 + Sqrt[3])*r*x^2)^2]/(2*3^(1/4)*s*Sqrt[a + b*x^6]*Sqrt[r*x^2*((s + 
r*x^2)/(s + (1 + Sqrt[3])*r*x^2)^2)]))*EllipticF[ArcCos[(s + (1 - Sqrt[3])* 
r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 + Sqrt[3])/4], x]] /; FreeQ[{a, b}, x 
]
 

rule 807

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m 
+ 1, n]}, Simp[1/k   Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, 
x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
 

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 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]
 

Maple [F]

\[\int \frac {\left (e x \right )^{\frac {4}{3}} \left (d x +c \right )}{\sqrt {b \,x^{2}+a}}d x\]

Input:

int((e*x)^(4/3)*(d*x+c)/(b*x^2+a)^(1/2),x)
 

Output:

int((e*x)^(4/3)*(d*x+c)/(b*x^2+a)^(1/2),x)
 

Fricas [F]

\[ \int \frac {(e x)^{4/3} (c+d x)}{\sqrt {a+b x^2}} \, dx=\int { \frac {{\left (d x + c\right )} \left (e x\right )^{\frac {4}{3}}}{\sqrt {b x^{2} + a}} \,d x } \] Input:

integrate((e*x)^(4/3)*(d*x+c)/(b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

integral((d*e*x^2 + c*e*x)*(e*x)^(1/3)/sqrt(b*x^2 + a), x)
 

Sympy [A] (verification not implemented)

Time = 8.06 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.09 \[ \int \frac {(e x)^{4/3} (c+d x)}{\sqrt {a+b x^2}} \, dx=\frac {c e^{\frac {4}{3}} x^{\frac {7}{3}} \Gamma \left (\frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{6} \\ \frac {13}{6} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {13}{6}\right )} + \frac {d e^{\frac {4}{3}} x^{\frac {10}{3}} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {8}{3}\right )} \] Input:

integrate((e*x)**(4/3)*(d*x+c)/(b*x**2+a)**(1/2),x)
 

Output:

c*e**(4/3)*x**(7/3)*gamma(7/6)*hyper((1/2, 7/6), (13/6,), b*x**2*exp_polar 
(I*pi)/a)/(2*sqrt(a)*gamma(13/6)) + d*e**(4/3)*x**(10/3)*gamma(5/3)*hyper( 
(1/2, 5/3), (8/3,), b*x**2*exp_polar(I*pi)/a)/(2*sqrt(a)*gamma(8/3))
 

Maxima [F]

\[ \int \frac {(e x)^{4/3} (c+d x)}{\sqrt {a+b x^2}} \, dx=\int { \frac {{\left (d x + c\right )} \left (e x\right )^{\frac {4}{3}}}{\sqrt {b x^{2} + a}} \,d x } \] Input:

integrate((e*x)^(4/3)*(d*x+c)/(b*x^2+a)^(1/2),x, algorithm="maxima")
 

Output:

integrate((d*x + c)*(e*x)^(4/3)/sqrt(b*x^2 + a), x)
 

Giac [F]

\[ \int \frac {(e x)^{4/3} (c+d x)}{\sqrt {a+b x^2}} \, dx=\int { \frac {{\left (d x + c\right )} \left (e x\right )^{\frac {4}{3}}}{\sqrt {b x^{2} + a}} \,d x } \] Input:

integrate((e*x)^(4/3)*(d*x+c)/(b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

integrate((d*x + c)*(e*x)^(4/3)/sqrt(b*x^2 + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^{4/3} (c+d x)}{\sqrt {a+b x^2}} \, dx=\int \frac {{\left (e\,x\right )}^{4/3}\,\left (c+d\,x\right )}{\sqrt {b\,x^2+a}} \,d x \] Input:

int(((e*x)^(4/3)*(c + d*x))/(a + b*x^2)^(1/2),x)
 

Output:

int(((e*x)^(4/3)*(c + d*x))/(a + b*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {(e x)^{4/3} (c+d x)}{\sqrt {a+b x^2}} \, dx=\frac {e^{\frac {4}{3}} \left (21 x^{\frac {1}{3}} \sqrt {b \,x^{2}+a}\, c +12 x^{\frac {4}{3}} \sqrt {b \,x^{2}+a}\, d -7 \left (\int \frac {\sqrt {b \,x^{2}+a}}{x^{\frac {2}{3}} a +x^{\frac {8}{3}} b}d x \right ) a c -16 \left (\int \frac {\sqrt {b \,x^{2}+a}\, x}{x^{\frac {2}{3}} a +x^{\frac {8}{3}} b}d x \right ) a d \right )}{28 b} \] Input:

int((e*x)^(4/3)*(d*x+c)/(b*x^2+a)^(1/2),x)
 

Output:

(e**(1/3)*e*(21*x**(1/3)*sqrt(a + b*x**2)*c + 12*x**(1/3)*sqrt(a + b*x**2) 
*d*x - 7*int(sqrt(a + b*x**2)/(x**(2/3)*a + x**(2/3)*b*x**2),x)*a*c - 16*i 
nt((sqrt(a + b*x**2)*x)/(x**(2/3)*a + x**(2/3)*b*x**2),x)*a*d))/(28*b)