\(\int \frac {c+d x}{(e x)^{4/3} \sqrt {a+b x^2}} \, dx\) [1603]

Optimal result

Integrand size = 24, antiderivative size = 1040 \[ \int \frac {c+d x}{(e x)^{4/3} \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Output:

-3*c*(b*x^2+a)^(1/2)/a/e/(e*x)^(1/3)+3*(1+3^(1/2))*b^(1/3)*c*(e*x)^(1/3)*( 
b*x^2+a)^(1/2)/a/e/(a^(1/3)*e^(2/3)+(1+3^(1/2))*b^(1/3)*(e*x)^(2/3))-3*3^( 
1/4)*b^(1/3)*c*(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)*EllipticE((1-(a^(1/3)*e 
^(2/3)+(1-3^(1/2))*b^(1/3)*(e*x)^(2/3))^2/(a^(1/3)*e^(2/3)+(1+3^(1/2))*b^( 
1/3)*(e*x)^(2/3))^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/2))/a^(2/3)/e^(7/3)/(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)-1/2*3^(3/4)*(1-3^(1/ 
2))*b^(1/3)*c*(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))/a^(2/3)/e^(7/3)/(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)+3^(3/4)*(1/2*6^(1/2)+ 
1/2*2^(1/2))*d*(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/...
 

Mathematica [C] (verified)

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

Time = 10.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.08 \[ \int \frac {c+d x}{(e x)^{4/3} \sqrt {a+b x^2}} \, dx=\frac {3 x \sqrt {1+\frac {b x^2}{a}} \left (-2 c \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {5}{6},-\frac {b x^2}{a}\right )+d x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {b x^2}{a}\right )\right )}{2 (e x)^{4/3} \sqrt {a+b x^2}} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.87 (sec) , antiderivative size = 1051, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {553, 27, 557, 266, 807, 759, 837, 25, 766, 2420}

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.

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

\(\Big \downarrow \) 553

\(\displaystyle -\frac {3 \int -\frac {a d+2 b c x}{3 \sqrt [3]{e x} \sqrt {b x^2+a}}dx}{a e}-\frac {3 c \sqrt {a+b x^2}}{a e \sqrt [3]{e x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {a d+2 b c x}{\sqrt [3]{e x} \sqrt {b x^2+a}}dx}{a e}-\frac {3 c \sqrt {a+b x^2}}{a e \sqrt [3]{e x}}\)

\(\Big \downarrow \) 557

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 807

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

\(\Big \downarrow \) 759

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

\(\Big \downarrow \) 837

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 766

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

\(\Big \downarrow \) 2420

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

Input:

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

Output:

(-3*c*Sqrt[a + b*x^2])/(a*e*(e*x)^(1/3)) + ((6*b*c*((((1 + Sqrt[3])*e^2*(e 
*x)^(1/3)*Sqrt[a + b*x^2])/(a^(1/3)*e^(2/3) + (1 + Sqrt[3])*b^(1/3)*(e*x)^ 
(2/3)) - (3^(1/4)*a^(1/3)*e^(2/3)*(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]*EllipticE[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]) 
/(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]))/(2*b 
^(2/3)) - ((1 - Sqrt[3])*a^(1/3)*e^(2/3)*(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])/(4*3^(1/4)*b^(2/3)*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])))/e^2 + (3^(3/4)*Sqrt[2 + Sqrt[3]]*a*d*(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[((1 - Sqrt[3])*a^(1/3)*e^(2/3) + b^(1...
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 553

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp 
[1/(a*e*(m + 1))   Int[(e*x)^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 
2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && LtQ[m, -1]
 

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 837

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

rule 2420

Int[((c_) + (d_.)*(x_)^4)/Sqrt[(a_) + (b_.)*(x_)^6], x_Symbol] :> With[{r = 
 Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(1 + Sqrt[3])*d*s^3*x*(Sqr 
t[a + b*x^6]/(2*a*r^2*(s + (1 + Sqrt[3])*r*x^2))), x] - Simp[3^(1/4)*d*s*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 
*r^2*Sqrt[(r*x^2*(s + r*x^2))/(s + (1 + Sqrt[3])*r*x^2)^2]*Sqrt[a + b*x^6]) 
)*EllipticE[ArcCos[(s + (1 - Sqrt[3])*r*x^2)/(s + (1 + Sqrt[3])*r*x^2)], (2 
 + Sqrt[3])/4], x]] /; FreeQ[{a, b, c, d}, x] && EqQ[2*Rt[b/a, 3]^2*c - (1 
- Sqrt[3])*d, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [A] (verification not implemented)

Time = 2.86 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.09 \[ \int \frac {c+d x}{(e x)^{4/3} \sqrt {a+b x^2}} \, dx=\frac {c \Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{6}, \frac {1}{2} \\ \frac {5}{6} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} e^{\frac {4}{3}} \sqrt [3]{x} \Gamma \left (\frac {5}{6}\right )} + \frac {d x^{\frac {2}{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {a} e^{\frac {4}{3}} \Gamma \left (\frac {4}{3}\right )} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {c+d x}{(e x)^{4/3} \sqrt {a+b x^2}} \, dx=\frac {\left (\int \frac {\sqrt {b \,x^{2}+a}}{x^{\frac {4}{3}} a +x^{\frac {10}{3}} b}d x \right ) c +\left (\int \frac {\sqrt {b \,x^{2}+a}}{x^{\frac {1}{3}} a +x^{\frac {7}{3}} b}d x \right ) d}{e^{\frac {4}{3}}} \] Input:

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

Output:

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