\(\int \frac {(c+d x) \sqrt {c^2-d^2 x^2}}{(e x)^{7/3}} \, dx\) [568]

Optimal result

Integrand size = 29, antiderivative size = 1088 \[ \int \frac {(c+d x) \sqrt {c^2-d^2 x^2}}{(e x)^{7/3}} \, dx =\text {Too large to display} \] Output:

-3/4*(4*d*x+c)*(-d^2*x^2+c^2)^(1/2)/e/(e*x)^(4/3)-9/2*(1+3^(1/2))*d^(5/3)* 
(e*x)^(1/3)*(-d^2*x^2+c^2)^(1/2)/e^2/(c^(2/3)*e^(2/3)-(1+3^(1/2))*d^(2/3)* 
(e*x)^(2/3))+9/2*3^(1/4)*c^(2/3)*d^(5/3)*(e*x)^(1/3)*(c^(2/3)*e^(2/3)-d^(2 
/3)*(e*x)^(2/3))*((c^(4/3)*e^(4/3)+c^(2/3)*d^(2/3)*e^(2/3)*(e*x)^(2/3)+d^( 
4/3)*(e*x)^(4/3))/(c^(2/3)*e^(2/3)-(1+3^(1/2))*d^(2/3)*(e*x)^(2/3))^2)^(1/ 
2)*EllipticE((1-(c^(2/3)*e^(2/3)-(1-3^(1/2))*d^(2/3)*(e*x)^(2/3))^2/(c^(2/ 
3)*e^(2/3)-(1+3^(1/2))*d^(2/3)*(e*x)^(2/3))^2)^(1/2),1/4*6^(1/2)+1/4*2^(1/ 
2))/e^(10/3)/(-d^2*x^2+c^2)^(1/2)/(-d^(2/3)*(e*x)^(2/3)*(c^(2/3)*e^(2/3)-d 
^(2/3)*(e*x)^(2/3))/(c^(2/3)*e^(2/3)-(1+3^(1/2))*d^(2/3)*(e*x)^(2/3))^2)^( 
1/2)+3/4*3^(3/4)*(1-3^(1/2))*c^(2/3)*d^(5/3)*(e*x)^(1/3)*(c^(2/3)*e^(2/3)- 
d^(2/3)*(e*x)^(2/3))*((c^(4/3)*e^(4/3)+c^(2/3)*d^(2/3)*e^(2/3)*(e*x)^(2/3) 
+d^(4/3)*(e*x)^(4/3))/(c^(2/3)*e^(2/3)-(1+3^(1/2))*d^(2/3)*(e*x)^(2/3))^2) 
^(1/2)*InverseJacobiAM(arccos((c^(2/3)*e^(2/3)-(1-3^(1/2))*d^(2/3)*(e*x)^( 
2/3))/(c^(2/3)*e^(2/3)-(1+3^(1/2))*d^(2/3)*(e*x)^(2/3))),1/4*6^(1/2)+1/4*2 
^(1/2))/e^(10/3)/(-d^2*x^2+c^2)^(1/2)/(-d^(2/3)*(e*x)^(2/3)*(c^(2/3)*e^(2/ 
3)-d^(2/3)*(e*x)^(2/3))/(c^(2/3)*e^(2/3)-(1+3^(1/2))*d^(2/3)*(e*x)^(2/3))^ 
2)^(1/2)+3/4*3^(3/4)*(1/2*6^(1/2)+1/2*2^(1/2))*c*d^(4/3)*(c^(2/3)*e^(2/3)- 
d^(2/3)*(e*x)^(2/3))*((c^(4/3)*e^(4/3)+c^(2/3)*d^(2/3)*e^(2/3)*(e*x)^(2/3) 
+d^(4/3)*(e*x)^(4/3))/((1+3^(1/2))*c^(2/3)*e^(2/3)-d^(2/3)*(e*x)^(2/3))^2) 
^(1/2)*EllipticF(((1-3^(1/2))*c^(2/3)*e^(2/3)-d^(2/3)*(e*x)^(2/3))/((1+...
 

Mathematica [C] (verified)

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

Time = 10.04 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.08 \[ \int \frac {(c+d x) \sqrt {c^2-d^2 x^2}}{(e x)^{7/3}} \, dx=-\frac {3 x \sqrt {c^2-d^2 x^2} \left (c \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{2},\frac {1}{3},\frac {d^2 x^2}{c^2}\right )+4 d x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{6},\frac {5}{6},\frac {d^2 x^2}{c^2}\right )\right )}{4 (e x)^{7/3} \sqrt {1-\frac {d^2 x^2}{c^2}}} \] Input:

Integrate[((c + d*x)*Sqrt[c^2 - d^2*x^2])/(e*x)^(7/3),x]
 

Output:

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

Rubi [A] (warning: unable to verify)

Time = 1.52 (sec) , antiderivative size = 1100, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {546, 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.

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

\(\Big \downarrow \) 546

\(\displaystyle \frac {9 d^2 \int -\frac {c+4 d x}{3 \sqrt [3]{e x} \sqrt {c^2-d^2 x^2}}dx}{4 e^2}-\frac {3 (c+4 d x) \sqrt {c^2-d^2 x^2}}{4 e (e x)^{4/3}}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 557

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 807

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

\(\Big \downarrow \) 759

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

\(\Big \downarrow \) 837

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 766

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

\(\Big \downarrow \) 2420

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

Input:

Int[((c + d*x)*Sqrt[c^2 - d^2*x^2])/(e*x)^(7/3),x]
 

Output:

(-3*(c + 4*d*x)*Sqrt[c^2 - d^2*x^2])/(4*e*(e*x)^(4/3)) - (3*d^2*((12*d*((( 
(1 + Sqrt[3])*e^2*(e*x)^(1/3)*Sqrt[c^2 - d^2*x^2])/(c^(2/3)*e^(2/3) - (1 + 
 Sqrt[3])*d^(2/3)*(e*x)^(2/3)) - (3^(1/4)*c^(2/3)*e^(2/3)*(e*x)^(1/3)*(c^( 
2/3)*e^(2/3) - d^(2/3)*(e*x)^(2/3))*Sqrt[(c^(4/3)*e^(4/3) + c^(2/3)*d^(2/3 
)*e^(2/3)*(e*x)^(2/3) + d^(4/3)*(e*x)^(4/3))/(c^(2/3)*e^(2/3) - (1 + Sqrt[ 
3])*d^(2/3)*(e*x)^(2/3))^2]*EllipticE[ArcCos[(c^(2/3)*e^(2/3) - (1 - Sqrt[ 
3])*d^(2/3)*(e*x)^(2/3))/(c^(2/3)*e^(2/3) - (1 + Sqrt[3])*d^(2/3)*(e*x)^(2 
/3))], (2 + Sqrt[3])/4])/(Sqrt[c^2 - d^2*x^2]*Sqrt[-((d^(2/3)*(e*x)^(2/3)* 
(c^(2/3)*e^(2/3) - d^(2/3)*(e*x)^(2/3)))/(c^(2/3)*e^(2/3) - (1 + Sqrt[3])* 
d^(2/3)*(e*x)^(2/3))^2)]))/(2*d^(4/3)) - ((1 - Sqrt[3])*c^(2/3)*e^(2/3)*(e 
*x)^(1/3)*(c^(2/3)*e^(2/3) - d^(2/3)*(e*x)^(2/3))*Sqrt[(c^(4/3)*e^(4/3) + 
c^(2/3)*d^(2/3)*e^(2/3)*(e*x)^(2/3) + d^(4/3)*(e*x)^(4/3))/(c^(2/3)*e^(2/3 
) - (1 + Sqrt[3])*d^(2/3)*(e*x)^(2/3))^2]*EllipticF[ArcCos[(c^(2/3)*e^(2/3 
) - (1 - Sqrt[3])*d^(2/3)*(e*x)^(2/3))/(c^(2/3)*e^(2/3) - (1 + Sqrt[3])*d^ 
(2/3)*(e*x)^(2/3))], (2 + Sqrt[3])/4])/(4*3^(1/4)*d^(4/3)*Sqrt[c^2 - d^2*x 
^2]*Sqrt[-((d^(2/3)*(e*x)^(2/3)*(c^(2/3)*e^(2/3) - d^(2/3)*(e*x)^(2/3)))/( 
c^(2/3)*e^(2/3) - (1 + Sqrt[3])*d^(2/3)*(e*x)^(2/3))^2)])))/e^2 - (3^(3/4) 
*Sqrt[2 + Sqrt[3]]*c*(c^(2/3)*e^(2/3) - d^(2/3)*(e*x)^(2/3))*Sqrt[(c^(4/3) 
*e^(4/3) + d^(4/3)*(e*x)^(2/3) + c^(2/3)*d^(2/3)*e^(2/3)*(e*x)^(2/3))/((1 
+ Sqrt[3])*c^(2/3)*e^(2/3) - d^(2/3)*(e*x)^(2/3))^2]*EllipticF[ArcSin[(...
 

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 546

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym 
bol] :> Simp[(e*x)^(m + 1)*(c*(m + 2) + d*(m + 1)*x)*((a + b*x^2)^p/(e*(m + 
 1)*(m + 2))), x] - Simp[2*b*(p/(e^2*(m + 1)*(m + 2)))   Int[(e*x)^(m + 2)* 
(a + b*x^2)^(p - 1)*(c*(m + 2) + d*(m + 1)*x), x], x] /; FreeQ[{a, b, c, d, 
 e}, x] && GtQ[p, 0] && LtQ[m, -2] &&  !ILtQ[m + 2*p + 3, 0]
 

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]

\[\int \frac {\left (d x +c \right ) \sqrt {-d^{2} x^{2}+c^{2}}}{\left (e x \right )^{\frac {7}{3}}}d x\]

Input:

int((d*x+c)*(-d^2*x^2+c^2)^(1/2)/(e*x)^(7/3),x)
 

Output:

int((d*x+c)*(-d^2*x^2+c^2)^(1/2)/(e*x)^(7/3),x)
 

Fricas [F]

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

integrate((d*x+c)*(-d^2*x^2+c^2)^(1/2)/(e*x)^(7/3),x, algorithm="fricas")
 

Output:

integral(sqrt(-d^2*x^2 + c^2)*(d*x + c)*(e*x)^(2/3)/(e^3*x^3), x)
 

Sympy [A] (verification not implemented)

Time = 10.80 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.09 \[ \int \frac {(c+d x) \sqrt {c^2-d^2 x^2}}{(e x)^{7/3}} \, dx=\frac {c d \Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{6} \\ \frac {5}{6} \end {matrix}\middle | {\frac {d^{2} x^{2} e^{2 i \pi }}{c^{2}}} \right )}}{2 e^{\frac {7}{3}} \sqrt [3]{x} \Gamma \left (\frac {5}{6}\right )} + \frac {i c d \Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{6} \\ \frac {7}{6} \end {matrix}\middle | {\frac {c^{2}}{d^{2} x^{2}}} \right )}}{2 e^{\frac {7}{3}} \sqrt [3]{x} \Gamma \left (\frac {5}{6}\right )} \] Input:

integrate((d*x+c)*(-d**2*x**2+c**2)**(1/2)/(e*x)**(7/3),x)
 

Output:

c*d*gamma(-1/6)*hyper((-1/2, -1/6), (5/6,), d**2*x**2*exp_polar(2*I*pi)/c* 
*2)/(2*e**(7/3)*x**(1/3)*gamma(5/6)) + I*c*d*gamma(-1/6)*hyper((-1/2, 1/6) 
, (7/6,), c**2/(d**2*x**2))/(2*e**(7/3)*x**(1/3)*gamma(5/6))
 

Maxima [F]

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

integrate((d*x+c)*(-d^2*x^2+c^2)^(1/2)/(e*x)^(7/3),x, algorithm="maxima")
 

Output:

integrate(sqrt(-d^2*x^2 + c^2)*(d*x + c)/(e*x)^(7/3), x)
 

Giac [F]

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

integrate((d*x+c)*(-d^2*x^2+c^2)^(1/2)/(e*x)^(7/3),x, algorithm="giac")
 

Output:

integrate(sqrt(-d^2*x^2 + c^2)*(d*x + c)/(e*x)^(7/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x) \sqrt {c^2-d^2 x^2}}{(e x)^{7/3}} \, dx=\int \frac {\sqrt {c^2-d^2\,x^2}\,\left (c+d\,x\right )}{{\left (e\,x\right )}^{7/3}} \,d x \] Input:

int(((c^2 - d^2*x^2)^(1/2)*(c + d*x))/(e*x)^(7/3),x)
                                                                                    
                                                                                    
 

Output:

int(((c^2 - d^2*x^2)^(1/2)*(c + d*x))/(e*x)^(7/3), x)
 

Reduce [F]

\[ \int \frac {(c+d x) \sqrt {c^2-d^2 x^2}}{(e x)^{7/3}} \, dx=\frac {-3 \sqrt {-d^{2} x^{2}+c^{2}}\, c +\frac {3 \sqrt {-d^{2} x^{2}+c^{2}}\, d x}{2}-3 x^{\frac {4}{3}} \left (\int \frac {\sqrt {-d^{2} x^{2}+c^{2}}}{x^{\frac {7}{3}} c^{2}-x^{\frac {13}{3}} d^{2}}d x \right ) c^{3}+\frac {3 x^{\frac {4}{3}} \left (\int \frac {\sqrt {-d^{2} x^{2}+c^{2}}}{x^{\frac {4}{3}} c^{2}-x^{\frac {10}{3}} d^{2}}d x \right ) c^{2} d}{2}}{x^{\frac {4}{3}} e^{\frac {7}{3}}} \] Input:

int((d*x+c)*(-d^2*x^2+c^2)^(1/2)/(e*x)^(7/3),x)
 

Output:

(3*( - 2*sqrt(c**2 - d**2*x**2)*c + sqrt(c**2 - d**2*x**2)*d*x - 2*x**(1/3 
)*int(sqrt(c**2 - d**2*x**2)/(x**(1/3)*c**2*x**2 - x**(1/3)*d**2*x**4),x)* 
c**3*x + x**(1/3)*int(sqrt(c**2 - d**2*x**2)/(x**(1/3)*c**2*x - x**(1/3)*d 
**2*x**3),x)*c**2*d*x))/(2*x**(1/3)*e**(1/3)*e**2*x)