Integrand size = 29, antiderivative size = 1079 \[ \int \frac {c+d x}{(e x)^{4/3} \sqrt {c^2-d^2 x^2}} \, dx =\text {Too large to display} \] Output:
-3*(-d^2*x^2+c^2)^(1/2)/c/e/(e*x)^(1/3)-3*(1+3^(1/2))*d^(2/3)*(e*x)^(1/3)* (-d^2*x^2+c^2)^(1/2)/c/e/(c^(2/3)*e^(2/3)-(1+3^(1/2))*d^(2/3)*(e*x)^(2/3)) +3*3^(1/4)*d^(2/3)*(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))/c^(1/3)/e^(7/3)/(- 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)+1/2*3^(3/4 )*(1-3^(1/2))*d^(2/3)*(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))/c^(1/3)/e^(7/3) /(-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^(3/4) *(1/2*6^(1/2)+1/2*2^(1/2))*d^(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))/ ((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+3^(1/2))*c^(2/3)*e^(2/...
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 = 0.09 \[ \int \frac {c+d x}{(e x)^{4/3} \sqrt {c^2-d^2 x^2}} \, dx=\frac {3 x \sqrt {1-\frac {d^2 x^2}{c^2}} \left (-2 c \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {5}{6},\frac {d^2 x^2}{c^2}\right )+d x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},\frac {d^2 x^2}{c^2}\right )\right )}{2 (e x)^{4/3} \sqrt {c^2-d^2 x^2}} \] Input:
Integrate[(c + d*x)/((e*x)^(4/3)*Sqrt[c^2 - d^2*x^2]),x]
Output:
(3*x*Sqrt[1 - (d^2*x^2)/c^2]*(-2*c*Hypergeometric2F1[-1/6, 1/2, 5/6, (d^2* x^2)/c^2] + d*x*Hypergeometric2F1[1/3, 1/2, 4/3, (d^2*x^2)/c^2]))/(2*(e*x) ^(4/3)*Sqrt[c^2 - d^2*x^2])
Time = 1.52 (sec) , antiderivative size = 1093, 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 = {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.
| \(\displaystyle \int \frac {c+d x}{(e x)^{4/3} \sqrt {c^2-d^2 x^2}} \, dx\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle -\frac {3 \int -\frac {c d (c-2 d x)}{3 \sqrt [3]{e x} \sqrt {c^2-d^2 x^2}}dx}{c^2 e}-\frac {3 \sqrt {c^2-d^2 x^2}}{c e \sqrt [3]{e x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \int \frac {c-2 d x}{\sqrt [3]{e x} \sqrt {c^2-d^2 x^2}}dx}{c e}-\frac {3 \sqrt {c^2-d^2 x^2}}{c e \sqrt [3]{e x}}\) |
| \(\Big \downarrow \) 557 |
| \(\displaystyle \frac {d \left (c \int \frac {1}{\sqrt [3]{e x} \sqrt {c^2-d^2 x^2}}dx-\frac {2 d \int \frac {(e x)^{2/3}}{\sqrt {c^2-d^2 x^2}}dx}{e}\right )}{c e}-\frac {3 \sqrt {c^2-d^2 x^2}}{c e \sqrt [3]{e x}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {d \left (\frac {3 c \int \frac {\sqrt [3]{e x}}{\sqrt {c^2-d^2 x^2}}d\sqrt [3]{e x}}{e}-\frac {6 d \int \frac {(e x)^{4/3}}{\sqrt {c^2-d^2 x^2}}d\sqrt [3]{e x}}{e^2}\right )}{c e}-\frac {3 \sqrt {c^2-d^2 x^2}}{c e \sqrt [3]{e x}}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {d \left (\frac {3 c \int \frac {1}{\sqrt {c^2-\frac {d^2 x}{e}}}d(e x)^{2/3}}{2 e}-\frac {6 d \int \frac {(e x)^{4/3}}{\sqrt {c^2-d^2 x^2}}d\sqrt [3]{e x}}{e^2}\right )}{c e}-\frac {3 \sqrt {c^2-d^2 x^2}}{c e \sqrt [3]{e x}}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {d \left (-\frac {6 d \int \frac {(e x)^{4/3}}{\sqrt {c^2-d^2 x^2}}d\sqrt [3]{e x}}{e^2}-\frac {3^{3/4} \sqrt {2+\sqrt {3}} c \left (c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}\right ) \sqrt {\frac {c^{2/3} d^{2/3} e^{2/3} (e x)^{2/3}+c^{4/3} e^{4/3}+d^{4/3} (e x)^{2/3}}{\left (\left (1+\sqrt {3}\right ) c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}}{\left (1+\sqrt {3}\right ) c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}}\right ),-7-4 \sqrt {3}\right )}{d^{2/3} e \sqrt {c^2-\frac {d^2 x}{e}} \sqrt {\frac {c^{2/3} e^{2/3} \left (c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}\right )}{\left (\left (1+\sqrt {3}\right ) c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}\right )^2}}}\right )}{c e}-\frac {3 \sqrt {c^2-d^2 x^2}}{c e \sqrt [3]{e x}}\) |
| \(\Big \downarrow \) 837 |
| \(\displaystyle \frac {d \left (-\frac {6 d \left (-\frac {\left (1-\sqrt {3}\right ) c^{4/3} e^{4/3} \int \frac {1}{\sqrt {c^2-d^2 x^2}}d\sqrt [3]{e x}}{2 d^{4/3}}-\frac {\int -\frac {\left (1-\sqrt {3}\right ) c^{4/3} e^{4/3}+2 d^{4/3} (e x)^{4/3}}{\sqrt {c^2-d^2 x^2}}d\sqrt [3]{e x}}{2 d^{4/3}}\right )}{e^2}-\frac {3^{3/4} \sqrt {2+\sqrt {3}} c \left (c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}\right ) \sqrt {\frac {c^{2/3} d^{2/3} e^{2/3} (e x)^{2/3}+c^{4/3} e^{4/3}+d^{4/3} (e x)^{2/3}}{\left (\left (1+\sqrt {3}\right ) c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}}{\left (1+\sqrt {3}\right ) c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}}\right ),-7-4 \sqrt {3}\right )}{d^{2/3} e \sqrt {c^2-\frac {d^2 x}{e}} \sqrt {\frac {c^{2/3} e^{2/3} \left (c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}\right )}{\left (\left (1+\sqrt {3}\right ) c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}\right )^2}}}\right )}{c e}-\frac {3 \sqrt {c^2-d^2 x^2}}{c e \sqrt [3]{e x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d \left (-\frac {6 d \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) c^{4/3} e^{4/3}+2 d^{4/3} (e x)^{4/3}}{\sqrt {c^2-d^2 x^2}}d\sqrt [3]{e x}}{2 d^{4/3}}-\frac {\left (1-\sqrt {3}\right ) c^{4/3} e^{4/3} \int \frac {1}{\sqrt {c^2-d^2 x^2}}d\sqrt [3]{e x}}{2 d^{4/3}}\right )}{e^2}-\frac {3^{3/4} \sqrt {2+\sqrt {3}} c \left (c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}\right ) \sqrt {\frac {c^{2/3} d^{2/3} e^{2/3} (e x)^{2/3}+c^{4/3} e^{4/3}+d^{4/3} (e x)^{2/3}}{\left (\left (1+\sqrt {3}\right ) c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}}{\left (1+\sqrt {3}\right ) c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}}\right ),-7-4 \sqrt {3}\right )}{d^{2/3} e \sqrt {c^2-\frac {d^2 x}{e}} \sqrt {\frac {c^{2/3} e^{2/3} \left (c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}\right )}{\left (\left (1+\sqrt {3}\right ) c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}\right )^2}}}\right )}{c e}-\frac {3 \sqrt {c^2-d^2 x^2}}{c e \sqrt [3]{e x}}\) |
| \(\Big \downarrow \) 766 |
| \(\displaystyle \frac {d \left (-\frac {6 d \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) c^{4/3} e^{4/3}+2 d^{4/3} (e x)^{4/3}}{\sqrt {c^2-d^2 x^2}}d\sqrt [3]{e x}}{2 d^{4/3}}-\frac {\left (1-\sqrt {3}\right ) c^{2/3} e^{2/3} \sqrt [3]{e x} \left (c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}\right ) \sqrt {\frac {c^{2/3} d^{2/3} e^{2/3} (e x)^{2/3}+c^{4/3} e^{4/3}+d^{4/3} (e x)^{4/3}}{\left (c^{2/3} e^{2/3}-\left (1+\sqrt {3}\right ) d^{2/3} (e x)^{2/3}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {c^{2/3} e^{2/3}-\left (1-\sqrt {3}\right ) d^{2/3} (e x)^{2/3}}{c^{2/3} e^{2/3}-\left (1+\sqrt {3}\right ) d^{2/3} (e x)^{2/3}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 \sqrt [4]{3} d^{4/3} \sqrt {c^2-d^2 x^2} \sqrt {-\frac {d^{2/3} (e x)^{2/3} \left (c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}\right )}{\left (c^{2/3} e^{2/3}-\left (1+\sqrt {3}\right ) d^{2/3} (e x)^{2/3}\right )^2}}}\right )}{e^2}-\frac {3^{3/4} \sqrt {2+\sqrt {3}} c \left (c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}\right ) \sqrt {\frac {c^{2/3} d^{2/3} e^{2/3} (e x)^{2/3}+c^{4/3} e^{4/3}+d^{4/3} (e x)^{2/3}}{\left (\left (1+\sqrt {3}\right ) c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}}{\left (1+\sqrt {3}\right ) c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}}\right ),-7-4 \sqrt {3}\right )}{d^{2/3} e \sqrt {c^2-\frac {d^2 x}{e}} \sqrt {\frac {c^{2/3} e^{2/3} \left (c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}\right )}{\left (\left (1+\sqrt {3}\right ) c^{2/3} e^{2/3}-d^{2/3} (e x)^{2/3}\right )^2}}}\right )}{c e}-\frac {3 \sqrt {c^2-d^2 x^2}}{c e \sqrt [3]{e x}}\) |
| \(\Big \downarrow \) 2420 |
| \(\displaystyle \frac {d \left (-\frac {6 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 )}{c e}-\frac {3 \sqrt {c^2-d^2 x^2}}{c e \sqrt [3]{e x}}\) |
Input:
Int[(c + d*x)/((e*x)^(4/3)*Sqrt[c^2 - d^2*x^2]),x]
Output:
(-3*Sqrt[c^2 - d^2*x^2])/(c*e*(e*x)^(1/3)) + (d*((-6*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[((1 - Sqrt[3])*c^...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
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 ]
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]
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]
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]
\[\int \frac {d x +c}{\left (e x \right )^{\frac {4}{3}} \sqrt {-d^{2} x^{2}+c^{2}}}d x\]
Input:
int((d*x+c)/(e*x)^(4/3)/(-d^2*x^2+c^2)^(1/2),x)
Output:
int((d*x+c)/(e*x)^(4/3)/(-d^2*x^2+c^2)^(1/2),x)
\[ \int \frac {c+d x}{(e x)^{4/3} \sqrt {c^2-d^2 x^2}} \, dx=\int { \frac {d x + c}{\sqrt {-d^{2} x^{2} + c^{2}} \left (e x\right )^{\frac {4}{3}}} \,d x } \] Input:
integrate((d*x+c)/(e*x)^(4/3)/(-d^2*x^2+c^2)^(1/2),x, algorithm="fricas")
Output:
integral(-sqrt(-d^2*x^2 + c^2)*(e*x)^(2/3)/(d*e^2*x^3 - c*e^2*x^2), x)
Time = 3.21 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.09 \[ \int \frac {c+d x}{(e x)^{4/3} \sqrt {c^2-d^2 x^2}} \, dx=\frac {\Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{6}, \frac {1}{2} \\ \frac {5}{6} \end {matrix}\middle | {\frac {d^{2} x^{2} e^{2 i \pi }}{c^{2}}} \right )}}{2 e^{\frac {4}{3}} \sqrt [3]{x} \Gamma \left (\frac {5}{6}\right )} + \frac {d x^{\frac {2}{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {d^{2} x^{2} e^{2 i \pi }}{c^{2}}} \right )}}{2 c e^{\frac {4}{3}} \Gamma \left (\frac {4}{3}\right )} \] Input:
integrate((d*x+c)/(e*x)**(4/3)/(-d**2*x**2+c**2)**(1/2),x)
Output:
gamma(-1/6)*hyper((-1/6, 1/2), (5/6,), d**2*x**2*exp_polar(2*I*pi)/c**2)/( 2*e**(4/3)*x**(1/3)*gamma(5/6)) + d*x**(2/3)*gamma(1/3)*hyper((1/3, 1/2), (4/3,), d**2*x**2*exp_polar(2*I*pi)/c**2)/(2*c*e**(4/3)*gamma(4/3))
\[ \int \frac {c+d x}{(e x)^{4/3} \sqrt {c^2-d^2 x^2}} \, dx=\int { \frac {d x + c}{\sqrt {-d^{2} x^{2} + c^{2}} \left (e x\right )^{\frac {4}{3}}} \,d x } \] Input:
integrate((d*x+c)/(e*x)^(4/3)/(-d^2*x^2+c^2)^(1/2),x, algorithm="maxima")
Output:
integrate((d*x + c)/(sqrt(-d^2*x^2 + c^2)*(e*x)^(4/3)), x)
\[ \int \frac {c+d x}{(e x)^{4/3} \sqrt {c^2-d^2 x^2}} \, dx=\int { \frac {d x + c}{\sqrt {-d^{2} x^{2} + c^{2}} \left (e x\right )^{\frac {4}{3}}} \,d x } \] Input:
integrate((d*x+c)/(e*x)^(4/3)/(-d^2*x^2+c^2)^(1/2),x, algorithm="giac")
Output:
integrate((d*x + c)/(sqrt(-d^2*x^2 + c^2)*(e*x)^(4/3)), x)
Timed out. \[ \int \frac {c+d x}{(e x)^{4/3} \sqrt {c^2-d^2 x^2}} \, dx=\int \frac {c+d\,x}{\sqrt {c^2-d^2\,x^2}\,{\left (e\,x\right )}^{4/3}} \,d x \] Input:
int((c + d*x)/((c^2 - d^2*x^2)^(1/2)*(e*x)^(4/3)),x)
Output:
int((c + d*x)/((c^2 - d^2*x^2)^(1/2)*(e*x)^(4/3)), x)
\[ \int \frac {c+d x}{(e x)^{4/3} \sqrt {c^2-d^2 x^2}} \, dx=\frac {\int \frac {\sqrt {-d^{2} x^{2}+c^{2}}}{x^{\frac {4}{3}} c -x^{\frac {7}{3}} d}d x}{e^{\frac {4}{3}}} \] Input:
int((d*x+c)/(e*x)^(4/3)/(-d^2*x^2+c^2)^(1/2),x)
Output:
int(sqrt(c**2 - d**2*x**2)/(x**(1/3)*c*x - x**(1/3)*d*x**2),x)/(e**(1/3)*e )