Integrand size = 18, antiderivative size = 540 \[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^{7/2}} \, dx=-\frac {4 b e \left (1-c^2 x^2\right )}{15 c d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x (d+e x)^{3/2}}-\frac {16 b c e \left (1-c^2 x^2\right )}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b e \left (1-c^2 x^2\right )}{5 c d^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {4 b \left (7 c^2 d^2-3 e^2\right ) \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 \left (c^2 d^3-d e^2\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 d \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{5 c d^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
-2/5*(a+b*arcsec(c*x))/e/(e*x+d)^(5/2)-4/15*b*e*(-c^2*x^2+1)/c/d/(c^2*d^2- e^2)/x/(e*x+d)^(3/2)/(1-1/c^2/x^2)^(1/2)-16/15*b*c*e*(-c^2*x^2+1)/(c^2*d^2 -e^2)^2/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)-4/5*b*e*(-c^2*x^2+1)/c/d^2/(c^ 2*d^2-e^2)/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)+4/15*b*(7*c^2*d^2-3*e^2)*El lipticE(1/2*(-c*x+1)^(1/2)*2^(1/2),2^(1/2)*(e/(c*d+e))^(1/2))*(e*x+d)^(1/2 )*(-c^2*x^2+1)^(1/2)/(c^2*d^3-d*e^2)^2/x/(1-1/c^2/x^2)^(1/2)/(c*(e*x+d)/(c *d+e))^(1/2)-4/15*b*EllipticF(1/2*(-c*x+1)^(1/2)*2^(1/2),2^(1/2)*(e/(c*d+e ))^(1/2))*(c*(e*x+d)/(c*d+e))^(1/2)*(-c^2*x^2+1)^(1/2)/d/(c^2*d^2-e^2)/x/( 1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)-4/5*b*EllipticPi(1/2*(-c*x+1)^(1/2)*2^(1/ 2),2,2^(1/2)*(e/(c*d+e))^(1/2))*(c*(e*x+d)/(c*d+e))^(1/2)*(-c^2*x^2+1)^(1/ 2)/c/d^2/e/x/(1-1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)
Result contains complex when optimal does not.
Time = 33.62 (sec) , antiderivative size = 407, normalized size of antiderivative = 0.75 \[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\frac {2 \left (-\frac {3 a}{(d+e x)^{5/2}}+\frac {2 b c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \left (-e^2 (4 d+3 e x)+c^2 d^2 (8 d+7 e x)\right )}{\left (c^2 d^3-d e^2\right )^2 (d+e x)^{3/2}}-\frac {3 b \sec ^{-1}(c x)}{(d+e x)^{5/2}}+\frac {2 i b \sqrt {\frac {e (1+c x)}{-c d+e}} \sqrt {\frac {e-c e x}{c d+e}} \left (c d \left (7 c^2 d^2-3 e^2\right ) E\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right )|\frac {c d+e}{c d-e}\right )-c d \left (6 c^2 d^2-c d e-3 e^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right ),\frac {c d+e}{c d-e}\right )-3 (c d-e) (c d+e)^2 \operatorname {EllipticPi}\left (1+\frac {e}{c d},i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right ),\frac {c d+e}{c d-e}\right )\right )}{d^3 (c d-e) \left (-\frac {c}{c d+e}\right )^{3/2} (c d+e)^3 \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{15 e} \]
(2*((-3*a)/(d + e*x)^(5/2) + (2*b*c*e^2*Sqrt[1 - 1/(c^2*x^2)]*x*(-(e^2*(4* d + 3*e*x)) + c^2*d^2*(8*d + 7*e*x)))/((c^2*d^3 - d*e^2)^2*(d + e*x)^(3/2) ) - (3*b*ArcSec[c*x])/(d + e*x)^(5/2) + ((2*I)*b*Sqrt[(e*(1 + c*x))/(-(c*d ) + e)]*Sqrt[(e - c*e*x)/(c*d + e)]*(c*d*(7*c^2*d^2 - 3*e^2)*EllipticE[I*A rcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] - c*d*(6* c^2*d^2 - c*d*e - 3*e^2)*EllipticF[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] - 3*(c*d - e)*(c*d + e)^2*EllipticPi[1 + e/(c *d), I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)])) /(d^3*(c*d - e)*(-(c/(c*d + e)))^(3/2)*(c*d + e)^3*Sqrt[1 - 1/(c^2*x^2)]*x )))/(15*e)
Time = 1.07 (sec) , antiderivative size = 516, normalized size of antiderivative = 0.96, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.056, Rules used = {5749, 1898, 635, 633, 632, 186, 413, 412, 688, 27, 688, 27, 600, 509, 508, 327, 512, 511, 321}
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.
| \(\displaystyle \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 5749 |
| \(\displaystyle \frac {2 b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^2 (d+e x)^{5/2}}dx}{5 c e}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 1898 |
| \(\displaystyle \frac {2 b \sqrt {x^2-\frac {1}{c^2}} \int \frac {1}{x (d+e x)^{5/2} \sqrt {x^2-\frac {1}{c^2}}}dx}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 635 |
| \(\displaystyle \frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\int \frac {-\frac {x e^2}{d^2}-\frac {2 e}{d}}{(d+e x)^{5/2} \sqrt {x^2-\frac {1}{c^2}}}dx+\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx}{d^2}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 633 |
| \(\displaystyle \frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\int \frac {-\frac {x e^2}{d^2}-\frac {2 e}{d}}{(d+e x)^{5/2} \sqrt {x^2-\frac {1}{c^2}}}dx+\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d^2 \sqrt {x^2-\frac {1}{c^2}}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 632 |
| \(\displaystyle \frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\int \frac {-\frac {x e^2}{d^2}-\frac {2 e}{d}}{(d+e x)^{5/2} \sqrt {x^2-\frac {1}{c^2}}}dx+\frac {\sqrt {1-c^2 x^2} \int \frac {1}{x \sqrt {1-c x} \sqrt {c x+1} \sqrt {d+e x}}dx}{d^2 \sqrt {x^2-\frac {1}{c^2}}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle \frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\int \frac {-\frac {x e^2}{d^2}-\frac {2 e}{d}}{(d+e x)^{5/2} \sqrt {x^2-\frac {1}{c^2}}}dx-\frac {2 \sqrt {1-c^2 x^2} \int \frac {1}{c x \sqrt {c x+1} \sqrt {d+\frac {e}{c}-\frac {e (1-c x)}{c}}}d\sqrt {1-c x}}{d^2 \sqrt {x^2-\frac {1}{c^2}}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\int \frac {-\frac {x e^2}{d^2}-\frac {2 e}{d}}{(d+e x)^{5/2} \sqrt {x^2-\frac {1}{c^2}}}dx-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \int \frac {1}{c x \sqrt {c x+1} \sqrt {1-\frac {e (1-c x)}{c d+e}}}d\sqrt {1-c x}}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (\int \frac {-\frac {x e^2}{d^2}-\frac {2 e}{d}}{(d+e x)^{5/2} \sqrt {x^2-\frac {1}{c^2}}}dx-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {2 \int \frac {e \left (3 d \left (2-\frac {e^2}{c^2 d^2}\right )-e x\right )}{2 d (d+e x)^{3/2} \sqrt {x^2-\frac {1}{c^2}}}dx}{3 \left (d^2-\frac {e^2}{c^2}\right )}-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2 e^2 \sqrt {x^2-\frac {1}{c^2}}}{3 d \left (d^2-\frac {e^2}{c^2}\right ) (d+e x)^{3/2}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {e \int \frac {3 \left (2 d-\frac {e^2}{c^2 d}\right )-e x}{(d+e x)^{3/2} \sqrt {x^2-\frac {1}{c^2}}}dx}{3 d \left (d^2-\frac {e^2}{c^2}\right )}-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2 e^2 \sqrt {x^2-\frac {1}{c^2}}}{3 d \left (d^2-\frac {e^2}{c^2}\right ) (d+e x)^{3/2}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {e \left (-\frac {2 \int -\frac {6 d^2-\frac {2 e^2}{c^2}+e \left (7 d-\frac {3 e^2}{c^2 d}\right ) x}{2 \sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx}{d^2-\frac {e^2}{c^2}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (d^2-\frac {e^2}{c^2}\right )}-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2 e^2 \sqrt {x^2-\frac {1}{c^2}}}{3 d \left (d^2-\frac {e^2}{c^2}\right ) (d+e x)^{3/2}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {e \left (\frac {\int \frac {2 \left (3 d^2-\frac {e^2}{c^2}\right )+e \left (7 d-\frac {3 e^2}{c^2 d}\right ) x}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx}{d^2-\frac {e^2}{c^2}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (d^2-\frac {e^2}{c^2}\right )}-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2 e^2 \sqrt {x^2-\frac {1}{c^2}}}{3 d \left (d^2-\frac {e^2}{c^2}\right ) (d+e x)^{3/2}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 600 |
| \(\displaystyle \frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {e \left (\frac {\left (7 d-\frac {3 e^2}{c^2 d}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x^2-\frac {1}{c^2}}}dx-\left (d^2-\frac {e^2}{c^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx}{d^2-\frac {e^2}{c^2}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (d^2-\frac {e^2}{c^2}\right )}-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2 e^2 \sqrt {x^2-\frac {1}{c^2}}}{3 d \left (d^2-\frac {e^2}{c^2}\right ) (d+e x)^{3/2}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 509 |
| \(\displaystyle \frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {e \left (\frac {\frac {\sqrt {1-c^2 x^2} \left (7 d-\frac {3 e^2}{c^2 d}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx}{\sqrt {x^2-\frac {1}{c^2}}}-\left (d^2-\frac {e^2}{c^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx}{d^2-\frac {e^2}{c^2}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (d^2-\frac {e^2}{c^2}\right )}-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2 e^2 \sqrt {x^2-\frac {1}{c^2}}}{3 d \left (d^2-\frac {e^2}{c^2}\right ) (d+e x)^{3/2}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle \frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {e \left (\frac {-\left (\left (d^2-\frac {e^2}{c^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx\right )-\frac {2 \sqrt {1-c^2 x^2} \left (7 d-\frac {3 e^2}{c^2 d}\right ) \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e (1-c x)}{c d+e}}}{\sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}}{d^2-\frac {e^2}{c^2}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (d^2-\frac {e^2}{c^2}\right )}-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2 e^2 \sqrt {x^2-\frac {1}{c^2}}}{3 d \left (d^2-\frac {e^2}{c^2}\right ) (d+e x)^{3/2}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {e \left (\frac {-\left (d^2-\frac {e^2}{c^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {x^2-\frac {1}{c^2}}}dx-\frac {2 \sqrt {1-c^2 x^2} \left (7 d-\frac {3 e^2}{c^2 d}\right ) \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}}{d^2-\frac {e^2}{c^2}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (d^2-\frac {e^2}{c^2}\right )}-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2 e^2 \sqrt {x^2-\frac {1}{c^2}}}{3 d \left (d^2-\frac {e^2}{c^2}\right ) (d+e x)^{3/2}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 512 |
| \(\displaystyle \frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {e \left (\frac {-\frac {\sqrt {1-c^2 x^2} \left (d^2-\frac {e^2}{c^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{\sqrt {x^2-\frac {1}{c^2}}}-\frac {2 \sqrt {1-c^2 x^2} \left (7 d-\frac {3 e^2}{c^2 d}\right ) \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}}{d^2-\frac {e^2}{c^2}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (d^2-\frac {e^2}{c^2}\right )}-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2 e^2 \sqrt {x^2-\frac {1}{c^2}}}{3 d \left (d^2-\frac {e^2}{c^2}\right ) (d+e x)^{3/2}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 511 |
| \(\displaystyle \frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {e \left (\frac {\frac {2 \sqrt {1-c^2 x^2} \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \int \frac {1}{\sqrt {1-\frac {e (1-c x)}{c d+e}} \sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}}-\frac {2 \sqrt {1-c^2 x^2} \left (7 d-\frac {3 e^2}{c^2 d}\right ) \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}}{d^2-\frac {e^2}{c^2}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (d^2-\frac {e^2}{c^2}\right )}-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2 e^2 \sqrt {x^2-\frac {1}{c^2}}}{3 d \left (d^2-\frac {e^2}{c^2}\right ) (d+e x)^{3/2}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2 b \sqrt {x^2-\frac {1}{c^2}} \left (-\frac {e \left (\frac {\frac {2 \sqrt {1-c^2 x^2} \left (d^2-\frac {e^2}{c^2}\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {d+e x}}-\frac {2 \sqrt {1-c^2 x^2} \left (7 d-\frac {3 e^2}{c^2 d}\right ) \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {x^2-\frac {1}{c^2}} \sqrt {\frac {c (d+e x)}{c d+e}}}}{d^2-\frac {e^2}{c^2}}-\frac {2 e \sqrt {x^2-\frac {1}{c^2}} \left (7 c^2 d^2-3 e^2\right )}{d \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{3 d \left (d^2-\frac {e^2}{c^2}\right )}-\frac {2 \sqrt {1-c^2 x^2} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d^2 \sqrt {x^2-\frac {1}{c^2}} \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}+\frac {2 e^2 \sqrt {x^2-\frac {1}{c^2}}}{3 d \left (d^2-\frac {e^2}{c^2}\right ) (d+e x)^{3/2}}\right )}{5 c e x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \sec ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}\) |
(-2*(a + b*ArcSec[c*x]))/(5*e*(d + e*x)^(5/2)) + (2*b*Sqrt[-c^(-2) + x^2]* ((2*e^2*Sqrt[-c^(-2) + x^2])/(3*d*(d^2 - e^2/c^2)*(d + e*x)^(3/2)) - (e*(( -2*e*(7*c^2*d^2 - 3*e^2)*Sqrt[-c^(-2) + x^2])/(d*(c^2*d^2 - e^2)*Sqrt[d + e*x]) + ((-2*(7*d - (3*e^2)/(c^2*d))*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]*Ellip ticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*Sqrt[(c*(d + e*x) )/(c*d + e)]*Sqrt[-c^(-2) + x^2]) + (2*(d^2 - e^2/c^2)*Sqrt[(c*(d + e*x))/ (c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e )/(c*d + e)])/(c*Sqrt[d + e*x]*Sqrt[-c^(-2) + x^2]))/(d^2 - e^2/c^2)))/(3* d*(d^2 - e^2/c^2)) - (2*Sqrt[1 - c^2*x^2]*Sqrt[1 - (e*(1 - c*x))/(c*d + e) ]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(d^2*Sqrt [-c^(-2) + x^2]*Sqrt[d + e/c - (e*(1 - c*x))/c])))/(5*c*e*Sqrt[1 - 1/(c^2* x^2)]*x)
3.1.68.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ )]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2 Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c*f)/d, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt [c + d*x])) Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] , x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[B/d Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp [(B*c - A*d)/d Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, A, B}, x] && NegQ[b/a]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > With[{q = Rt[-b/a, 2]}, Simp[1/Sqrt[a] Int[1/(x*Sqrt[c + d*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > Simp[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2] Int[1/(x*Sqrt[c + d*x]*Sqrt[1 + b*(x^2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && !GtQ[a, 0]
Int[((c_) + (d_.)*(x_))^(n_)/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp[c^(n + 1/2) Int[1/(x*Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] + Int[( (c + d*x)^n/Sqrt[a + b*x^2])*ExpandToSum[(1 - c^(n + 1/2)*(c + d*x)^(-n - 1 /2))/x, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[n + 1/2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( (m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^ (q_.), x_Symbol] :> Simp[x^(2*n*FracPart[p])*((a + c/x^(2*n))^FracPart[p]/( c + a*x^(2*n))^FracPart[p]) Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + a*x^(2*n ))^p, x], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[mn2, -2*n] && !I ntegerQ[p] && !IntegerQ[q] && PosQ[n]
Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol ] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSec[c*x])/(e*(m + 1))), x] - Simp[b/ (c*e*(m + 1)) Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x] / ; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1617\) vs. \(2(491)=982\).
Time = 8.78 (sec) , antiderivative size = 1618, normalized size of antiderivative = 3.00
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1618\) |
| default | \(\text {Expression too large to display}\) | \(1618\) |
| parts | \(\text {Expression too large to display}\) | \(1642\) |
2/e*(-1/5*a/(e*x+d)^(5/2)+b*(-1/5/(e*x+d)^(5/2)*arcsec(c*x)+2/15/c*(7*(c/( c*d-e))^(1/2)*c^4*d^3*(e*x+d)^3-6*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c* (e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),(( c*d-e)/(c*d+e))^(1/2))*c^4*d^4*(e*x+d)^(3/2)+7*((-c*(e*x+d)+c*d-e)/(c*d-e) )^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticE((e*x+d)^(1/2)*(c/(c*d -e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c^4*d^4*(e*x+d)^(3/2)-3*((-c*(e*x+d)+c *d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticPi((e*x+d) ^(1/2)*(c/(c*d-e))^(1/2),1/c*(c*d-e)/d,(c/(c*d+e))^(1/2)/(c/(c*d-e))^(1/2) )*c^4*d^4*(e*x+d)^(3/2)-13*(c/(c*d-e))^(1/2)*c^4*d^4*(e*x+d)^2-7*((-c*(e*x +d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticF((e* x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c^3*d^3*e*(e*x+d)^(3 /2)+7*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2 )*EllipticE((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c^3*d ^3*e*(e*x+d)^(3/2)+5*(c/(c*d-e))^(1/2)*c^4*d^5*(e*x+d)-3*(c/(c*d-e))^(1/2) *c^2*d*e^2*(e*x+d)^3+2*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d +e)/(c*d+e))^(1/2)*EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d +e))^(1/2))*c^2*d^2*e^2*(e*x+d)^(3/2)-3*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2) *((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticE((e*x+d)^(1/2)*(c/(c*d-e))^(1 /2),((c*d-e)/(c*d+e))^(1/2))*c^2*d^2*e^2*(e*x+d)^(3/2)+6*((-c*(e*x+d)+c*d- e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticPi((e*x+d)...
\[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\int { \frac {b \operatorname {arcsec}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
integral(sqrt(e*x + d)*(b*arcsec(c*x) + a)/(e^4*x^4 + 4*d*e^3*x^3 + 6*d^2* e^2*x^2 + 4*d^3*e*x + d^4), x)
Timed out. \[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e+c*d>0)', see `assume?` for mor e details)
\[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\int { \frac {b \operatorname {arcsec}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+b \sec ^{-1}(c x)}{(d+e x)^{7/2}} \, dx=\int \frac {a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^{7/2}} \,d x \]